Particles, Sources, and Fields

Volume I

David Pines, Series Editor Anderson, P.W, Basic Norions of C o d w e d Mater Physics Bethe H. and Jackiw, R., Inte ee awnturn Mechanics, Third E&tim Feynman, R., Photon-Hdron lnternctions Feynman, R., Quantum Elect~odynamics Fewman, R., Statistical Mechanics Feynman, R., The Theory of F u d w t l t d Processes Negele, 1. W. and Orland, H., Quantum Many-Pa~tickSystems Nozi&res,R, Theoy of Interncting Fermi Systems Farisi, G., Statistic$ Field Theory Pines, D., The Many-Body Probkm Quigg, C., Gauge Theories of the Strong, Weak, and Ekcnomagnetic Interactions Schwinger, l., Particks, Sources, and Fields, Volume I Schwinger, J., Particles, Sources, and Fields, Volume II Schwinger, J., Particks , Sources, and FieLls , Volume III

aOURCES, AND

ULIAN SCHWINGER late, University of California at Los Angeles

P E R S E U S BOOKS Reading, Marsaehusetts

Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. m e r e chose designations appear in. this book and Perseus Books was aware of a uademark claim, the designations have been printed in initial capital letters. Library of Congess Camlog Card Number: 98-87896

Copyright 43 1998, 1989, 1970 by Perseus Books Publishing, L.LC.

Ail rights reserved. Na part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any fclrm or by any mans, electranic, mechanicai, photocopying, recording, or othenvise, without: the prior mitten permission of the publisher. Printed in the United States of America. Perseus Books is it member of the Perseus b o k s Group Cover design by Suzanne Heiser

1 2 3 4 5 6 7 8 9 10-EB-0201009998 First printing, September 1998

Editor's Foreword

Perseus Books's Frontiers in Physics series has, since 1961, made it possible for leading physicists to communicate in coherent fashion their views of recent developments in the most exciting and active fields of physics-without having to devote the time and energy required to prepare a formal review or monograph. Indeed, throughout its nearly forty-year existence, the series has emphasized informality in both style and content, as well as pedagogical clarity. Over time, it was expected that these informal accounts would be replaced by more formal counterparts-textbooks or monographsas the cutting-edge topics h e y treated gradually became integrated into the body of physics knowledge and reader interest dwindled. However, this has not proven to be the case for a nllrnher of the volumes in the series: Many works have remained in print on an on-demand basis, while others have such intrinsic value that the physics community has urged us to extend their life span. The Advanced Book Classics series has been designed to meet this demand. It will keep in print those volumes in Frontiers in Physics or its sister series, Lecture Notes and Supplements in Physics, that continue to provide a unique account of a topic of lasting interest. And through a sizable printing, these classics will be made available at a comparatively modest cost to the reader. These lecture notes by Julian Schwinger, one of the most distinguished theore tical physicists of this century, provide both beginning graduate students and experienced researchers with an invaluable introduction to the author's perspective on quantum electrodynamics and high-energy particle physics. Based on tectures delivered during the period 1966 to 1973, in which Schwinger developed a point of view (the physical source concept) and a technique that emphasized the unity of particle physics, electrodynamics, gravitational theory, and many-body theory, the notes serve as both a textbook on source theory and an informal historical record of the author's approach to many of the central problems in physics. I am most pleased that Advanced Book Classics will make these volumes readily accessible to a new generation of readers. Bavid Pines Aspen, Colorado

July 1998

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Vita

$dim Schwinger UdverSity Professw, University of CafoAa, md Prafessor of Physics at the Angeles since 1872, was bosn in Nevv York City m U~versityof Califoda, fxis Ph.D. in physics fm F e b m q 12, 1918, Profwsor S Cdvmbia U~versityin 1939, He has itlm raived honotq doctorat= irr from four iastitutions: hrdue U~versity(19611, H m a d U~vergity(19621, Brmdeis University (19731, and Gustavus Adolphus &Bege (1975). fn addition to teach8 at the U~versityof Califoda, Profesmr SGhwinger h trruCyht at hrdue U~vmsity (B41-%-$31,and at H m m d U~versity (1945-72). Dr. Sch~ngerwas a Rmewch Asmiate at the U~versityof C a E f e a , Berkeley, and a Staff Member of the Mlllsachwetris Institute of Twbolow hdiation Laboratoq. In, 1965 h o f a s ~ r rwipient ( ~ t f iEchwd F e p m n m d Sin; Itiro Tomon in Physia for wark in qumtum d e e t r d p the C. L, M q r Matwe of Li&t Awad

Scimw Awud for Physics (1964); a Hurnboldtt Awad (1981); the a di Castiaone de S i ~ (1986); a the M o ~ A. e Fmst Sips Xi. Awad (1956); and the Amerlcm Academy of Acbevement Awmd (1987).

vii

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Special Pre

Isme Newon used his newly hvented method of f l d o u s (the cdculus) to mmpare the impfiations of the inverse square lrtw of paGtation with Kqler's empihcai laws of p l m e w motion. Yet, when the t i m c m @ to ~ t thee Brincipia, he resorted entkely to gwmetfial demonstrations. Should we wnelude that cafculus is saperflwus? S~arce&----to wEch the w n c q t of renom is foreign-and the =me a s w a s renornaked operamfielil t h e a ~have both been foland problems (wh_ieh &sappoints some pmple who would prefer that wurw thmq prduct: new-md wrong-answers), Should we waclude that source t h w q is thus superfluous? t same respnse: the shpler, mare intuitive fomaBoth questioas m e ~ the tion, is preferable. 'This &ition of P m i c l a , SOUTC~S, crnd Fields is more extmsive &m the o ~ @ n atwa t volumes of 1970 and 19'73. It n w e o n t h s foux atSditimd seetims that f ~ s hthe chapter entitld, ""EeetrOaryn&~csH," "ese s a t i o n s e t t e n , h 1973, but rem~nedin pwtidl_yt w d f o m for fiftan y w s , I am indebteb to Mr. Ronafd B o b , who aged to d&pher my fa&ng and mmpletd the typmfipt. Pargcular attention should wted to Section 5-9, where, h a context sommhat lwger than &ex-trsd Trzetwen saurw and operator field t h w q fin* ~e~ first acqu;zintma ~ t sour= h &mq should wnsdt the Appendix in Volume I. This Appm& contkns srrgestions for he&&g m's s w a y though the sometha &uttered pages,

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and it is a textbook, It is the r m r d of a hi@y hi& e n e w peicle physics. The iopdients personal reation to the were: fmstration with the matthematid mbiguities and physical rmoteness of aperator field thmq, dissatisfa~gonwith the overly mirthematid attitude md spwulath philomphy of the supposedly mare physied $mat& thmv, outrage at the pretension of cunmt dgebra to be a fundmentd descfipGon rather thm at low m e r p phenomenolow. The result was a point of Gew and a tecwqae that empha phenomena. The physid sour= mncqt, upon w ~ c h ticd prwursor in operam field thmq. But it was not w M e tackng a Haward grduate mm=, &at I. how the phenomenolo@caf soww wncept w d d be f r d from tmcture md used as the- basis fm a mmpletely hdqendent development, with much domr ties to experiment, rapidly, at U C M that e rmnsfmction of efec ,and during a repelition at W=, iastead, devotd to the n m approwh, Developments in pion physics that (11966-1x71, in wEch the new most sumssfay &ppl,i&dtwns in mathematical shpgc=itymd me, if no one else, of the wncqtuaE c l ~ t that y its use bestowd. The lack of appraiation of tbese faets by others w w dqrashg, but undersbndable. a i y a d e t d d prsenation of the ideas and methds of sour= theory could chmge t b t situa~on.The writin& of

*&

As a textbook, this voXum is intended for use by my s n~melativisticqamtum mwhdcs, who fishes to fern m ~ h ~ cI sthi& . it of the utmost h p o r t a n ~that such acquaint== with the Berating ideas of murw t h w q w a r befare expasure to one of the current s f i h d a ~ e hm s w w & him past the elastic bait. In the Preface to a volume on authar spe&s of the desirabiEty that the student have ai (opmator) field thmq. X echo that ~ s t f ucall, l but ce to hcXude S-matk thwq. f have m d e no attempt to supply the traditiond my m who dlegdly first did wbt, whea. Perhaps I to the distoaions fierent ia the shplistie asswiac But there is a mre iagportant thn of ideas m& xllethods with s p ~ i f i hdividuals. reason, general &tique of eistiaf: attitudes is mwntid in motivating this new vi , it would have b n too distrrtcthg if mnstanr reference: ta &Mqu@s for wGch obsolesmnce is intended had ammpanied the development proach. The expert comes ready nade ~ t ~pinlons h about what has done. To the student d l that matters is what is new ts b md X hope that he will fmd much in these pages. X m gatefd to Wss Cuanane and Miss Jeri Ingersan who, at different pfiods, devotd1y aided the: burden of t p h g the t. The book would never have been wmpleted (I hold the world's r w ~ r dfor the lagegt number af unfhishd first chapters) without the patience and understandkg of my wife. Xt is thmefare apprap~telyddicated t s the. C,G.S, system. Belmonl, Massachusetts Oetobet 1969

Contents

l --I 1-3 1-3 1-4

Unitary Transformations Galitmm Relatiuity Einsreinian Relatiuity Oitique o f Particle Theories

Spin O Particles. Weak Source Spin O Particles. Strong Souree Spin I Particles. The Photon Spin 2 Partictes. The Crauiton Particleg with Arbitrary Integer Spin Spin $ Particles. Fermi-Dirac Slatistics More About Spin iPanicles. Neutrinos Particles of Inleger + $ Spin Unqication o f All Spins a d Staristics

3-1

3-2 3-3

The Field Concept. Spill O hrticles The Field Concept. Spin f Particles Some Other Spin Values

Mtcltiqinor Fiel& Action

Inuariance Transfarmabiaprs and Flues. Charge Inuariance Meebtanica me Electromagnefir:Field. Charge Quantriath. MW$ Prtmitr'ue EIectrom@@dicfnteraetiom and Source M d e k Extended Sources. Soft Photons Ifiteraclion S k e l e f ~Seatteri~g ~. Cross Sections Spin iProcesses Sources m filcrrterers H-hrticfes Imtability and Mgltiparticle E x c h ~ g e The Grauitational FiekJ

Appendix: How to Read Volume I

If you ~@~'fr_iiorin"em, beat "m,

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Sources, S

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PARTICLES The concept of the p&icle has undergone drastic changes and generalisations in the c a m e of $he hisb6cal development that l& to the! atom, to the nu~leus, sad then 4x1 subnuclear phenomena, This htts aka been a progesrjion from es~eotialynonrel&ivistic behaGor to an ulkra-relativistic dometin, It is interesting b appmciafe how mueh of the kinema.tical particle a t t ~ b u t e is s implied by the mumeb stmretuxe!of %herel8btivity goup of krm~fi3rnEfrtio~1~ among equivalent coordinate sy-stem%, In preparatioxr Ear this dimurnion we fimt review some propt?r&iesof qusntum nneehanical unitary t r d o 1-1

UNITARY TRANSFORMATIONS

Quantum mechanics is a ~Mlbolioexprwisn of the laws of micramopic m@asurt?med. Stabs, situ~tiogsof optimum infornation, am represented by vmtom in 4; and physic~lpropdies by a earnpiex s p w [left vectors ( A]. The freedom in physical Knear H e r ~ t i a nomratom . on de~riptiancome~pondsto the fretzciom of matlternaticaf repr6wntat;ian wswiahd with unitary operators. These are defined through the Hermitisn adjoint operation p by trtu = uut = I (1-1.1)

b t us tramform df vectom and operators weording to

-

- - .C- - -

=( U

(

=

)

X

=

v-'xu.

Then d l numerical and adjoint relations among vectors and operatarn are umhanged, We rredfy that

( 2 )= a The adjoint relation~hip,

,

($1

(2lXIF) = (a'lxlb'). =

(1-1.4)

Id)'

i~ trsn~fomedinto

shows that the Hemitian operator A is mapped into the Hermitian operator a. 1

A eomplete get of states (a'[ f o m s a b a i s or coordinate system ilz the state space, Any veetor 1 ) is represenled by its components mlative ta this bmis, (dl ). Amther basis is produeed by a unitary transformation: and the gven vector has a eammwnding new mt of csmpnents, Thwe numberg are albmatively dmribed m eompanent~,rel&tiveto the initial basis, of the new w c b r U1 ), An anabgous relatian for matrix ekments of operator8 k (1-1.10)

If two suecwive transformations rtre w d o m e d on the bwis, the net change in tlre componenfs of a vector is given by

1%is produeed in one s$ep by the unitary operator U%&, in vvhich the multiplication order reflee& the sequenee of tr~nsformations,The appasih sequence is repremnhd by U1U2, and the two are eomgamd by defining the unitary aprator that h needed to convert the mead equence into the first,

An infinikaimaf unitaq %ransfornnationis a transfarmation in. thr?:infiniksinral neighbarhood of the identity. It is represerrbd by

where C is an bfiaik8imaj Ifemitian operator, When two m& tramformatians are compared we find that Ut121

E

1 4- i@1121,

(1-1.15)

where

at121

-s

-Gall

(l/i)lGt, @zl

(1-1.16)

intrduces the commutator of Gl and Gz. Ttte effect of an infinitesimal uai%&q tr&nsformationon an aperator is given by where

sx = (r/z">[x, q,

An equivale~tfarm ia

X

=

u-lxu = X - &X.

If we compare alternalive evafuatiaw of

When pre8enkd in b r m of daubfe commutatorr~, this is reaogniad as the Jscobi id~ntity. Now let us conisider a group af unitary tram formation^ i L h n real, continuous ptitmmeters k, a -- 1, . . , n, which m d m i m f e collectiively as X. If U(Xr,%) are typi~aloperators of the group, it is requird Lhet

.

where are the parameters of another element of &hegroup. For unifary operabm the existence of the irtvem ancl, of fhe identity is awumd. An i n f i k h i m l $mm formation of the group with parametrtm &X, i~ comtmeM from

*ere the n finite Hemitim opclrators Qa are cailed the &c3mratomof the &roup. One is free Lo redefine Cbt: generators by real nowindrtr Einmrr frm&~msbdiom, with eorrmponding redefinitions of the prametem. On subjmtbg the infinitttsimid transformation operator U(6X)Lo an mbitrav unitav tradomatioa of the group, we mu&$obtain another infirzibsinnal frawfarma$iorr. TKi irr expresged by ~ ( k ) - ' ~ ~ v= ( hE ) @.a(&)@&, (1-1.26) b

where the numbem uaa(X)sre real, We shall also use a matrix m t a % i oin ~ Ghe a-dimensional parameter ~paef3trnd write The unitafy transformation is presenM e~lbmativefyad3

The two Bets of matrices are related by where T degnatea matrix transposition. Note that this is equivalent to Hermitian conjugation when applied to the u matricee, since they are real. The correspondence established between the unitary operator U(X) and the matrices U@),d(X) is maintained under multiplication. Thus,

and

~(~~)U(XI)GU(X~)-'U(~~>-' = U(X~)[G~(X~)IU(A~)" = @m2)4(X1).

(1-1.32)

Since the unit operator corresponds to the unit matrix, we write

where Ba

= -ga

T

.

T*

= ga

This gives [G, Gbl = gbG = -G4b and, if the irnsginary elements of the matrix gb are designated ss

we get the explicit commutation relations of the group generators:

We see, incidentally, that In view of the multiplicative correspondence between U(X) and u(X)[&(X)],the matrices ga[la]also obey these commutation relations,

The latter are a set of quadratic restrictions that must be obeyed by the numbere g* the so-called group structure constants:

This cyclic structure also follows immediately from the cyclic form of the

+ [W&, Gel, G,] + [[G,, G,],

(l-1-41) = 0. The ~tmctureeomfants speeify the compsition propedies of inhibsimal par~mekm. Let blXPIXah the parametrsrzl of Ithe infinihsimal fmnsformation that conoeetrs the two mquences in which Lhe tramformation8 labled by and b2X, can be applid, Accordjng to the commuhtions relations of tho ~ o u p gener~tom,they %regiven by [[Gat

@blF

Gel

Gb]

In the discussion ta Eollow, group composition propedies am supplicsd by georxretfied considerations. It is important to recopize that the associated unitary g~oupm ~ not y be an exshet image of the underfying geometrieai group, This is tt eoasequence of an intPinsic arbitmriness 5f any quantum mechanical description whereby all stak8 can be change$ by a commm phase facbr, whieh is the unitary transformatka generakd by the unit operator. Consider, for example, the commutative (Abelian) group of translations in a tw&dimensional apace. Let the parameters of the two independent infinitmimal displacements be writLea &zl, 8z2 and the eomi3ponding He tian diqlwement operators be denoted 'by p 1, p2? SO Ghat The fact that succeSsive displaeements are insensitive to the order in which. they are t?;pptiedshould imply the eontmutalivily of p1 and pa, But d l that h requi~ed of the commu"ttoor is that if generate a unitary transformation without physical eomequences, Accordingly, with a suitabEr: normaliaation of the dkplz~cementoperators, Now the eEect of %heunitary transformation U"(6x)on the operators p1.2 is given by

whioh shows that the displacement operators also sew@as coordind o p r a b r s We recognizie the propertim 5f the q,p phase space wsoeirtted with. a single qurtntum degee of freedom. Translations in this two4imemional @pace are describd by a three-parabmeter unitary group. This is explicit in the f o m

G

X

p liq

m

q lip

+ &PI.

(1-1.47)

The onr respondence between. the unitary operators U(&) and the finite matrices %(X), &(h) doe8 nst nece~sarilyincl& the mitary ehsraekr of the

Istfer. [Note that if the %(X) m;:unitary, or real odhagonaf, matrices, we have %(h) == B(%), md then the Eiernnitian, or ima&ary mtkymnzetnisal r n a t ~ m g. = ia.] Since the structure of the g matrices csn be altered by changing the generator basis in the parameter @p-, it is useful to have a basis-independent ith which to judge the pomibi1iC;y of exhibiti I;ian g matfices %(X)rn%tfiw. If the mt of rr 42 operabrs i~ by tht!linearly cornbimtiom XG, the g nnatficm ux1,dergo the 8ame liniear %rangtber with a similafity transfomation p r d u ~ e dby the nonsine;ular m a t k X. Bw&usethe trmt: of nnatdx produels is unchanged by the Xathr $randomation we somider the real quadratic form

which must be positivedefinite if the g, are transformable into linearly inde rnatri~es. The failure. af that propedy implies Lhat m& s and unitav %(X) nnatfiees do not exist. This i s the ~3itu& tim, in the ~xannplewe have just di~cussiedl,of the gene.mbmp, p, I. There the qudmtia f o m ia identicdly lie ce the unit operator is repremnbd by Lbe nu11 r n a t ~ while , the ma;tri i a M with q and p haw only single nondiagonal entries in such a way that all matrix products vanish. The positive defixlibnm of the real rsymmet~ematxrix ry but alaaa sufficient for &heg, to bc? equivdexrt to linearly i n ni tm . Since the elements of the r nnad~xare unhanged by the aimilafity transformation

A red symmet~aalpsitivedefinitr? matrix can a l ~ ~ abe y swritten as the square Then of another mch matnk, which we designate as r

which makes exphcit the similarigy transformation that introduces u n i t a r ~ ~ (arthogod) %(X) matrims and Hermitian (anti~ymnetrical)g matrims, In the new bmis, r is a m~ltiple~f %heunit matrix, Let tm suppom that the o p r a b r s G4 pomes a finih4innemitbnaI linesrly hdewndent Hermitian mtrix reaiizatition. That inrplbs the e r i ~ b n wof the m1 synnmetfic pogitivedefinib m a t ~ r

Ealilsan relativity

1-2

7

The invariance of the* numbers under unitary transformations on. the operator8 leads again to the form rf= u ( x ) r ' ~ ( x ) ~ , (1- 1.54) with the implication that the %(h)can be pre~entedas unitary matrices. The corresponding Hesmitian matrices g,, whi~hare linearly independent if we exelude the uninteresting possibility that the group has an Abelian poup as a factor, arc? an example of a finite-dimensiod realization of the G,, m the %(X) provide a finite-dimensional unitary realization of the U(&). Conversely, if the h unitary [Hermitian] realization matrix Tab is not positive-definite, no s u ~ finite of the U(&)[&)can exist. A finite-dimensisnd realization of the group meana that a finite number of staks can be found which are transformed among themselves by all, operations of the goup. frz general, the aetioxl of zt unitary operator on tz s t a b introduces new states, and the repeti-l;ion. of the operation continues the prmess of producing additional states, This can terminae with a finite number of states only if thztf repetition eventually cea~esto provide new operatom, that ia, if the group paramebr sp~ect;is cio~ed. The distinction between a closed and an open group manifold is most familiar in that between rotation8 and Lranslaf ions. XE the matrices g, are Hermitisn, the stmetum constants gabc are antisymmetrical in a and c, M well as a and b, which implies antisymmetry in ;b and c. This cornpieh antisymmetry can only be realized with n 2 3, For n 3, EL, suitable normalization brings the im.a@nav structure constants to the unique farm gabe ==z i e ~ b c t (1-1.55)

where Ecrbe i~ the totally antisymmetrical symbol specified by resulting group comrnufatian relations

[G., Gal

=i

El23

= 4-1. The

C e.acCc, e

are familiar in the Lheo~esof three-dimensional anmlar momentum and isohpie spin, The Lhree-dimensional g matrices satisfy these commutation relatians, and r is a multipie of the unit matrix,

Space-tiw coordinghs Ebpwar in qu~zntummechmies &B an abstraction of the roles of the maerascopie measuremenL appara,lus. All evidence eonfirms the equivaleace af two coordinab ;systems %hatdiEer in any or aIl of the follopring ways: a tretnslation of dhe ~patialoPi@n, a translation of Lhcs time ori@n, a rotation of the spa% &X@@,t3r eonstant relative veloeity between the two ~y@Lf?rn~.

These trangformations cortstitute the relativity group, or rather the ~ubgroup sf transformations that are eontinuousXy eonnectd with the identity, When Ezfl particles move slourly in comparison with the speed of light, the time coordinrate has an absolute significance md is affeckd only by displacement of if8 origin, This is Galilean re1ativit;y. It is oharachrieed by infiniksirnal coordinate $ran& formations r, t -+ F, f, where and 6r = 6e

+ 6w X r + 6vE.

Note that the sign conventions are appropriak to the significance of 60, my, as the displacement of the origin of the spatial coordinate frame to which a dven point is referred. If it is the point that is moved by Sr, its new position relative to the fixed reference system is r 6r. m e graup camposition properties of this 10-parameter group are gpecified by compafing the wquence of tramformations

+

with those in the opposite order. The result of pedoming the .tramformation sequence 1, 2, l l, 2-l, or, equivalently, for infinitesimal transformations, I-', 2-l, 1, 2, i s

where

The irlfinitesimal unitary transformtzeisn, U = 1 an infinitesimal coordinate transformation is given by

+ iG, that is induced by

The quantum unit of action, h = 1.0515 X 1 0 erg~,,c, ~i s henceforth ~ replaoed by unity on adopting suitable abrnie units, The generatam P and J are conventionally odled the tinear and angular momentum operators, while H ia the merw or Hamiltonian operzktor. The generatom af infini-imal velocify changes have had no verbaf desigattCion. But, in %hisroeket conscious age, the finite! veiocity tmnsforrnations have eome ta be ~ a l l d"bm~tS."Perhapg one s h d d tern N the booster. We need to specify group composition praperties for the

new sealsrr p%r&meter6 9 . A genera1 bilinear form for Bfxzlp= - 6 f 2 1 1 9 is

where K, I;, M are constants. The Jaeobi identity, iltpplicsd Co thrm mt8 of infinitesimd treznsformatians, implies that

One ewily verifies that the eae6cienf of M vanisha identie~lly,bu& thorn of K md L do not. Henee the latter eoeEelt.nts must be aero, and we have, simply,

Not as a proof but as a mnemonic, we mention, that 6 0 be and 60 really pseudosedars, while 6e 6v is a scalar, The full set of generatar earnmutation. relations is

-

&v are

where we have adopted the summation convention for repeaM indices; here, the index m = 1, 2, 3, The eommuttllor of two generator8 ean be inkvreted, in two dbrnative wntys, m the effect of an infinitssimal unitav transfomatian upon an operabr:

The eommutaltors involving the anwlar momentum apratior, for emmpIe, can be w ~ t k nans . sUJ= (X/.i)fJ, J * 401 = 6c;~X J, (1-2.13) 6,P = fI/z'>[P,3 Gccr] == 6 0 X P, BUN = (I/i)[N, J 6 4 -- 6 0 X N, which state the rmponse of a vector to infinitesinsaf rofstions, and

-

&,B= (1/$7[H, J 401 = 0,

(1-2. r4)

which charaeterires H as a rotational scalsr. Analogous linear momentum and

translettioa, rmponse equations are

d the nstum The response of the anwlar momentum to tramlation is in m ~ o r with of angular momentum as the moment of the linear momentum, and indicabs the exisknee of a pasition, vwhr operafor R. such that (l/$)[R,P . 6e] = 8e

or

1% Pll = a'Skl* We therefore write where S is( a translationally independent contribution to fhe ~tn@ar marneaturn. This is inbrnat a n d a r momentum, or spin. The carrect rotratianal re~ponmaf P is w u m d by this con~truction,and that of R will follow if the components of R are mutually commutative, and if S commubs with R weEl ss with F. To produce the proper rotational hbavior of S, these: operators must the obey the snljgufar momentum eonnmut~tionrelations, which; we c m abo w r i b a ~ a S X S = HS.

(1-2. a0)

The translational respan% of N indieahs that it can be idenkifid d t ; h -MR, together with a translationally invariant term. Since a boost ia3: a tmnslatiorz that grows linearly with time, wt., infer that All contmutstors among J, P, FiT are reproduced by these eowtmc-tisw. If I ) is tt dynamically powible ~ t a t eo is U1 ), where U rep tivlfy transflrrnation, since the veebr ) b s the same eomponenb ars dwe3 I ) in the transformed description. This means that the relativity group generators are constants of the motion, Such dso is the conclusion of %hecommutation rdations invol\ring R, if it is uadersfood th&t R, P, S are not, explicit funotion of t. Thus,

Of courBe, H is not; an. explicit function of 1 for an kolahd dynamical syskm. The c a n ~ w a t i o nof Pil also appears as

which eleslrly identifies the part3lmede;r M as the i n v a ~ h i emass of the sysknn, The position vector R moves with constant velocity:

The stmcture of H is determind somewhat by the various eon~errrationlaws. We note thrtt dP - - ~ [ R , H I =aH -- = - 1 [P,H f = - aH -- dt 2' d~ -p ' dt a ~ = 1 M " (1-2.26) and

The consequence is

vvhieh gives the decomposition inta energy of motion of the whale rs;vstem and internal enerw, The latter will generally involw indernd dynamkal variables, which commute with R and P, combined in sueh a way that Hint is invariant under the rotation generahd by the inbrrral angular monnenlum S. An dementary particle is a aystem tk'ifiout inkrnd enere, or a t lea& one for which inkrnal enerw is effectively ined under the limikd physical eircumstances under consid64ration. Let us consider n elementary gafiieles, each described as above by variables re, p,, Sa and mms m,, a = 1, . . ,n. The operators msoeiated with digerat pafiieles commuh, The krretie traxrsformation generators of the whole sysknt are then obtained additively as

.

=

where and

(pat -- m.~,) = Pt

-- MR,

12

Psnl~les

Chap, 1

The oprators for the total sy~teznhave the required propeeiesi. Note that the inbmal vafiables introduced here are not l i n e d y indqendent :

which is dm conveyed by the eonnmukation relation

If the v ~ r i a u spafZ,ieles are dynamically kolabd, the energy aperator is also dditive, More generally, we d e s e ~ b einkracting systems by

where the inbrnsl enerw of the system is

and V is a malar fwction; of the inkrnal coordinaks r, - R, p, - (m,/M)P, and possibly others. The number of particles being degcrihd cannot be a dynamical variable apart, from rather speial cireuxxrstanee~,Let there be several digerent t y p s of pa&icles, with m m e s m,, Then

I,,

where N , is the nunrbr of parvticles of Qpe a. Since there am generally no rational relations among the mttssm of diffefent particles, Che invariability of M implie8 the comtancy of N , for e a h t y p , An exception oecum; for unstable pa&icles, as in the a-instabilitgr of nuclei (a-particle kinetic energies can be suficiently small to ualidak the Galilean regime), Here the mass of the unsl;able nucleus very c10mly equals the sum of a-particle and residual nuclear masses. The i;eneral charackrization of inkraeting sys&ms enables one to give a simple description of the behavior of a particle that is influenced by a macroseapie, canlrotlslbfc?enviranmttnt. Since a clwical t h a q of such interactions underlies the measurement of free paFticle properties, a Le& of mlf-consishncy is &so involved. Let the Hamittonian operator of a system of par-t;icles be; divided into two parts: H,, comprising all terms containing the variables of a given particle; H-,, being all other terns, describing the residual system after the partiele of interesk has been removed, We rtssume, for simplicity, that the interaction terms in H, are no more than linear in the velocity and in the spin of the particle, but we da not inelude bilinear or spin-orbit coupling terms. AC

1-2

Oatilsan relativity

13

though the interaction is not necessa,I.ily electromagnetic, we use a notation designed to facilitate that identificafion :

It is understood that the noncornmuting opemtors p and A(t.t) arc?symmetrized in rnultiplieatisn to produce a Merrnitian product. The explicit time dependence appears as an effective replacement for the actud dependence on the variables of the external system, as indicated by

The equations of motion are

and

where

In deriving the lmt equation we have omitted such commuta;tors as [(e/e)A, ecp]. This is La be validated, not as a clmsical approximation, but through the negligibility of the dynamical reaction back an the externat system. We also note the commutation relation indicated by

We do speeialiae to electromagnetism on equatiag F with a multiple of H,

which is the identification, of the inkrnal magnetic dipole moment,

Analogous electric dipole moments have never been observed. One value of the gyromagnetic ratio g has a special property. In a homogeneous magnetie field the veloeiLy and the spill vectors precess about the field axis. The two p r e ~ w i o n mtes are equal if g =.I 2. The observed values of g are very sligfitly in excess of 2 for the electron f2(1.001tM)] and Lhe rnuon [2(1.001166)], but differ widely for other padiele~.

We shall find it interesting to consihr a spidess charged parLic1e that move8 in the magnetic field of a distant stationary magnetic charge. Let the ~oordinate origin be plseed at the position, of this charge (we now use the letter g to denah its strength in Gaussian units), so that

This system is characteri~edby the equation of motion

where a symmetriaed produet is understood, together with the connmuta;tioxl relations [nit (MV)II = i & k t t (1-2.47) and The last equation is in~onsistentunless the particle can be controlled to remain distant from the magnetic charge, in. the weak sense of r > Q, That; follows from the Jacobi identity: O = [[vt, v2], v,] $ cyel. perm. = m3c

1. r"

The equation of moments, which uses symnnetrized multiplieadion, is

But since the Hamiltonian is no more than quadratic in the momenla, symmetrized multiplication enables one to write d

f (r) =

1

dH = v Vj(r), C~P

[f (P), H ] = Vf(r)

--

and thereby recognize the consewed angular monnenlurn veetor ~=rXmv-eQL. C

r

(l-2,52)

One easily verifies that it is the rotation generator. Them is an impadant conmquenee of that fact.: Consider the coordinate wave funetion repre~entixlgrt p&rticular sLaLe, (rt[ ), and perform. a coordinak system rotation about the axis provided by r, An infinitesimal rotation ifs given by

and the corresponding generator is simply

1-3

Einstginian relativity

15

The response to a finite roti-ttion is therefore

and the known limitation to single or double valuedness for rotation throul,Th 21r radians implies that eg/e is either an integer or an inkger plus i. As a discussion of magnetic charge and its implications this is quite incomplete. However, the special opraLor system given in Eqs. (1-2.471, (1-2.481, and (1-2.52) will soon be encountered again in a very difierrznl physical context.

The new feature associated with. the finiteness of c, the speed af light, is the abandonment of absolute simultaneity, It is replaced, for infiniksimal trarrsformations, by

where &c0 is the displacement of the origin for the variable cl. We now designate the space-time coordinates collectively by zp = et, r, where z0 = --X@ = et and zk =r= a = r k . The infinitesimal coordinate transformations of the Einsteinian relralividy g o u p are

where ady=

- &oVIL.

The six independent parameters of this four-dimensional roLation are relakd ta 6w and 6v lay li~k= t E ~ ~ ~ h O~ k = w Bv~ ~/c. , (1-3.4)

The composition properties of the 10-parameter group are specifid by

The generators of the unitary tramformation, induced by an infinihsimal eoodinah transformation are comprised in

The eonrespondence with the GaliEean generators is

and c ~ =o H

+M C ~ .

16

Chap, t

PartZelss

we 8hall reco@zle &ho&lythe neeemity far the &M% henera o ~ &b$wmn $he relativktic and nomlati~&icdomaim (to ug@;the aanventiond l%b& for the two relati~tie~). As to the compo@i&ioa for tht? wd&r pararnekr 8p, it i8 interesting thaL no vanear s c a k S1121~"= - - Q l a l l ~can. km fomad in flre four* dimensianal MinkowsE spMe hm.the wectom b l ,pep and the '&mm B1s2WP". ALccordbgly, 4IZXP == Ot (1-3.9) and the full, mt of eommut~tomfor the geaemtom is

where QC(* i~the metric bmor specified by The coxulmuta%amcan &im

?Mpremnkd, as

indic&tingthe respom of veetom and bnsam to infixuSdc3.siml Lorentz robtiom (comp~~ing three-dimemional rotations and bmts), and

which give8 the traoslaLiontnl rwpom of these apratom, m e a w ~ t b nin %hrm-dinoemionalnotation, d l the~ecornmatabm reprduct? the G a m n with t w exeepdions ~ : In Gdilean relativity, then, J/cZ is neglected snd H is negIeeM mhtive to Me2, giving the effective replrteement of the operator Pale by the number M. Eencefo&h we d o p t a b ~ unih c in, which c = X. There is one obviow reali~atiorrof aff the commutation relatiom for the 10 geger&~om* It ig P I > = Q , J"" )=O, (1-3,162 which d m ~ b the g fatal ilrtva~anceof the s t m e t u m l ~vmuuna. Any by PO > 0. The scalar formed from the framlatiomlly invafian* PB, is invariant under dl operatiom of the Lomndz waup (we are

i ~ ody g tbs

Elnstelnian relativity

1-3

17

trawf~rmstionsthat are continuously connected with the icfetatity, the p r o p r orthochronous Lorentr group). According as M' is positive, zero, or negative, the four-vector P@is timelike, null, or space-like. With a, timelike momenhm, is an invariant prope&y. For M = 0, too, remains valid under Lorentz tran~formations. But the time component of s ~ p a e l i k eveetor can. be given either aigxl by appropriately choosing the eoordinste system. Thus M 2 < 0 i8 of no interest for physics. The nonnegative ig the mass of the system. quantity (M2)'/' Another tramlslionafly invariand object is the peudovectarr where *JP@ ~

L C ~ ~ J . ~

forms the tensor dual to J"" with the aid of the totally antisynmretricaf ternor srpcified by e0128 -M (1-3.21) E

Thk invwrietnm prowrty fsllows from the tr~wlationaIresponse of J,x, aad the antisymmetry of c'"', We also n o h that &W" = 0. The sealiar

W 2= WPWr 2 0 is invariant under aEE Lorentg tr8nsformations. As indicated, the vecfor W', being orthogonal to PC,eannot be time-like. The @ommudation,relations among the components of WIrare The behavim under wordinate disglaeements that is pregexlkd in the equations (l/i)[J, P 6 4 = 6r X P, (l/i)[N, P be] = -- drPo, (1-3.26) again indicates &heexistemer: of a position operator R, obeying

(Qne muat stifie %beimpukrc!to introduce a tirne operator complementauy to PO, That w d d contrdict Lhe physical nature of the enerw s ~ c t m m . ) A particular realization of J and N, in which additional displacement independent quantities do nof occur, is J -- R X P,

N .- PX@--- PR,

(1-3.29)

where aymrrretrizd multiplication, is used for %henoncommuting operrthrs R and P', (l/i)[R, P'] = aP"/aP = PIP'. (1-3.30) The eomeet three-dirnexlgiona1 rotational hhavior of slX the operators con~ d e r e dis ob%%ined if IRXRs0. (1-3.3 1) The other charaehri~tiecommutator of Einsteinian relativity here reads

lit is obyed without fureher ado, since i[P"Rk, P0Rl] =. RkP1 - RIPk.

(1-3.33)

The simplicity of this result, despite the presence of symmetriz;ed gradueds, depndg on the fact that cammutztforg of R ~ t functions h of P introduce no fudher commutator^ and are nee~marilycscnceled by the Hiermitian symmetriza, %ion, I n this si$uation the informftt;iszlabout %heenergy operator Lhat can. be deflved from the eanewation of N,

+

is already contained in the relation PO = (p2 M 2 ) l i Z . Now let an inkrnaf a n w l ~ momentum r be added:

As such, S m u ~ commuh t with R and P while itself obying the a n v l a r mos i also necessary to supplement P4 irt order to mentum commutation mles. It r generab the spin h r m of the commutator

A suitable form is

N

=

-- P'R

+ a(P")S X P.

(1-3'37)

The ealeulettion of N X , N involves

+

-~P@R X [a(S X P)]-- i[a(S X P)] X P'R a2i(s X P) X (S X P) = ( d a / d P ' ) ~X ( S X P ) 2P0& - a2[pZf5 - P X (S X P)], (1-3.38)

+

and the r e q u i d resulL is obtained with

We cone1ude that

a(Po) = (PO the alternative choice with ( P o - M)-' The 6nsl form is J=RXP+S,

N

=

PZO

+ M)-', being singular a t P = 0, P' = M.

+

- PR

+

M SXP,

whieh incidentally shows that S' is B Lorenta invariant. It is worth pointing out the converse, that operators with the slat& properties of R and S can be eonstructed from the Lorentz generaton (zO= 0):

The exceptionat position of M = O is evident here, The components of the pseudovector Wfi are given by

or

w 0 =P+, W = P@S-

(l-3,451 +

M P x ( S X P ) = M S + ~ , + P~ P - S .

The last relation can dso be written as

There is a eonnee-t;ionamong the several i ~ v a ~ a n t s :

W 2 = M'S'. This discussion refers generally to M 2 > 0. We next consider the limit ss M' -t 0 for fixed .'S The resulting relation

can be $ven the cova~antkm where X = P *SIP0

is a Lorents invariant. This quantity is the component of the spin along the direction of motion, or the helicity of the particle. In view of its invariance, a phy~icalsylshnn need exhim only an@value of helieify, or, if ~pwemfieeLion parity has a meaning for the interactions of that system, the pseudoscalar h osn have two values, fs. The photon, with s = 1, illustrates the latter situation, white for the neucfinos, with g = 3, X = +s and -a refer ta emntiaUy dZ~?ren% p&&icfes, f f only one helieity vatue is meanh@ul, or with s 2 1, even if h = trl and --s are both resliaed, not all of the 28 1 spin magnetic quantum number ststes exist. Accordingly, the operator S cesses to be defined (with two excep tions) in the limit M -+0, and we must introduce m w vaGabfes for this eircumstance. f n. order to deleb S we define the new position vector

+

wbieh i~ such that

k XP= Then

R

X

P+S

J=~ZXP+X(P/P@),

-- PPeS/(P0)2. N=Pz@-POR,

(1-3.52) (1-3.53)

and to complete the verification that only X appears explicitly, we give the contmutabr

This is the operator system that we anticipated in discussing magnetic charge. The co~esps~dence is and the restriction r > O is here validated as the Lorentz invariant energy property P@> 0. The absence of certain helicity values is now manife the noneommutativity of componenh of R, This htrimie ~oxzlocalityof m lew particles is de~cribedby the unm&ainty principle

or, for a nnomentum s b b with some d e g e of dimtionality,

indicating that rm average wavelen@h, roughly mk the scde of c m spec%eability. Incidentally, when the explieit constructions of J snd N am inw*d in the fomulm for M R and MS, thme exprwions do v&nbh, m dom

M&* I n the situation we have just discussed, W' = 0. There is another logical possibrtity. With X = P SIP@assuming any accessible finite value, let S ' -+ a, as M 1 -4 0 to produoe the limit

The chetr"acf;eristimof S Gve t h w opemtors the follovving p r o p d i a : and [h, T2]= 0.

The h v a ~ a n t

1["2 , ,

can be assbed any pagitive v a b . The componenb of T act to change h by &l,and t'hk Pvithout Itimi*. We now have The commutation relations among the oomponents of W,, which am mtisfid t ~ v i d l ywhen T == 0, hem demand that ( l / i ) ( p o4 , = T X P,

(l/i)(hP

+ T) X (XP + T) = P-,

(1-3.63)

and t h w are valid stakmenk aboiut T. We continue to urn the wt~ifionveclor

R =R

.

but we must be; camful h no& tbaL

S X p/(PO)'

M

f 1-3.M)

0,

"IKe now find, ss %hecounhmsrt of MS -.,T, ths&

MR

= T X P/(P')'

and R ' --t m with vsnishing M. However, M R = 0. The e htion mhfiom tN X N = J continue to be obeyed despite the bLrodue1ion of the T

brm, since

QX

+

(T X P / ( P ~ ) ~ )(T X P/(PO)~)X

= 0.

(1-3.69)

This involves the commutator wfiieh is also used to verify that J generates the rotations of I", The significant obmnta;tion is that X has eeawd to be ft Lorentz i n v a ~ a n t :

This faet, together with the unbounded nature of the h spectrum, ranginf; over all inkgers or all integem indicates that physically acwssible states would exist for which (bk12is arbitrarily large. We suggest the following verbal principle for massless particles: A zero mass padicle is not completely XoealizabXe, but a finite degree of localizability exists. The principle has the: following valid consequences. There is na aginless zero mass partieb, for the eomrnutative position veetor R would be available The same reasoning exelude s = 3 massless pa&ie1w for which space reflection parity is meaningful. And the systems we have just discussed, with W 2> 0, are condemned wholesale by the existenw of state8 that are unlwalized without limit. There is a simple pattern for the kxtown or strongly caxljectured maesless garlieles; their spins are given by 8 = Z4, cr = -1, 0,+X. The conwpt of elementary particle in rellitivistie mechanics remains an operational one, Lhat under the eonditiorzs of physicafi exeifa$ion available, if is consistent to @sign a unique vazlue ta m-, spin, and ather ehttracterisfic invariant a(ltributes of the system. Far a set of n noninkraeti~gp & ~ i ~ the le~, Lorentz generators of the whole system are giwn by the additive forms

++,

The operators R and S for the total system must be obtained from the conatmctim (1-3.43); one is not likely to produce them by an a pfiori definition, Consider, for example,

1-3

Einstainian relativity

23

We approach the topic of interacting particles by giving first a relativistic generalisation of the nonrelativistic treatment of a particle moving in a macroscopic environment, Xn order to make covftriance more explicit we define a proper time derivative:

with the usual symmetrization understood in the last term. Thus, for a singleisolated prartiele, we have

With attention restricted do a homogeneous electromakgnetic field, the covariant generalilizlations of Eqs. (1-2.39) and (31-2.40) are stated as

The constraints

are compatible with the equations of motion, a t least to term8 linear in the field strengths; this involves only the commutation relations for a free particle, Why bid we not begin with a general theory of inkrating particles, specified by variables r,, p,, S, a = l, . , . , n, and then proceed to follow the mation of one particle under the infiuence of the others, as in the nonrelativistic discussion? Quite simply, because no such &;eneraltheory exists, Apa& from the obviously formidable algebraic task of stating the relativistic conditions on in_leraction brms (smdl, deviations from nonrelativistic behavior pose no problem), the atternpt founders on the failure of the assumption that there is a fixed number of parlEticles, The relation between reiati~sticand nonrelativistic energy can be exhibited as

1x1 fhe nonrelativistic limit where changes in H are small compared to each ma, the conservation of P' generally demands, first, the conservation of each N,, and then, Ghat of H. But if the kinetic and interaction energies contained in H

b o r n eotnp&mblewith in&edud m, valuw, one ean no longer oonclude th& the Na mmsin c o ~ s b n t . It is tbe charak6stic featwe of mlativbtic padiele dynamim that pafticles can be @matedsnd wanihilahd in high e n e r e en@ounkm. 1 4 CRITIQUE QF: PARTICLE THEORIES

expriment abundantly confim, that the concept of le objeet k unknable under pronounced relatietic ~ b k r a ~ t i o n c~n.&%ions. s Them have Bwn two extreme reaction5 frO t h i situation, They rwpond fd) the f&lum of a detailed space-time description in. particle language by: (1) bisting on the pomibilily of a detailed ~pacie-timdescription but in h r m of a concept more fundamentat &an pa&icle; (2) rejecting the possibility of a detaifd space-time deseriptian by denying that any concep% underlies that of particle. We shall give brief derseripliom of these attitudes. carfiers of physied prop&ies an? the 2. More fundamental *an parlticles volume elemenb of thrm-dimensional space itmlf, If the sped of light l i d b wery means of commrunication, disjoint wlumw at the Bame time are phy~icatly indewdent and &odd contribute additively to the tot81 emrm and momentum. Using an evident Iimiking prmedure, we v v ~ k

where TOO(%), ~"(z)are functions of the dynamical variables a t time z0 that eonvey the physiGal situation in the infiniksimi neighbarhood of the p i n t x, The d y n a ~ e avafimtbfes, l ss operator funetione, of space and time eoordinsb~, sm oprstor fiefds, and the approach we: am describing e m be callled operator field theory. As the above notation suaesb, covariance can be made mpli~it by idenfifying the vohme element (dz) urith the time oompnent of a direckd element of sres on s plane space-like surfaee in four-dimensional space. This

which i n t e p b are independent of the surface tr according to the consemation of P". Qn h t h g the null diRerenee of two such i n t e g a l a an e q ~ v a l e avolume t

we recognize the suficieney of the load condition

Critique of particle theories

1-4

26

The conservation of the six other Lorentz generators, regarded aa moments of momenta, J ~=' ~ U A ( Z L T ~x'T'~), ' (14.5)

/.

is assured if TP'(z) = T'"(x). The three-dimensional form of these operators is

The tensor transformation response of the stress tensor Tp'(z) to an infinitesimal Lorentz transformation is given by

The possibility of producing the new operators

by the associated unitary transformation implies the commutation relations

Integrations over a space-like surface, employing the stated properties of Tp', will reproduce all commutators for the 10 Lorentz generators if one uses the following integration theorem for a system that is closed in space-like directions:

The commutators of quantum mechanics express the mutual interference of measurements on the two properties involved. The physical independence of volumes in space-like relation thus requires that

When the coordinate system is so chosen that xO= xO', an everywhere-valid expression for such commutators must involve b(x - X') or a finite number of derivatives of this function. For the energy and momentum densities, which are the Tpvcomponents used to construct the Lorentz generators, the implied form

of the equal time eonnmutation relations is

The brms involving two or more derivatives are such thizt l-hey do not conffibule wbem intepations are pedormeb to eonstmct one of $he Loresnde generafors, We have indicated only the minimum n u m b r of d e ~ v a t i v erequired; ~ more generd possibilitks are introduee-d by appropriitb generaliaa;tion of f , 8, snd h, The% three functions are symmetrical wikhin each pair of indices, as iZlu~tr~ted by (1-414) f n r r f p p - f nm1plr = fna.rrq ~fhile f and h are antisymmetrieaf under an. exchange of the pairs, as in Anokher relation is

-aOf m n * p q ( z )

Bmnrpg(Z)

Q ~ ~ * ~ ~ ( c c ) .

(1-4.16)

There is a simple example of a system for which none of the additional derivative k m s appear, We begin with the energy and rnornentum densi%y expressions thrtt are identified with the classical electromagnetio field:

Thrt attempt to r~producethe e n e r g density cammutator

sueeeeds with. the commutation. relations

if the mornen$um density spertator is interpreted as a synnmetrized P ~ ~ u G1& n . a r ~ v i n gst the desirt?dEorm we have used the formal ddLa funetion property

The commutators among the momentum density components also eoxltain no higher derivative terns, but to reproduce the required structure it is necessary to impose the following conditions, V-E(z>=O,

(1-4.21)

V-W(Z)=O,

which are compatible with the commutation relations. The commutators between energy and momentum densities then follow the anticipakd pattern and supply familiar expressions for the stress components Fkr;in particular, Although we have hewn wiLh a suggestion from classica,l physics, this discussion is a self-contained verification of a Lorent~invariant quantal system. Other properties are now derived from the stmcture of the Lorentz; 15c;nerators. From PO Byededuce the equations of motion of the field operaton, which are the homogeneous; Maxwell equations. The Lorenta transformation behavior of the field strengths is that of the antisymmetrical tensor FP.. As an example, consider

(1-4.23)

Then, since

(lli)[Ek(x),T0@(2')1= cttmar 6(x we get the infinilesinral response ~ E ( x )=

-&V

- xP)Hm(z'),

(zOv+ xaO)E(z)- &V

X

H(%).

(1-4@24) (1-4.25)

We add brief comments about more realistic system^, in ~ ~ b i the c h electromagnetic field interacts with other dynamical va~ables,If we are to maintain the geometrical transformation properties of F,., the added terms in Toomust not alter the computation just performed. That excludes from (Ex(z) , To'(z')] any additional single derivative of a delta funetion, giving the general form,

f d is no longer necessarily tme that E is divergeneeles~,and, on. w ~ L i n g

V E = j',

(1-4.27)

3%

Particles

Ch~p.3

Among the consequences of these relations that are produced by integr,ztiot~l;; j ' , j) identified over X' are: the inhomogeneous Maxwell equations with P = ( as the electric current vector, the local charge consewation law, and a Lorentz; transformation response, affirming the four-veedor status of j', Ex~mplesexist of interacting syskms for \vhieh the very singular terms of the energy density commutator do not appear, but there are evere restrictions in the choice of dynamical variables. Basically, only scalar, vector, and simple apinor fields are permitted. At issue here is the consistency of the operator field hypothesis, that meaning, albeit ideali~ed,attaches to the physical properties awoeiakd with a sharply defined geometrical volume. To examine this queslion we consider various weighM averages of the e n e r a density st a dven time,

and ecmstruet

This is the basis for an uneeftainty pfineiple stattlment about the aceuracy with whieh values of T 1and !F2can be assigned in a even state. We firgt consider an application where f m"nnppa does nod enter, in comquence of the antir~ymmetryin. the two sets of indices. Let T 1and ZT2 be partitions of Lhe total e n e w operator, so that (1-4.32) @I(x) 4-@ 2 ( ~ zz=) 1. Since derivatives of

v1

and of

v2

diaer only by a minus sign, we find, simply,

Now choose vl (X) to be a, unit step function, defining a semi-infinib redan which shares a, surface \dth the complementary volume defined by etz(x). With clS an element of area directed from the latter volume, we get

This gives a eorreet acmunt of the rate a t which the enerw in each p r t i a l volume changes, owirtg to the enerw flux acroas the common surface. Irxeidentally, if the domain8 defined by v, and v2 had been regarded as disjoint but approaching contact in a limit, the value abtsinied for the Il.igh&band side would have b e n zero, while, if they had initiaEXy overlapped the eventual boundary and then the cornman. volume had approached zero, the limiti~~g value of the

Critique af particle thaoriss

1-4

2r)

right-hand side would have been twice the stated one. Thus, an alternative evaluation uses the average of the two limiting definitions. Ar~stherchoice of weigh%functions is @ (X) .l @(X), VZ(~) xkv(~), (1-4.35) which gives

When @(X) is a unit step function that defines precisely a finite volume, the operahrs T t , T 2are the asmciated energy and its first moments* But no meaning can then be assigned to the products d,, d,v d,u, which calls seriously ints question. the eonsiskncy of any operator field theory for which. $nanlpS1 (2) @ 0. This gives a pPivileged position to that limited clslsl; of fundamental field variables for which f does vanish, The impact of this result is only slightly weakened by the follojving property of physical systems that have vanishing The funetion fmnlpq.

cannot be zero, and it is correspondingly impossible to specify simultsneousfy, within ally finite precision, the total enerlSy. and a, component of total momentum that are associated with a shsrply defined volume. [The term physieal system occurs here as a reminder that the vacuum state, with a11 its atkndant properties, must be compatible with the assumed eharacteristi~sof the system. I n particular, the zero enerm and momentum invariantly assigned to the vacuum state 1 ) require that (TO0)= 0, TO^) = O (1.--4.38) and also (T") = 0. There is some freedom to adjust the definitions of the TB"by additive constants, but, as inspection of the [ ~ ~ ' ( z )To"(z')] , structure will confirm, it is limited to a multiple of g"^". A nontrivial requirement is thus given by I n the example of the electromagnetic field, tvith Tkr = Too,it is impossible t o satisfy 4(T0') = O since TO' = +(E' H%)is a positive-definite operator; the uncoupled electromagnetic field is not a physieal system. It is a t least coneeivable that the vacuum. properties so cireumseribe the possible dynl~micalvariables and their interactions that the real world is selected.] We consider the cornmutator of the energy density f'unctional

+-

with its time derivative (dx)v ( X )

TO'

(g),

which gives the vacuum expectation value (dx)ak~,v(x) (gh*")a

p

= ~(TP'T).

aCIu(x) (1-4.43)

We recognize the necessarily positive expeetaticm value of the enerw in the nonvacuum state I" ), Tbc numbers thus form a positive-definite matrix, but there is no guarantee that these numbers are bounded, fL is clear from this discussictn, however, that a statement h u t momentum drtnsity is also one concerning the time derivative of energy density, and this additional dynamical specifiability may be unnecessary to the self-consistency of the theory. The particle, in operator field theor?., is a derived dynamical concept. To eonst~lletfrom a few fundamental field variables a rel%tivelylarge number of Btable or qumi-stable excitations-particles-is the ambition of this viewpoint. A elawifieatian of particle spectra is produced m follows. Let X(%) be an algebraic earnbination of the fundamental field variables, so devised that it has an eleme&ary response ta Lorenfz transformations. This includes the requirements that, st x = 0, the rotational behavior corresponds to a definite intrinsic angulsr momentum or spin, while translaLional msponse is pa~ametriz;edby fbe eoordinates zp, (1-4.44) F(z)t &I (1JfJ apxcIz>E

The finite unitary operator presentation of the latter is

P""z@ and X is evaluated at the coordinrtte origin. The state x(z)[ ) where P;E: is produced from the vacuum state by a localized excitation. To study the pfdSele aspects of this excitcltion tve examine its spaert-time proipltgation characteristics through the correlation %vithan analogous excitation having a digerent localization : =2=

The unitary operator fhnt describes the displacement from z' to z can be exhibited in terms of i t s eigenvatues and the associated non~egativeHermitim projection operators,

where ( d p ) = dpo d p , dpz dp3.

The values of fl that contribute to the integral, thwe for which F(p) f 0, must conform to the physical sp~etral.requirements, With a given threcdilnensional momentum, (dp012 = d 3 f S pso that

The di&rentjal dcJ, is an in~rztriantm o m e n t m space m s u r e on the hypersurface --P2 = M 2 . This gives

is a real, nonnegative function. The state X / ) selects from P(p) the subspace with the angular momentum propexrlies implied by the rotational behavior of X , snd f ( p ) j.I. 0 at - P 2 = M 2 a~sertsthe existence of an excitation with those physical parameters. Merely for simplicity, we only consider a scalar field X , which limits f ( p ) to dependence on the scalar --p2. There are three qualitatively difierenl possibilities that can be realietxi in f(p) = f(MZ).

a. An isolated mass value appears in the spectrum,

For a given spaLial momentum, the time dependence of the field correlation function contains the isolated frequency 'p = +(p2 m2) This excitation i s s stable pareticle. We n o b thaL should ~ ( sobey ) a finite-degree differentid equation,

+

ul(-a"~(~= ) o,

and f(di2) is eomposed entirely from delta functions. b. There is e. pronounced inerease in f(M2) above a smooth background, which is cenbred a t M = m and hss a mass width measured By T" << m. For a specified momentum, the time dependence that is associated with this port;ion of the mass spectrum is given by f t ='2 -- zO')

32

Psrticlss

Chap. Z

in which is the energy computed from mass m. Owing to destructive interkrence, the amplitude of this oscillation will drop substantially below its initial value when (P~/~)(I/P)* (1-4*37) This is an unstable particle decaying into several other particles with a proper lifetime -l/I".

c. The function f ( M 2 ) varies smoothly. In ths't part of the spectrum several gartieles are present, with no tendency to hcome associated in a single unstable particle . Some aspects af field equal time commutation relations can be extracted from the comelatiorr funetion. Thus,

where exp[ip.

(X

-- X')].

(1-4.59)

If the field ~ ( zis) a fundamental dynamical variable, its equal time eornmutrztion relations have a, kinemaitical bwia, I"cs typical af a sedar field thaL and the fatter implies the sum rufe

Imagine no\' that the field X ( % ) is uncoupled from all others, and then. obeys a linear digerentirml equation that gives f ( M 2 ) = b ( M 2 - m:), Supipom that when the physical couplings are restored a stttble particle still exists. Its mass will be shifted by the interaction, m. -+ m, and f ( B f 2 ) will have multiparticle contributions in addition to the discrete mass term: f 6 ( d f a - m'). The sum wle thus requires that f < 1. If we are not inkrested in the details of the particular excitation used to generate the psrticfe, but wish only to describe the physical particle itself, 11-ediseard the eonfinu~usmass cantribution to ( X ( Z ) X ( Z ~ )and , comespondingly adjust the scale of the correlation; function by removing the factor f , This example has supplied the designation for the general proceduw that transfers attention from the fundamental dynamieal geld variables to the derived phenomenalogical ptarticle level. It is called renormalisation, More elaborate field earrelation functions give information about the inkraetion of particles. Consider, for example,

~vfierethe various fields are the algebraic combinations of the fundamental dynamical variables that contsin the pecrticles, a, b, c, d, respectively, in their excitation spectra. Xn order to refer to the pa&icular rertcticm. c d -4a b (we do not consider here properties like electric charge that i n t r d u w the distinction between partielee and antipsdiclw), the regions in. which the coordinates are placed mmt be appropriably &own. The points x and z q i e far in the future of z" and xf"', while z is widdy separabd spatially from z' as ia zrr from X'". Under these circumstances the renarmalization prowdupes that isolate the physical particles can be applied independently, and the resulting function of particle properties supplies the amplitude for the physical reetetion. NOWrecall that the dynamical variables of quantum mechanics fa11 naturally into two sets .tr.hich, at a given Lime, exhibit commutstivity or antieommutativity, respectively, between a, pair of variables referring to diEerent degree8 of freedom. I t is points in space-like relation that play the Xathr role in operator field theory. -If x and s h r e suaieiently separated spatially that detailed cornposite structure is not involved,

+

+

tvhere n,, nb are the respective number of anticommuting fundamental field variables used in constructing the operatom ~ ~ ( $ ~1~, ( z ' ) )Only . the even or oddnegs of the inkgers is signifiesnt, and the sign fwtor is 3-1 except when both n, and nb are odd. Should a .and b refer to the same pa&icle type, the field operators amciated with a pair of spatially separated poinb eammub or anticommute according as the number of anticommuting fwdamental variablw is even or odd. The two-particle state produced from x(z)x($")l ) by renormalisation is correspondingly symmetfical or antisymrnetricaf in the pa&icle variables, qpropriak to particles $hat obey Bose-Einstein or Fermi-Dirac ~tatissfies, respectively, The same statir~tics-labeledproflrties enter in rel~tingreaeliom in which the rofes af some initial and final pa~iclesare inferchanged, Thus, the field correlation function

+

+

can be used ta derive the rewtian amplitude for b d -+ a e, If the two eamelslion functions we have mention& %re known throu&oul the mulLiple spaee-time domain, they are known for the re@onswhem xt aand 2" are in like relation. But there they are equal, or diEer by a minus sign, depending upon the statistics af particks b and c. According.Ey, the two function8 are space-time extrapolations or continuations of each o$her, The impfied connections among digereat reaction amplitudes are usually referred to as eroming relations. Dynamics is explicit in oprrtlor field theoq, It is conveyd by the nonlinear s t m ~ t u r eof the field equations obyed by Ghe fundamelltal dynamical va~ables, They, in turn, imply equations carmect;ing the vafious &Id eorrelatiort funckions

84

Particles

Chap, l

from which the latter can be constructed, in principle. Two radically different situations occur, in practice. In the first, the interaeticlns are suaciently weak that the particles of interest appear in the excitation spectra, of the fundamentstl variables themwlves. This is the ~zssumedsituation in quantum electradynamics where the particles are the photon and the electron (or muon). The equations obeyed by the field correlation functions can be solved by ~ r t u r b a t i v eor iterative methods bawd on the mallness of the eharaeteristi~coupling constant a = 11137.Q.8. The result@may be presented in, twa different ways, a t thct unrenormrtlized field stage, or at the renormalized particle leveicl, The fieid version contains divergent integrals, the renormalized stabmenti3 are finih an;d in exceptional agreement with experiment, The fairly rapid convergence of the renormalised expressions means that experiments do not probe to very high moment%or very short distances. The underlying hypothesis of operator field theory concerning the conceptual possibility of descriptions at arbitrarily small distances remains unksted by the available evidence. This hywthesis is involved in the unrenormalized, field results, but whether &henonexi~$enceof the expressions signifies the failure of the hypothesis, or merely the inadequacy of the perturba'cive ealculational methods that are used, is premntly unknown, It may be that opemtor field theory is unnecessafily dognatie about the physical silpificanee of arbitrarily small volume elements. Totafly new concepts might enter a t very large momenta, without altering the practical succesws that have been obtained. The other situation is that of &rang interaction physics. Here %hehypothesis that whole families of particles are the dynamical manifestation of a feu?fundamental field variables excludes the possibilit;y. that the excitation spectrum of the latter contains the known. par%icles. These objects must b generakd by combinations of the basic variables, Being iporant of the underlying dynamics of the fundamental dynamical variables and lacking the computational methods that could @v@the consequences of that dynamics, if it were known, one musk fall back on speculations concerning the composib field structure of the known partieles. And such speculations become interfwind in the more immediate problems that are presented a t the phenomencrlo@cd level. Is it not possible to separate particle phenomenolqy from speculations about particle stmcture? 2. If the particle is the ultimate structure, no detailed description is passible in

regions of intense inbraction where the eharzteteristic additivi* of independent particicle conhibutions eea~esto be valid. All that can bc3 done is to compare Lhe state of noninteracting parcrtieles afker a collision with the s t a b of the generally diffe~entnumber of noninteracting pa&iele~prbr to the collision. That relation is an abject of calculation in o p e r w r field theory. With the present viewpoint it is the fundamental quantiLy which, through its postulated prope&ies, gives the?eomplete statement of the miera~~opic dynamics of particles. When we write

Critique af particle theories

1-44

36

we introduce a transformation from one complete set of noninteracting particle states to an analag~usset. Hence the operator S is unitary, 8%

S

88"

1,

(1-4.66)

It is invariably referred to as the S(cattering) matrix, perhaps because one is unable to handle more than a small set of matrix elements. Thew matrix elements are explicit in the probability amplitude The absolute square f (i is interpreted as the probability that the transition 2" i will oceur during an unlimited time interval. The abhorrence of any vestige of a detailed .temporal description restricts the obs62med padieles to stable ones. In the same vein, all refexrenee to spatial desehption is rejeeted and only the momentum speeificatioa of states, together with spin and other properties, is admitt&, Xf the particles h a g p n not to inkract, the appropriate S is the unit operator. Thus, more interesting than S is S -- 1, which o b y s the unitary restriction in the form Since a collision must respect the overall conservation of energy and momentum, we write

thereby defining the transition matrix T. Only the momenta have been made expfieit here, and the delta function is four-dimensionsf, The resulting farm of the unitarity condition is nonlinear, relating matrix elements of i(T - T?) to products of T' and T matrix elements. The probabilities of transitions must have a Lorentz invariant significance. Xt is therefore asserted that the S, and the T,matrix: elements must be invariant functions of their arguments, This is partieuIarty simple %.hen slf parlicles are spinfess since it requires that the ' I ' matrix elements be a funetion only of the independent sealars (we ignore the possibility of pseudoscafars) that can be formed from the N = n -t- nhomrnenta, which are individualy subject to the particle mass relations, -p: = m:. The number of such scalar combinations is 3N -- 10, where the subtracted number counts the Lorentz trrtnsformatioh parameters, Thus, in two-pa&icle reactions, where N = 2 2, there are only two indeCo energy and scattefing anlSfe in the pendent scalar variables, c~rre~ponding cenkr of mass reference frame. The constructive principle of S-matrix theory is the gostufab of analyticity. It is =sunned that the physical reaction amplitudes, in their dependence on the

+

scalar variables, are boundary values of analytic functions of corresponding complex variables. Since analytic functions are specified by the nature and location of their sinwlaridies, the dietemination of the I8tter encompasses all the physics that is admitted in S-matrix theory. Here are the words of some enthusiasts: "One of the most remarketble discove~esin elementary particle physics has been that of the existence of the complex plane,'' ". . . the theory of functions of complex variables plays the role not of a mathematical tool, but of tb fundamental description of nature inseparable from physics . . . .' T r o m the are distinguished tts viewpoint of analytic functions, the elements of T and boundary values of the same analytic function that refer to opposite sides of the real axis for the relevant complex variables, The resulting discontinuity statement is the condribution of the unitarity condition toward determining the aingularities of the transition matrix. But the postulate of analytieity also widens Lhe scope of the unikarity condition to include the so-called crossed reactions. The conversion,of an initial or incoming particle into s final or outgoing particle is formlly exprewd by the substiLution pp + ---p@, as judged by the contribution to the net energ5r-momentum balanee. This is to be achieved by rtnalyt;.rc continuation, and the unitarity condit;ions for the various reactions that are in. crossing relation give singularity information. a b w t me analyLic function in various domains of its complex va~ables.As to whether this kind of information 8uEces: "The S-matrix is a Lorentz-invariant analytie function of all momentum vafirtbles with only thow sixrmlarities required by unit~rity." There is na explicit statement of dynamics in S-matrix theory. And the p~sibilityof regardiw some partieles as fundamental and deriving o t b m r;ks b u n d states is rejected as an urraeceptabfe stmeturing of the padicle concept, distinguishing elementary and composite particles. To prevent just such a distinction being made, it is proposed that, no matter which particles one uses to construct composites, the same total particle spectrum emerges. This view of dynrtmi~slself-consistency is usually rehrred to as the "bootstrap" hypothesis The discussion of Section 1-3 indicates that S-matrix Gheory is too dogmslt3c in digmiming all reference to microscopic space-time description, Whether or not one wishes to recognize it, the! structure of the Lorentrr;group itself gives meaning to spatial loealizability and temporal development, outside redons of intense inkrzletion. The very nature of a collision involves a memure of spseetime causal control, and the existence of even a limited microscopic space-time dewriflion indicaks Lhat cacusality is not likely to be restfieled to macroseopis circumstances. It is widely recognized that the intuitive physical property of causality in spaee-time must underfie the abstract mathematical asartion of analytieiQ, Should one not be able to exhibit and exploit causdity as s eoustructive principle, thereby relegating analytieity to a secondary, derived role? And as for the b8sic hypobbesis of S-matrix tl.leory, that the particle is the ultimate unandyzable entity, we again ask: Is it not possible to separate particle phenomenology from speculations about partide structure?

SOURCES The critical eornmentrs of the last section set the stage for the introduction of a new theory of particles. I t is a phenomenologieal theory, dmignd to describe the observd particle@,be %heystable or unstable. No speculations about the inner stmeture of pa~ictesare introduced, bu"che road to a conceivable more fundamental theory is left open. No abstract definition of particle is devised; rather, the theory u8es symbolic idealization8 of the reali&ie procedures that give physical meaning to the particle wncepL. The theztry is thereby firmly grounded in %p&ee-time,the amna within which the experimenter manipulates hia tools, but Lbe question of an ultimate limitation to miero~eopiespace-time dmchption is left open, with the decision re~mvedto experiment. Correspondingly, no operator fields are ursed. The compbmentary momentum-space description plays an important role, but the possibiliw of ultimate limitations on this space is not exciudd, and there is no appeal to andytieity in momentum space. The constmetive principles of the new thwry are intuitive one&causality and uniformity in space-time. What emerges is a thcsory inkrnedict-te in position. between operator field theor;)r and 8-matrix t'fimry, which rejects the dogmas af each, and gains thereby a caleuXationaE erne and intuitiveness that make it a worthy coatender ts displace the mrlier formulations. The range of the term "particle" h.m been systematicalfy extended by expe~merrtaIdiscovery. From the stable electron. and proton, to the Iong-lived neutron, do the rapidly decaying ?r and A, to the highly unstable p and N*, there has k e n a progression to more energetic and sfiorbr-lived excitations* It Is now the normal situat.ion that a particle must be creawd in order ta study it. And, in a general sense, that h also true of the very high energy stable particles produced in accelerators. One ean regard aff such creation acts 8s cofli~ions. The emence of such. a collision is that it occupies a finite, and to some degree controllable, space-time reeon wherein other particles combine to transmit to a particular one thee properties that call it into exisknce and uniquely characterizie it. If i s psrt of the experimenter's creed that a new resonance not be admitted Lo full status as ra particle until it h a been ob%rved wifh the same charae-t;eiristies in a number of diflerent reactions. Thus, if a pa&icle is defined by the collisions that er-ettb it, the details of s sgecifie reaction are not; relevant and one ean idealize the role of the other particles in the coIlision, recognizing th& their funetion is solely to ~uppjtythe needed baiance of properlies-they eonstitute the source for the particle of inderest. What survives in. the idealization is a 37

38

Sources

Chap, 2

general specification of the regioxr in space-time where the source is effective, lvith some numerical measure given by a function $(X), and a statement of its ability to produce various momenta, as measured by a functian S(p). The ttvo source functions estlnut be independent but must convey the quantum mechanical eomplementarity bettveen these descriptions-the more detail that one possesses, the less is permitted t a the other. We have spoken of particle creation, but equally important is particle detection. This is invariably achieved by transmuting the parLieXe% properties into other more easily handled forms, I n iil. general sense, the pa&icle is annihilated by the process of detecting it, Were too are collisions with their controllable space-time aspects which, in principle, involve the same mechanisms that create the particle, The receiving radio set unavoidably radiates, the r-meson created in nucleon collisions is captured in nuclei. Long-lived p a ~ i c l e s may decay, and thus be detected, by mechanisms too weak to be useful in creating them, but this option can be overridden a t the choice of the experimenter-the neutron is not generally observed through its 6-deeay. The callision procgsses used to detect a particle can be ideaiiaed as sinks wherein the particle's properties are handed 0x1, in zt, tvay that permits some memure of spsce-time zznd momentum descriptiox~,but sink and source are clearly digererrt aspects of the same idealizrttioxr arid we unite them under the general, heading of '4~ouree.p' We now proceed to give the source concept a quantitative frannekvark, beginning with the simple situation of stable, spinless particles. 2-'l

SPIRI Q PARTICLES. WEAK SOURCE

The elementary acts ta be represented as the effect of a source are the creation of a single particle tvhere none existed previously, and the annihilation of that single particle. Since the actual presence of other particles in realistic collisions is abstrackly portrayed by the source, the s t a k s appearing in corresponding and ( o + ~ I , ) ~are: , /Q-), the vacuum quantum mechanical amplitudes, (1,\0-)~ state before the operation af the source K; (Q+[, the vacuum s h t e subsc;quent to the operation of the source: (sink) K; and (l,j, II,), describing a single particle state in lvhich the momentum is specified rr-ithin a small volume element (dp). The connection of this discrete labeling with, the continuous variable specific&Lion of momentum states is The individual creation and annihilation acts are not analyzed; the source is defined as a measure of the whole process, as suggested by (~veanticipate a particular variable factor)

Spin O particles, Wwk source

2-1

31)

Subscripts appear here, kmporarily, to distinguish sources efieative in emission or absorption. The designation "weak source" means that the definitions are appropriate %?henprobabiliw amplitudes referring to creation or annihilation of several particles are relatively negligible, We procecf t;o make these definition8 mare precise. The states ( p ! , Ip) refer to a particular time, or, more covariantly, a spacelike surface, f f the origin of the space-time coodinate frame is diqIaced by P, eorrmpanding staLrss are produced by the unitarry transformation: Since these states;play the analogous role in the new eoordinah ssystern, they are assaciated with a space-like surfaee that is displaced by XQn the initi~lcoordinate system. But all that is significant in the probability amplitudes ( p l ~ - ) R and ( ~ + j ~ is) ~the relation between the space-like surface and the space-time localization of the murees, for the vacuum staks are invariant. Equivalent to the displacement of the surfaee by X" is the rigid dilsplaeement of the source by -Xp. This is expressed by K --r X , where

R(x) K ( z -4E

X),

(2-1.4)

or

X(z)=K(z),

Zfi=z@-X'.

(2-1.5)

= eiPXK,(p),

R.(p) = @ - ' P X K a ( ~ ) ,

(2- 1.6)

Now,

which shows clearly that the relation between the complementaryr coordinate and momentum source desriptioas is given by Fourier transformation:

The space-time coordinates in these exponential functions are referred to an origin located in the space-like surface, but we shall not usually make this explicit. We consider next the behavior under hornagenctous Lorentl; tr;amformations. The respans af the single particle states to infinitesimal transformations of this nature is given by = i ( p l ( G o *J Sve N), &[p)= --ifsw J Gv N)lp), where [Eq. (1-3.29)] g

+ +

In exhibiting N we have set z0 = 0,sinee this is the origin of time, and used an alternative form of the symmetrised product of r with The coordinate

SQ

Sources

Chap. 2

opersltor in the momentum dese~ptionis represenkd by (2-1.10) and therefore

with an analogous formula for (p0)"21p). H ~ v i n ganticip~tedthe square root fmtsr, we transcribe this as

vvith a similar statement for K,@). The implied infinitclsinnd change of H,(%) or Ka(x)

if3

+

6K(z) = 160 r X V 6~ * (ra, = 6%" (~,K(z)~

+ Z'V)]K(Z) (2-1.13)

where 6z" = 6 d r x , .

(2-1. f 4)

fZ(z) = K ( z t 8 4 ,

(2-1.15)

This result, or

R ( z ) = K ( ~ ) ,~ " = z ~ - 8 z ~ ,

(2-1.16)

when conzbined with the displacement response, assrrrb that the soume functions of qinless pafticfes, K ( z ) , behave .dsr scaler functions under the tramformalions of the Larente poup. An important corollary is that the ehoiee of K(z) as a real flanetion would have a Lolrent~-invariandmeaning. That is in, sharp earrtrast with the nonrelativistic situation, where N -+ -mr and'p + p2/2n. Then, if we consider on1y boosts for simplicity, and The implied form for finite transformations is

R,(r, t ) = exp[v . (= exp(--imv

imr f tV)]Ke(r, t ) r -- i&mv2t)~(r vt, t )

+

(2-1,19)

R.(F, 1) = exp[-i(mv. r - fmv2t)]Ke(r, t ) , F = r - vt, Evidently, a red emission or absorgtion source \.~louldhave no Galilean invariant rne8ning. Incidentally, ia carryiw out the evduation af (2-1.19) we have used

2-1

Spin O particles. Weak source

41

a simple example of the formula ( p = - i d / a g ) : The precise relationship betiveen the ernissioai and absorptioxr abilities of n source is disclosed by combiriing the arthogonality af the vacuum srtd single particle states, prior to the intervention. of sources, with the completeness of the various particle-specified states that refer to the final situation. Tltus,

in which the additional terms are negtigitale urrder weak source conditions. Furthermore, it suffices to use source-free values for the factors of ( 0 + [ 1 , ) ~ and (~-jl,~ = )( ~ I ~ # \ o - ) ~namely *~ expressirrg tlre ixlvarianee of the vacuum state, and apart from phase factors \\.hick serve only to ensure dllat, in the resulting relation, ( o + ! ~ P= ) ~-- ( l p l ~ - ) ~ * , (2-1.25) both single particle states are referred to the same space-like surface. Tliis connection bettveen ereation and arlnihifatioxr probability amplitudes earl also be presented as i(o+i = [i(lpl~-)K]*. (2-1 .xi) Thus, with a permissible choice cf arbitrary phases, the source functioxls K,(s) sad K,(%) are reciprocal complex conjugates. The simplest possibility, and thc one with r~hiehr,ve begin, is a real function, We. now unite the various detaiis and state our explicit defini-

@P) 1 K , = ( ~ U , ) " ~ K ( ~ ) ,dw, = ( 2 ~ 1 32p0 and The experimenter's bbasic tool is a beam of particles. A very weak beam of ~pinfeljsparticles has the following causal reprrasentation. We begin .rvith tbe vacuum slab. Then rt weak soume IK2(x), occupying a finice apace-time region, goes into action. It most often does nothing, with the associated probability

4

Sources

Chap, 2

-

amplitude ( 0 + 1 0 - ) ~ 2 1, and occasionally produces a single particle, as characterized by ( l , l 0 - ) ~ 1 . After the emission source has ceased to operate, the re~ultingvacuum or single particle state persish unehawed until we e n k r the space-tim region of an absorptian source K l ( s ) . Its effect in detecting a single particle is described by (0+j1,)~1, and we thus return to the vacuum state. The eompleee pmeess is represented by

where vacuum state sulbsc~ghdegignating causal aecyuence are slws?ys relative the ixrdieakd sources. The individual vacuum amplitudes h&vethe form On, making explicit the single particle creation and annihilation probability amplitudes, we get

where rt, refers to the weak 8ouree limitation. The functions K l ( z ) and K2(x> are the disjoint parts of the tots1 source in this situation, which is given by There should be nothing in the overall description to distinpisk one component part of the source from anodher, aside from reference to the space--time region that it occupies. This is space-time uniformity. It implies that ( o + ( o - ) ~ depends only upon K, and in the manner made explicit by the hilinear structure in K l and K 2 . Accordingly, we write

The displacement invariant funetion b+(z - z'), as the kernel of a quadratic form, can be chosen symmetrical with no loss in generality, The two equivalenL ~ontributionsof the type &;K2 then aupply the ~trueture of h+ for a caustzl arrangement:

= [We recall that @' is the energy-momentum veetor of a particle, so that +(p2 m2)1'2.] From these characteristics of h+(%- g'), we deduce that

+

Spin O particla. Week sour~s 43

2-1

The explicit constructions of A+(z - s" may appear to refer only to c~lusal or time-like relations between the points z aad d.But in fact the_ygive meaning ta this funetion. everywhere, The only possible dificulty would be that, when z rand sf are in spsee-like relation, where causality hits no invariant meaning, different values might be obtained depcfing Upon the choice of coordinate system. This does not happen. Since dw, and eke"'-"' are invariant structuresp there is no harm in choosing a coordinate system for which z0 = so', and

is indepndent of the ambimous sign, for the integsal depend8 only upon (X - X')%= (z - z ' ) ~ .As a result, there is no longer any indication in (2-1.35) of the initial causd arrangement of sources, and that stmcturr? is applicable to an arbitrary disposition of ~ourcm. This space-time extrapolatim must meet a mvem k~t, however, We are nomr able to compute the probability that, despite the intemention of the sources, the vacuum state persists, It is (dcc)(dz")K(x) h ( I/i)b+(z - z")K(z'),

(2-1.40)

where is vdid e-rrerywltere,and the reference to the real part is uranecesssry since it is implicit in the symmetry of the quadra;tic form, But probability considerations also demand that

The challenge is successful1y met, for

There is one conceivable modification of b+(z - X') that would appear to retain the necessary physicd charaeterislics, It is the addition to &+(X - z') of a red funetion, which difiers horn zero only when (z i~ space-like intenrd. That would contribute nei$her to the causal exehange of partiGIes between saurces nor the computation of the vacuum persistence probability. The hypothesis af space-time uniformity, forbEdding the existence oT speeigl , relatianshi~pebtween: source-s, excludes that possibaity, In .(ih.b ~ o n k x t OM can give the- ukformity hypothe~isa more precise, if rather ab~traet,form by considering the four-dimensional Euclidean space that is a t i t a d d to the Minkowski space through the ttomplex transformation

Chap. 2

Sourcsa

There is no andowe in Euelide~nspace to the Minkowski distinction between lime-like and space-like inkmal~. AecordingIy, special space-time stmctures would l x rejected if one insiskd that the invarianf vaeuum amplitude Chat dmeribea a compfek physied process continue to be meaningful and invaxciant on, mapping the Minkowski spaee onto the EucXidean space, This is the Euelidesn postulate. We recagniz-e that the Euclike~npa~tulateis a n8tural one by noting Ghat 6 4 % - z') has the require$ properties; there is an assrociatd Euelidean invariant function, AE(s - 2') which. exist6 &most everywhere (z # &). X t i s obtained from the i n k g a l reprewntation

by the 8ubsdilut;ion ilzD -

-4

lz4 - z:I,

\vhieh requires that the ordering of the real numbers zO,zO'is mapped into the same ordering of the real numbers Q, X:, We remove a factor of i in &fining

An explicitly Euclidean, invariant form appears on using the inkgraf relation

in which Ihe notation, ignoring any distinction betwwn contravariant and eovariant eomponenls, ennphasi~esthe Ertctidean; stwcture. With Lhs recopition that AB($ -- r*)is a Euelidean invariant function, degenden%only upon

can r e t m to (2-1.48) and cltome the Euelidean coordinab syrstem ta get the real positive expression

whieh i s one among a variety of single-parameter intepal representations. This

2-1

Spin O particles. Weak eourca

46

one immediately supplies the two limiting forms

Mote aho the simple inequality

Even better, since it reproduces the eorrecf limiting forms, is the inequality

One can; connect the Minkowski and Euelidean. descriptians by equating the source strengths associated with corresponding voluple elements

while maintaining the reality of the souree function. This gives

and the right-hand side is a real number, which, is less than unity. The physical vacuum amplitude can also be reeo.verc3.d from the Euelidean version, by the earnpitex substitutions 2 4 -r

iz0,

p*

-+ -ipop

(2- X. 58)

provided they are understood to mean the limit of complex rotations angle approaches lr/2 from smaller values, z',

p4

-r

M

the

exp

Sueh cautiaxr is necessay since the resulting M i n k o ~ ~ sstructure@ k have singularities: (z - 2'j2 = Oj the light cone singularity in coordinste space; p2 m2 = 0, the partiole mass shell singularity in momentum space. We find tfiat

+

(p,)2

+ m'

-+

p2

+ m 2 - P:(I + 2 i e ) = ppr f m2 - ie,

(2-1 *W)

in which, despite various scale changes, E retains its meaning as a, prameter thaf

46

Sources

Chap, 2

appromhes Eero through positive vafue@i. The resulting four-dimensional mpmntation of A+(x is

(2-1 .fix)

where

intmduces the Cauchy principal value for integrals. evaluatbn

The contour integral

reproduces Eq. (2-1.45). A limiting form in coordinebk spa= is

Asymflotic h s appmpriate to large 8eparwLion will be statecE for space-like intervals, [(z -- z') = E > 0,and for time-like intervals, [-- (z - z ' ) ~ ] " ~ = l' > 0,although they are eonneeted by the substitutions Ii! ++ il':

We turn next to the mom general situation in which Kfx) is a complex function. Nour the sources eRwtive in ernimion and sborption art? reoiproeally e ~ l e conjugate x funelions, If we did no mom than introduce that featwe into tht? previous dictussion, the single-particle term in the constmctim of the eomplek vacuum amplitude would bmme

But this is dearly incomplete, far the implied soume stnrc%ure,Xinear in

2-1

Spin O particles. Weak source

47

dso requires the eontributictn of the causal term

referring to the ernis~ionand subsequent absorption of another kind of particle. What is the mass of this particle? If the two masses were unequal, the stmcture of the new h+(%- z') function in the vacuum amplitude

would still be given by (2-1.37) and (2-X.38), but differcent masses would appear in the two causal forms, Then we could no longer conclude that B+(z - 2') had a unique edrapolation into spaee-like regions. I t is the principle of spacetime uniformity that demands equal masses for the two kinds of particle, which are identified as particle snd antipartick. The Euclide&n postulate produet38 the same conclusion through. the absence of an invariant distinc-t;ion btween 24 - Z : > O and 2 4 - 2; < Q, which permits only one mass pammeter to appear. In view of these remarh the definitions that relah sources to single-particle production and annihilation prObabi1iw amplitudes mu^& be extended ta where & distinwish parCicle and ranfipadicle, and Note csrefulf y the distinction between

Accordingly, we have

Thus, the explicit appearance of the p or --p Fourier transform, representing the energy-momenhm balance, distinpi~hesenniasion or srbsorptcion, respectivety, while K and K* identify particle and antipadicle in emimion, but csnversely in absorption. The function of K is to ereate particles and annihilate antiparticles, while K* creates antiparticles and annihilates particles. In analogy with the way that sources act to increase or decrease the amount of energy in the system, we can conceive of K and K* aeting to increase and decrease, respectively, the quantity of s. property which must assume opphsite values for particle and antiparticle. This is a familiar ohamebri~tieof eleotric

Chap, 2

Sources

48

charge, and we rwognize that some chargelike p m p e ~ ydways dbtinguighw pahicle and antipadiclt?. The formal countemad of them remarks &ms from the invariane of the vetcuum transformation funetion under phwe transformations of the c o q l e x Bources,

If we examine the response of the probability amplitudes (1,*10_)'

to the= p h s e transfomstions, combined with a rigid displacement of the source, by X@,we get ( ~ ~ * p--,-ef) i@eipX(lp+lO-)X, ~ (2-1 '77) whieh makes explicit the mehanieal and "charge' attributes of the singleparLiele ststes. An dterrtative presentation is obtained by replacing the complex souree K ( z ) with two real murees, a c c d i n g fa

K ( z ) = 2-1'2[K,i~(z) - iK,,,(z)I,

+

K t ( z ) = ~ - " ' [ K , ~ , ( z ) iKll)(z)b (2- 1.75)

This @ves

We have now exhi*bit& twa indepadent murces, with their wsoeiated particles, But the fact that these particles have the same mass (and spin) implies that the dwompo~itioncan be done in an infinib variety af ways, camesponding to the p h w transformation8 of eomplex sources, which naw appear as two-dimensional Euelidean rotations :

The Xratkr can dso be wfithn in matrix notation m

and

identified as the e h ~ r g matrix. e Nob that it is imaginary and antir~ymmetrieal. Its eigenvalul~tsare k l, and the complex sourees K(%),K*(%) are the colme-

i8

Spin O partirzles, Weak source

19

sgonding eigenvectors. The real sources

do not produce single-particle states of definite charge. They refer to the eomplementary property of charge symmetry-the states turn into themselves or their negatives, respectiveIy, when positive and negative eharges are interchanged. A matrix presentation of this transformation is

K(s)

+

rqK(z1,

where the real matrix has the property TqQ

:T"":

-Qrq.

The symbol C is often used for this charge refleetion operation, When two-component matrix notation is used, the vacuum amplitude has the same formal expression as tvith a single real source,

This remains true of its Euclidean counterpart:

which ettn also be written in terms of complex sources,

Euelidean. transformations decompose into two conneeted pieces, distinwished as proper and improper trsnsforrn&ions. In, contrast, the full Lorent~ group contains four connected pieces, owing to the discontinuous causal dislinction between z0 > O and z0 < O. The wider invariance introduced by the Euclidmn postulate thus enables one to perform some discontinuous LarenL~, transfornnations through the intermediary of continuous Euelidean transformations, The mosL important example of that is

a proper transformation, which is rz time refleeticm transformation in Minkawski space, The formal invariance of the vacuum amplitude under the transformation is an immediate consequence of the symmetry

fSO

Sources

Chap. 2

but it is the Euclidean postuiaite that suppliea the general btzsis for this inv a ~ a n c e , The reflmtion of the time coordinate inverts the causal order of sourem and inbrchangrzs creation and annihilation. This is evident from the momentum form of the source transformation,

K(P)

+

K(-P)?

and thus

K,,*K:-,

K,--K:+

or According to its csnstmction W a time refleetion (I')and a space reflection (F-arity), which also hm the egect of inhrchanging parCicle and %ntipa;rticlea charge reflwtion (C)-this process is often known rts the TCP opration, but one is Hkely to encounter any other permutation of the three constituents. Perhaps it should be called the Shell g a m . 2-2

SPtM O PARTtCLES. $TRQNG SOURCE

The experimenter's beam cont2tin~many particles at a @ven time, which exisf under conditions of effective noninteraction since they are widely s p a c d on the sede set by nniera~opicinteraction distances. A beam of dectricaIEy charged W i e l e s is sn exception to this, in grineiple, but in practice the disturbance by long-range inkractions ean be made sufficiently smdf by controlling the barn. densi$y. We give a fheoretieaX Lranscripfion of this situation by exploiting the dimetiondity that sroureea possess as an aspect of the complemenla~tybetwen the K($)and K ( p ) repremntation~. A Source that i8 spakially diEused and suitably phmed (to use antenna language) ean p r d u w a highly directional h a m , s f i a ~ l ylimiting the possible lacations of a detection Bouree if eAt"f:ctive caupling i s to be achieved. We visualige an arbitrav number of such paim of directional weak emiisaion and absorption sotrmes, operating side by side with negligibk cross coupling. If roughly the slsbmrt clbusal amangement is us& far all the pairs3 of gourees, ~ \ n ' ( 3have produced a situation in which, a t mme time intermediate htween the emiaion slxld absorption regons, an arbitrary number of padiclczs can exist in eireumgtanees of noni~terae%ion,owing to the 8p%%ial wpslr&tionsamong them. We first ~ o n ~ i d real e r sources and de~ignateby &(g), a -- 1, 2, , the individual weak saurce~that correspond one self-contained emiwion aad absorption process. The physi~alinclependence of &ese various aets, which has been achieved through our control of the sources, is expressed by imply mutliplying the individual probability amplitudes to produce the v w u m amplitude for the complete anangement :

. ..

(d.)(dg')K,(z)d+(z

- xF)Ka(rF)

2-2

Spin O particz-lss. Strong murm

@l

The sourees Ka(sif are di~jointp&& of the tot81 8ouree

The principle of space-time uniformity requires that no specific distinctions among the campomds of K ( s ) be admitM. I n short, the vacuum amplitude must depend only upon R($). This b r d i s e d by imrpording the property

which asseds fhe &bf~enceof coupling between diEerea6 singk particle exchange re@ons. Then, since the individual sources are weak,

The slam form ztpplies to two-component real sourees, and for empXex source8 if bwomea

We accept t h e wponential vaeuum awliLude ~&rueturcss as dmriptive of any arrangement of sources, with arbitrary gtrengLh, subject only to the festriction thaf the psrficle~have na efiwtive interrtolion. To tat the consbte~cyof this sssedion we corrsrider a- gimple causal arrangement, e x p r w d by in which. \cte maintain. $he convention that Kl refer8 60 p h y ~ i ~ &et8 d that occur a f b r the cornpletlon of those repwanted by Kz. For the situafion of red sources WIEShave

where, tzceording to the causal disposition of the ~OUPGW,

The ems& arrangement also ennsbles us do analyz;e the eomplete proem into an initial multiparticle emission act, represented by the probability amplitude

((E)l 0 _ ) ~ 2 , and a subsequent absorption proeess, described by (O+l { n ) ) " ~ ~ where {n) indicates the collection of physical sttributes that distinguish the various n-partiele states. The resulting causal snalysis of the vacuum amplitude is (2-2.10) ( ~ , l o - > ~= C (O+ W n))Kl({E)IQ-)Kz* In 1

To display, in this form, the explicit structure (o+Io-)~ = (o+[o-)~'exp

(O+~O_)~.,

(2-2.1 1)

we have only to introduce the mpansion of the exponential funetion

This supplies the required identifications

where the multiparticle label is realized by the collection of integers {B,). The avidenf interpretation of n, is a particle o e ~ ~ p a f inumber o~l a~aciftt;edwith the indicated physiesl properties. This is confirmed by the response of the multiparticle states to the source translation K(,) + K(% X), nvhich gives

+

((3 Io-)~

eiPX((n)Io-)~,

(Q+/ (n))K

+

(@+l{R)

(2-2.14)

The btal energy-momentum thus oblained,

displays the additive contributions of the particles present in the state under consideration. The probability amplitudes must meet the following total probability or complekness test :

and indeed

2-2

Spin O particles. Strong source

63

No& how the vacuum amplitude has been used in two distinct ways. Through the consideration of a causal arrangement, relative multipartick annplitudcs are obtained :

and the assumed complekness of the multiparticle states leads to

Then the vacuum amplitude is applied directly 8s a probability amplitude, with consistent results. The extension to a pair of real sources, or the equivalent complex source, is immediate, The summation over momenta in (2-2.12) is norv sutpplemenbd by zz summatian over the two kinds of particles and the results are analogous, as in

where q == &l is the charge label that distinguishes partiele and antiparticle.

A combined source trainslation snd phase transformation changes these states as follows : ( {n)1

0 ~ ) --+ ~ eiQveiPx( $4 Io-)~?

(O+I jnj

)K + (O+I {R)) R e - i Q p e - i P X

(2-2.21)

where

Q

=

C %,,g, PQ

=

C PQ

%p,$

(2-2.22)

exhibit; the h t a l charge and energy-momentum attributes of the mulLiparticle s t a k labeled (R).

The momentum labcling of individual particle states is naf the only passibility. A spherical or angular momentum specification is introduced by the transformation K, = (dn)'I2 Y~,(P)KP@Z~P

C

where

Im

and jr, Vz, are standard symbols for spherical Bessel funetions and spherical harmonies. The discrete angular momentum quantum numbers

cation * G b h the infiGhiaaaf ~ I i m d gXw da. Thus

md one has anly to ohan@ the labis in. (2-2.13) ia obdsin tfie souroe mpm wnbtion of tbe ntsw mdtipartielie &W. The p n e d i s a t i o ~fr, wmplex is &km immdab. A m d n g to the uthd and@depnde~liwsf which is em(a"w), the sowee robtiorr indie~Mby

&pIiays the bbI mam&k quangurn numbr of the multipartide

The TCP rttfation btwwn e ~ s and b a h v t i a n now appam tw wEch 8 b M h r F-1 mwem; o t h e k w 8 refemnce %a@hawmfietion is dd&* There is an aial de~c~ption, which is abbined fmm

whae XJ. iS the projwW cmrdinak w b r in the plane pt3vn&calar b the: and p indim* appmp~atemimufhal angle8 abut thw &g, The TCP opmtian &km the fom The mehawe of

a m y e&nnaturdly b aXm. Dimt power wriw mpamion @ves

spin O particlea, Strong murca

2-2

Bg

(d.)(dd)K (x;>&+(z- z')KZ(g')

where ~em(sfi+(zi- ~2)=

C

n! perm.

A+(zl

-

g

* *

A+(%

--

) . : g

(2-2.37)

defines the pemsnent, a determinant without minus signs. Clearly displayed here are the e h t i v e sources for n-particle emkion and absorption, together with the funofion thaf represent8 the noninbrac$ing propagation of the n peiebs, The fatbr ia syrnmetfigd among tfirt spwetime m r d i n a b ~of the particles which, together with the unrestricted occupation numbers a, = 0,1, 2, . . . , proclaim8 that we are dwribing pa~iclestbspt Bo~eEinstein stathties. The f a ~ l i acharaokristics r of this statisdies are dso appwent in the answer to $he follawiq qumtion. Whad are the vduw of the general prob&bility This is to ask how the effeetivenegs of a Bource in emitting or absorbing particles is influenced by the prior presence of particles. We cornider the f o l l a ~ g gc ~ w %~i6uationl A- strong source Kg(%)aeLs first to creab pa&icles, vvhich are subaeqwntly idueneed by the probe souree Ko(z>, afkr which the particles are annihilated by the detmkian source ,Kl(z) :

The causal situation is &henrzlbr~ativelyconveyed by

fmm which the d e ~ ~ probability ed amplihdes, refer*g to the probts wwee KaI can b ofitfaind. We fimt eo~sidera weak proh &ad accordingly relain o d y linear b r m in KO, Then, aince

MS

Sourccas

Chap. 2

which refers very immediately to the rnonornial structure of t h w probability amplitudes, we get

which generalize the initial definitions (2-1.28) while retaining the weak ssuree limitation. I n particular, the probability for the emission of another partide,

show8 the additional stimulated emission, that is characteristic of B,E. statisf ics, As a preliminary to picking out the pneraX transition amplitudes, we construct the probability amplitude ( { R ) +l {n)- ) K , which has the same initial a d final configuration and, in. that sense, is a generdi~afiontof $he vacuum amplitude. This object is extracted by retaining only equal powers of K:, and K z , in the expansion

We are going to introduce a useful simplification here by recognizing that for sufficiently small do,,higlrer p w e m in this series are neglgble. XL is only the dependence on the probe source that is at issue:

We also remark that

which applies to each momentum cell independently. For the process of inter@@%, then, (iK:,iKop iKgPiKs,) -+ [l iKgpn,iKop]

c P

+

P

+

and we conclude that

({n)+i{n)-)K= exp where

The last term is rso bvritten in order to maintain the symmetry in x and 3'. The

Spin O particles, Strong source

fiT

explicit causal forms are

Note that the probability amplitude ({R)+l {nJ-)K reduces to unity for K = 0, which means that the initial and final multiparticle staks are r e k m d to the same time or spaelike surface, as is appropriate to a reasonabIy loealiged probe, To find the probability amplitudes in which initial and final ~ t a h are s no8 the same, we do not return to the general eonstrmeLion given in (2-2.40), but directly, in the manner of the vacuum amplitude. Thus, use (in)+l In) coxzsider the ectusal sou=@arrangement

K(z1 =

(4 4- K 2 ( ~ 1 ,

tvhich implies Chat

The coupling between the component sources is now given by

The explicit terms indicated for a given momentum describe the mverd processes in which, respectively, no ehange in particle number oecurs, an additiond particle is emitted, an incident particle is absorbed. Higher p w e m , containixlg more complicated procemes involving several particles, are relatively negligible for suficiently small do,; the probability for emitting two particles into the momentum range do,, for example, is - ( d ~ , ) ~ But . we must not let this apparently innoeent sirnplificaLion pms without comment;. The infinikgimd chzbracter of h, will be vitiated if there is an inordinate sensikivity to p produced by the appearance of very large coordinate intervals (e*'X). T o put it more physically, we recall that we are dealing with a beam of parlieleg interwting with a probe source. What we have done is eorreet if the probe is placed well in the interior of the beam where there is no ~ignificantposition dependence. It trill fail if the probe is outside or near the boundaries of the beam. This is a

68

Chap. 2

Sources

reminder that underlying any momentum description is s n appreciation of the spacetirne causal situation. Having understood i b limitations, we now use (2-2.52) ta identify the individual probability amplitudes for processes in which, independently for any number of momentum cells, a single particle has been added or removed. The result is indicated by

where the products refer to the various particles that are (e)mitted or (a)bsorbed. The two statements are equivalent, if it is &mitt& that ( { n 11, + I {n f 1,) -)K and ( { n )+l{n)J K differ negligibly, owing to the infinitesimal nature of do,. Now we face the consistency test associated with the alternative uses of the probabiiity amplitude. From the completeness of the fins1 or initial multiparticle sfates we deduce

+

n

~ ~ I ~ W + I {= ~ I - [ )l +~ (np I ~

+ ~ ) I K+~ ,I I~K ~ I ~ I

whereas, by direct calculation,

I(In)+lh)JKI= exp [-

1(dx)( d d ) K ( z )Re ( l / i ) A ,

ln +(X

-z3~(2')] (2-2.55)

and Re ( 1 /i)A l,

+(X

- X')

+

= ~ e / d o , ( 2 r c ~~ ) e ' ~ " - ~ * ' .

(2-2.56)

The test hm been passed successfully. The extension to complex sources and charged particles is ~traightfon~ard, and generally the introduction of the charge label q, supplementing the momentum index p, suffices to produce the required result. We o d y remark on the following detail. In constructing the probsbility amplitude ( { n )+l { n ) we are led, as in (2-2.46), to the factor exP

[gGPQ%~K~P,](2-2.57)

and the result ((n) +l{n)

[I

,

- &)K(x!)] - 23 has the

= exp i (dz)( d z ' ) K * ( z ) ~ ,,+(X

Now, however, the propagation function Al.)+($ meaning :

(2-2.58)

following

Spin O particles, Strong scccrcs

2-2

fig

which is no longer neeemarily symmetried in r and x'. There is still a TCP symmetry, in. ~vhichzfi -+-9 is c~rnbiil~ed ~vithcharge reflection, The explicit causal stmeture of this function is given by

It should be mentioned that the propaga;tion funetion is symmet~ealin z and 2' if the incident beam is neutral at every momentum, a,+ = B,-. Then, and and associate two only then, can one introduce the, real sources Ktl,, KZ,, independent particle types ~viththem, We have seen causality and space-time uniformity ~vorllcingas ereative principles. The physical requirement of completeness, or unitsrity, has then been verified; it is not an. independent principle, We shall nou7examine this relationship in more detail. But first we return to the vaeuum persisknce amplitude far real Baurces and consider it8 e~rnpfex conjugate :

o_)"* = (o-Io+)~ = exp

(dx>(~Z?K(Z>A-(Z

-xf)~(zP)

To ~ v a euniform presentation of the tvrpopropagaLioni funcfcions A*, everywhere, the positive and negative frequency functions A' as

which are connected by TXle tiro propagation funGtions Lhien. appear as

We note the everytvhere-valid ~ l a t i o n

\VC?. define,

The momentum inbgral defived from (2-5.45) 'by complex conjugation,

f e d to the same Euclideam. function EMbefore,

by means of the gubg%ituthn

Tben we have

whieh is the same Euelidesn form that is obtained from (o+~o-)~,as the strong wurce genera;liga$ion of (2-1.57). The Euclidean version of the veteuum amplitude is a red number lying in the i n t e n d htween 0 and 1. The vacuum. amplitude (o-Io+)~ i s regained through the substitution

Pot

which, incidentally, supplies the invariant rqresentation

Another canneckion between the reeiproealfy complex contjugak vacuum amplitudes come8 from the t?xi~%ence of the common Euclidean tranmription. Pmceeding &rough the Euelidesn h m as an intermedimy, we have

(I.

+ i15)pa

-4

-(f

- if.)^^,

snd, in respon~e, A+(z

- 2') -+---A-@

--- z'),

(0+

2-2

Spin O particles. Strong souroe

61

To verify this directly, we note that with the usual djuslable scale for r. One must also remember that the ordering of variables is retained throughout the transhmation. Limits of integration do not change, therefore, and

(2-2.77) while

(2-2.78) which confirm in alternative ways the stated transformations. We shall now use the causal structure of the theory to give a complete derivation of the unitstrity praperty. This is done within a very limited physical context, of course, but it is clearly a general procedure. For our present purposes we replace the causal labling K l , K z by K[-,, K,+,. (While this may seem .to be still another use of the overburdened signs, it will turn out to be consistent notation.) Let the time T be located between the regions defined with the by the two component sources. Introduce a new time coordinate for z0 > T by reflection at T, sncl then transform this time interval in, the manner just diacussed;

z', which is earlier than T to the same The immediate effect is to replace extent, %hatthe original time exceeded T . The transform& Kt-) souree also occupies a refiected position, earlier than T. Before this ontjration is p r f o r m d , the vacuum amplitude has the following composition:

where the appearance of the last term indicates the causal arrangement. When the transformation is carried out, the quadratic K(+, term remains unawarc! of what happens letttw, the quadratic K(-, term responds in the known manner [h+ -+--A-] without reference to the other terms, and the last term changes only by a minus sign arising from

the influence of the em"' factor, which is not compensated by the refleetion that is al80 being used. f n the latter the inkpation limit8 are reversed to maintain

82

Sources

Chap. 2

a positive measure. The result is ( O + ~ O - )+ ~ exp [-+i/(dz)(dz1)K(-)(Z)A-(X - d ) K ( - ) ( X ' )

+ / (dz)(dz')( - ~ ) K ( - ) ( z ) A ( + ) (-z z 1 ) i ~ ( + ) ( x ' ) ] (2-2.83) which does not depend upon T, that being any time after both sources have ceased operating. The physical meaning of this combination follows from exp [/(dz)(dzl)(-~)K,-)(Z)A(+)(X - zt)iK(+)(z')]

[F

= e . ~ ( i ~ ( - ) p )(W+)$] *

for, on using the fact that

({n>Io-)~'

= (0-1

(2-2.85)

we get (0-10+)'(-) exp

[/(dz)(dz')( - ~ ) K ( - ~ ( Z ) A ( + ) ( Z - z1)iK(+)(x')]( o + I o - ) ~ =

C+)

c (0-1 { n ) ) K ( - ) ( { nIOJK(+) ) (nl

( O - I O - ) ~ ( - ) ~ ~ ( +(2-2.86) ). As the notation indicates, the picture has become that of a system evolving in time from the initial vacuum state under the influence of the source K(+)(%) and then traced back to the initial state in the presence of the source K(-)(%). It is not the physical system that goes back in time, of course. What is reversed is the causal order of the states that are being compared. If the two sources are identical, we must regain the initial state; that is, K ( - ) ( z ) = K ( + ) ( z )= K ( x )

(2-2.87)

implies (O-l{n))K({n)10JK = 1,

( O - I O - ) ~= ~~

(2-2.88)

inI

a statement of completeness, or unitarity. According to the exponential structure (2-2.83))this is true if

where the last terms appear in that form to produce the necessary symmetry in and X'. We recognize the identity (2-2.67).

X

2-2

Spin b particles. Strong sourea

63.

The full @tatemeatof u n i l a ~ t yemergee on using the sources Kt*) to generate arbitrary multipsrticle states. We write where K z and .Kz#act prior to the source K, snd intfoduce the earresponding cauwl analysm : ({a)10JK'+' =

C

( {n)+l

What we mu8t verify is that all refemnce ts the source K(%) disappeam from (2-2.862, l e ~ ~ on1 n gy

for that is the u ~ b r i t ywertion about the eEeet 01 the source K:

Beyond the condition we have almady used, Eq. (2-2.591, w h d is required is

But, under the given causal circumstances, L\- --+ --id(-) and A+ -+i~'+', which completes the verification. is a1w useful for the direct The probability amplitude (o-Io-)~[-I*~(+~ eornputation of various expectation vaium. L&

for emmpb, which mplztees the unit evaluation for X = O by

This is the expectation value of eiPX for the states produced from the vaeuum by the action af the sourn K, Sin= only the relative dhplawmenl of the tvvo muram is sipif cant, we have

&,(eiPx

-- I ) ~ K ( ~ )= / ' exp

[T

(e'

-- 1)

On considering infirritmiml digplacements, we learn that

or, with an obvious identification,

The total number af psrtieles i~ iindiercled by

N = E n,.

(2-2. la0)

P

Thus, the werage total number of particles creakd and the vasuum persisLence prab~bilityare &implyreletted, according to (2-2.17) :

The discu~~ion of Auetuations is fwilitated by writing (the vafious indiees arc; omitted)

The simplest example is

which we can also inkrpret as

One consequence af the latkr is

Statements about tbe total nunrbttr of particles aw derived directly by eansidering %hesourea

2-2

Spin O particles, Strong murcs

86

Aceordin&to the hmcture of the rdative prob~bilityamplitudes, we have

(2-2,107) One digerentktion with re~pectto X, at X = 1, @ves

The coefficient of kX in the summation (2-2.107) is the probability of emitting N particles, without further identification. Comparison of the power series expamion supgliw iL ax (#jN @--(NI* p(N,@lK = (2-2. f 09) N! The fluctuation, formula (2-2.105) is a f a ~ l i a rchsbrackrigt-tic of thig Paissont &tribution, All such p m p e ~ i e sare deriyd from

by differetntliation with respect to X:

The generalization of this discussion to the amplitude ({R)-1 { R ) -)Kt-l*Rt+} only requires introducing the function A +(z - 2') and its pabrtners :

The causal relatioxls among thege functions are the same m in the v ~ e u u m situation although b\$)(z - s'), for example, is no longer an exclusively positive .frequency function, I n deriving expectation values we mu& note the$ the amplitude ((B') +l {n)-)K responds to the translation K(%)-r K(z X) with the factsr @x~ti(P(%') )XI, (2-2- 1113)

+

since?both initial and final s l a k s are now mlevant, Some results are (2-2.1 14)

and

(fib%;.) - (nL)(nk.) = bPPf((nk - ~ ~ ) ) ( 2+ 1 11). ~

(2-2.115)

IW3

Chap, 2

Souross

The treatman$ of oomplex rsourees and charged par2;iclw k quife a ~ a l ~ g a w . The vacuum amplitude deaaribing the time cycle ir~ (dz)(hf )K:-,

( o _ I o - ) ~ ~ - ' ~= ~ ~exp +'

(~)a-(z- z ' ) ~ , - ) ( z ~ )

-- zf)K(+)(z'f

(dz')K?+,(z)d+

( d & ) ~ ; -(z)A'+' ) (z- z f ) K t+)(a;') ( d z ' ) ~ ~ - , ( z ) ~ ~+ 'z')K;+,(z') (z

wGch r d u e e ~.to unily for Kt,,

(a;) =.

Kf-,(z) = K(z),

;K(+,fz):), By choa~iingthe, somew

K{+,(z) = h e i @ ~ (fz X),

(2-2.117)

with X real, we obtain

We ea0 also intmdum the total numbr of pa~itivelyand negatively chargd padiales, N.+== -C- Q), N- Q), (2-2.119) X

and rmxpreBs the expseation value formula as

Accordingly, P

~,+I"

,M-):

=a

P

IK@-I~,

(2-2.121)

whib individual prcibrabilitiw are given by

A simplified formula, designed to answer questions about electric oharge only, is (eg4@); = exp[(ei@ - 1)(N+)

+

(e-ip

from which wcz deivci:fhe.individua1 prababilitiers:

- 1)(N-)j,

(2-2.123)

Spin 1 particles. The photon

2-3

67

on. using a familiar Berne1 function. expansion. The introduction of the gropag* tion function (2-2.59), with its atkndant stmcturers, generalbw (2-2, X 16) do the probability amplitude ({B)-1 {n)- ) K ( - ~ ~ K ~ + ~ . 2-49 SPIN 1 PARTICLES. THE FNOTQN

Before developing tbe general source regresent&tion,for padic1es of arbitrav spin, we shall give an dementary discussion of some: gpecial examples whiah are of great physical importance, The exponentid form that has b n mhbiished for the vacuum amplitude, within. Ghe eontext of spinless pa&iclm, embodie~ the physicaf pas~ibiIityof producing any number of independent w t s af sin& particle emimion and abwrption, These gpace-time properties &re independent of the spcific spin of the particle. The latkr ean only influence the mare detailed stmcture of the source. It is clear that, if spin O p~r$ialf3~~ are described by a; scalar-source, sources transforming as vecbrs and tctnsar~of va~orrsranks must; refer to pttrticles of unit and higher spin. A vector source, designated m J P ( % ) , is the obvious candidate to de8cfibe unit spin particles. T h m m certain obstacles, however. This source ha8 four component^, iin contratst wiLh %hethree independent sources one should associate with f i e three spin pawibilities that; are accessible to a nongero mass particle, Thics pre~umabliymeans that J'(z) is a mixture of a unit spin source with a souree of spinless particles, corresponding to the possibility of forming a scalar function by differentiation, a,Jg(z). And, independently, W ob~ervethat should we do no more than replace the resll scalar murw K(x) by the real vector s o w e P(&

(o+Io-)~

= exp

- z"rJ,(x")

fdz) (d~"J@(z)A+(z

we should eneaunder a serious p h y s i ~ ddificufty, for

is aat marantmd to be l= than. unity, since can wsume either sign. Both diEculties are overcome by the following invariant slructure, which is appropriate ta a particle af mass m $ 0:

In wduaLing the vacuum pemhknee probability,

now encounter

68

Chap. 2

Sources

Since this is an invariant combination, it can be inspected conveniently in the rest frame of the time-like vector v,where

f l rest frame:

= 0,

= m.

(2-3.6)

The components of the symmetrical tensor that appears in (2-3.5) are then given by p=v=O:O, (2-3.7) p = k, V = 0: 0, ~ P D (1/m2)ppp~: p = k, V = I : bat.

+

The result is simply 151*,which is positive, and which contains three independent source components, transforming among themselves under spatial rotation, as is appropriate to unit spin. We note that (l/m)@ is a unit time-like vector, which can be supplemented by three orthogonal space-like vectors, 4x, obeying 4?eppxr = 8xxn. pp4x = 0, They give a dyadic construction of the metric tensor,

(2-3.8)

The symmetry of g@' indicates that complex conjugation of the three e$, produces a unitary transformation on the set. With the definition the vacuum persistence probability appears as

We now consider a causal source arrangement,

+ JS (4, exp [/doput (P) *(gpv+ m - 2 ~ p ~ v ) i J ; ( ~(o+Io-) )] " Jp(z) = J';(2)

which implies (o+~o-)' = (o+~o-)'l

= (o+~o-)'l exp

[Ci l : p ~ i r 2 p x ] PA

(o+Io-)~~.

This standard structure identifies the multiparticle states

where n,x = 0, 1, 2, ... again indicates B.E. statistics. The consistency between the two uses of the vacuum amplitude is obvious.

2-3

Spin 1 particlas. The photon

69

One can choose the unit spacolike vectors 6~to be real. The orthogondity requirement 0 (2-3.15) P epk = P 0epx displays the role of p in providing a reference direction. If epx is perpendicular to p, the time component e$, vanishes. Let epl be such a real unit vector, Then is another one, and the set is completed by = (p"/m)(p/lpl), We note, incidentally, that eps

0 epa

= Ipl/me

(2-3.18)

Alternative, complex, choices are suggested by angular momentum considerations. The response of the vector JP(z) to the homogeneous infinitesimal Lorentz transformation ?EP = xp 6&*x, (2-3.20)

-+-

For a three-dimensional rotation, this becomes

and, equivalently,

Now let us consider a rotation about the direction of the momentum,

60 = ~V(P//PI). We realize a single-particle state of helicity X :

110

Chap. 2

Sources

Zero helieity is aehievd with e parallel to p. Accodingly we relabel ego, (2-3,28)

The helicity ~ t a t with e ~ X = rh= l correspond to the complex combination8 which are so chown that +l

(6o* S)& lfe:kt

--e& X 6 0 = i

(2-3.30)

k1=-1

give8 dhe ~tandardunit spin m a t ~ xelements, A clwifica%ionof murees and pa&iele ataks in, relation to btal anwlar momentum ean be introdued. As a preliminary shp, we emulab the eero spin procedure and define

X

( d ~ ~ ) " ~ ~ ' ( p )( d ~ ) " ' y t ~ ( ~ ) J > t m ,

(2-3.31)

ilm

where

Nothing more need be done for the time component J@,which is s %h sianaf scalar fumtian. But the fhree components of J refer to $he uni%spin, which mast be coupled appmpriately with the orbital a n d a r marneaturn to produce total anmlar momen%umstates, This is raccomplisbed by the following inkmduction of s veedor o&honormml system, mpIaeing the scdar spherical h a m o ~ mt, e

mid the reder is warn4 not 4x1 confum the letter m, u d in subseript~to dtcrtnoh a rnaeetic quantum. numhr, with m, appearing e1cpXieit;lg in i f s mle ws particle m m , The above ~tmetureis mch that

2-3

Spin 1 particles. The photon

Tt

We also note the relation

(2-3.36) On combining the various contributions, we get the required form: dadp(p)' (g,

+ m-'p,p.)JY

Jpjm hJp0jmh,

(p) =

(2-3.371

pDimx

where h = 1, 2,s di~tinpishesthe three excitation8 with total snlgular mamea%urnquantum xlumbrs j, m, These sources esn. be exhibited expjicidly. The vector orthor~omality praperty enables us to evaluate

for example, and this ean be converted into

where nolv

L = X X (lJi)V, (It is unfortunate that the cornbination of two well-established no.Latioaal conventions produces things like ji.) Incidentally this type of aourct: van&he?3 for ji = 0. Similarly, we find that

which also vanishes for j = 0, and

It is seen that the sources designated as X = 1 , 2 depend only upon J(z), and that in the form V X J(z), while far the Lhird @pe of source we have, effectively,

There is no dii-ficulty in implementing the same generalizations thak were discussed for ~ e r ospin particles--charged particles, multiparticle initial and

"112

Sources

Chap, 2

final stabs, eyelie time development-but the details are too s i ~ l a do r merit reptition. We turn in~teadto sn important spcial situation, thc? limit of sero mam, M realigd by the phokon. It & evident from Eq. (S3.4) that the sero mass8 limit does not mist unfm aPJ@(z) vanishes. One ~ g he t t e m the gouree of a nassfess gero spin and identify K(%),in the limit m -4, particle. The latter would be oompletely independent of the residual photon muree, however, and since m .= 0, s = O particles sre unknwn experimentally, in %ny event, we only eansider photons in stating the source dessc~pfiorm:

(o+Io-)~= exp (2-3.45)

W"(%) =: 0.

We uge the symbol D+ Lo indic%tethe redrietion tr, gem mass, The murce associakd ~ t a hpadicular psrt;ielc? is an abstraction of the realistic procmw that ereate or annihilate %hepadicle. It retdns what d l w ~ h mctchanjsms have in wrnrnon and ignorw the ~pecifieehari%cteristicsof indi"ViduEtl mechmisms, Any generail rest~etionon m u r ~ e sthat is implied by speeid h&tures of the particle must be eammon ta all nnechebnisms and fhus sfaks a gemfa1 law of physics. I n the situation of the photon, we have dc?ldue&, from its zero m a ~the ~ ,nc~ceseityof a reslrietian on the v e c b ~ asourct?, l It m u ~be t divergene less, which is the local statement af ai commafion. law. We? are in xla doubt; about the identi%yof this consewed physical propedy. It is elee%~c charge, The loss of one degrw of excitation far masslem psrticbs is evident in the vsfious ways of labelins particle staks. Thug, aa nz Q in (2-3.191, under the rmtriction P ~ J @ ( P= ) 0,

(2-allats)

we a e v e a t Jp3

S

0,

(2-3.47)

and the two remaining sourses Jg1,2~ f e to r the two transverse linear pol%Gzgtioas accessible to photons. With helicity labeliw we have, equivalently, r and J P k l reprwnt the two circular pola~aations,Turning to a n ~ f s monsendam s h b , we have analagously, from (2-3-43), Sincej = Q dws not appear in the twa other soum Lype~,t h k is the counkvarl af the abmnce of rtero helicity. We have arrived a t the ratrietion fo two pola~zationor helicify s$alw for Lhe photon by a limiting procedure that began with massiw unit, spin padides.

Now let W obtaixz this result directly, by trsiag the photon source desc&ption @venin (2-3.45). The consideration of a causal arrangement, J""(3;")

impfies (O+IO-)

= (O+IO-)

Jg

J!(z) -I- J$(r),

(2-3.W)

(o+[o-)".

exp

(2-3.51)

The dysdie represenlalion for g,, given in Eq. (2-3.9) is not ap;propriak here dnw p"" is now a null vector, (2-3.52)

pg = 0.

Let FP be obkained from p"" by mversing the rnotion of the * o h ,

+

Then pl" 'p and p" -- pp am,rmpectively, fa timelike and spmelike vector. They are ~uppbmentedby two orthrtgonal uniL spae-like veebm @gx,

to give the d y d i ~ coxlatrwtion I -

(P'

+ p'l)~ (PP' F3-pp)_ (P'

-- )'P (P' - p')

gm

+

C @'PX X

@V*

P&

We now urn the photon gaurce reskriation,

which k the desired partide exchrzngs form. If is also impGcsd by (2-3.54) that the two have aero time component and, as spatial vecton, are perpendicular to p. This k s self-contained desefiption of the k m tramvem8 witatima that art3 pemithd to photons. It bm been recomized earlier that a s concept that is invarirtnt under proper orthoehronous Lorentr transformations. One &odd be able fo make mare evident thL wpwt of tbe photon helioity @ t a b .A d we should like to under~bndwhy the M e i t y slates fasve appeared psired afthough no overt reference fa e?gatiai refieetion h= been made, b t us begin with the remark that the conservation eonditiop imposed on JP($) is sadisfied identiaally if J@(x)= dPMp"z), (%a.58)

where MP"(x)

We

=

-MY"(a;).

introduce the eoncepd of the dual to an anti~ymmetricsl

in which P"& is the totally antisymmetrical tensor that is n o d i a e d by The opration of forming the dual hss the foXXa~ngrepegition proprfy,

The dual t e m r is used Go m i b

The 1aGbr property indieabs that eaeh of them objects hm only three inde

pendent components, as i l l ~ s t r ~ by %d

T h m coxrrpnents %re complex numhm, of eoum, and The deeompo~idion@ven in (2-3.63), and indicated by ia an invarisnf one

far as eontinuouta change8 of coardiniFtte aysbms are conmemiond nsbtion, we have t k 3 expucit 0~)mtm~tian8

The eEmtiveness of dhese gources in mdiating a phofon of heficity X is measured

of makhfag the value! of X k % hf f. 8haw~ s~cor&ngto (2-3.m). The n w that the =t;l 1 a b h OD.the Bources da indeed mfer Lo %hemiqare heli~iti.esBf the photons that are ennitM ar abwrbd by %he~e compnent rnww~,

Spin 7 panictrs, The photon

2-3

76

Why can one not modify this phobn dwriptim by odtting J',x(z>, a y , and themby prduee a theory with only positive hdicity phsbna~tFor tbe same reamn that one cannot have a theory in which only p~itivelycharged p~&icltls occur; it would violak %hepfinciple of hipwe-time uniformity, To d k u m fhb point in more detail, consider the cantribution to %hev ~ u u r nanaplitude ated with the emi~sionand submquent abhiavtian of a p 4 t i v e helicity photon

where the c a u d 1abds 1, 2 have been displaced for &reah1~" elafi*y. We are jwgified in writing g,, &nee the e s q u i v h t palariakbtion vector eummation rduces to the appropriafe positive helieity hms, in vi&ue of (2-3.71). The compbte sortme coupling should b linear in

and in

J;s(~= ) J;l(z)t

+ 5;1(2)~

(2-3.73)

(2-8.74) J$~(z):t- J'+s(z):. One might try Lo resist the iderenccs that there is another eaupling involving (g')l and J'/ ,($)g by introducing the ~paoe-timeextrapalation of (2-3.72) with an addil;ionsl factar, 1, z0 > z", q(zO- zO')= 0, x0 < zO', J'+312)'

E

which is d e ~ i m dta elirninak muree arrangementhi whem dhe m d ralw of J;, and df are revemd. This step function d ~ have. g an invsnisnt meaning when x and z h r e in timelike or nuli relabion, but it is not invdan6 like intervals, and its introducfion would vialab the principle a f 8 uniformity, We cannot avoid r e ~ ~ p i ~ i%he r t gpreence af the &di%io coupling term (2-3.X)

and the particles dehieribd here mwt also be of n;wo m- if a unique splleetinre extrapoltbtion is fo be acbievd. T h w antipa&iclr?s am t;he negative helicity photons, J'+s(~)'= Jtl(z), (2-3.77) and the additional krm ean be rewfi6bn as (dz)(dar')JI", (z)T

(2-3,78)

$vl,

Furthermore, the analogow structures involving JLl and JeltJ h equal zero since one ar fhe okher factor in the pda~a&fionvecfor gummatian will v a ~ s h .The rmdt is fa recanstitub the real soume

76

Sources

Chap. 2

appeafing in the coupling

which ia mogniaed as the @ a w lp 8 ~ i c bexchange krm in

AB a comllary of &is dicscumion we no& that the soure=

afifl give an equivsled dmcription of photon ernimioa and absorpfion. This new mume is ~ p r m n M by *JR(Z)= a,

(2-3.83)

*MC~(.

Tbe nature of the tmndormation is afso indieaM by

1/2(~/1~1)

(2-3.M) JW, which makw expli~itthaG Lhe pola~ratianv ~ t o r shave been r o b b d thmutgh the angle ~ 1 about 2 the photon direction of mofion. If fhe mation angle k p, *JP&

=

(d@p)

X

&@ :

*

the tr&~sformatiorr. hcomes

+

(2-3.S) JP(%) 4 Jg(x) cos (a *Jfl(z)sin 9. The subsfieuLion of *P for JP &so has a Elimple eEect upon the an&ar momentum labled BWTCW, J p a i d . W@fimt remark that

+ a. *M(z) -+V X M(z) - ip'

J(z) = V X M(%)

*M(z), (2-3.86)

whew %hefatter ~ub~ti%ution indicaks the eBectivcr; value in the in@pab that; compBe J p ~ i , h * Sifilarly, we have (i[pa)vX J(z) = V X *M(%) ($/pa)(VV -v')M(s) 4V *M(z) i p a ~ ( z ) , (2-3-87)

+ +

-

which uses the eflective value --vZ --t 'p = (P@)' and the pmperty L V = 0. The ~ubatitutiianJfi 4 * P , which is equivalent to M -+ *M,*M -., -M, interehangea these vectorial structures and tnrmforms the two souroes w o r d ing to J p ~ j n r l -* JlOjma,

Jpojm%

--JpOjml*

(2-3,s)

The m m genersl sub&itution (2-3.85) gives the rotatian

In tha general spacetime form of the vseuum amplitude, Eq. (2-3.451, rsoumw need not ernif and 8 k r b photans and indeed m&ybe incapable of doh&

Spin 1 psrticlcts, The photon

2-3

7'7

so if they vary too slowly in time, I t i~ Che principb of space-time uniformity which thus asserts the physical unity betwen collision meehanisns &at do liberaLe enough energy to create a particle and those otherw2se analogous mehenisms that h~tppenta have an insufficient energy supply. To illu~tratethe new physical information that is obtained in this manner we consider photon sourctts Lhat vary very slowly in time. The wrty is preipared for this limi$ by writing

where, as a consequence of (2-1.451, rOD

This struetum indieaks that $he scale of significant T variation ia set by Ix - X'[. If the sources vary little, in the time intervals that are msociakd with the distances characte~sticof the instantaneous digt~bution,one can i p o r e the r dependence in J@(x,z0 & 47) and evaluate &P)+(X

- X',

7)""

-

d~

0

sin polx

- X'\

PO

This ~ v e the s foXIowirrg form to the vacuum amplitude: (0+j0-}~= eup

(2-3.93)

where

One reeognism here the aeeumulakd phmf?;change of s state that has a time vafiable energy, E(xa). When a stesdy-state regime is established, we are led to asgociste with it the energy value

whieh is a s&aternentof the Coulomb and Ampbrim laws of charge and current interactions* This shows haw the principle of space-time uniformity provides the logical connection befvveen the properLies of photons and the chsmekristics of quasi-stationaq charge distributions. There is one suLtLleLy hem we should not overllook. One cannot produce a complettlly arbitrary statie charge distribution. The local conwrvatioa condition

78

Sour-

Chap. 2

aBJ@= O impXim comrvaCian of the htal ehsrge

pm~4d the sourn is wnlind fo some finite ~patialre@on. Being wro in the i&tbl muurn s h b , the bbl ch~rgemmains Piera, We m y picture initially p ~ i t i v and e negative charge distribution8being separaM, maved mmpe anb then m o m b h d . But them is another way of viming the abut in4duction of a charge &t~butionintto an emp%yredon, It requirw rmomil;ly than is wusl tha$ a physied dme~ptionrefern only to the iaig m fi~b re@on which h under the exp~menbr'smntrol. The iniCial and final vacuurn ~tatesp&ain to s boundd three-dimemionaf re@an, W e gon, oubide the walls- We thus &pp~ci&%e that an arbitrary charge di* Critouticrn can b produced by the t r a a ~ p dof charge acrom the boundary, into the mdon of inter&, and that thil charge distribution can be dismanlled uldirxrtabIy by withdradng id across the boundaq. 2-4 SPIN 2 PARTICLES, THE ORAVITON

Next ia complexity afkr scalar and v w b r mumas is the red ~ymmetriealtensor 80uree Tp"(z)= TF@(g). (2-4.1) X t has k n eommnentF3, But they include the 3

and the sealsr wurce P(%)

+

1;

component vector saurce (2-4.2)

gPyT@*(jc).

When thew are removed, the residual multiplicity af five is the anticip~tLedone for spin 2 pl.ticIerr of noneem mass, m. T o c&rvout this p r o p m we exploit our @ x p ~ e n e*th @ unit spin p&&ielesand wride direcffy the physically am88hbq ~$mc%urrt for the vscuum pc?rsi~bnwprobability, It is do,F'v(p)*~r.(p)g.k(p) Pk(p)

where pp.(p) =

and

+ (l/m?)p,p.t T@'(p)

pYB&.(p)

0,

= TL*(p) - 3yrpg@@(p)Twfp)

38

(2-4.4)

(2-4.5)

0by8

B ~ ~ ( P ) T ' ' ( P ) Qv S

(2-4.Q

In Lhe rr?rst frame of the momentum p,&,(p) prajeets ontcl thrm4imenaional spa=, m detailed in (2-3.1). Aeeardingly, the only socompnent~thaf contribute in (2-4.3) are the six T k l , whieh have a vanishing diagonal sum,in view of (2-4.6), Here is the fivefold multiplicity asso~iatdwith spin 2,

Spin 2 prrrttctm, The gravtton

2-4

78

An alkrmtive writing of (2-4.3) is given by P v ( p )*F~~(P)P~x(P~~'~(P) = TLp(p)*~,~,~&(p)T'~(p)~

(2-4.7)

in, which

Some properttiw of tfie latter are

The projmlion

chaxacbr of q , , , , ( p ) Xeada to the dyadie repreaemfation.

and are five in numbr. The sources for specifie slates are then identified as When hdiciv states an? uwd in the veetor & d i e mnstrueLion

+ e(lph6e;h)- B(-

= f (GheLh6

1)'6-~h#

C (-

1) "e:kle;-hl

(2-4.161

X1

obeys the relation

C (--I)~G-~X X

S

+ @So= 0.

-26+x',;l_l

(2-4.17)

The spin. 2 hctlici-t;ysLaks art3 then identified as

The detdleui ~tna6tureof the vaeuum probability amplitude that %heprczbability (2-4.3) is wrfitten out as

I d 8

to

with

In order that this expression continue to exirsf in the limit aa m + 0, we must have where Jp(s)and H(%)are independenk sorrrce~at m == 0, The partiadar linear combination of the two scalar 8aureeg K(%), T(z) echosen to eli coupling between them. This is evident in the limitjng form

We Bee before us %heinvsrianL decomgositian. that &hefive heliciey &ate@, accw~ibleto a mamive pa~icle: of spin 2, undergo as m -+ 0, falling into the thrm poups *2, &L, 0. The massless particle of helicity f 2 vvil be identified as the graviton. Its source da~riptionis given by

We insert a treatment of gravitons that begins with this characterization. The causal source &mangemen% 8ves the asud factoriealioa of the vacuum amplitude:

where each component murce o b v s %Tpv (p) = Q.

(2-4.27)

Thig eource restdction snd the dyadic form (S3.55) are combin& to abtain the eflective rep1&cement 3($'gP" gl@g'@- g"gP') --t chh?e$$:, (24.28)

+

X&'

where, on using helieity s t a b , h = &l,

The three independent tensor%mntrtined here: rare and which represent the two helieiLy states of the p v i t o n . The graviton is unknown, as yet, to exprimental science. Nevertheless, we shall accept it and ib conjectured p m p r t i e ~as the promr starting point for %hetheory of pvitational phelraamens, ju& as the photon with it9 attribubs initciates the ttteory of e h e h r n ~ m e t i ephenomena;. The ~ ~ d e n efore the existence of the gaviton is indirect, but imprmsive, To indicah its naturcct W present the following psrable: "The laws of quantum mechanics and relativity have b e n wctll wdsblighed, but the intermtion pmpertiea of electfie eholrgm are known only under quasi-static conditions. Two physickb, Max Stone and Ichira Ido, point out that all such properties would follow from the postulated exisfence of rt certain particle, on using Lhe principtm of source thmw. They gnedict that the particle will one day be discavered. Othem dismka this ~ J U Q ~ W %ionas untvlbrr~nfed~ p c d a t i o n .The is~ueremains undecidd. " The postubbd existence of thei gmviton le& first snd foremost ta the s o u m re~triction$T@"(z)= 0, which, as in Ihe phobn discwssian, &at= the aislence of a geneml physical fw. It is a con~ma-tionlaw, con~%aney of the vector (24.32)

The noerttion already indicates that them is only one conceivable identification of this v e e t a ~ a lproperty-it is enerw-momentum. Unlike photon ~ources, which hme a unique measure through the electric charge inhrpretation, grraGton gources are eonfranted with an independent sale ori@naling in Lhe mechanical significance of T,.. We provide an empirical conversion factor by writing

is intrinsically positive. Again in contrast with eleetrie charge, energy or The establishment of a g r t ~ t o nsource dk%fibutionin m iaitial vacuum sifurttion oa;n only be realized through Lhe t r a m p & of enerm and momentum infm the mdon of i n b r e ~ t ,through the Esoundaries th& deEdt this domah. We

~onsidera slowly vaqing distfibution of graviton m m e g and deduce, as the anabgue of (2-3.94), the e n e r e

where the maehanicd memure of paviton sources is used. In the folloPi.ing a~tronomieal &pplications m are eoncemed vv3fh the inkr&ction betweerr twa bodim, oxre of which (the "Sun"")m dimemions that are effectively negligible and is charaekfizd by the single saurce oomponent T0'(x), such th&t (24.35) The interaction energy between the Sun, staLioned at fBe o ~ G nand , the second test body with source distribution l,,(x, zO)is given by K

E ~ ~ ~ , (= z '-) GM where

Whm a body that move8 ~ $ & yhas momentum p@, where cr i s an inv&ant measure of the mms distribution, For s ~ t a t i o n s vbody of m w m, with dimensions th& are xlegEi&ble eompared %a32, the distance from the od$n, we get

Thh is idb?RCifi~bfem $he NewLanian potentid e n e r e of attracting nzw~e8, where~ / 8 a= G = 6.67 X 1 0 cma/& ~ ~ seca = (1.62 X l ~ am)2. ~ ~ (2-4.M)

The second ver~ianrefers to &ornic units, in whieh & = C = It. We shall now use elennenta~eon~iderationsto reprduee the four O ~ B ~ W L L . tionral hsts of the Eimteinisn mdificatim of Nedonian theo~).. 1. A slowly nnoGng atom of mass m hafs the total enerw nz -- (GM/R)min the neighborhwd of the body with mass M , The enerw relewd in an inbm8l ~ the ~ 8 v i t a tran~formationis thus reduced by the halor 1 - (GMlR). T h i is dlorr%lred &hi&.

2-4

Spin 2 paaiclesr. Ths graviton

83

2. Let the test body be a light beam for whieh t = opZ= O. The interaction enerw with the 8un thus exceeds it%Newlonian value (replacing mass with total enerw) by a faetor of tw, That is afso the increase of the deBwtion angle of light over the Nefftonian wlue, whieh is Einstein's result For a direct cdcrtlation we compare the acquired transverse momentum Mrith fhc: longitudinal momentum of the h a m , which pmses act a distanee p from the Sun. The deflee-

3, The same inkraction reduces the speed of light by the factor l - 2(GJ(r/R), d n ~ the e energy of a phobn is pi (l - 2GMIR) and differentiation with respect fo p gives the velocity. This &et has b e n oherved by mmuring time delltys in radar echoes from the inner pl%neb, We eonsider the superior eortjunetion of a planet, with the line of aight from the earth passing at distanee p from the Sun. Then the abntieipated additional time delay for the echo is

whem z, and z, are the distances, from the point of closest approach to the SW, to the earth and the planet, re~pectively. The coefficient in the differential relation

hm h e n verified with fair accuracy. 4, The mosC subtle and inkrwting test is, of eourw, the perihelion precegsion.

of planetary orbib, We fimt consider the carreetion to the Newtonian ptential enerw k" = -GMm/R (2-4.44) that is produed by the motion of the plan&. fiar small speeds, we have

where

8rC

8ources

Chap. 2

Thee eEects eorrect the Nwtonian p t e n t i d to

There is a earnparable relativistic madificwtion of the kinetic energ, given by

And, finally, there is the contribution to the energy density to@that is assoeisted with the gravitafisnal inLeration between the planet and the Sun. This is not locslilred on eilher mass, but is distributed in space in a way that can be esl~ufibtedwiLh suEaientP preeisisn from the Newtonian field stren@h:

The inkraetion s n e r e density is proportional Lo the mutual h r m in fhe sward field strength. XL is ~ v e by n

as one verifies by inkgration:

The energy of inkraction between, the mass M and this distribaM m a s is

Ail sddilional interaction terms are exhibited in

This can be simplified by using the nonrelativistic enerw rdcbtisn, E = T -4- V, which enabbs T to be eliminated in fslvor of V. An additional eonstsnt multiple of V' does not produce wrihetion preewion; it only changes slightly the scale of the orbit. It is the V' term that gives the significant deviation from the

Particlss with arbitrary integer spin

2-6

85

Ne~vtonianpote~ltietl,and perihelion precession. The resulting eEeetive potential is Yefr. = V - 3 ~ ' l n . (2-4.55) Now, the equation of ;rtn orbit can be wriLten

is the angular momentum per unit planetaq mass (it is often defignated and by h). Here we have

and

We see %batthe essential deviation from Ketvtonian behaviar is a reduetion of the angle seale by the faetor 1 - 3G2MZ/&:, which requires that p increase by more than 2n between sueeessive perihelions. This perihelion precession angle is which is mactly Einstein" result. [He gave it in krms of the semimajor axis a, the period T", and the eccentricity e, The connection is G2tI/L1 = 2 ~ ( a / T ) (1 ez)-"'.l 2-5

PARTICLES VVliTCI ARBITRARY INTEGER SPIAl

I n discussing unit spin and spin 2 particles, it was natural to replam the s ~ a l a r soltree of spinlms particles by vector and knsor sources, The response of rc veetor source, for example, to a homogeneous infinite~irnalLorentz transformation is given by (2-3.221, which we write as

Cleetrljr exhibited here is ~h four-dimensionsl version of orbital and spin angular momenta. This is a particular infinitesimal il1ustral;ion of the general, linear response of a, multicomponent object to Lorentz transformations, S(2) =

L(i)S(z),

(2-5.2)

where 2Z?C = EPpxY -

rp,

IF,l;t,,E"x =

(2-5.3)

86

Chap. 2

Sources

details the typical inhomogeneous Lorentz transformation. One can always choose the elements of a suitably multicomponent source to be real, and this property is maintained by a real transformation matrix L(1). Corresponding to the composition property of successive Lorentz transformations,

-

name1y

-

=

-

.F" = (I~~z)'Az'- (€1'

2' =

+ 11Pve2"),

1 2 v A~ Ae2', (11l2)"~= IlP.I2'~,

(2-5.4) (2-5.5)

\S-ehave

3 )=L and

~

S

, S(a) = L(I,)s(x)

-

S(?) = L(l1)L(l2)S(z) = L(llZ2)S(~).

(2-5.6) (2-5.7)

It is in this sense that the finite matrices L(1) give a matrix representation of the homogeneous Lorentz group. For infinitesimal transformations

+ 8wY,. Qw,,, = -6wrr, L(1) = 1 + 3i 6dVs,,,

l', = 6': wve wvrite

(2-5.8) (2-5.9)

which are imaginary when L(1) is and conclude that the matrices S, = -S, real, obey the commutation relations that express the composition properties of the six-parameter homogeneous group [cf. Eq. (1-3.10)] :

The complete infinitesimal response of S(x) is

Comparison with (2-5.1) gives a (4 X 4)-dimensional example of the imaginary spin matrices S,,,, (2-5.12) = (l/%? (a)gwK - 6bp,>. (spl)A~ As the follo\ving illustrations show,

the matrices at1 are antisymmetrical and therefore Hermitian, while the see are symmetrical and skewv-Hermitian. It was preordained that not all the S,, matrices could be Hermitian, for the discussion of Section 1-1 shows that the open structure of the Lorentz group precludes any finite-dimensional realization of the group. This injunction is not applicable to the attached Euclidean

2-6

Parti~Ieswith arbitrary integer spin

87

group, and indeed the correspondence (zr = iz') 80k

= -isx4,

(

S

(

4

P

(25.14)

do= give Euclidean spin matrices, (QII~)XI

(l/i)(6~paYK

(2-5.15)

~ X Y ~ ~ ( X ) I

that &red l imaginary, antisymmetrical, and Hermitian. Such diEerentid operator-matrix .realizations of the Lowntz generators &re in striking contrast with. the operator &ructures given in (11-3.42) or (1-3.72), for there the finite skew-Rermitian matrices sok are replaced by Hefmitian sprators that are functions af momentum. Horv do we reeonciie these very differrdnt hms? There mu& be a eonneetlng transformation that preserves the commutation, relations but does not mai~ltainHermi-ticity and therefore is not a unitary transformation, A. suggeslion of ~vhatis required comes from tho follawing Lransformstioxl, ~vhich is appropriate d s t-2 sforvly moving particle (p' -- m):

expfk(p * s/m)](-- mr) e x p t (p ~ - slm)f = -rnr & fr, p sj +

illtustmting how the momenlrxm dependence is removed, a t the expense of introducing ske\lrrHermitiars. operator^. The follol~ingis the sn,zlogous stn;temenl for arbitrav momentum, @xp[=tr@(p * s/lpt)I

e x p [ &(p ~

s//pl)] = -rp0 & is,

(2-5.18)

\{?here sinh B = lpllrn,

cash @ =

(2-5.19)

Its vmificatian proceeds m s t simply by eonsider-ing cornponeas parallel and pewendicullar to p. The former reduces to the defining digerentiai equation

and the lat%erto

d@(lpl)ldIpl= l/pO,

88

Soureas

Chap, 2

which describes the behavior of s under the "rotaition" "specified, by 8, Incidentally, if a particle moves ~vithmomentum p, aXong the third axis, the t r a w formation to its rest frame is

z3 = g3 GO&8 - z0sinh 8, Z@ = --re3 sinh B -f- xo cosh 8, The appearance of &isk in the role of ?tk

=

8%

is emily understood. The commutation relations for the sp, are simplified by introducing the linear combinations indicated by

and their cyclic permutations. All four-dimnsional commutation relations are sumrnsrizled by the three-dimensiontzX commutation properties of the tiro independent rznmlar momenta, s(E)

X

Stl)

&(l)

sC2)

X

S(21

= is(2)

I

(2-5.25)

Conversely, Ire have the construction (8'2 = 83, . . .) ~ = ~ " ' + ~ ' g ' ,

m=i(s"'-s'2'),

(2-5'26)

*''

where the use of conventional Hermitian matrix representations for s" gives Hermitian s matrices and skew-Hermitian R rnstI"ice~. What ive have encountered in (2-5.18) is the special situation in \'hieh sC2' = 0, or 8"' = 0. To deal with the more general possibility presented in (2-5.262, we musk eonsider s ta be the resultant of other spins. And, since it is often convenient to build the latter out of still more elementary spins, we present the following general theorem, in wrhich the h, are any commuting objects that obey

and

a superposition of independent *ins.

The theorem is

2-5

Prrrtictes with arbitrary integer spin

89

If is verified as before by using the individual relations

The general matrix construction

clearly satisfies all the relevant commutation relations, including We note that the skew-Herrnitian character of n is maintained with the use of Wernitian matrices for the X,, as well as with the numbers =f= l. We have not described explicitly the symmetrization between r and p0 since it would repeat the spin O discussion. The infinitesimal Lorentz transformation response of the state (p! is now written as

where spin operators have been replaced by spin matrices, as symbolid by (PIS = S(PL (2-5.35) in which the matrices act upon unwritten indices in (pl. Next, we perform the transformation (there are constant factors to be specified) (p0)"2(~lo-)S

B(P)S(P)~

(2-5.38)

and derive d8(p) = [da (p X

$ + is) + 6.

(p0

$ + in)] ~ ( p ) (2-5.37)

On writing S(p) = this becomes &!$(X)

=

(dz)e-'~'~(z),

(2-5.38)

3 GC#'(X~~~ - X& + &,)S(X),

(2-5-39)

where the matrices S,, are those given in (2-5.32). The analogous construction of (0+lp)' is [cf. Eq. (2-1.25)] ( P ~ ) " ~ ( ~ + I PS(P)*B(P), )~ (2-5.40) which involves the Hermitian character of B(p). The infinitesimal Lorentz transformation behavior of S(z)* is like that of S@), but with the matrices W

90

Sources

Chap, 2

--6:. replacing the ..s, The use of matrix representstions with imaginary .8, is re-quired to h consistent with rest S(%),as we have mentioned before, The compact notation used in \vriLing (2-5.36) obscures an essential point, We are desc~binga particle of definite spixl, but embed it in a larger system when we employ constructions like (2-5.26). tlceordingly there must h premnt atl the left of B(p)S(p), say, an explicit election of the states of interc;st, We shall illustrate this, and a t Ghe mme time @ve a simple example of the eonncct b n bt\vef?nfhe pre~entmatrix approach and the earlier pro~edures,by Gfioosing

The re~ultantof the two spins of 4 is either s = f or s = 0, f f \re trish to describe unit spin particlea we must sdect them from the larger system. The familiar s = f triplet spin funetians ean Ibr; trfiften as

An stlkmative vergion, \\"hieh &ISOinvolves the reality and sy mmeky of t h w functions, is i I (d(2)I f T s * IQ( 1 1 ), (2-5.43) where G e i l = 2-"'(@1 icp), g. = 2-"2(-gl ig2),

+

age;

+

== Q.

(2-5.44)

The three spaee vt;?etorathus defined sre orthonsrmal in the sense We also n o k the singlet funetion: which exploits the antisymmetry of the Pauli matrix cz, as (2-5.43) depends upon $he symmetry of the three ~ 2 ~ The k . latter proprty is also expressd by A corresponding decamposilion of the four-component mume into singlet and &ripjetfunctions is conveniently written las (X) # s q ~ ~ ~ . ~ 2=) (p 2-112 ) Jp(p)g21a'2'), (2-3.48) where (2-5.49) C F = ~ -g@ = 1. We now examine the unit spin parliele source structum: in whieh appropriate factors have h e n supplied. A eonsishnt use of the matrix

Partfelss with arbitrrrw integer spin

2-5

91

notation gives (dop)-1'2~p~ = 4 t r I@ e:b(p)#Jyl (P)~(P)] = &:Jp:~,(p),

where

b(p) "- =p

l---P@ pi'lpll*

We first note that

e',;

=

tr [c.e?b(p)crPb(p)]

has the follorving property: PP$!

m

(2-5.54)

0,

sinee pPa, == m(cosh 6 3- sinh Ba * p/lpl) = m[b(p)-1]2,

(2-5.55)

trak = 0,

(2-5.56)

and Next, we consider

in which the form of the second factor depends upon the HermiCian naturt? of b(p) and the a;, The following identity expresses the role of the four matrices 2-v2g, LLBan orthonormal basis for 2 X 2 matrices: (trapX)(tra,%f)= Ctr @X) (tr o V ) --- (tr X)(tr )'l = 2lt.r (X Y ) - (tr X)(tr Y ) ] = det ( X - Y ) - ded ( X Y). +

+

(2-5.58)

The rnultipli~stionproperties of determinants, and the remark Chat

whieh follows from (2-5.561, shot%?that dl reference to b ( p ) diwppears from (2-5-57'), giving eg:epphp = f tr (G et. eh') b ~ ~ f . (2-5.60) And, finally, let us consider =

tr (cpbZ) 4 tr (qpb2)$ +[tr (gp@') - (tr 8)(tr Q')].

The individual traees here are

(2-5.61)

92

Sources

Chap, 2

and

- (tr bJl)(trG')]

*[tr (rN@')

= gfiV,

(2-5.63)

tvhich gives

We have nmr reproduced all the covariant p r a ~ r t i e sof the three polarization vectors for unit spin, When the third axis in (2-5.44) is ideatifid with fhe direction of the momen't;urn vector p, the explicik expresions obtained from (2-5.53) are just the heficity l a b 1 4 vectom (2-3.28,29). LneidentalXy, an using the singlet rather than the tripjet functions, we get the farm which is the anticipated scalar combination. AS the basis for s corresponding treatment of arbitrary integer spin, we consider the spin combinations

where the individual makriees set upon the appropriate index of %hesource

&l1). ...pQ'," ....p(p).

(2-5.67)

Since all the matrices ,'2sl:@ a = l, . . . , R, appear OD the same footing, we impose tt perrmis~iblesymmetry. reskriclion by requiring that (2-5.67) be unchanged by any permutation of the a indiees, in which'':c and @hzbZ'are regarded as a unit. Thus, for n == 2, we have 8,yf C%E1)"(2) 12) : , S e a l ) (11 ( 2 ) ( 2 ) . I *B "1 02 @l (2-5.438) The simplest procedure is to replace each four-valued index pair @L1'@:'' by a four-vecdor index in the manner detailed for unit spin, This gives the equive lent souree S F l - *.@@(P), which is unehangd by any permutation af the ar indices* The knsor of rank 7% a8 intmduced here describes a larger system than s, particle of definite spin, Part of the nrscesstzry reduction is produced by the projection faetors &,(p) that appear separately for each. v e ~ t o rindex in the coupled source structure urhere the secand form refers to the rest frame of the momentum f . The number of independent components possemed by the symmetfical threedimensisnal namely i(n I) (n 21, agrees with the number of states extensor Sh,...ks, hibited by a 8ymmetricaX collection of n unit spins. The total spin quantum

+

+

Partitles with arbitrary integer spin

2-5

93

numbrtr ranges from s = rtr through s = n - 2, ?t -- 4, . . . , terminntirtrg a t f. or Q as n is odd or even, and (2s 1) == $(n 1)(11 3). A eombinntion of two urlit spins into a, nuH resultant earrcsporlds ta Eormirkg the trace of :I,pair of indices, as in Skkka*.*k,. T O remove this passibility srld themby %lee$ o ~ f y s == n l we must make S k , . . . k , traceless. The subtraction of tire appropriate number of restrictions gives the independent component courlt

+

+

-+-

as expected for s --- n. The resuit of subtracting successive traces is indicated by thc symmetrical fo m

where zZ=

(gk)'

8nd i t i~ required that

Xn view of the btal symmetry of the tensor, this property guarantees the vanishing of the trace for any pair of indiees. The problem thus p a s d is a familiar one. The polynomial of degrcse 1~ given iu (2-5'72) is a solution of Laplace's equation according to (2-5.73). With z2 set equal to unity, it is a spherical harmonic of degree n. T o identify the coefltieients Gnmt it S U % C ~ S to consider the single nonvanishing component S33...3 = 1. With 'g = I, = p, we encounter the poIy nomiaI which mu& be proportional to Legendre" ppolynomirtf, P,(p). Hence,

The reference to the rest frame is removed in

S*&...p l ( p ) ~ *@ ~* z L n = S @

P n ( p ) ~ @* ~zP.

This generalizes the construction glven for n = 2, Eq. (2-4.5), and produces symmtfic hnsom of rank n that obey

(2-5.77) provided sueh tensors are used in the egeetively three-dimensional context of Eq. (2-5.70). ~ ' ' ' ~ ~ ~ ) ~ f i ~ ~ ~ Q? - ~ ~ f i , ( ~ )

The etmefure of the coupling between gources can now be p r e ~ n b dalternatively aa

The form of the prajeetion Lensor TX is given by

where, for exampfe, 2

and r*

=X

Y = flPp,(p)zY

- $?l[(% ~ *

*

) 1 ~ ~ ~ *

The ddition theomm of spheric4 harmonics provide8 the fabatorisation

atthough we here use the s ~ r m b ~ Y ,lx to designah mlid harmonics. We infer the d y ~ d i ccorntruelion

The solid h a r m o ~ c sare being used in a somewhrcf symhlio way. They e m be removed by introdueiag the generating funetion

whem, in. two-eomporrend mabtrix notation, For ~implieily,we ilntradu~ethe abbreviation

F a r t i ~ l s swith arbitrary integer spin

96

and obtain, for arbitrary n,

and, for n = 1,

Accordingly, we constmct the polarization vectors for spin n, from those hlonging to unit spin, by

The known re~ultsfor n =. 2, given in (2-4.18)) erre immediately reproduced, One can verify the orthonornzality of the 2n I polarization tensors by multiplying one such expression with the complex conjugate of another, in vrrhieh. zMiig replaced by a/dx,. This gives

+

from which. we infer that rl.--r*, eph ept...p,phG

(2-5,921 It may be coneluded that the source effective for emission into the specific particle state Isbeled pX is ab~~,,

The complete description of multiparticle emission and absorlption processes for these B. E. particles is contained in the vacuum amplitude

(o+jo_)'

=

exp [iW(S)).

(2-5.94)

I n order La present the structure of W(S) ss compactly as possible, we use the four-dimensional momentcm apace version of &+(X - X') given in (2-1.61) and obtain

The tensor f I ( p ) retains the algebmic form represented by (2-5.79) and is an even polynomial in p of degree 2n, The corresponding coordinate spaw structures are illustrated, for n == 1 and 2, by Eqs, (2-3.4) and (2-4.201, reswetively. All the generalizations discua~dt3arlier in the context of special examples can be developed for the arbitrary integer spin situation. No reference hass been made to parity as an independent ~peeificationof particle staks. That is because the particle sdates we have construehd are

88

Sources

Chep. 2

rtutornatically endowed with a defitnite psrity. The gmmed~caloprsration that reverses the positive sense of the three spatial axes is reprwented by the unitary operator R,. Its effect upon the individual particle operators r, p, B izs I?;iven by The transformed sin@@pa&icle state refers to the spatial momentum. --p. Only for p = 0 can one exhibif an sigenvector of R,, a state of definik parity. As in the discuion of continuous Lorente transformations, what is relevant to the probability amplitude ( 1 ~ 1 0 ) ~ is the relettionship between the description of the padicle state and the c h a m terization of the source. The tr%nsforaned padiele sLateeis repmsented by a correspondingly transformed source which illustrabs the gentjral linear response

The reflection matrix r, is required to be mal if real sources are u s d . It acts upon the spin indices to egect the geometried transformation

or, in view of (2-5.26), The corresponding action of re upon S,;~I.. .,!jl;n).,!r,

is the interchange of the

oA1' and ck2'labels, spa& from the option of an additional minus sign, whieh

is compatible widh the simple geometrical properly rs2 = l.

(2-5.101)

The pt3rmutation of a single pair of spin indices afiets ogpasitely the singlet and tP.iplet combinations, comesponding to the opposite behavior, under spafisl rdection, of the time and space components of a four-vector, We have expressed it this way, sinee if Xeaves free the choice of overall sign in the refiee%ion response, which is the alternative between a vector, and a p~eudoor axial vector. The behavior of the tensor S,,...,s is that implied by the several vector indices, together with the overall i factor. The concept of parity refers to the rest frame where the surviving source components are Sx,...rs, which aet as s unit under spatial refi.~?etion.When standard =tor behavim is eansider~d, the parity is ( - T ) n . This gives a sequence of integer spin particlw with deernating parities, as symbolieed by of, I-, 2+, . . The other aequence is 0-, l+,2-, . . . Although the only known or conjectured massless particles of integer spin have already been discussed, we shaH nev&heless pre%nt a unifirtd treatment

.

..

Partletss with arbitrary integer spin

2-5

97

of aH integer spin massless particles. As in the special examples, it is clear that the limit m + O in (2-5.95) cannot be performed unless is valid ad m = 0. Were we to c a r e out the limiting process in the manner sXfeady illustrat;t?b, we would be tracing Lhe decomposition of the 2% J- I spin states into the helicity pairs X = &n, & (n - l), . . . , =tz l, and X = 0. This time, however, we shall directly extract X = An. The invariant form of such. a source coupling is ...v ~ i ~ * * * ~ ~l .(. .pgm,@ ) * nPrr8iE @ n(~), (2-5.103) where the projection tensor fX has s structure indicated by.

The produets formed from x b n d yp are four-dimensional. Any u s of the vector pp, as in (2-5.79), vcpould give no contribution in, view of the source restriction (2-5.102). We now exploit that fcact to replace the tensor TX with another that is equivalent to it in the context of (2-5.103). This is accomplished by the following substitution, applied to both z p and y6",

in which 'F is any null vector with"p 00, such that p p f 0. The absence of any change when p*%, O assures the ewivalence of the two structures for the application of interest. The new version of II: is given by E=:

where, for example, g

Y = s"Pp,(l-",P)xP

and

Tn the discussion of the exchange of a massless particle, p@ is dso a null vector and qp,(p, p) projects onto the subspace orthagonal to and pp :

+

Considered in the rest frame of the time-like veetor pp $P", the orLhogoasl veehr pp -- p@ has only spatial components, doubling the parlicle's momentum, and we recognize that. the subspace dected by 8,, is the two-dimensional Euelidean plane perpendicdar to the momentum of the particle. If only hcslieitiw X = fn are to be represented in the source eoupliing (2-5.103), the tensor n must be irreducible with respect to forming traces in

98

Chap. 2

Sources

the t~vodimensionalEuclidean space,

This is equivalent to asserting that, as a function of the X variables or of the y variables in the plane, (2-5.106) is a solution of Laplace's equation, which is homogeneous of degree n. The required t~vodimensionalharmonic function is

.

X)(Y

[(X

where

(2-5.111)

. y/[(x.X ) ( y

y)]'I2 = COS 4

(2-5.11 2)

Tn(p) = cos (ncos-' p ) = cos n4.

(2-5.113)

P =X

and T n ( p ) is the Tchebichef polynomial

From the coefficients of this polynomial we learn that dnm

m54n:

( - l ) m n (n - m - l ) ! J m! 4m ( n - 2 m ) !

(2-5.114)

and, in particular,

n

2

dnl = -4n.

2,

The value of -3 obtained for n = 2 is in agreement with (2-4.24). The identity COS n4 = +[e'n~e-'n~'+ e - i n ~ e " W ' 1, + = ~ p - ~ ' ,

(2-5.116)

provides the relevant addition theorem. It implies the dyadic construction n ~ l ~ ~ ~ ~ n . v= ~ ~ ~ u ePl.,-*n V L . . . ~ . PA ~ P A

A-f n

where

-

e ~ ~ ~ " ". x Pxfln l =

(+X

3

n f inp .X)(112)n.nf z e .

(2-5.1 17) (2-5.1 18)

The phases are so chosen that, for n = 1, l 1 2 .l*l

e'pilxfi= ((ti.2)

z

e

*
reproduces the conventions of (2-3.29). We now have

and the explicit construction ePl"'Pn

pin

- eyhl -..e2il,

which generalizes the n = 2 result, Eq. (2-4.31).

(2-5.119)

2-6

Spin

particles. Fermi-Birae staristics

$9

The massless particle of helicity 3 is represented by the space-time source structure

where and Ordinary matter possesses no conserved physical properties that could be identified with the ones described by the laeal conservation law (2-5.1231, or indeed for any n 2 3. The inability to construct their saurees strongly sfirms the empirical absence of the particles. Bud perhaps one should not rejeet totally the possibility of eventually encountering such properties, and the associated padicles, under cireumstanees that are presently unattainable. 2-43 SPIM

g

PARTICLES.

FERMI-DIRAC STATISTICS

There are two simple alternatives for constructing a spin in the sense of Eq. (2-5.261, namely

4 particle description,

The two possibilities are interchanged by a reflection of Lhe smtial coordinates. This indicates the convenience of a, more symmetricail treatment in. which both t a b part. It is sIso advantqeous to replace the complex sources upon which the 2 X 2 Pauli matrices aet by equivalent real sources, These remarks point to the utility af a spin -$ particle description that employs four real sources, In order to retain the symbol cr for use in the new context, we designate the initial 2 X 2 matrices as ~ k and , use 7; for an independent set. Real, antisymmetrical matrices can be constructed from the .irk by rcplaeing any explicit i by the algebraically equivalent real arltisymrnetrical m&rix hi. Thus, which are indeed real, antisymmet~cal4 X 4 matrices. algebraic properties af spin matrices :

4(qk,cl] = Eikll

i g 1 2 ' ~ 2= i~3 l,

They preserve the

(2-6.3)

and we identify sk =

$C&.

(2-6.4)

Initialy, in the role of n k we have ih*rk, where X is now a 2 X 2 matrix that commutes with the T & , and has &l as eigenvalues, When the transformation

1W

Chap. 2

OOsour~ftil

T E -+ Q& is

introduced, the m t ~ c e that s could be used for i k are j u ~ fthree in

number : i

~

~

~ = i h~ $ipg~

~ T ~ T Q ,

(2-6.5)

They are the analogues of igk with the T and 7' matrices inbrchanged. The two sc?h of three anticommuting mnatfices arc? mutually commutative. These six antisymmetrieal mtri.ces, Q,p&, and the ten symmetrical matrices 1, orpg, provide s basis for a11 4 X 4 matrices. Since the three px, are on the same footing, we arbitrarily identify h with p2 and writet where the &rereal, symmetrical matrices. We note their algc?br&e pmpedies:

the last statement b i n g the ma-lization of Eq, (2-5.33). Since space reflecLion induces n -, --n without ehanghg s, it is r e p r e m u by a matGx that eommutes with ar and anticommutes with pz, The only matricea with those eharacteristics arc: pl and ps. We choose the lathr arbibrarily and multiply this antisymmetrical matrix by i to get the real space reflection m a t h which obeys The spme refleetion matrix appears in another role on considering the real matrix ms~ciated'VVith an infinitesimal Lorexltz; tr&nsfamatianfef. Eq. (2-5.9)f:

According to the symmetv properties of the matrices, transposition hrt.8 the fallowing @Beat, L T = 1 - i 6 ~ - -4 6 ~v - & r r , (2-6.12) whereas L-' = 1 -- i 6 0 - fo+ & v - +a. (2-6.13) We express %his,through the aclion. of re, &S

The validity of this statement for the finite tran6farmations of the groper orthoclrronous group is assured by $he composition prverty of succeiclsive

2-6

Spin ) particles. Fermi-Dirac statistics

101

transformations, ( L ~ L ~ ) ~ T .= L L~TLTT.LIL~ ~L~ = L;~.L, = 7..

(2-6.16)

The relation (2-6.15) also holds for the space-reflection transformation, since r.Tr. = 1

(2-6.17)

combines the antisymmetry of r. with the iterative property (2-6.10). The appearance of the matrix r, in (2-6.15) exhibits it in its fundamental metric role. It is the analogue of the metric tensor in F,g,,lPx = g,x or, using matrix notation, lTgl = g, for (f)g, which attributes opposite signs to time and space components, is also the space-reflection matrix for vectors. Another aspect of the infinitesimal transformation matrices (2-6.11, 12), in relation to the real symmetrical matrices a k and (2-6.20)

a0 = 1,

is given by LTaL = a! - 8w X

a!

- 6va0,

which are united in LTapL = (6:

LTaOL= a0 - 6v a, (2-6.21)

+ 8d',)av.

This is the response of a vector to homogeneous infinitesimal Lorentz transformations. The repetition of such transformations yields the finite transformation law L * ~ L = rVay, which is also valid for the improper space-reflection transformation generated by L = 7.. Note that the symmetry of the a' and the antisymmetry of r,, as well as their reality, is maintained by the Lorentz transformations. We now consider the coupling between sources associated with singleparticle exchange, where the individual emission and absorption acts are represented by (2-5.36) and (2-5.42), with The spin 3 particle has been placed in a larger framework, as evidenced by the existence of the three matrices pk that commute with a. Two of the four components must be rejected by interposing a spin-independent projection matrix between the two B(p) factors that are associated with the individual acts. The possibilities afforded by the three pr are really only two in number, depending upon whether the p matrix used commutes or anticommutes with a. In the

Chap. 2

first situation, we have

while the =@andone is iflustraM by

Spin 3 particle saurces will be designated m ~ ( z ) ?(p) , or m m explicitly qr(z), ar(p). The space-time extrapolation of the source coupling takes two alternatirts forms:

where, s s we have verified on wverd occasions, the use of tbe propagstion function d+(z -- z') is required to maintain space-time uniformity, or the EucXidean postulate. The= are examples of the qnadrati~stmeture

As the irreducible kernel of a quacffatic form, .Kgt (zt "i)~houldrwpond as a and X'. This is not unit to the act of transposition, intercharrdng 5. and true of the first possibility, (2-6.271, since 1 and p2 k h a v e opgosibly under transposikion, Accordingly, the projeckion factor f -l-p2 is spurious since only one of the krms contributw ta the quadradic farm. The second kernel cfm act M a unit under the general transposition: f

c',

[(mps - #(l/i)aL)~+(z'-- z)lT = -(mp3 - olh(l/i)a,)h+(z - z'). (2-43.30) f t is antisymmetrical! Onr? might try ta convert this kernel to a symmetrical structure, without upgetting the spin description, by invoking partricles and etntiparticlt3.s. This insertion into requires an additioml twwvslued source index, and permits the kernel of the antisymmetritt~leharge m s t q.~ The resulting kernel iis gymmetrical but indefinite, sinee q is eonveded into -q by a charge refieetion. That is in fiat contradiction with the physical requirement on the vacuum persisknce probabiliky, which demands %]h& the ima@naq part of the quadratic f o m be positive, I,iWlz = ,-zrrnW 1 (2-6.31)

Spin # particlss, Fermi-Dirac stetisties

2-6

163

The conclusion is unavoidable that spin -& presents ta totally new situation. Only one coume is open. Insbad of trying to modify the symmetfy cfismckristies of the kernel to suit the algebraic properLies of the source, we must adapt the algebraic prope&ies of the source to the antisymmetry of the kernel. The comparison of the two equivalent versiom af (2-6.29) with the andisymmetq property (24.32) &ts(ztIX ) = -Ktlt(xp X') will cease to be a paradox and become sn identity if

+

We sre thus forced by the charaetedsties of spin to abmdoxl the ordinary numerical, commutative sources of Bose-Einstein stfttisties and introduce s new kind of source and a new statistics. It will be verified shostfy that this is Fermi-Dirac statisties. The symmetry aspects of this discussion have been faeilitakd by the use of matrices with definik symmetv, the symnnetricd the antkytnmetricd p%. In later developments, however, uniformity of algebraic properties and Lorentz trtansfarmadion behavior are more significant. f t is alllgebraieal~llyawkward that the anticommuting ar commute with ;'ro the representation of a Lorenta transformation on the 'a ws LTapL is not a similarity tramformation, and aga" doe8 not have tensor transformation properties. To improve latter situation one must replace LT with h-'. That is accomplished by the relation (2-6.14) which gives the new vector transformation form

lit is convenient to define im~dnary-matrices 7' = ir;la"

that; obey

L-Xr"L= P,?", together with.

L-l?@rgL= l @ g i p h ? E ~ h , and so forth. The algebraic property r: = -1, along with 'a = 1, shows that and which afso gives the identification The r matrices do not have a common symmetry, The: definition (2-6.35) hplies that fpT --i@~r;I$

104

Saurcs

Chap. 2

This restates the antisymmetry of rO,which commutes with r. = iyO, and shows that the ~k are symmetrical, skew-Hermitian matrices since they anticommute with the space-reflection matrix, Algebraic relations among the r k are obtained as

*

(yk, 7 1 ) =

-4

(at, W)=

-&I.

(2-6.44)

The various characteristics of the r, contained in (2-6.38), (2-6.43), and (2-6.44) are united in B{YP?v)= - ~ H v * (2-6.45)

This unified algebraic statement i s maintained by Lorenta transformations, according to &-l+ jr', 7 , )L = -l~,lvhgab = -g". (2-6.46) The Y matrices also give unified expression to We first write a = ( 1 / 2 i ) a X a as crr =

1

[at,ad =

&PI, 711,

(2-6 -48)

and then note that gok = iak =

~YOY~.

(2-6.49)

These matrices are united in @-P*

= ~+EY,'IYVI,

which transforms as an antisymmetrical tensor, L,-'aPYL = 1",lvhaKx.

The symmetry properties of the imaginary a,, are given by which affirms that the akl are antisymmetrical and Hermitian, while the a o k are symmetrical and skew-Hermitian. The process of multiplying different r matrices together terminates with

This matrk i s real, and since

Alkmative factadsations of r5are

whiekr allso supplies the identification iYs = pz.

fZ-@*F;a)

The Lorentz trszrrsformation behavior of rSfollows from (2-6.54) as

L-'v,L

=

1°,1'.12,13k~"gkrS = (det orS,

(2-8.61 )

which c h a r w h ~ z e rS s m a pseudoscalar. It is invariant for proper transform tionss, det I = +I, and mvems 8ign for improper transformra;tions, or reBec~ from the anticannnnute~. tions, det I = --I. The latter property f ~ l l o wdirectly. tivity of 7 and Y &S ~peeifi~slly noted in (2-6.55). Let us also obsenre the pseudo or axial vector efiarz9cter of ~ T P Y ~ , Q

L-'irfirSL= (det I)P,irPv,. (2-6.62) The components of irprs eomp~sethe four ways of mulLiplying together three diAFerelat r matricea The I6 independent elements of this CXiRord-Dirae etlgebra am organizd through their Lorents tramformation bhaviar into the five

+ + -+

for which the count is l -f- 4 6 4 1 = 16.. Clsmly reXat& but &mtinccf is the organization by symmetq properties. As suwwted by the emstruction we consider ror, where r refers to any of the sets exhibited in (2-6.63). Then, =

---rT~O = -ya~;lrT~I,

(2-6.6s)

sad the various equivabnces begween transposition md Bpwe reflection sfiow that them mstriees have a definite symmetry. Indwd, the 16 m t ~ e e given s by

106

sources

Chap. 2

+

comprise the 4 $ 6 = 10 symmetrical matrices ror,, Y%, sad the 1 4 $- 1 = 6 antisymmetrical matrices r ', rOir,r, r O r ,. All the matrices are Hermitian. The vacuum amplitude far an arbitrary spin +,four component spinsr source 42) will be stated with the matriees p8 and a",in (2-G.%), replmd by the appropriate r matrices:

and the source8 arc: totally anticommuting real objects, which constitute the elements of a, Grmsmann or exterior algebra. Let US analyrte the causal source arrangement

It is important to notice that even combinations of the totally snticommuting sources arc! commutstive objects, and that the Lwa terms involving ql and 9% are equd sinee the anticommutativity of the sources matches the antisymmetv of the kernel. Acc~rdingEy,we get

and therefore

The matrix factor that occurs here is ~ u s (2-6.26) t in a

nebf- notation,

where, it will be recalled, cosh B =

+

sinh B = Ipl/m.

(2-6.75)

The projection matrix +(l TO) is constructed from any ttvo orthonormal eigeaveedars vh, f 'vr = V ~ U = ~ ' 6hy, (2-43.76)

Spin f partictea. Fermi-Dirrc awtistics

2-6

in the dyadic f o m

+(l

+

=

F

107

v*u!.

A multipXicity cheek is provided by the trsce of this matrix equation,

where the relevant null trace of ra expresses its antisymmetry. A more general remark follows on nsting that

A specific ehoiee of the v& can be made @%Y6 3 , daVo = @Vu,

t;ts

eigenvectors of a cornponenf of a, (2-6.80)

& 1,

Cr

We dso ~ltrantthe% eigenveetors do be refa&$ by standard spin operations:

Other statements, expressing the use of imaginary ro and ar matrices, You:

=

--v:?

@,v,* = -av:,

are satisfied by

-*(ex f

wz)~$= v&,*

* -- Z"QY&Y@*

(2-6.82) (24.83)

v-,

Since v: is an eigenvector of Y @belonging to the eigenvslue -- 1, there are corresponding orthogonaliky properties, = Ot

**

v62;lUf

Q,

(2-6.M)

+ Y')

On inserting the eigenvector construction for &(l

in (2-6.74), we get

where

which involves the anticommutativity of Y'Y with Y' and the eigenvector significance of v, relative to 7 ' . The same properties are used in verifying the artlraonormzzlity of these vectars in the form

* 0Up.' ZLpuY No%?,according to Eg. (2-6,59),

= V:@,? =

[email protected].

IOS

Gouross

Chap, 2

which, combined with the hyperbolic relatiom,

This form shows the utility of defining the v. with respect to p ss s spin reference direction. Then @ = p/lpl can be replseed by the eigenvalue a, which i~ now a helicity 1Plbel. On employing the relation (s6.83) these veebrs borne, simply,

They are altso emneekd with Lheir complex conjugates by Q ,-:

= ior5up.

When the falowiog vemion of (2-6.851,

is combined with the orthonormality statements (2-6-87], we recognise that this non-Hermitian makrix has the algebraic projection p r o p e ~ y 2

- m - ?P. 2m

This is equivalent to (m

- Yp)(m4- ?p) == 0,

which is directly ve~fiable,since We also learn that up, and u;@rO obey Let W return to the source coupling (2-6.73) and write

the consistency of the two definitions conveys the Hermitisn nature of 7'. These are the precise definitions of single particle emission and absorption sourem, nrhich have k e n built up from vari~usfachrs, Thus B(p) i s contain4

Spin 4 partiicfss, Fermi-Dirrec statistics

2-63

109

in uOzr,,. In the rest frame of the parkicle, U,, reduces to v,, which is an eigenThus the veetor of TO and therefore of the space-~flectionmatrix F, = g?', single-particle states have a definite, i f imaginary, parity. Incidentally we did not prejudge this question by using the same matrix, in. defining r, and the projection factar +(l ps). I t is now clear that the latter also performs a parity selection, and that the refieetion matri.x must be defined accordingly. The particle sources v,, and v;fl, as linear functions of the qr(z),are also totally anticommutative,

+

O;rpa, qpfua'

*

{%t

*

*

~ p ~ c = r ' {(7)1~#, ~ p ' r r r )

Q.

(2-6. X W )

In particular, (VP.)~

= 0,

(v;@)%

= 0-

(2-C5.101)

The commutafivity of even source functions is used to write

All this is quite the same as with, B. E. statistics. But; now the power series contains just two terms: a,, = 0, 1, for, on reversing the multiplication order of two elements, we see that

and the whole series Germintlttes at n,, -- 1. In this limitation. to a maximum value of unity for what art3 clearly particle oeeupation numbers we have a statement of the Exclusion Principle, which, is a chartzcterislic feature of F. D. statistics, The catlsa;l situation is conveyed by the causal analysis of the vacuum amplilude,

It is indeed possible to factor the coupling terms in the desired way, but strict account must be kepG of the minus signs that are involved, This is facilitated by the falEo6ng procedure, which we illustrate with two pa&icle sla$es, fabeled a and b,

By always displacing sourees Lhrough an even number of factors, one avoids the explicit appearance of minus signs, In this way we a r ~ v eaL a facbri~atiaa where the emission sources are multiplied in some order, r e d from left to fight,

110

Sources

Chap, 2

while the absorption sources appear in the same order, but read from right t o left, It is given general expression by the following identification. of multiparticle states :

(in)IO-Y

= (O+IO-)~

n (i,.)%g

nT

in which symbolizes the opposite multiplication sense from and any standard sequence can be used for the denurnerably infinite number of particle states. As in the B. E. discussion, the pa&icle occupation number interpretation of n,, is supported by the response to source translation, 4%) -+ rt(z X), which gives

+

where

shows the additive contributions of the various particles that are present. The completeness requirement on the muttiparticfe states is stated alternatively as (2-6.109) where (0-1

s a d by

(72.1)"

((C4IO->'?

(2-6.110)

with

We have been at pains to bvrite these more carefully than in the B. E. situation, since we are now dealing with functions of antieornmuting numbers, No pre~ a u t i a n sare needed for the vscuum amplitude, which is an even function, and we present the two completeness statements as

where we have omitted the compensating factors of z" and -i. The comparison of the two forms suggests a rule of complex conjugation for F, D. Boureeg thaG we shafl find is a, consistent one: complex conjugation also inverts the sense of multiplication, as illustrated by

Then, the single statement of completeness is conveyed by

which essentially reverses the faetorizzztion procedure of the causal analysis. We must confirm this implication of completeness with a direct computa( O + ~ O - ) ~ / ' . It is importsnt to recognise that the complex conjugation rule for F. D, sources implies that the product of two red sources is imaginary,

z 'real ) sinee YO is imaginary. This is another aspect Consequently, q ( z ) ~ ' ~ (is of the matching of the statistics to the spin. Sirlee the matrices (I/i)rF are real, the only eomplex quantity in W is &.+(X - 29,and

(2-6.118)

The relation (2-6.1 X 9) then gives

q(-p)ro(m -- r p ) ? ( p ) Up

=

C )l(p)* y~ U ~ @ ~ ; U ~ O V ( ~ ) Q

Re C s;,np.. P@

(2-6.120)

The injunction symbolized by Re is redundant, since

which makes essential use of the complex conjugation rule. This resull,

is the verification of eompiteteness. The Euclidean postulate was introduced as a sharpened vergion of the principle of spseetime uniformity. It has new and interesting implications for spin particles, if it is interpreted to mean that the Euclidean transcription may contain no indication of the original Minkowski space. All reference to

112

Sources

Chap, 2

Miakotvski space doe8 disappear in the Euelidean descfiption of i n t e p spin pa&icles, but spin htroduces a new situation. In dbussing unif spin, for example, we ohsewed that the Harnnitian, real, symmetrical m t ~ c e eb4 a .= igolr; are conveded to Eerdtian, inn+nary, a~tbymnnetricafm a t ~ c m thus , uniting them with the srl, by me&nsof the tnrnsformation assooisted with J4 = Q@. N o b that it i~the sqmre root of the spetcerefieetion m a t ~ x or , itie negative, that m k r a thi8 transfomation, To pedarm an an%lagausaperation, on the real symmetrical matrices err = ~ O Y ~unifying , them with the ima$nary antiaynrmeLrieal ~g -- *&%, we mu@% find a suitable unitary transformation, one . lslthr set. The only poasibif ties a;vailable far the unithat cornmules ~ t hthe

But all t h e ~ em $ t r i ~ eare . ~ real, and cannot ehange fhe reality of the ~ 4 AccordingIy, an inspection af the redity, or symmetw of the c,, p, v ==: 1, . . . , 4 , leave8 no doubt about which Euelidean axis is related fa the Minkovrtski time =is, This is a viof%tioaelf the EueXidean postulate. We have slready remarked that the symmetry of m ~ t ~ ~ean e t gbe rever~ed, witbout d t e ~ n gtheir spaeatinne charackr, by u ~ i n gan independent a n t b m metrical nnatrh

which acts upon stn dditiowl %WO-valued source index. Its introduction enables us to form e, eomplex unitary matrix by multiplying the real r. = g'/' by Lhe i n r a a a q q and then taking the square root, in a n a l o e ~ %thehunif &pin procedure, The explicit tr~nsfsmationis -(r i/41q7'

e(ri/4)g~' E

@&v@

and indeed

t

(2-6. f 25)

are all i m a g i n ~etntisymmetGea1 ~, matrices. The detailed tfansfomatim from Minkowski Lo Euclidean source is @yen by Whm this trantgfismaLion is performed in, the vacuum amplitude, one enoounters the following rnatrk (note that a is symmetFioa1) :

are given by

(r4= iro) at = r0vX,

ad = pyoe

Spin 4 patticteat, Fermf -l)iirsc atetistics

2-43

113

They arc3 all real, symmetrical matrices that obey and The regulting Euclidean tmn~criptionof the spin

vacuum amplifrxde is

which is a real struetum when real Euclidean Isources are ued. The implieation of the Euclidean poshfah, that every spin -& ptbdicle possesses a charge-Xike attribute, is entilrely compatible wifh the empirical situa%ir>rr,Although we must give speeial attention Lo the massless neuCrinos, it is a gcneral inference from the data thaL every hmnion (F. B. particle), including electrically neutral ones, has its antiparticle counte~parL,while no electdcally neutral bo~on(B, E. particle) shows sueh duplexity, The charge label q = &l. is added h the spin 3 state8 by edztrging Q, and up@Lo bc? eigenvecLors af the charge matrix with the eigenvalue g. Since the charge matrk is ima$;inal.y, complex conjugation introduces ---g., and some correspondingly modified statements arB (2-6. f 34) U: -, -,= GY6~,,, and

The relatd particle source definitions are 1/23

*

(2mdo,) ~ , . , ~ O v ( p ) , = (2mdw,) "'s(p) *you,,,. (24.138) This dbcus&on. of the Euclidean postulate brinp %heTCP operation t;o mind, Thmu& the attached Euclidean p u p wa produce the tramformtion zi" = ---g^ (2-6. 137') qpgg =

&S

~ ( 3= ) rstq(~)p

(2-6. t 38)

where =

e(ri/2)elle(rii2)c.4

e(w~/Zlszge(ri/l)s~, e ( r i f 2 1 @ i i e ( ~ i / ~ ) * = 4 --iT5

( 2 4 .f 39) de-l&ifs the ewivslent raLations through the angle r in turo peqrtndieular planes. This matrix is antisymmetricd, imaenary, and obeys

The invariance of Che vacuum amplitude is vtsrified direekiy on uging Lhe mlstion

This is accomplished, hors-ever, a t the expense of replacing the real by tan imaginary q, sinee that is the nature of rat* If we insist on s r e d T , as in the transfarmation s(Z> = Y 5 1 ( z ) , (2-6.142)

W turns into --W, But this sign change can be compensated by reversing the multiplication order of all sources, which is in sceord with the representation of causal sequence by mu1tipEieativc?-position. The eflect, on the individual emission and absorption, sources, of the substitution ?(P) Y~v(-P) (2-6.143) +

= (2m do,) "2q(p)*70~,-@(-icr)

(2-6.144)

or ?pug

+

and

-i@vF

v;uq

-@

iflvP -@

(2-6.145)

-g,

(2-6. f 46)

-g*

The resulting correspondence between multiparticle emission and absorption processes is

where = E,

-@

-qt

and the source transformation that constitutes part of the FCP operation produces the required reversal of multipliestion order,

2-7

MORE ABOUT SPIPJ

$

PARTICLES.

NEUTRIMQS

A8 a preliminary to discussing the angular momentum specification of particle states .u;e review the addition of orbital angular momentum with spin 8. StaLes of total angular momentum quantum number j -- 1! =i= are selected from the subspace with orbital quantum number I by the Hermitian projection operaton

+

They obey f

j

j

j

j

CMrj=I i

More about spin g prr2icles. hlautrlnor

2-7

115

and have a trace appropriate to the multiplicity, We define ordhoxrormaf spin-angle functions

which are given explicitly by

The two functions are also connected by s n ~ p e r a b r : where n is the unit vector that supplies the angle variables of the spherical harmonics. The fofiawing properties of a * n are involwd: i t commutes with the total angular momentum veetor, but dters %heorbital angufar momentum. by uniw; it does not change the ortkonormaIity of the sph-angle funclions-it has uniL square. All thia shows that the left- snd righ6hand sides of (2-7.6) are the same, to within phase constants that cannot depend upon m. Xt then sufliees to set n parallel to the third axis and choose m == l$. The unly surviving harmonic, Yto = (21 1/4?r)"~,selects v+, and (2-7.6) is confirmed. The ~trvctureof the soufce coupling produced by single-particle exchange is (causal subscripts are omitted for simplicity)

+

restating Eqs. (2-6.73,74). We intraduce the preliminary transformation ( 2 dwP) ~ "2?(p) = where

C ( d ~ ) Yim(~)rlp@tm, 1m

(2-7.8)

also carries an unwritten index, expressing that of the multicomponent ?(X). The projection matrix &(l ro) makes a. selection of these components, and 4(1 rO)irS = i ~ & +( lTO) makes a complementary selection. This will be indicated by adding subscripts and - to qpar,. The residual spin multiplicity

+

+

+

116

Sources

Chap. 2

is coupled with the Ylm(p)to produce the spin-angle functions, as in

C (dQ)'l2Ytrn(~)~*p0~m = C (d~)~"~ljrntl*potjm?

(2-7.10)

ljm

Im

where we rely on context to distinguish the orbital magnetic quantum number m, which assumes integer values, from the total angular momentum magnetic The specific combination that quantum number m, which is an integer appears in (2-7.7), for given j, m, is

++.

COS

36 C ( d Q ) " 2 ~ l j m ~ + p ~ljm sin 3 0 ~(p/lpl) C ( d ~ ) " ~ z ~ j m q - ~ o t ~ m I

I

= C ( d ~ ) ~ ' COB ~ Z t 3 & + P ~ l jm E

- sin +6q-p~ijrn], (2-7.11)

and its complex conjugate, where = 2j

- 1,

indicates the orbital angular momentum change that is produced by cr p/lpl. The orthonormality of the Zum in the subspace selected by +(l 7') gives the d t i n g form of (2-7.7) :

+

where

and the charge label is left unwritten. On combining the various transformations, these single particle suurces are exhibited as np0l jmq = ( d ~ ) + p ~ Z j m q ( ~ ) * ~ ~ ~ ( ~ ) ,(2-7.15) with

wherein the Zljmqare constructed as in (2-7.5) from the eigenvectors v,,, and the spherical harmonics refer to the angles of the unit coordinate vector. The comparison of (2-7.13) and (2-7.15) with the left-hand member of (2-6.73) supplies the identification

and the antisymmetry of r0G+(x - X ) extends this to

More abaut spin 4 particles. Nsu.trlnes

2-7

117

Unlike linear momentum states in general, angular momentum states permit a specification. of spaee-refiectim parity. The response of the parficfe sources to X) -+~ Y ~ ? ( Z-I) ~,

Imvolves the transformation behavior This follows h m the homogeneous nature of spherical harmonic&, and the significance of Z l j m g as an eigenvector of 7' with the eigenvelue +l. While i~$Zrjmqassigns the eigenvalue -1to YO, this sign change is compensated ( =- The result is which exhibits space parity as a product of two factors, the intrinsic parity i, and the variable orbital parity (- 1)'. The label 1 in $palimQ should be understood as (-l)', the exact parity quantum number, for both orbital angular momenta, 1 and 1, are present in this funetion. In the spin 3 situation the two states with comrnon values of j, m can be distinl~;uishedby their different parity values. For spinless particles, aeeording to (2-2.24), parity also appears as the orbital parity (--I)', multiplied into an intrinsic parity which is +l for s. scalar, --1. for a pseudoscalar source, Eere, p a ~ t yis superfluous as a label, being cornptetely determined by the angular momentum quantum number. With unit spin particles, however, parity is insuffieienf to identify d l three at8tes of specified total angular momentum. In addition to an intrinsic p a ~ t yfactor, -4for a vector, +L far an axial vector, the state dese~bedin (2-3.39) h@ the ofiitaf parity (-l)j, representing I = j, while the two states of (2-3.41,42) have the whieh is common to 1 = j & 1. But for the massless orbital parity --(--l)', photon there are just two types of 8ta;t.e~ of a given aagufar momentum wanLurn number j 1. The photon state with source has parity --(--I)~, and that created by has parity (-- l)j. The two kinds of sources are conventionafly cdled magnetie and electric multipole moments, respeetivdy. Before i n v e s t i g a t i ~the e k t of the TGI, operation on anwlar momentum states, we examine the reality properties of J . p o r j m g ( z ) . Let us ant note that

>

Z$ma =

(-l)'+"miy~l

5 -m

-pp

(2-7.23)

which uses the spheric& hsrrnonie property and the complex conjugation behavior of v,,, being Eq. (2-6.134) with'p = m. the additional minus aigns that are produ~edby the expficit appearance of z' may be compe988tc3:d through the

On foming the complex eanjugate of

space-time reflectTon zp --+ -9, The canneetion which this relxttion es%abEghesbetween the two causal forms of 6+(z -- a") is conveyed by the invari~neeprope&y

in agreement with (2-6.141). The eEecL, in Eq. (2-7.151, of the substitution

(&)h" jmP(-~)*To~ 5qf~= ) i(- 1)

~p*a E J-m

-Q,

(2-7,28)

and then jmg

--B

-i( -l ) E + j + m rlpar j -m

(2-7.29)

--g-

This gives the detailed canrespondence bebeen single-padicrle ernisgion and abmrptian aets. The multipadicle correspondence is analogous, wi&hthe reversal of multipfication order etppmring as an aspect of the TCP transfarmation, A, spaee-time description of the multip~rticleexch~ngebet\veen sources is produced by f he power series expansion:

where the discrete indices an sources and prop~gationf"unc.tionsare regarded 8s combined with the explicit spacetime coordinah~. In contrmt with %hepermanene of (2-2.37), the symmetry of which conveys the commutativity of B, E. SOtXrceg, the antisymmetriczal drtteminant det,,, r"@+(zi-- x$) = nl perm.

e jl . . . j,~'~+(zl - zj,)

a

r"@+(z, - 25,)

(2-7.31) expresBeS the anticommuta;tivity of F. D. sources, We aee hercl: the simlple and necessary connection between the symmetry propedies that clnaraebrige the two statisfcics and the elemenlary algebraic propedie~that; distinguish fhe two kinds of soureek Appropriahty symmetfized prorfuels of individual praprtgation functions give the space-time representation of the noninteraeting multiparLi~Iesituation. Let us diaeuss now those generalizations in which the t e r m i d vacuum states sre replaced by multiparticle ~t%tf?s.A causal situation is considered, containing emiwion gaurce gz, probe source q0, detection source q1 :

More about spin 3 partiefes. Neutrinos

2-7

l 19

The vacuum amplitude is given by (O+lO-)L (0+l0-)'l+'~ exp i (dz)(dxf)ql(X)? Oc+(x - xf)qo(zf)

[/

+ i / ( d z )( d x 3 ~ o ( z ) 7 ~ @-+ bx')~a(xO](o+[O-)'O where the index r represents any set of single-particle labels, say pcrq. The causal analysis of this vacuum amplitude is

from which the detailed effect of the probe source can be inferred. To describe a weak probe one must interpret the product isr({n)10-)?. If the single-pebrticle state or mode r is initially occupied, n, = 1, the result is zero, ( v , ) 2 = 0. This is the Exclusion Principle, forbidding the introduction of an additional particle into an already occupied node. Otherwise, where n,, counts the number of occupied modes that precede r in the standard sequence, which is the number of source factors in ((n) 10-)"hat iqt must be moved through in order to place it in proper position. Similarly, and we get the weak source results ( (n

+ I t ) +l(n)-)Q(-

X)n
{ { n - lp)+l(n}-)q Si (-I)n
To construct the probability amplitude ((n)+[{n) -)V, one must retain only equal powers of v f , and q 2 , in the expansion of (2-7-33), exp

[F(isrdrlo, +

iv;,inzr)] +

IJI[1 f i q ~ r i q o h & i ~ 2 ~ 1(2-7.38) .

In contrast with the B. E. situation, the series terminates with the indicated product. On referring to (2-7.35, 36) we see that

where the factor n, indicates the absence of the term n, = 0. The effective substitution is, then,

= exp

[F

iYwn.iq~,]

(2-7.40)

920

8aure8s

Chap. 2

The linear relation between ? ( X ) and emission and absorption sources for any type of mode specification can be w i t t e n 88 (dz)~ ( z ) r ~ ~ . , ( z ) .(2-7.41

Thus, (2-7.42) and Eq. (2-7.16) supplies another example. The related canslmction of the propagation function, is that illustrated in (2-X17, 18) :

$,.@(z) = (2mdw,) "2u,,,e'P",

We now ge.t; ((n)+1 in)->L exp with

The form of the second term assures the antisymmetry of TOG~,,+(~- S'). Explicit causal stmctures are

The eannpa~on*tb (2-2.49) emphtzsiaes the essential role of the sta%isticsin stimulating (B. E.) or suppreming (F. D.) additional particte emission. We return t;o the vacuum amplidude exprm~ion(2-7.33) and ob~emethJ, in gexrer&l,

X

[l

+

i q Z i q o r i s ~ l i s ~ r l(2-7.47) ,

which converts (2-7.33) into

A typical term of the product

n, in (2-7.48)

(2-7.48) appears as

where the two sets of modes labled a and e are disjunct, sinee the individual mode facton are linear in ql1 and q f . If B nonvanishing t e r n is to result in

More about spin 3. particlsrr, Neutrinos

2-7

l21

(2-7.48), it is necessary that n, --. n, = 0. Then the a(bsorbd) modes art: tho~e,occupied in the initial atate,

th& sre not oecupied finally, while the e(mitted) modes are those oeeupietd find1y,

~phichwere initially unoccupied. The outcome is

(in

+ l.)+l{n + L)-)"

({%l+{a)-)"

[(-l)n<'i~rl G

nT

[(-~)'<~iq:l,

a

(2-7.52) whieh i~ the generdizlation of (2-7.37). In order to test $ha?eompledenctss of the multiparticEr? state8 in this general context, we multiply (2-7.52) on the left by its complex conjugate and prrtst;nl the ~ m l int a farm that reinstates (In) m an arbitrary initial 8taLe:

The summation. over a11 Gnsl states is represented by

keeping in mind the antieommutativity af sources. (2-7.4, $51,

But, according to (2-7.55)

and on utilizing the analowe of (2-6.122) for a, general mode specification:

e (2-7.M) equals unity. To derive?(2-7.56) This confims that the bfl-hand ~ i d of direcfly from the propagation func6ion eonstmction (2-7.43)we no%@ that, as a

statement about individual elements,

is valid for df

;e

- z', But irl(z)rl(z') is red, and fherefor~

(0+/0-)7' = exp

(dz)(dz')q(z)~'$,(z)$,(z') *r"v(z')

as antieipatd. The reduetion of unitarity to causality for spin i$ particles imitates the pattern alresdy established with spinless particles. We follow the development of the syskm. from the initial V ~ ~ U Ustate, H ~ under the influence of the source qls)(~), and then trace it back to tlne initial strtte, using the source g(,,(z), The stru~tureof the propagation function 6+(z - z') governs %hisevolution and we get, as the analogue of (2-2.83),

G-(z - 5') = (m - ~ ' ( ~ / i ) a , ) ~ -_z') (,s G"'(z - z') = (m -- Y@(I/~)~,)A'&'(~ --

(2-7.6 l)

Some relations among these matrix funetions are:

and

When arbitrary mode functions are used,

G'+'(%

G[-)(X- X') - zp)= C +,(z)y~.~(~')*r~, r

=

--C$r(~)*h(z')~o. T

(2-7 -64) The various functions are dso eanneeted by the identity

According to the causal analysis

a cheGk of completeness s r unitarity is prformed by verifying that (2-7.M) reduces to unity on identifying ll(,)(z) and ?(+)(X). This i~ just the content of the identity (2-7.65), combined with the third statement of (2--7.62). The generalization to the amplitude ((R) - (B) -)sc->eT(+) invalves the replacement of C+(%-- X') by +(z - X'). It is expressed by retaining the same set of relations but based on the new definitions,

As in the spin O discussion, tfle general unitsrirty proof uses the sources q f f ) ta generate arbitrary terminal st&s. The complotc3 removd of reference to a subsequently ~letingG O U T C ~~ ( 2requires ) the additional relations

w7hich are comect statements under the assigned eausd circumstances. The eharge property that is required by the Euctidean postulate played no role in the initid treatment of spin particles, This raises a question eoneernixlg the possible existence of other kinds of spin g parLictes for which the ebarge matrix dam make an explicit appearance. We shall take for p a n k d in this discussion that Ghe charge attribuk remains ulnalkred during the travel of %be particle between emission and &sorption sources. That permit8 a, dependence of the propagation funetion upon the charge matrix g, but not upon the chargereflection n a t r k r,. The generd f o m of such a pmpagstion function is

which lacks only the matrices g,, = --CF.,. Since TO and ror, are antisymmetrical and r0up symmetricd, the antisymmetrical matrix q eennot multiply them. But p must be used to reverse the antisymmetry of Y @ Y ' ~ Y ~ . This structure shauld fit into the mode function pattern detailed in (2-7.633, M) since the latter refern only to the combination of individual emission and absorption wcta An essential aspect ia the positiveness property

) an arbitrary complex-valued numerieaf funetion. X t a implieation where! ~ ( 2 is

124

Chap. 2

Sources

for (2-7.69) is the following Hermitian matrix positiveness requirement associated with any particle momentum F, which incidentally asserts the reality of ml,mz, a, and X, When a particte of mass m > 0 is viewed in its rest frame, this condition reads The three matrices ipT5, Y', ror5anticommute and are of unit square, from which we infer the numerical requirements

+ + m:]'"

ma i [ ( m a ~ ) m ~:

2 0.

(2-7.73)

In addition to the conclusion that a is positive, we note that the zero value must be attained if a projection matrix is to be produced, and accordingly m2a2(1 - h2) = m;

+ m:.

(2-7.74)

Throughout the open interval h2 < 1 , it is permissible to normaliee a by a2(1 - X')

with the consequence ml2

This is represented as a = cosh 8,

QX = sinh 8;

= 1,

+ m: = m2.

ml = m cos ( p ,

m2

= m sin cp.

(2-7.77)

It is then easily seen that y OG+(,

,t)

= ,-c

l/ 2 ) ~ ? 5 1/~ ~ )( @ ~ Q T s

X [r"(m

- r*(i/i)a,)a+(x - Z t ) j e ( 1 1 ~ ) e 4 ~ 6l el 2( 1 ~ 1 7 (2-7.78) ~.

and the symmetry of ((v5,these matrix fa* In view of the antisymmetry of tors can be consistently transferred to the two source functions in W. Thus, despite the initial appearance of q and of T5, they disappear after suitable redefinition of the source and we restore the structure of (2-6.67,68). The remaining possibility is X*=

1,

r n ~= m2 = 0.

If we now chwse a= there emerges

4,

2-7

More about spin

+ particles,

Msiutrinos

725

The first version ixrdicrttes that we have regained (2-6.2q, with the objection to the antisymmetry of p2 =. iyti removed by the pmsence of the additional antisymmetrical matrix Xq. The second version is related ta the standard form of raG+ by symmetrical matrix factors, which could be transferred to the sources. These are sinwlar projection matrices, however, and the new sources will be subject to the restrictive condition

A source constraint states a universal characteristic of all realistic mechanisms %hateoxltribute to the creation or annihilation of the given particle, Whe$her the iateraetioa mechanims of spin. particks might be compatible with such a restriction cannot be examined at this point, save for one exceptional class of spin $- particles-the neutrinos. Only one k n d of neutrino interaction has been observed, processes in whi~h. thr?y are created or annihilated in company with a charged lepton (electron crr muon), The neutrino astsoeiated with an electron is a different particle from that agmciiated, with a muon. The masses of the neutrinos are small on the scale SM?L by their leptonic partner, but they are not known to be zero in the same sense as are the photon and graviton masses, where inverse masses must ex~eedvery Iarge macrosqic distances. Both neutrinos are observed with a unique helieity, which is detemined only by the electric charge of its partner and reverses sign vvith the latter, The spin 4 possibility just discuswd provide8 a natural franework for the representation of these properties. (Indeed, it was in essence proposed long before the experimental disclosure of the two neutrinos, and conerLiLuted a prediction of that fact.) To avoid confusion with electric ebargc?,let the ehrzrge prope&y carried by the neutrinos be designakd by 1, the leptanic chrtrge. The alternatives contained in k2 = 1 give two kinds of particles which are distinguished by the corresponding projee(tion factor:

+

On considering a neutrino of momentum ,'p the other matrix factor in (2-7.81) becomes -YOY'~, ='p - i r , ~p. (2-7.M) which usea %heeffective equivalence between i Y t j and &l %h&is enforeed by (2-7.83). Wben the neutrino has an energy that is large in comparison with its mass, which need not be zero, nor the same for the two neutfinos, a unique helieity is selected : (2-7.85) The conservation of leptonie charge requires that the efeetrically charged lepton accompanying a given neutrino carry a Ieptonie charge opposite to that of the

128

Chap. 2

Sources

neutrino : l

+ b.

lept.

= 0.

We now put forward the natural hypothesis that one role of Ieptonic charge is to distinguish, and label the two leptons with a common electric charge q: Ech. tept. =

(2-7.87)

TB*

Its consequence is the empirical equivalence between neutrino helicity and the accompanying electric charge, In the interest of compIeteness we shall exhibit the sources for specific neutrino states, under the simplifying assumption of zero neutrino mass. One uses the dyadic construction

where ~ =

~

and

*

Up1

6 r~ ~ 1 ~ ~ ~

~

~

l

= Up-l.

These eigenvectors also obey

The sources for this kind of neutrino are and those for the second type are obtained by the reflection where r~ is the real symmetrical leptonic chargereflection matrix. The TCP substitution d P ) W?(-P) +

interchanges the neutrino emission and antineutrino absorption sources (2-7.93) in the folfowing manner: Additional minus signs appear for the other neutrino type. Finally, here is a brief comment on the Euclidean transcription of the vacuum amplitude associated with (2-7.81) (=i=l replace Xq). It is

~

Particles of integer 4-

4 spin

127

extends the set of red symmetricsl, anticommuting matrices of unit square. Unlike the Minkowski form, las is mtisymmetrical, and correspondingly mticommutes with the a,. The explicit appearance of the imaginary matrix I means that the individual Euc-lidean forms are not reai. But if the mmsw of the two neut~nosare the same (zero?), complex eonjug&fioninkrchanges the two Euclidean structures, Then one can regard the two n e u t ~ n osources as obtained by prclljeetion. from one general source and the complete Eucfidean vacuum ampli%udeis red, being (2-7.97) without the factor +(l 2a5>. 2-8

PARTICLES OF fNTEGER 4-

3

SPIN

One can d e s c ~ b espin particl~sby combining the four-veclor treatment of unit pia with the four-component spinor mpect of spin +. The resulting vectorspinor source qf(z) has l 6 eomponents, apart from additional charge multiplicity. ?'fie reduction of this larger system fo Lhe one of inkrest is partly produced by the projection. matrices approp~sdeto the constituents; pp,(p) for spin l, m - ~p for pin 4. On, eonside~ngthe murce coupling wssoeiated with ~ingle-padicleexchange, in the rwf frsme of the particle, this procedure s u p plies the effective source +(l T @ ) ~whieh ~, hrw six eomponents. The final reduction to the four components characteristic of spin # is accompfislted by fhe projection mabix (2-?.Q, with I = 1, j =" g. In the conkxt of threecomponent vectors and two-component spinors it is represenkd by

+

The projector character of this matrix is equivalent ta the properl;y

and its specifie identification is confirmed by evaluating the trace over the sixdimensional space. The resulting form of the source eouplimg in the rest frame is rl:*(1

where

+

r0)(6k&

W

"ii rc

*@k@1)'11 ~rlk

= q:B(1

+

7')~kt

&Q~@ZVZ

(2-8.3) (2-8.4)

obeys Qk-?k

= 0,

(2-8.5)

evhich make8 explicit the rejection of the spin & composite s p k m , The remoml of the rmt frame specification is facilitatd by wfiting and the straightfomard pneralizittion of (2-8.3) (war%from a &tor of 2m) is

128

Sour~et

Chap. 2

The second term is given a somewhat simpler form by noting, successively,

+

The first rearrangement restates the commutativity of 1 ra with e r ; the second invoke8 the prope&ie~of and the Iwt uses %heprewnee of the factor m -I-. ? p to substitute m for rp. An sltc?mat;ive form replaces % (I/m)p, with --(ilm)$hpk. The implied expression for the vacuum amplitude

+-

1vriCten in four4imensional. momentum spaw for conciseness, iis given by

The kernel af this quadratic form is antisymmetrical, under transposition of the mat^ and vector indices combined wifh pp --* - p p , which dernazxlb a eorresponding smticomnnutativity of the sources or F. D, statistics. To identify psrticlet emission and absorption mwces, ~pffieihcsllythose referring Lo the four heXicity staks msociated with % given momentum, we consider a causal arrangement and marnine the coupling term in i W :

in which we bsve returned to the vemion given in (2-g.?), and u& $he dya&c

construction for gtr,(p). The introduction of the dyadic spinor realization for (m - r p ) / 2 m gives the form

which ~ervesto idenkify the four eigc3nvectom g$&,X =. 3, , . . , -8, In order to proprty, exhibit them explicitly m use the follo~~ing

\vhich is to be undembod irl the limited conkxt, tr = & X , This formula is easily checked in the rest frame, where the lefbhand side reduees to et .v@ Its genera! validity is mured, acearding to (2-6-86), if

-

It will be rrzcopiet3d that this is equivalent to the constru~tion(2-6*53),with the a matrices replaced by the algebraicslfly indistinguishable set i ~ ~ a , The explicit, forms of the helicity Xabefed eigenvectors are

which are standard combinations of states for unit and Their orthonormality propdies arc given by

anwiar momentum,

The resulting souree identification is

to which a ehafge lahf can, be added. The complex conjugation pmprties of the eonstifuents supply the relation = (-l)ta/z~+'ir,u;h.

(2-8. l S)

The TCP substihtion q"(p)

therefore: inducm VPX

i(--f )

(812)-X

,*

+

~JV~(-P)

?.ph

-+

-i(- l ) t 3 1 f ) - X q p - ~ ,

(2-8.29)

(2-8.20)

which can be suppXemenM by a charge index, in the usual way, The treatment of dess spin 8 pafti~lesgenerafly fol1cbtc.s the unit spin pattern. On writing ars"(x) = mq(x), one recopizes that, a m -+ 0, helieity rt=# decouples from hetiGity &g, which is represented by the spin source, ?(2) - +i~,q"(z). Mmsless particles of helieity =tr# are der;cribd by (dg)(dx')

where

?'(z)r0[scu (-7 X(l/i)ax)D+(%- z')

- *% ( - ~ ~ ( 1 / i l a h ) ~ + (-z Z~)?~]~@(Z'), a&q"(z) = 0.

1343

Sources

Chap 2

The coupling term in z"W for tt causal alrrangement is

One ewn replace to helicity k X :

evevwhere by jmt the two krms of the dyadie that refer

SlnGe

PP.Y(P)= a,

We also use

m

0.

(2-8.26)

rO(-rp)= 2p0&(1 -- irsa p/Ip

*

= 2~"

(2-8.27)

@~,~eup,t,

U'=&

where --. @ " ) . U ~ ~ P =

(a6-E- Q

' ) u ~ ~ ~

(2-8.28)

and the algebraic ps~perties

The out~omeis the replaeemexlt of (2-&.g) with

in which +(l. tionis are

-+ XB)

~eleCLsonly the staks of helicity

&#. The two m d e fun* (2-8.3 I)

= ei+l@~k,

and the corresponding sources are given by

As in the spin fj- n e u t ~ n odisewsion, one can introduce an. rtdditioml decornposition in which helieity is coupled to charge in a unique way. Preparatory to generafi~ingthis approach to all parliclw of spin s n -f- 9, n = f , 2, . . . , we return to the r e ~frame t spin projection =I..

Vk

== Ptk

(2-8.33)

$@k@l171

and remark that ilk

S @ p n t p . ~ ~ @ ~ ~ t ,

(2-8.34)

where a k p , t l l -=

*(8kt8,,

-4-

akrl8ta)

m

"=.kp6tQ

(2-8.35)

is the rest frame vemion of the n = 2 projmtion tensor that is defined generally

Pattieies of intsgsr

2-8

+ 5 spin

'l35

by (2-5.79). The properties of this tensor assure that

Here is the generdization of this resdframe treatment to symmetric tensarspinor sources :

Although it is evident that the garded as a consequence of

i j k , ...kR

are traceless, this property can be re-

@klBkl.-.k,

=z

(2-8.38)

0,

according to 0

2=

@kflk,qk,kzkg*.k,

(2-8.39)

== qkkks..sk,.

I<eeping in mind the restriction to tu-o-component spinors, we see that the count of independent components is

+

consistent with the description of spin s == n ij. The numerical ftletor in (2-8.37) ean be derived by noting that the latter should be an identity if qr,..,~~ is replaced by ql, ...l . I n that circumstance the projection tensor reduces to the symmetrized unit matrix appropriate to rz 1 indices. There are two classes of terms; those that select p .= q, which are R! in number, and those with p = L j , q = ki, being the remaining %(R!) terms. The firs6 set is muftiplied by ( @ p ) 2 = 3, and the second set by 2, since

+

and olfir,...r. = 0. Accordingly, I

apnkl,..knp,~I...2,ct@q~ll...l,

f3(n!) -4- 2n(n!)]qk (n 4-l>!

as stated in (2-8.37). Alternativeiy, one ean verify that the trace of the projection matrix that is defined on the spaee of n three-eamponent vector8 and two-component spinors h a the required value of 2(n +) 1 :

-+ +

since the trace of the projection matrix that refers to 7t equals 2(n 1) I.

+ +

+ I thme-vector indices

132

Sour~aa

Chap. 2

+

A particle of spin s = n $- can be describd by the symmtricd tensorspinor source $ 1 . " ' e ( 2 ) . The four-dimensional momentum space version of W i.s

This is not t o be taken fiterally, however, far an wlgebraie simplifiestion should be pedormed hfore the space-time extrapotation embodied in the fourdimensional vemion is carried out, In the i n i t s cau~alsituation, ~vhere( m rp)/2m is a projection rnrttrix decling ?p .= m, two powers of the momenlum, appearing in the form ?'p.(m rp)phrh, can be removed. Thus, the matrix polynomial in p thffiL occurs in (2-8.44) is of degree 2% I = 2s. That ig iifluslrakd for n =. l, s = #, by (2-8.10). f n. the dbect applickttion of (2-8.4) Lo a, c a m 1 arrangement, the introduetion of the dyadic construetiom for the spinor and tensor projection matrices must supply the tensor-spinor dyadic

-+

+

+

(2-8.45)

where

is consisknk ~k-ith,the proprties of this structure. It is not diEcuft to pick out the bmm of highest hejicity, X = s,

which apwwrs on the Iefbhand sidti?of (2-8.45)with the coegeknt

The other heliaity functions are produced nnos"cimp1y fmm this one, by rotation, ss effectively rewliad in. the algebraic construction

where $,&(E) is defined a8 in (2-5.87) but with n replaced by s, The rwults for = 8 that are &vent irr (2-8.15) are immdiakly repmduced in this w8y. The

i;:

2-43

Particles of integer

+ f spin

133

sourees for the helicity labeled stzt;hs of thew F. I). psdieres arts identified as

+

T o close this seetisn, we consider the masslms particle8 of heiieity f (a +), . . (although no example comes to mind). It seem evident that the

n = X, 2 .

mecewary tensor-spinor restriction

must be accompanied by the corresponding projection tensor, which is described by (2-5.104), and inded the general form is

one verifiw Ghat (2-8.22) is reproduced, while becoming aware of the egtrivalmt form

The coupling hrnn in 2"W for a causal arrangment is obtained from (2-8.52) as

where the introduction of the new projection tensor, defined in (2-5.106), is justified by the properties

The dyadie eonstruetion (2-5. f f 7), combined with (2-5.1.21), conveds the tensor-matrix of (2-8.55) into (the tensor indices are raised, for clarity)

where, utilizing (2-8,291 and (2-8,27),

134

Saurcm

Chap, 2

The f m b r 4(1 f a') locks the spin 4 hdicity to the olfiw, and we re-co@;nige the genemEsaLion oE (s8.31) for s = n 4 :

+

vy.-.v,

= epkn with the w m c b t d wmce definitions r/z ; P ~ * . . P * , $p& ( 2 ~ 'h p ) %p& ~rl...r,(~)$

(2-8.59)

h = As:

2 4 UNIFICkTIQM OF ALL SPINS AND STAT4STlCS

The proeedure~we have M b w d for descfibing the v&dow spin possibilities exploit elementav anmlar manentum pmp&ies. The spin s (n = 2,3) . . . em be csmpoundd Irom n unit spins.. And it s u E c e ~to add s shgle spin of 8 to produce the aquence s n +,a == l, 2, . . . . But all a p b possibilities c m be con8tmckd by combining the fundamental spin system a sufficient ven for intc3rger i pia, odd for ixlhger % spin, Aceordlngfy, m replace the tmmr or multivector, expressing the composition of unit spins, and the [email protected];etdkmor-spinor, by the; umiver~~1 muf"tispinor t h h i g appropriate to the compo~itimof a numbr of spin 3 canstituen$~. A muftbpinsr saurce will be explicitly written as Srl...rm(~), but the indices will often be suppressed. All connponenL aping are on %hesame footing and %dditioaalsymmetq mquire men& can be i m p o d on the multbpi-nor. The mast import@ntof them is %he r e q ~ e m e nof t tattit1 symmetq : ET:

=-b

+

+

.

where . . . ar, any prmutation of 1. . .n. The mul%kpimor refers Ito a larger system than desird and prajwtion m ~ t f i are ~ e req.tl_ired, ~ even ets in the simplest ~ituakiann = 1, s = Indwd, it s ~ m tos use m& spin prajectisn m ~ t ~ con e se ~ pino h or index to abtain $he mquird rduetion t;o the phyftieal ~sysbnrof spin s,

a,

+

for a, aymmet~ealmultispinor. In the re86 frame of a mwive p&icXe, th& projection matrix ia ?B

where? a desigasbs the spinor index on which the corrmpc>n&ngm t r h acts. Its @Beeti% .to reduce dbe r a g e of each, spinor index .t.o two values. A symmebrical funcLion of n twwva1ued indice8 has s nunnbr of intctependen* components

Unification af all spins and statistics

2-8

f 35

as anticiggted, The spin vafues obtained in this way are Only

8=

0 is ~ s u i n g . For that it suffices to consider n === 2 and choose the

An &ntisymmr?trieaI function of a, pair of h c t i v d y two-valud indices has anly one independent component. The general expression of these remarks is given by the foIb.eviw vaeuum amplitude, where, wing four-dimensionaf momentum space, we have

The h m e l of this quadratic lorm ha9 a definite wsynrrrnetq under matrix tr%nsposition combined with the substitution p"" + --fl,

Aecorifingly, if the dgebraic propertim of the source are to mahh the symmetv properlim of the kernel, we must have n even, s == integer: [S(x), S(&))]= 0, B. E. stsdistics, (2-9.10) n odd, s = integer 3- 9: (S(%),&(X')) == 0, F. S). &&%&ties,

which is the general stakemen%of Lhe connection betwrmn spin and ~tstistics. This proof will be eompjete, however, only when we h w e shmn that any a t tempL to revem thme natural connections does violence to Lhe completeness of the multiparticle stabs, Let a8 eon~ider-the causal srrangement B(x) = Sl(2) 4- 82(~),

which implim

(o+~o-)~= (O+IO-)'Z

exp

[?'(m - 7p)IaiS2(p)

d w , i ~(p~)' a

(2-9.12)

Ollr usirtg (2-6.93) far each spinor index, we have

which, in general, must be projected onto the spsoe of symmetrical spinors. Employing he1iciQ labdeli spin functians, for definiteness, we recognize that

136

Chap. 2

Sources

the highest helicity contained in (2-9.13), X = *n, is represented by the function n

%P.

=

n (up+).,

(2-9.14)

a=l

and the whole set is generated by

I n the special situation of the antisymmetrical spinor with n = 2, the single eigenvector is up = 2-1'2[(~~+)~(~~-)2 - (uP-)~(uP+)~~ (2-9.16) The orthonomality of the helicity functions, in the form

is derived from (2-9.15)as

With the definitions

Eq.(2-9.12) becomes

which uses the fact that even functions of the sources are commutative for either statistics. The causal analysis

(~+lo--)~ = C @+l l* 1

leads to the identifications

ns'(

n)l~-)~'

(2-9.21)

Unification of all spf ns and statisgitiicra

2-9

137

where opposite multiplieatian order is used in the two pmductcs. It is only through the implieit, alvbraic prope&ies of the sources,

B.E.: F* D.:

fSPx,SP3~f=0, (&x,

=

&lx?>

(2-9.23)

o,

that the two statistics are distinguished. In particular, the algebraic property

F. D.:

(2-9.24)

=O

leads to the ckraehris%icF. I).limitation, n,k = 0, 1. The two expremions of completeness,

Become, respectively, 1,

(2-9.26)

and

Then the single stakment of eomplekness is given by

For a direct computation of / ( o + / o - ) ~ we/ ~ return to (2-9.8) and note that ccomphx conjugation interchanges S(p) and S ( - p ) whife reversing their muE tiplkation order. Therefare,

according to the Hermitian nature of each propefiy gve:s

ro(n- rp) matrix. This reality

~r-here,as a. statement about integrals,

+

= f (p2 m2)u2. The t~v\.oterms are interindicates the restrietion to changed by the substitution pp --+----@, under ttihict.1 the integrand of (2-9.31) remains unaltered. AecordingXy, with pp designating a physical momentum, p' > 0, n-e get (all this is the four-dimensional momentum space equivalent of an often repeated space-time computation)

in conformity -with the requirement of completeness. N o ~ plet us examine how this consistency would be aEechd if we intervened in (2-9.5) to reverm the natural connection between spin and staki~tics,by injecting an anti?aymmelricaImsttrix

which a c b on an independent index and thus preserves the spin classification that has been achieved. The identification of multiperticlie s t a b s from rt causal arrangement proceeds analogously, with the helieity vectors up& exbnded to u,hql q ==; &I, The result is given by the follotving replacement in (2-9.22):

where the product of the additional p h a ~constants reprodueeg q, I n the difect consideration of completeness, hotvever, these phme conatants disiappem along lvith the factors of i, and the outcome is just (2-9.29) tt-ikh the q index added, (2-9.36) Turning to the vacuum amplitude itwlf, we obseme thak the realiky property (2--9.30) persists 1viL1.1 the Hermitian matrix q inserkd, and that matrix t3survivtls in (2-9"33)to give

The clem contradiction .cvi%h(2-9.36) completes the unified proof of the cannection between spin. and statistics.

Unification of all spins and statistim

2-9

*r39

The TCP opemtion, is defined for every spin by the substitution

combined with reversing the multiplication order of all sources, The egect of the substitution on W comes down to the minus sign induced on eseh r@ by the r, transformation, and thus W is multiplied by (- 1)". The reversd in the sense of multiplication introduces a plus or minus sim, in accordance with the st~tistics. Through the connection between spin and statistics, W, (and the vBeuurn amplibde), is left invariant; under the compie%eTCP operation. To study the eEect of TCP on individual emis~ion~ n absorption d sources) we first notice the generalization of the spin complex conjugatioa property (2-6.92), which depends ugan the multiplicative eornpasifion of the u,~, Then we find that to which a charge index can h tzdded in the h o w n way. The eorrwponding multipsrticle transformation is

where = n, -x.

We have been discussing particle rcspectg in which unification is achieved, the specific nature of the system being implicit in the pa&iculsr value of n, the number of multhginor indices. But when we turn to the Euelidean postulate in the context of multispinor sources, %hefundamental diBerence betwen the sttztistics, or between, integer and integer 4 spin, beeome~explicil. The

+

where p is a n n e t ~ xto be specified, replaces %hekernel of (2-9.8) with

The r@m&trices, which mirror the indefinite Minkowski metric, must be removed in the tran~form&ionto the Euelidean description. This is ~.ecomptished, for n even, by the symmetrical matrh

which is such %ha$

IrK)

Souram

Chap, 2

while the matrices

we &l1red,antisymmetriettl matrices that obey

in which we eon%hueto designate tfie Lrmsfomed m a t ~ c e sas 7,. When thia is combined ~ %thehtransformation af momentum integrale,

we get the camespondence

The possibility of producing the transformation (2-9.46) is contingent on the lefehand side being a symmetrical matrix. For n odd, it is an sntisymm e t ~ c a matfix, l If tbe latter is to reprme_nt the Euclidean metric, it must be unaltered by Euclidean transformations and is therefore in the nature of a charge matrix q. The Euclidean postulate requires that every integer 4 spin particle carry a chargelike attribute. Were (2-9.45) applied unaltered with n odd, we would get pTp = i, which does not eliminste the r@ matrices. The appmpfia"tedefinifion of p for odd n is

+

snd now

The tr&n~formedh matrice8 are W£

which continue to be real, antisymmetrical, snd governed algebrsjcdly by (2-9.48). Thus the Euelidean correspondence for odd n ia

Unification of all spins and statistics

2-9

141

Incidentally, for n = 1 the connection with the real, symmetrical a, matrices of (2-6.129) is a, = -iqT,, (2-9.55) where these r, matrices are the transformed ones of (2-9.53). The space-reflection transformation is defined generally by S(a)=r,S(r),

2O=z0,

%=-xr,

(2-9.56)

with n

r8 = ( z t )

(ir3.

(2-9.57)

a==l

Some properties of this real matrix are given by

which distinguish integer from integer

+ 4 spin, and the generally valid

The uniform selection of p' = f l in the rest frame gives the definite parity (f)P, which is real for integer spin. With n = 2, the alternatives of antisymmetrical and symmetrical spinors give the spin-parity properties 0-, I" and 0+, l+,corresponding to the sign option in (2-9.57). Otherwise, with the general use of symmetrical spinors, integer spin particles fall into the two sequences of parity (&)(-l)'. No rest frame is available for massless particles. In this circumstance, the kernel of (2-9.12), referring to causal conditions, becomes

Now it is the values of the individual helieity matrices U p/lpf and the asssociated Yg matrices that specify a particle state. For a systematic classification of almost all helicities, using symmetrical spinor sources, it suffices to identify the value of every ir6 matrix and thereby of the individual helieity matrices. This is sccomplished by inserting the following symmetrical real projection matrix:

Then we have

142

Sources

Chap, 2

The limitation to a pair of helicity states is confirmed by waluating the trace- of the lefthsnd side in (2-9.62), for whieh one can use the full $"-dimensional multispinor space: 4'(1/2')(1/2"-') = 2. The list of all helicities obtained in this wrty, X = f+, &l,&g, . . . , only lacks X = 0. For that, one can choose n = 2, replace the r, projection factor by -$(l- i ? , , i ~ ~ and ~ )use , an antisymmetrical spinor. The emission and absorption sourees are identified as

Although this discussion applies to n both even and odd, the necessary existence of a charge propedy in the latter situation, of h an integer 9,invites ra further classification in which the helieity is tied to the charge value, This is produced by replacing (2-9.61) with the symmetrical real projection matrix (the common =t sign gives two alfernatives)

+

For a given value of q the trace of the complete projection matrix now equals 4"(X f21Z)(X/2") = 1. Thew tare only two states, labeXed by q = & l , and the helieity is == ( ~ ) 4 $ @ , (2-9, M) where the sign option refers to the alternatives of (2-9.65). Pn each situaion

This treatment is k3ss general than the earlier neutrino discussion for R = 1, sine@fph%t did not require the msumption. of zero masg. We &all close this section by examining the connection h t w w n the muftispinor description and the tensor treatment of integer spin particles, in the simplest situation of a second rank spinor Silt. It is convenient to regard the l a t k r as matrix, and to correspondingly rewrite the structure of W as

Unificaticn of crlt spins and M a t i ~ t f ~ 143 ~

2-9

The general e~ntisymmetriealand symmetrical matrix can be prewnted nt, re~pectively ,

+

2S.(p) = iroS,( p ) -tiy,r0Sz(p) ~ S ~ ~ T @ S # ( P ) ? (2-9,70) 2 ~ . ( p )= rpr@S,(p) ioB'r0SPu(p),

+

in which the individual matrices are real, As a useful algebraic rearrangement, \re note that tr [(-v12

- ~ ~ ) S ( - ~ ) ~u rp)S(p)yO ~ ( r n ] = -(pZ + m2)tr S(--p)TraS(p)70

+ m tr ( S ( - P ) ~ ~ @S(p)yOl) [~P, + + tr ( [ r ~s ,( - P ) ~ ~ ~ I [ Y P~ ,( ~ 1 7 ~ 1 ) (2-9.71)

\%-here,for the tiro symmetries,

The evaluation is redueed to computing the traces of n&rices formed by multiplying linear combinations of the Dirae matrices. These 16 mrallriees are

orthogonsl in the senw of the produet defined by the trace. Their normafizaGiaxrs v a v in sign \r.ith the Xfermitian or ske~v-Herrnitiannature of %liematrix, as dictated by the space-time metric, Thus, the algebr&ieproperties of the Y, imply that (2-9.73) 4 tr r,r, = 4 tr YrYt;YuYS = --(firu, ~vhile PEA g,rt@rh S r x Q v a * (2-9.74) The results are E

with

K($) = 2"'(mS2(z)

+ a,P(x)),

(2-9.78)

Chap. 2

and

The K and J stmetures are the anticipated onm for spin O and spin X. There sre additional terms, however, ~vhiehmodify the vacuum smpfitude by the typicat. factor (S stands for SE,8 2 , S,, S,,)

This is an equivalent description. The additional phase fetctor daes not change the vtacwm persisknee probability nor does it contribute to the coupling between sources in a causal arrangement. And it has no implication for the obsem&blewpeets of the energy asociakd wikh a quwi-static source distribution, for they refer to the effect of relative displacement of two disjoint parts. Physical eonsideratiom that arc? sensitive to such souree overlap terms can appear only in the fudl-ter development and specialisation, of the general souree f ormlzlism, Far m == 0,unit helicity parlieles should be selmted by inserting the projection matrix Its aetion upon the second-rank spinor is given by the matrix trans~riplion The two terms in the symmetrical spinor of (2-9.70) commute and anticornmub, respeetivefy, with Y ~ .Only the XatLer is retained by the projection nxat~x, which eEecti-vely sets &(p) egual to zero. As we recognize from (2-9-78), Lhe divmgence of the veebr source JP(%)then vanishes identically and the photon O in deseription is regained, It would not have suffi~edto merely let m (2-9.78), since it is dsa necessary that (I/m)dJF+ O, We have remarked that $he antisymmetricat spinor should be supplid andogously with a rSprojection faetor that digem from (2-9.80) in the relative sign of the two Lerms. This selects terns in S, that commute with rS, which is uniquely the axial vector contribution of (2-9.70). Nsw, ho~\rever,it is sufficient .t;a set m = 0 in the eEe~tivesoume (2-9.76). It seems to be a specific property of the second-rank spinor repremtaticm that the source of massless spin 0 pa&ides acquires the special form of the divergence of a vector. -+

FIELDS &-3

THE FIELD CONCEW.

SPIN O PARTICLES

Sources are intrdueed tia give an idealized description of the creation. and the dcrtection of particles. But the puwose of this activity is to study the proprties of the particles, snd this takes place in some region inkmmedictte htti-een the locations of the terminal acts of creation and debction, Thus one n e d s a convenient measurn of the strength of the excithion that is produced in s w i o n that may be far from its sources. Ttre natural way to obtain such a memure is by investigating the eflect on a probe or test source that is introduced into the region of inkrest. Accordingly, con8irleeng spin O particles and their real scalar sources, a represented by

we examine the efict of adding an additional weak source 6K(x), L t is given by

where This combination of source and propstgation, function, measuring the eEwt of preexisting sources on a weak k s t source, is the$eEd of the sources, 1%is defined in an analogous tvay for any type aE particle, as indicatd by

or, equivalently, by the four-dimensional momentum inkegral

We now recognize that it obeys a simple inhomogeneous differential equdion. That is mast evident from the second expression since the application of the differential operator that produces m2, when wting on exp[ip(z - X')],

+

146

146

Chap. 3

Fields

cancels the denominator and leaves the four-dimensional delta function

AIternativeIy, one uses Eq. (3-1.5) and notes that

+

since p 2 mZ = 0 in these integrals, while the discontinuity of the time derivative across x0 = zO',

is equivalent to the presence of the four-dimensional delta function in Eq. (3-1.8). The differential equation that it obeys identifies A+($ - X') m a Green's function of the differential operator --a2 m'. It is the particular solution that has only positive frequencies for X' > zO' (e-ipO'O, > 0) and only negative frequencies for xO < xO' (eipO"O).This boundary condition is more simply stated by considering the associated Euctidean Green's function. The latter obeys the differential equation

+

+ m 2 ~ ~-E23( =~ ~ E ( X- X'),

[-

(3-1.11)

where (d.)

6(x - X')

C-,

(dx) b(x

- x')]g

(3-1.12)

c-, SE(x

- X')

(3-1.13)

or (xq = isa) ( I / i ) 6(2 - X')

restates the correspondence

Unlike the Minkowski situation, the two fundamental solutions of the Eudidean differential equation are sharply distinguished by their asymptotic behavior: -e*"R. Thus the requirement of boundedness, for X # X', uniquely selects one solution, the one that is produced automaticaIly by the Fourier integral solution of (3-1-11],

and A+(x - z') i s recovered by the previously explored procedures. The alternative methods of imposing boundary conditions can also be applied directly

3-1

The field concept.

Spin 0 particles

147

to the differential equation that describes the field of an arbitrary source, Other kinds of fields and Green's functions are introduced on considering the time cycle description that is associated with an initial vacuum state: ( ~ - I O - ) * ( - ) * ~ ( +It) . is characterized by

Now the test source response is written,

where the minus sign of the second term recalls the opposite sense of time development that is involved. The two fields encountered here are

(3-1.19)

Let us examine these fields for the particular situation in which K(-)(x) = K(+&) = K ( 4 .

(3-1.20)

Then

and, on using the relations

> x0': in which

- i ~ ' - - ' (~ X'), ( X - 2'1,

(3-1.23)

we see that +(-)(X>

where

=

#(+)(X)= +mt.(x)

0mt.(4= / ( d x ' ) ~ret .(X

-x')K(~)

(3-1.25) (3-1.26)

548

Fields

Chap, J

As the last property proclaims, this is a retarded Grwn's function. It is real, since complex conjugation interchanges the functions and A'-'. The three Gren's functions A+, A-, bt. refer La the same inhomogeneous differential equation sinee A'+' are solutions of the homogeneous equation

(--a2 + rn2)h'&'(z- x') = 0, and therefore

(-a2

+ m 2 ) ~ , . t . ( z=) K(z)*

The retarded field of an arbitrary source ean be found by solving tthia equation \vi%hthe boundary condition that the field be aero prior to the intcjrvexrtion of the sources. 1%is remarkable that this classic boundary conditbn requires the device of the closed Lime path for its appearance. fneidentsfly, the f o m awumed by &Wfar the cimumstanees stated in (3-lam),namely

Is a reminder that MI = O for &-,(S) = lYc+t(z),and this property will persi~t if the equality of the soureea K(*,(z) i s maintained by the test source. To return do the general situation Gven in Eq. (3-f.19), iL is seen that thwe fields obtjty the differenlid equations

The solution is eharaeterized by the following boundary conditions, Before any saurees are operative, & " ( + f ( ~ ) contabs only negative frequeneie~and +[-,(x) only posithe frequencies; after all sources have eeased to function, @(+)(2) and (6(-)(z) are equal. The initial boundary conditions are msde explicit by writing

And, if it is observed that a+(z--~r)+e-[~-~1)=a,,,.(x-z1)+dsdv,(2-~'),

(3-1.33)

Tha field concept.

3-1

Spin O partictes

lrC9

where

one gets

s the final boundary condition. which m ~ k e expliciG Still other kinds af fields md Green" functions appar on replacing the vacuum state with a general multiparticle state. Rather than USE? any specific one, we consider a pararnetriaed mixture, as in

C (m+l @ - ) K ~Ia ( 1 = exp[iW@(K)l, tn!

where

(3-1 -36)

and 19, is an arbitrary time-like vector with > O* OR reviewing the discussion of ({B)+l{n) J K , pa&icuI~rlyEq. (2-2.46), we recognize that this probabgity amplitude is linear in eaeh occupation number n,, which is merely =$ace$ by an average value in (3-1-36), namely :

(3-1-39)

with A6(2

- g')

= 6+(1:- X') f-

doP (n8 )8[e'p'z-"t'

-+- G-'"~-

"'1.

(3-1 ,401

By expanding exp[iWg(K)Jinto the form (3-1.36) the individual ({R)+l {E) can be recovered. The Geld (3-1.41)

which

defined by (3-1.42)

obeys the same differential equation, since A@fs-- 2') is anather

--a2

+ m2.

green'^

function of the diEer~ntiaIoperator

To marnine tbe boundary eonditianrs that ebwracferke this Green's function, we? write5

where

are relabd by On noting that

ah-'(~ - X@) = AB(4-1(Ze

(%,)a

(3-1 .del

2).

+ 1 = gP(np)sp

(3- 1.47)

we can as=& the formal connections A ~ ' (z 2') = ~I;)(z - 2' - ip),

ab-)(~- g')

= a;+'(%

- 2'

+ ia), (3-1 +48)

and khese are d l combined in

which i s a, real function. In analogy with the dransfarma$ianfrom the Euclidean to the Minkowski description, if is useEul to introduce an extrapolation to a real time-like displaeemerrt, i @ p --+ X@, (3- 1.50) where .XQ > 0. (3-1.51)

For the restriekd domain specified by

0 < z0 - zO" or --X0

which are united in

.XXTD

< z0 - zO' < 0, -- zO'/< XO,

/2@

the relations (3-1.48) become statements about the propwation function namely This is an &=&ion of periodicity, which is the boundaq con&%ionfor t h w fmnetions,

Spirt O plilrti~lsr l81

Ths f iefd concept.

3-1

If one wishes to verify that the periodicity condition. does produce the de~iredsolution of fhe Green" function diflferential equation, it is convenient ta adopt the rest frame of the time-like vector X@,with X' = T,and satisfy the periodicity requirement in z0by using Fourier series while retaining the Fourier intepal. treatment of the spatial coordinates. That gives the Green" funetion repreentation

The Fourier series that occurs here is not unfamiliar:

12 T

-

i

0

-

}

i

--

+

@(l i2)ip0~e-ip'lr0-."/

(pO)z - (211~n/T)Z 2298

-

EI ~ ) i ; p o ~

@--(X

1/ 2 ) i p 0 T ipOlrO-so' i

e

IP I ~ ~ O T

(3-1 -58) and the substitution inverse to (3-1.50) in the rest frame,

followed by removal of the reference to the rest frame, does indeed produes Aa(z - 3'). The same results are obtained directly from the digerential equrttion for +(z) by imposing the periodicity boundary condition

To extend this discussion. to the time cycle function,

we must dso cornider the function that replaces d,1,j Am@(%--

2"

-(2

- 3') :

-ihk--'(X - g'), = z0 > sop: f +f zo < 2" : -ia, ( X - %'C').

(3-1.62)

The designation th& we have given it exploits the following form& property of the averaged oecupw;tion numbers,

@,)-a

==

-((%>a -C- I>,

and thus (4-1 L'%-@ (z

- XI)

= --rhk-)(z

-- zt),

*'-" -e (x -

57

= -A# (4-1( X

- X/). (3-2.64)

The required generalization of (8- I. 17) is

Chap. 3

The fields defined by ~ W B ( R , -M,+,> ), =

(dg) SK,-)(z)~iat-lI2)

(3-1.66)

are

(dx')~;-~(l;'(z - zf)K[-, (z'),

(dxf) ~ . a ' ( x - zf)Kt+,(xt). (3-1.67) I n the particular situett;ion t h e ~ efields become

K(-I(%)== Kt+,fz) .=: KQx), @@r-r(z) a t S,@rt,(z)= atmt,(z),

since sX1.the eausall relations among the var;ious bc func-diom are hdependexrt of p, tas is

according eta (3-1.49). o p r s k to produce

Removing the rmt~ation.(3- l,GB), the ssme reason8

One esn also vv~b

or an equivalent expression using A'-' functions. Ln c;bbinixrg these regullts directly from the diEerentid equatiom continuity between the tm funetions is required after the sourcm have clt3wsed operalion. Psior to the aotion of any source, the two fidds are connected by whieh is the 8pgropriak form of the periodicity eoxrditian. Some of this diausjsion retains its form1 appearance when charged partieks are comiderd and de~cribedby a pair of real sourem, sixree the field i~ iseorrwpondingly gene;r&lized. But the extension to terminal mdtipar;iele state8 is most conveniently prformed when complex soure= are used. Aecardiady, we prewn6 $h& treatment in, 80nne detail. Begnning ~ %thehvseuum smpfitude,

Ths Cietd eonospt.

(o+lo-)"

=

Spin Q) parclclas

exp[iW(K)], ( ~ z ) ( ~ L . ~ ) K * ( + ) A--, ( xx')K(zF),

163

(3-1 35)

the introductiorl of probe sources defines tlvo fields, according to They are

One must not be misled by the notatiori and conclude that these fields stre in complex conjugate relation. That is s correct rts%rtiort about the differential equations they obey, but t;hese equations are to be solved with the same boundary coxrditiocls-th& of ouLgoing waves in time, by ~vhichiu meant positive frequencies in. the futufe and negative frequencies in the past of the source, Let us examine the structure of these fields in the two mympbtic time regions. If the fields are evaluated at a time after the sources have ceased operation, that causal circumstance is expressed by replacing P+(z - z') witfi h ( - X ' Thus (z > K suggests the causal arrangement):

and z

> K:

according to the definitions (2-1.72). In the other situation, the evaluation of fields prior to the functioning of the source, A+(% - S') iis r e p b e d by i ~ ( - ) ( z- z') : z

< K:

X

< K:

K *(X') (dup)"2e-'Pzi~~+

= P

IM

Eiefdd

Chap, 3

The two eausal evaluation8 of the fields are assseisted with particle emimian and absorption processes, respectively. They assign the field (h,) 'j'e'~' to an individual emission act and the field (dw,)v2e-'p' to an individual absorption act. As in. the interpretation of complex sources these field8 produce definite charge chsngw. Depending upon the causal situation, +(z) de~cribesenrrit;td positively charged padieles or absorbed negatively cha-ed padicles, while +*(g) rcpre~enh emitilt;ed negatively charged patticleg or absorbed positively charged par-t;icle~. The time cycle vacuum amplitude is

) ( d z ' ) ~ t(z)bL+' , (z - zf)K,+,(X')

Fielh are defined by

&W(Kf-,, The ernlicit forms are

=

(dz)[s~:+,(z)4,, ,(z) 4- &~,+,(z) +:+,(X)

-

a

~,(z) +,-,(X) t

- a~,-,

+L,(X)].

(g)

(3-1

(3- 11 '84)

and

If is seen that the field struedure already given in Eq. (3-1.19) is duplicated here, and the earlier discussion can be applied, enlarged by the substitutions K + K", cf, -+ $*. In particular, when which implie8 the analogous complex coxrjugate cqurtdions, we have

where the d m retarded fields are complex conjugates since A,,t;.(x - X') is a real function. One implieation of this property is that any small deviation af

The f isld eoneapt.

3-1

Spin O particles

186

W ( K [ - ) ,KC+,)from zero is real,

GW(K(-,, K,+,> =

(dx)l (~K:+,(Z>- Q K ; - , ( ~ &.t.(z) ))

+

+,(g)

-- GK<-,(X))

= &W(K,-,,K,+,)*.

+,et

(3-1 239)

ThairL is a special example of the following statement, rvhich comes immediakly from the interpretation of the dime cycle vacuum amplitude that is given explicitly in Eq, (2-2.86):

The replacement of' the vacuum state with a general rnuZdigarticIe s t a k can be parametrizcd, as in (3-1-36), with tbc tveight funetion

The averaged oceupatioxl numbers are, analogouslly,

and d+(z

- x f ) becomes

according to (2-2.59).

We note again that this function is not syrnrnetrieskl in

z and x" but there is a symmetry in \\.hich. positive and negative charges are also

interchanged,

The latter is accomplished by the parameter transformation

a 4 ---a, and

das(z The functions &;$'(X

X@)E

ACt--aa(~' --- z).

- X'), defined as alivays by

(3- 2.94)

Other relations that falllow fmrn

+ 1 = e"'BP(gp+)a~, +1 = (@P--)agP -- 2') = ea~L~',;;)(z - Z' - $S), A~;'(z. - z t ) = e-olhb$'(z - 2' + ip).

,at*}

(Z

The periodicity propedy is eomespondingly modified to h,fi(z

-.-

X')

)

- X' - - X)

&e~4B(1:

= e-Qa8(z

- 2'

+X),

(3-1.rw)

Although we shall not; @ve the details, this boundary condition on the propa-gaLion funetrion her&(% - 5'1, in conjunction with its driaerential ewation, dow reproduce the original funetion. The counterpart of the multipiarticle replacement of h+($ - z') with &,@(S

- 2')

is

~ - ( x-

--t

- g)*

= h-,

me(%- X').

(3-1.1QI)

The time cycle function.

Nob that if we wished to write the last term in the aldernative way that uses the h'*' function, it is necessary to change a into --a,according to Eq. (3-1.97). The fields defined in the manner of (3-1.83) are now given by

Tha field concept.

3-2

Spin 4 particles

1S7

and

This i~ the structure already premntt3d in (3-1.67) with all d h e l i o n s carrying the additional index a or -a, which arc, interchrtnged m K -+ K', t$ + +*, By adding the index ar to the fields, the slakments (3-1.89) and (3-1.71) beome applic&ble,togelher with the resulb of the substitution K --+ K*, 4 -+4". The index a can be introduced everywhere. in (3-1,72), buL it must be mpfmed by --a in A~$)(z- z')~when sources and fields are =signed an asterisk. Finally, we mfe that $he field boundary condigions to be imposed on the &Rerentid equt3ctioa~are

3-4 THE FIELD CONCEPT. SPtN

4

PARTICLES

The vacuum amplitude for this system is

where

G+(z

- z') = (m - ? @ ( I / ~ ) ~ ~ ) A + (Z z ').

(3-2.2)

Let us obeme innmdiately that t h e algebraic prope~iesof the Y" matrices imply :

(r@(~/i)a, + n)C+(z - 2')

= (-a2

+ m2)a+(z - 2') = a($

- g').

(3-2.3)

This identifies G+(% --x') arj. a Green" function of the D i m mat^ diffaential operator. Aoeording to the stmcture af &+(X --- z ' ) ~it iei the one that obeys outgoing wave time b o u n d ~eondi$ions. The field definition fo be used here is ~

The presenee of the antisymmetrical matrix ro compensates the anticommutaof the tivity of the probe source Ciq(x) with +(S), which is formed from and i~ same nature as the totally antiaommtr-Gingsaurces r l ( ~ ) ,

as follo5r.s directly from the definition, or by using the n~ltisymmetryproperty

The solution of the Green's function differential equrttion (3-2.3) is given by the equivalent four4imensional momentum integrals

where has s different scale in the trvo expressions. Alternatively, the Green's function is constructed from solutions of the homogeneous Dirac equation,

G+@

z0 > xO': iC'+'(z - z'), - z f ) = z0 < zO': iG'-'(z - z'),

=

C

(X)$,@,

P@@

W )*TO,

G'-'(~ -- g') = =

-C +,.s(z)

'+p.p(z')rO,

PSQ

and [Eq. (2-7.42)] +p@4(2)=

(2mdo,)'izzl,,,e'ps.

(3-2.1 l)

A charge label appean since this is a general attribute of spin particles. The inhomogeneous term of the differential equation (3-2.3) is equivalent to the time diseontirtui$y

This requirement is obeyed by the explicit momentum integrals of (3-2. IQ), and is aIw expressed by

which is a statement of completeness for the navefunctions of positive and negative frequency, $,,(X) and $(,z) *.

f he f istd cancept,

3-42

The evduatian of the fields in esussf situations-after functioning, or prior ta itts introduction-is given by

The x

<

Spin 5 psrtictsa

159

the source has ceased

structure also emerges directly from (3-2.6) as

The field that follows the action s E s source describes the previously emitted vvith an individual emission partielea, and associtstes the tvsve function rt,,,(z) act; the field that preeedes the action of a source describes subsequently absorbed with an individual abptthieles and associates the tvave function \t,,(z)* sorption aet. It will be noticed that positive and negatively charged particles have been given a uniform treatment, That, is becaum we used reat sources, and assimed the task of selecting la specific charge t a the multicompsnc?xrQIup,, or jjpcrq(s). This is natumjt, since, u n l i h the spin 0 situation, spin already demzzxzds the presence of the faur-component U,,. One ean, however, also follow the pmocedure of praelecting the charge by using complex sources, From the pair of fourcomponent real. sources ) ) ( X ) ) ?<2) (X), \v"i"~anst~tct 2-312

I

( -1

2

+

1 ( ~*) = 2-112 ( 1 ( 3 ) ( 2 )

i9~2)(2))-

(3-2.17)

Then the vacuum amplitude is represented by

which uses the definition

9(s) =

and fields are defined through

rl(z)"y0,

Chap, 3

They obey the differential equations

+(z)a,T = -a,$(z),

and the obmwation that

To confom ~ t the h new notation, we mow ~ t c ;

and

apip. = %;@ro. We

m~8$ also fib

which restabs the property (3-2.7). The fields in the two causal situottions are obtained W

=

with

E +P@(%)r0iv@@-, Pip.

The field concept,

3-2

Spin f particles

181

and

=

where P

22 P@

+Pu(s)iT;u+,

,

S:,-

=

The specifications of the particle sources follow from the earlier discussion by identifying $(z) and $(X)?' with the projections of the eightcomponent field o n h the positive and negative charge space, respectively, To use the fields #(X:) and $(x) is surely the most familisr and the most pptrlar waty of appfyhg the Dirac equation. Neverthdess, we regard the asymmetry of the forms (3-2.30,32), in contrast with (3-2;. 14), as justifying, in general, the employment of the msl sources and the multiieomponent fields thab are defined in charge and spin space, rather than the pairs of complex sources and their sssoeiated fields. The time, cycle vacuurn amplitude

The fields defined by the diflerentiaf expression 6w(~[-),

tlf+))

=

are given as

Since G-(z

- 2"

obeys the same inhamogeneaus diflFerential equation as

@+(z - z r ) , according to [Eq, (2-7.65)]

these fielh ssfjlsfy the digerential equations An etlte-mative prment;ation of the solution &

I n t r d u c d here are the red Green" functions

1% is evident from these forms t h h , prior to the introduction of %heBOWCW,

-,

+,+,(z) contains only negative frequencires and (s)only positiw frequencies, and that Jlt+,fs) = $(-,(2) afkr the source8 have ceased operation. Them are

the b o u n d a ~condition8 that accompany the diRerexltial equa%ions(3-2.39). ZncidexztaXly, ans a glance at, Eqg. (S1.27) and ($1.34) will eoxlfirm, the retarded and dvanced Greenp@ functions are also con~;;-lrueted W

f n the ~peehlsi$uation ~f(-j(z)"'-- 8 [ + ) ( ~ X ) 9f~)

(3-2-45)

one evidently h= J.c-1(2)

z=z

1L(+j(x) =.;. J/rct.(z>,

(3-2,46)

with (3-2.47")

appearing as that; red solution of the differenbial equa%ion

Tha f ietd concept.

3-2

Spin 8 parOZctass

163

whiel.1 vanishes before the sauree earnrss into action, The form of W far smdX deviations from the situation (3-2.45) is (&~tt.,(.)

- a~,-,(z)) u"$m,.(z)

(3-2.49)

When the multiparticle mixture given in (3-1.91)is applied to w F, D. -stem, where n,, = 0, 1, the a v e r w d occupation numbers are

On referring to Eq. (3-7.45),

W

see that G+(% - z" iis replaced by

(3-2.51)

and tlsis Green" function. appears in ( d r )(dx')q (X)?'G.&

(X

- z') q(zf)

(3-2,521

to determine

C ((4+I

-)'pa&( ( R ) ) = exp[ilva8(s)l. (3-2.53) Far sirnplicitJI, no para-meter hsls bwn introduced to distinguish tfie w i o u s spin

stales, The funetions defined by

are, explicitly,

and

+ (1 (-4-1 ' [r0Go@ (Z

)l?',

~ ~ p q ~ a ~ ) *$psq(zf + p ~ g ~ ~ )

(-1 z)lT = -7 0Go@ (Z --

X'),

(3-2.56)

\\-hieh expresses the necessary antisymmetry of Y ' G , ~ ( ~- X') under the complete transposition of space-time coodinabs and discrete indices. The relation

leb

fields

Chap. 3

t~herethe symbol q now indicates &hertntirjymmelricd charge matrix, The corresponding Green" function property is

Gafl(2- 5') = -eaQG,B(x ==

-X) -e-aqQa4(.~ - ZI + X).

(3-2.59) If cr is mt equd to Bern, or, more generally, is accommodated in by a coordinab dependent rdefinition, the boundary condition on - 2') ~ppf-t~rs ass a s i w ebange in respan= to a coordinate displacement by X"; this property might be -lied sntiperiadicity. Concerning the t i m cyele generalization of d h w rmulLs, we shall only remark that the mulidipartieXe repbcement for O,(x 3') i s El, -@(z - z'). This statement is equivalent to the relations

ag(r

---.

-, -@(S - 2')

G%'

-- G:~)(Z - 27,

(4-1 - 2') = -GoB (X - X'), (3-2. (30) and they follow from the aver&geoccupation number property (--J"""

=

(npp)--a

-.B

1

-@(X

- (@pq)a~

(3-2.61)

Spin 1. According to Eq. (2-3.4), unit spin particles of mass m s" 0 am deseribed. by

+ (I /m2)a,~p(~)a,(~- x')~EJ'(z')]. The consideration of

a,

(3-3.1)

test source defines a vector field:

(dzf)A,($

- ~')aa'(z~). (3-3.3)

The divergence of the vector field is

and this defived scslar field vanishes outside $he region oecupied by the murce. The differential equation that is infemed from (3-3,3),

3-3

Some other spin values

166

on using the relation (3-3.4). Another version of this differential field equation is

+

where

a,G'"'(x) m2@(x) = J"(X),

(3-3.7)

G,v(x) = - G ~ ~ ( x=) a , 4 , ( ~ )- a,+,(x).

(3-3.S)

The differential equations that relate the vector field to its source conversely determine the field when appropriate boundary conditions are added. The divergence of the vector equation regains the relation (3-3.4), and thereby the form (3-3.5). The soiution of the latter with the outgoing wave boundary condition is just our starting point of Eq. (3-3.3). In this and other examples of B. E. systems, different boundary conditions can also be used, in straightfonvard generalization of the spin 0 discussion. Spin 2. Massive particles of spin 2 are described by [Eq. (2-4.20)]

+ (2/m2)asTp'(x)a+(x- xf)aX'T,x(xf) + ( l / m 4 ) a f i a , T ~ v ( x ) ~-+ (x')a:a'x~"~(x~) x - 5(T(x) - ( l / m 2 ) a , a , T v ( ~ ) ) x A + ( ~ - - x ~ ) ( ~ ( x ~ ) - ( l / m ~ ) a : a ~ ~ ~ ~ ( x(3-3.9) '))], in which T ( x ) = g,,Tpv(x).

(3-3.10)

The symmetrical tensor field that is introduced through 6W ( T ) = / ( d x ) ~ T " ' ( X ) ~ , ~ ( , ~ )

(3-3.1 l )

is obtained as

- ( l / m 2 ) a ./ ( d z r ) ~ + ( z xt)aA'~,~(x')

+ (l/m4)a,av/(dxr)a+(x - ~ ~ ) a : a : ~ ~ ~ ( d )

- *(g,,. - (1/m2)a,,av)/(d~')A+(z - x t ) ( ~ ( . r r-) ( ~ / m ~ ) a # i ~ . ~ ( ( z ' ) . (3-3.12) The divergence of this tensor field is the vector which vanishes in source-free regions. That is also true of the scalar field

and of the combination The digerential equation derivd from Eq. (3-3.12)

On regwi~gthe vecbr and scalar wume combinations by the equivalent field B % ~ G L u ~thi8 , dserential equation, hc~nnes

Or again, on using the infarmtian supplied by ifs trace, the latbr can be pn;serrled W

Oflrer vemions of this digerenfilal field eqtrafbla are

Convemly, the diRerenfiaf field equafiow, gupplemenkd by the outgoing b u n d a v esndition, have the unique solution dven in (S3.12). The explicit gpacethe relstion betwen field and 8omce ia stjll q ~ manb ageable for spin 2, but bwonnw inerwingly mwieldiy for higher spin valum. TKm i& to some exbnd by using faur4imemiowl momen%umnobkion, as we first 3lmtmGe for spin 2. The appropriah ~pe~ialbation of tbe gene& exprwioxl (2-5.95) is

Some other spin values

where S,W(P) = g,.

167

+ (llm2)P,Pv.

Some properties of this tensor are:

The field defined bv (3-3.27) appears as b . @ ) = p2 +

1

- ig

(P)

[ ~ P K ( P ) ~ V A

- &~I*(P)SK~(P)IT"'(P). (3-3.28)

The derived vector and scalar fields are indicated by 1 P'~PU(P =) 2 [ P ~ ~ P A ( P ) fpUgKr ( P ) ]F'(P)

(3-3.29)

1 1 O(P) = ;;;i[;;;iPKPI- 38~k(P)]T"(P)-

(3-3.30)

and

The algebraic combination its trace, ( p 2 f m2)+(p) - P%'+~A(P)= - - ~ S ~ ~ ( P ) T " ( P ) , (3-3.32) and the additional combination

then lead directly to the equation

which is the momentum space transcription of (3-3.17). Spin 3. I n three examples, of spins 0, 1, 2, the use of the structure indicated generally by Eq. (2-5.95) has produced fields that obey second-order differentia1 equations, with an inhomogeneous term that is just the source function. This ceases to be true for higher spin values, in the sense that derivatives of the source function cannot be completely eliminated. There is, however, the possibility of restoring the simpler situation by using the freedom to modify W by adding real terms in which sources are multiplied together a t the same space-time point.

As we have noted in Section 2-9, such contact terms contribute neither to tlte va~uurnprsisknce probability nor b the coupling of causally disposed sources. It is only through the additional con~idemtioninvolving the structure of field equations that a reason for their presence and specific appearance can be addue&. The maintenance of the inharnogene~usfield equation form that we now regard as standard nlso requires the introduction of certain auxiliary fields, ivhieb vanish outside the regions oceupkd by sources. The discussion of spin 3, which is intended to illustrate these remarks, \\-ill be: facilitaM by first examining the sirnpfer situation of m-less pa&icles. We have not done this for spins 1 and 2, since the* physically important examples will receive individud and exknded treatments. h t it be remarked, however, &at the appropriate field equations are & h i n d merely by settiw m = O in, my, Eq8. (3i-3.6) and (3-3.19). Also, the fiysically neeessaq soume restrictions, af vmisbiw divergences, are algebraic consequencm of the field equations, ars we em recognize from the m -. O limit of Eqs. (3-3.4) and (3-3.15). But more of thk fater. The massless spin 3 situ%tionis represenkd by [Eq. (2-5.122)I

and the follolving restriction on the sj~rnmetrical-t;hird-rank tensor: The I&der implies a lack of uniqueness for the field defined by

ssinee any additional term containing p&, p,, or p, ss Erzctors will not eontribuk in (3-3-38), oiving to the souree restriction (3-3.37). Hence, the genwal form of the field is

in rvhich the cy~licallyrelated sets of terms are required by the total symmetry +h,,. The new symmetrical tensor h,+)is defermind by the murce restriction. In order to urn the latkr, we first note thzzit

of the knsor

Some other @pinvalues

3-3

16s

Multiplication of Eq. (3-3.39) with p9then introduces just the combination that is evltluaLed tts *(#h --- p&+), and we get

An equation for +(p) is produced by combining p hp pp v@ h p . = ~ P ~ P @ P ~ ~ ~ ~ P

with P~P'P~#FV

pkp@p'+rp. - p2pk+*

namely (p2)'+ = p2pk+k

--

+

$pkp'pB+hpr.

(3-3.43) (3-3.44)

(3-3.45)

The momentum space version of the field equation obeyed by ~ b ~ ~is~ f p ) now obtained as

The construction of +(p) given in (3-3.45) is derived directly from this equation by multiplication with phpppv. It might seem that we have failed to meet the objective of providing second-order differential field equations. Three dePivaLivers act upon Q, and the scalar field obeys a fourth-order diEerenti%lequa;t;ian. But notice that the field equs.tion involves only this combination of fields:

The combination given in (3-3.47) is an %cceplablercsdefimition of (lixpv(p),which, means that +(p) can be trmsformed away. Thus, our final set of field equations is (3-3.46), with g, = 0,or equivalently,

where dots represent the terms that are genemted by cyclic permutation from the gven ones. The following is an algebraic consequence of this equation,

it is consistent with the vanishing divergence of the source, but doe8 not imply it. Xow let us consider a mwsive particle of spin 3, fir& using maltere$ the description given by Eq. (2-5.95) :

I70

Chap. 3

Fiaids

where, according to (2-5.79) with nX,.,kt,p.t

c31

==

+

5;

= g h h ~ j l p , ~ g vv ~ + [ g ~ , g ~ . ~ g r ~gppyhhgg,ppfi ,~

+

~ ~ h ~ ~ , ~(3-3.52) g ~ ~ h ~ l

in whielt the necessary symmetriaation in h f , p', v' h= xlsL been made explieid. Some propertim aE this tensor are

4-

Phhfigpfrt 4-

PhP,' 7

8 . 8 ~ f

The field defined by is obtained as

(dP> -&sApp(-p) qh&v(p) ( 2 ~ ) ~

(3-3.55)

The presence of the symmetrical source function enforces the symmetrisrttion in X', p', vf. To avoid doing this explicitly for X, p, v, we intrsduee an auxiliary sxmmetrical tensor sb', and display the field ss

The combination suggested by the lefehand side of (3-3.46) (with + = 0) is

Some other spin V ~ B I U 171 ~S

37-23

Quite? apart. from the explicit appearance of numerous derivatives acting on the source, the eoegeient of Sx does not have the value given. in. Eq, (3-3.46). Thak is rectified, however, by considering

for then

-

sbp[grh'8,ptgpp,

-~g,vg~~fgp~w~]~x"'pf

+

c, (3-3.82)

with

And, from the equation for

we get

As a differential equation, (3-3.62) contains third derivatives of the auxiliisv scalar field 21;. If we reduced this .to second derivatives by regasding the gradient of the scalar fidd as the appropriate auxiliary field, the meoxrd-order differeatis1 equation that the veetor field obeys eontains second derivatives of the source function. This is the background for the addition to W of a specific contact eaupling term:

The ~ s o c i a t e dsupplement to the field is given by

21-r. 4hld.Y

cont.

-P~PRP,

+

p2

+

?a2

gpvpr p8."

l

2;i;;li ~

r

y

~

h

(3-3.66)

and its contribution to the field equation is

x

Chap. 3

where

The addition of these terms to the righGhand side of Eq, (3-3.62) removes the third der-jivativea of 2, and replacm them by fimt derivatives of @. And %her@ is an additional contribution in (3-3,M), which can be d d e d Lo the lefbhaxld side as l -2% 2 P 2z+-ZOmflp2(m2 - P ~ ) P ~ S ~ . (3-3.69) =f

eont ,

When Z; is replac& by (P in that equation, all source terms disappea~ This successful realieafion of our proganz is displayed in the pair of diEerentid equations :

N o b that the malar field effectively vanishes as m -+ Q, and (3-3.4fi) is recover&, When supplemertted by outgoing w a w boundary eondi$ions, %het; unique solution of the set (3-3.70) is the field 4&,, ~ v e in n Eq. (3-3.56). Spin 4. According to Eq. (2-8.10), this system is described by

The field defined by (dsp) m4

61'(-

p)r0$@(~)

IS, therefore,

Two derivd fields are P'$@

1 = 2 ; i b - Yp)p,

+ *YP(~'YYf p.)lrlV

(8-3.72)

Some ather spin values

3-3

173

and

which. supply the Iineatr combinations

The urn of either of Lhe identities

+

- p,) (m - Y p ) , (m - ~p)(m?;- ~n,),

(m -4- Y P ) ( ~ Y , p,) = (my, (my,

-C-

4- ?P)

==

(3-3.78)

prrsduces the equation

(rp 4-

m)& =

4-

&r,

qv

1 +- 3 pp,Ip. - S(mr. +

PP)~?',

(3-3.79) from which we immediately obtain Like the field equation, for spin +,this is (the momenlum space e q ~ v a l e nof) l a firr~t-orderdigerentiaf,eqtr%%ion ~viththe source appearing m the inhomogeneous hrm. The solution of the equation under outgoing wave boundaq conditions i~ the field given by (3-3.7'3). As is indicsted by the m -.sl O limit of (3-3-76}, the necessafy source resl.riction for massless particles,

is an snlgebraic consequence of the m = O field equation. There is a marka ably compact way of prewrrting Eq, (3-3.80), whieh is reached by commuting ,?' with (m -- rp) and noting that

With the examples of spins 2,3, and # b&m us we can recognize the possibiliky of simple dgt3braic redefinilions of the seurces Lhat preseme the general stmcturtt, of the field equations, but introduce or modify contaet brms in Lhe expression for W , Thus, let

which has the inverse

On introducing this redefinition into the expression for W ( T ) ,say Eq. (3-3.241, the additional g,. terms supply only contact contributions. The explicit statemen&is

where W(2")has the same funetiond form as W(T). The implied field transformation inferred from

Since field and souree are frtznsformed linearly, and Ioeally in spacetime, the daerenlisl field equation mainfains its general farm, buL with changed caeEeierrts. This is illu~krakdfor a = 4, where

and similarly for the sources, by the diAFerentia1 equation

Xf this is applied to massless particlm, the diRerenlia1 equation demands the vanishing divergence, not of the source TL,,but of the combination T;, -- *@&PT'= TpP.

(3-3.92)

Concerning spin 3 tw rernark only that; the redefidion hdieated by

urhich maintains the general structure of the fidd equations but & a n p the eontael germs, cannot be aged, to Iremove the latbr, As we ohmwed, second~rderdiffewntial equations lacking in source derivstive~are not obtained if canttbet term8 are o ~ t ; k d ,

Some other spin values

3-3

176

Returning to spin 9, we note that the lineaf souree transformation produces contact terms :

The field Lransfomation implirzd by

ean be written as

+L

or

= $g

+ av,rpr(..

where The transformed version of the field equation (3-3.80) is

(?P $

-- a(pPyY$:-/-

in whieh a' = 3u(a - 1)

+ 1,

-!~-~ ( a '-ma ' ' ~ p ) ~ ' $ :=

(3-3.100)

+ 1).

(3-3.101)

4(3a2- 2a Eatice that there are just two situations in which a" =

The fimt gives the original field equation, and the second produces

Another simple choice is e=O,

a=*,

(3-3.105)

whieh gives

-4- *%Gm - Y ~ ) Y=Yg;.~ (3-3.106) For m = Q, the field equations imply the vanishing divergenea of a quantity funetion. that is the transformed statement of Lhe ori@nd 8 a u r ~ (rp -4-

%.

Spin Again, we turn first to the massless situstion in order to get the simplest indication of the field equation structure. The appropriate specialisation of the

176

Fields

general form given in (2-8.52) is

where

- f~"(-~)~'~~(-rp)r~4;~9(~) -- f s ( - - ~ ) ~ ~ ( - ~ p ) s ( p )(3-3.107) l, ?(P) a Q ~ W V ~ ~ ( P )

(3-3. XQS)

and 12,1.l"""(~) ==

0.

(%3. 109)

The field defind by (dp)

&g"'(

--P)$&P(P)

(3-3 * 110)

is not unique, 8.lnce $he source is restricted by (3-3, tW), ftAs general form is

where the vector-spinor It, is to be dehrmined &rough the ssume reatrictian, We shall not detaiX the extraction of the field equation, exwpG to rean&rk%h&, as in the spin 3 diseumion, higher derivatives geem fx, appear but c ~ be n removed thraugh a pernnigsible redefiniticm of Ghe field, namely The final form of the field equafion is

The alge'lzraie eoxlsequenee of %hisequation,

is eonskbnt with the souree restriction, but dws not imply it. By f o l l a ~ n gthe ins6ruetions given in SecCion 2-8, we a ~ v ade the f o l l o ~ n g scrum dwripdion for a,masgive. padiele of q i n #:

Le%us also @ t a bhere the dditiorrd contact brms that are req&ed Lo bfing

the field equations into first-order fsrm without source derivatives. They are derived from

The field equations are

wrhich reduce to (3-3.113) as m --, 0..The diligent r e d e r a k a is deairoui3 of confirming the statemeal of Eq. (3-3.117) could solve thew field equations in conjunction with outgoing wave boundary conditions and reproduce the field that is derived from (3-3.115) and (3-3.116), together with the following expression for the auxiliary field,

3-4 MULTISPINOR FIELDS

Tht! multispinor description provides a unifid approach do a11 spin values, wirth the first-order digerential equation af spin set.ting the pattern af field equations. I n order to reafiae t h i ~sfftndard, ho~sever,contael Germs must be introduced in d l situations save that of spin +. Consider, for example, tho descriptian of unit spin thad L provided by $he symmetric spinor of the second r&nk2as contained in.

The appropriate contact term has already been inlrodueed, and the soume differ8 in normaXiz;%ianfrom that used in (2-9.8), as indieat4 generally by q(p) = (2m) 2'"-"S(p).

(3-4.2)

The field definition is wnd

Now we observe Ghat

(rzp and addition gives

+ n)J.(p)= Z;;E [m - ?'P + m + ~,PI?(P), I

f*(r";" t-r'2>?1,t-. 4rl.l~) = ==(P)* (34.Ci) W d t h n in coordina;tte spme, this is the first-order digerentid field equtttion Another consequence of the equation, pair (3-4.51,

is also contali~edin (3-4.Q, and the use of

ss revealed by multiplication with rlp - Y z p

Crr

~ >

( ~ 2 ~ 1 2e

(3-4.9)

The simple algebraic property just recorded provides izxl e1ementtaw snd generally useful eantrol over our procedures. The eigenvttlue relEtLZLOnship~, TIP = f r,p, are invariant statements of the restframe possibilities, r! = f 720, where the plus si= se1ecl-s the appropriak aubspsee for the dwcfiption. of the particle, Correr~pondinl;Sty,setting ~~p ==: ? % p in (3-4.6) reduces the l%tterto the form of the spin Dirac equation, while the choice r2p .- -rip effectively removes the caordinizh de~vativmand. supplies a field %h&vankhes outside %hesource, The ~trueturegiven in (3-4.1) b c o m e ~mare obvious, for, with

+

TzP = TIP,

which puts into evid~neethe contact term and rrormli~ationthat are needed do attain (3-4.8). The effectiveness of such considerations becomes clearer on turning t a the nnulti~pharof rank 3, where the choice h p = Yzp = Yap leads to the wad=tion of

Multispinor fields

179

This indicates the proper starting point:

(m - TIP)(^ p2

- 72p)(m- 7 3 ~+ ) 3m

+ m2 - ie

l

- f(YlP+ 72P+ 7 3 ~ )*(P),(3-4.12) and displays the field

+(P> =

+ 3m - f C 7.p]

n(p)

(3-4.13)

that is defined bv

There are only two general alternatives open to the ?,p. Either they are all equal, or one of them has a sign opposite to the other two. These situations are characterized by

and, for example,

An expression that covers all contingencies is

it can be verified directly. Note that the field statement of Eq. (3-4.16)can also be presented as the unrestricted equation

Then, with the definitions illustrated by

which is antisymmetrical in the indicated pair of indices, we arrive a t the system of first-order differential equations,

ll&0

Chap. 3

FOsids

where the latter illuslrahs the set of three equation8 th& are relatd by eyelic permu% a%ion. It is aIso inbre~tingto eliminate the three auxiliary fidda and pm~entthefield equation in the form

f i r e the dolls indicah the two analogous exprmianr, p r d u c d by ~ G L permu~ c &atian. To v e ~ that y thist ~ingleequstGion pernits the reeo~%mction of the orie;inal field, it suBees to examine it in h a situatiorms:

wbieh do inded eantain the results of (3-4.15) ~ n (3-4.16). d There i~ another way ta convey (3--4,21), whieh foll~wsfrom the obmrvatioa that

I d ;isJ the smonderder differential equation

or, slbm&tively,

The Iakter may be earnpared with the second-oder di8erent;hl eqamiLio.on for unit spin : [ P ~ + , ~ - - - { P ~ + ~ ~ c Y ~ P ) ~ I (ImJ -. =+C.,,,

and for spin i:

(

~

(pZ+. m2)$ = (m - 7p)v.

The dk~usaionof the fou&h-rank multispinor begns wi& the hi,lg&r&ie ~bhrn@rz_t

But now there is ambiguity in giving (Yl p ) 2 s more general inkrpretation; shall it be --p2, or Q ~ . p ~ @ pIn? fsct, we shall use s ~pecificlinear combination of the two, 80 ~hogenthat p W e mof moment&in 3/ are held h the minimume

~

The actual expression is

The following indicates the options availabte to the ?,p : Y1p = Y2p = Y;@ = Y4p:

The particular structure adopted in (3-4.29) is designed to simplify the field, for the situation of (3-4.31), by eliminating a possible p2 term. The three examples of field equations are synthesized in

Another unifying statement, with its obvious generalisations to other index amangements, is

Using the definitions of antisymmetrical funetions that sre illustrated by

182

Fields

Chap. 3

one then wribs %hesyskm of firsborder differential equations:

where the lwL Llvo are repre~entafsi71.eof gets of ~uehequtatiom. The elimination of $he auxiliary fielda produces the sinhyle (multkomponent) field equation

kvhere the summation over a l paim of indices is indicated by s rc?preseatative term. The various dternsktives are iflustraZIed by

which m ~ t a kthe field expreaiow of Eqs. (3-4.30,31,32), Another form of the equation is

in whieh $he last sunnmafiion is extended over $istinet pairs, ar < 6,"a p', tit: art6 @ p', with ~ 1 repetilious 0 counting. This is equivalenf to s faurt;hh order diR6rendial equaf ion, Turning ts fifth-rank multispinors, we first note that

a

Ambigfuitieshave bee-n msolved in stating the field as

'The 8vaila;ble options are indicated by

The following art? generaEly valid statements:

where repetitioue counting of double pairs is avoided, and

in which the stmnmations m%rked .than a and 8. The auxiliary fields

X' are extendd over all index values other

and, for example,

then lead to the field equations

To complete the system we need a. differential equation for the + g a ~ I I . ~relating ~~lr them to the $rap1. Now

while

snd thus no immediate connection exists, owing to the inc?:s@ap&ble fact that h. It suggests, however, the introduction of another set of auxiliary fields, illwtrated by X (3-&52) Xr23jg4~11= f 12m2) rlpg(r2p - yap)*(y4~--- ?BP)?,

+g

p

which obey (TIP

5m)~[a@l[a"~~ -k Yl~$ladSlfla'~"

0,

(3-4.53)

aad enable one to write

The full system of firstorder differential equstions is given by Eqs. (3-4.49), (3-4.53), and (3-4.54). The auxilisry fields can be eliminated and a single equation for J/ constructed in various equivalent forms, but they have heorne too p n d e r o u ~to be worth recording, These procedures ean be extended to higher-rank spinors. Without exhibib ing the solution of the general problem, we do want to incorporate all available results into the larger framework approprittte to a multispinor of rank n. Generalized from Eqs. (3-4.6), (3-4.20), (3-4.36), and (5-4.49),the first two equa-

tions of the system are

C'

a'
+(Y~'P

Y@'~)$[aflj[a~@~l = 0,

(3-455)

which imply

The next in the sequence, ~vhicfnis as far s s it shall be developed here, are

where the k r m s left unwritten are those referring to three index pairs, This set has the property that

'~,t'hichis appliceble even to n = 3. The totollly symmetric muitispinor of rank n == 1, 2, 3, . . . gives a description for particles of spin s =. 412 = $ , l ,#, . . . It has alfeady b e n noted t h a t the list is eompleti-ed with s = 0 on considering n = 2 and replacing the gymmelt~calspinor by at1 antisymmetrical one. AI1 the equations for n = 2 continue to apply with tbc ehaxlged symmetry, But this is a, completely pneraf, remark. The system of equations have been diricwsing involves operations on the fields that are entirely symmetrical among the apinor indices. Aecodinhr;ly, t h e

.

186

Chap. 3

FIatds

field will acquire whatever symmetry characteristics the souree 72 posssesSct)s. The significant description of the particle is given in the subspaee Y! = * 0 referring to the particle rest frame. The specific value of the particle spin wit1 therefore bf? determind by the symmetry that is common to the source and the field. If there is total symmetry, the spin is s ==.: *E; if there is one arrtisymmetried index p&, and tatal syrnmetfy among the rest, the spixr is s = +(n -- 2) = In -- 1; and so forth. By appropriately choosing the symmetry, then, the third-rank spinor equations can be applied Lo s =I: -# or 3; the fou~h-rankequations to a =. 2, 1, or 0; and so on.

A syrnbolie transcription of our dis~ussionis contained in the following set of equations : (3-5.1)

That is, beginning with the quadr&ic source expression for W, ~vherer is the approprirzk repregentation of the metric, fields X are defined through the consideration of an in6n;itmimal test souree, The noalacal space-lime relation between field and source that is conveyed by G is then converted into a local differential one, which is symbolized by the operator F. Alternative expressions for W are and

Of particular importance is the linear combination

(Srx - gxrFx), To this point, the field X has been a derived quantity, a eorrvenient shorthand for M, It now acquires independent status, in the fallowing ~ense,Forego the knowletfge of any connection bedtvwn X and S, and subject them to independent vetriation in (3-5.7). This gives

X , the additional term should vanish. Indeed it does, since But 6W is ~ ~ S Y and I"x = S. This means that, consider& as rz funetions1 of X for pwscribd S, the

expression (3-5.7) has vanishing first variations or is stationary at the unique field eonfigumtion selected by the field equations, in conjunction. with Boundary. conditions, The quantity W is thereby invested with the adtributes of action, producing the field equations through the principle of stationary action, We hall no\%v iillustrate these general remarh in the context of specific spin values. Spin 6. The field equation is

and the action expression. (3-5.7) reads

There is a more symmetrical form that contains only first derjvatives of the field. No addit;ional surface integral term is assigned to the partial inkgration. This can best be appreciated with the aid of the assoeiakd Euelidertn description where fields decrease exponentially at large distances from the source, sinee that is the characteristic of AE(x - X", m 7C O. Even for massless pa&ieles the (z behavior is sufficient to suppress infinitely remote surface integral eorttributions. Accordingly, rre ~vrite

urhere the field dependent quantity g (4(z)) = -&[ae$(z)a~+(z)4-

@ (@(z)) l2

(3-5.12)

will be called the Lapange funclion of the sytstem. We shall note here that the field equations can a180 be presented m the firstorder set : The elirninalion of the vector field 4, gives the second-order dieerential equa%ion (-8'

+ ?n2)&(z)= K ( r ) - a,Kp(z),

(3-5.1 4)

wyhich exhibits the same kind of efieetive sealrtr source that Etas already been encountered in. Eq. (2-9-76)- Tbe corresponding action expression, is

f f the first equation of (3-5.13) is regarded as a definition, (6, loses its independent position and we recover the action %pression (3-5. XI), ~viththe effective scaler source indicated in (3-5-14), and the additional contact term - ~ ( d z ) + l < ' ~ , .

188

Fields

Chap. 3

Spin 1. The field equation leads to the action expression

with the Lagrange function To obtain i t one uses the rearrangements and But the latter could have been

and this produces a different Lagrange function: the restriction to first derivatives does not assure the uniqueness of the Lagrange function. Generally, two Lagrange functions that refer to the same system are connected by a relation of the form since the local divergence term does not contribute to the integrated action expression. I n this situation, converts (3-5.23) into (3-5.19). A system of first-order equations is

a,$,

- a,4, - G,,

=

nr,,

=

-nr,,,

a,G"'

+ m2+w= J ~ . (3-5.26)

They imply the action

with the Lagrange function d: = -3Gpc(a,+,

- ad,,) + +(3GrVG,,,- m2g4,,).

(3-5.28)

On regarding G,, as defined by the first equation of (3-5.26) we recover (3-5.19) with the effective vector source J' d,MM', analogous to (2-9.78), and the added contact term - ~ ( & ) & i l i p v ~ f p v .

+

Spin 2, The fieId equations are

and om possibility for the Lagrangc? function, in the aetiorr (dz)ITfiV~pv 3- $1,

(3-5.30)

with f&e option of replacing the last tern by

or by any tveighbd average of the; two. Fimf-order difirentiliaf equations ean be iutrodueed in two different ways, One such see is associated with the action expremion

where K&,, ancl G&,,are antkymmetrieal in X and v, and Gx = GXyP.

(3-5.35)

These field equations are

ahcpPh - apG, + ma(+pp + +g,.+)

= F.,

-

(3-5.361

with the left side uxldergtood to be symmettiged in p and v, and

The effective source of the lattw equalion:

Kip, == *(Kx,, K& = KhFp,

+ Kgkv -- KgFx

K ~ s ,-4, Rpgk,),

(3-5,s)

incileate~the rearrangements that are =quire& to produce the form (3-5.37) from Lhat ykM& by the stationary acLion principle. 1%is also uwfui to remark, in conneetian with (3-5.36), that

iS demoted to a dc?rived qu~ntity,one abtains a Lwange funetion %h&$ If is an equally weightt3d average of the tvvo possibilities d e ~ c h b din Eqs, (3-5.31, 32). Playing the role of source in this aetion is T,. ahR;.h, where symmetrisatisn in p and Y h understood, and there is an added con_tatedkm.

+

The se~ondsystem of equalions is represented by

"

where H,,r is symmetrical in p and v , ~vhile H & = X"',. H

(3-5.42)

The field ewatians are

ahH,,h

- avHL+ m2(+,v+ +gr&)

= Tpv-- +g,.T,

(3-5.43)

vvith the left side (specifically aVHJ syrnrnetrized in p and v, md

afl+rvf

a&hB

m

W

H,v&= L:.,,

(&5*4)

in which

When &,h ia considered to be defined by (3-5.M),the Lagfange funetion turm is into the one that uses the term given in (3-5.32). The effective sour@@ and there are EldditionaiX,contact terms. Spin 3. It b s been. seen that diRerent vemions of +he Lagrange fuxrctian. ean be introduced, We shall be content, however, ta record only one in this rejatively complicated situation. It is (3-5.47)

where the amiliaq function tionary. action principle.

is to rweive indepndent variation in Lhe sta-

Spin t. Little need be said here. The field equation

trP(r/i)a,4- mIJ/-(s)= v @ )

(3-5.49)

implies the action (3-5.50)

Action

3-45

t91

with the Lapange funetiorr There is an apparent asbitr~riness,for the derivative coufd bi? transferred from the riglilthand field to the left-hand field, with a minus sign. "She t ~ v oalternstives are identied, however, since the rorpare symmetrical matrices and +(z) anticommutes with a,$(t).

Spin

4.

Vector-spinor field equations are

lyA(l/i)ah4- m]$,#- (l/i)a&yp$P- T,(l/i)ap+P 4- ?,[R% - ~ ~ ( l / % ] a k ] r ~=$ @ V@, (3-5.53) and they are embodied in the actiorl (dz)[9'r0$,

+ $1

with the Lagrange funetion

The second and third terns effect an explicit symmetrization between the application of the derivatives to the ~ g h and t to the left, This is _automar;Cicfor the first and fast k r m s of (3-5.55).

Spin 8. The symmetrical tensor-spinor field description : (dz)I.l"r"+#.

c = -+($@yro[r"(l/i)ak

- 2#ppr0rp(l/i)a"k.

provides the following aetion

+ C],

(3-5.57)

+ m]+r. - 2vyr0((l/i)8,rk#hv + 2pvror,[m - ru(l/i)d.]th+h.

+ + r @ r ~ ( l / i ) a ' ++~~~* ' v ~ r , ( l l i ) a ~-$ *$ro[rh(l/i)ah + m]$)

+ am(#rOq - *rO+) - f*rO[rA(l/i)ah - 3m]!P.

(3-5.58)

The auxiliary spinor IP is to be varied independently in applying the principle of s t a t i o n ~ vaction.

....

Spins Q,*, 1, g, 2, #, Under this heading are collected the multispinor descriptions, Thus, the following ig applicable to sipins 0 or 1 :

It shaufd not b fargotkn thst these even-rank spinors are commutative quttntities (B. E. statistics), matching the symmetry of ?!?g and the antisymmetry of Y : Y ~ ( Y ~f 79. The second-order differential equation (3-4.26) can also be adopted as the bmis of an &ion princble. Let us write it as

Then we comtmct the aetion

where the fachr 112m is supplied to make the two action expressions direetly compar@ble;otherwi*, it esn. be abs~crrkdinto a common scale factor for field ~tndsource. The value of the W that is implied by (3-5-62) can be pre~entedm

which a s s e d that the two actions diger in eontent only by a contact term. The Iatler removes the contact term that was added in Eq. (S4.1). Another such remark is based on. the eornmutativity of the s y m m e t ~ c dmathx i ~ ~ ~ 2 " with ?!?g, as well as with the matrices (ror@),. On decomposing $ and into the aetion component8 with the aid of the projection matrices +(l expression (3-5.62) completely separates into two independent pshs. Thus, it is possible to use a reduced form crf the action. principle which contains only one of these field components and its assaeiaLed source* The latter aetion should be multiplied by two in order to retain the same scale for sourees and fields, That the above procedure only changes eontatzt terms is vel.ifid by considering

Action

3-5

An analogous description can be introduwd for spin considering the second-order differential equation

193

+,incidentally, by

and the action

+

(dz)[r(z)7@16(z) C (#(2))1,

Again we observe that

diEerv from the original action only by a contact term. Next, supposc? that one of the two projection matrices

multiplied by the faetor 2, is introduced into (3-5.66) so that it multiplies everywhere on the right. Then, since

YO

consistent projectioxi of field and source onto a subspace has bwn brought about. The signifieanee of this new aetion is given by

a,

only the contact term has been altered, and the same physi~alsystem is h i n g described. But ~ ~ h e f hthis e r second-order formulation,af spin has any pra&icaf. merit will not be diseu~edhere. One remark is in order, however, The r6dependent contact term in (3-5.70) i s imaginary (roir,is antisymmetrical and real) and should be subtracted from the second-order action to: prBerve the! detailed physical equivalence of the ttvo descriptions, Sinee thia subtractive term is given contact form through the use of the source v, and not J, it emphas i ~ ethat s the ~eeond-orderfomulatian could not be adopLed m the fundamental spin description.

+

+

7M

Chap. 3

Fields

Spins -4- or

m represented by

where

r=

: 7 = 7:y;yg

and

In dditian ta $, the three 8uxiliary functions $[23 . . . are varied independent1y in the action principle. A. second-oder formulation is provided by

in which

c = [m - i$ C r:(l/i)a,ln*

(3-5.75)

For this situation, we have

where the last term can dsa be wfitkn [cf. Eq. (3-4.19)l

It is of eontaict type sinee the are c~IlfinedtO the interior aE the s a w . And only the contaet terms lare cbnged if r in Eq, (3-5.74) is evemwhere rrmltiplied on the right by onc?!of the mall-iees I =tr: rl[(i~~)~.T h i S~( ? C O ~ ~ - O ] P &formu~ lation is physi~alXyequivdent to the first-order dti?h&ption. We shall bypass the dbcumion of foudh- and fifth-mnk spinors La p r w n t the action principle formulation for spinors of rank n,a f least in ids earlier stages. This incompfek Lagrange function, witten vvithout the symmet~sationof

&rivatives that is wed in (%5.73), for emmple, is

All repetitions sre to be rejected in performing the summations over pain, of indice@. It is intere~tingto sfudy the explicit connecll;ons between, the multkphor wtion formalations and those employing tenmm or kngor-~pinam. Only the to&, seeond-rank PnufLispinors will be cowidered. Sginom of the acond rank srt? wefutly kreated as m t ~ c e for s this purpose. Some % m m ~ ~ p t i of o ws i p a e a n d eombinations are

snd the action. (3-5.59) is correspondingly rewfitten eils

The symmetricd field and ssource spinors of unit spin me dven general form by

which has ~ f i p p ~ r ebefom, d in other notation, as the wcond line of Eq, (2-9.70j). The commutators and trace evaluations sLaM in Section 2-9 apply here and $ve immdiakfy :

(da;)fJ""br, 4- +M""%,

+ cl, +

e = -tGp'(a,#r - a.+F) - t+@a'a'c,. ~@@'G,.- +m'##p.

(3-5.82)

This .is the firsborder form (3-5.27,28), with the derivatives symmetr.rzed in application; to vtt~-t;orand knsor fields;, Similarly, the 8ntisymrnetricttl spinam af zero spin are presented rias

and one obtains

-do be compared with Eqs, (3-5.15, 16).

The nnla;trirr Lranscriptioft @ implies thad a brm of the matrix 3, that commutes (anticommuties) with i l ~is an eigenvector of 1"iiY52 with the eigenvalue -1 ($-l), Both (?igenvalu@&re reprwented in the expressiom (3-531) and (3-5.83). In applying the secondorcler aetisn form (3-5.621, it is permissible to use projected vemions af the field and source, as illustrated for spins X and 0, re~gectirrely,by ~ r n - l / ~= + rryO+,,

2rn-Iizt = r@ro(J, apM,.)

+

(3-5.86)

-- a,Kr),

(3-5.87)

and 2n-'l2$ = ir9r0@, 2rn-"'t

= ir&~'(K

where the f structurefs are the appropriate projections of (3-5.61). With thee rduced fields and sources understood, the aetion (3-5.62) becamm (a factor of 2 is supplied)

The trace evaluations give directly the action expressions (3-5.18, 19) and (3-5.11, 12) with the effective vector and scalar sources, explicit in (3-5.86,87), th&t have been s t ~ t e dpreviously. One can also mske the opposite choices in these projections, and we record those action form which are, for spin 1 and 0,

Actian

19c7

and

The effective sources that appear here are M,. - (1/2mZ)(arJ. -- aJ,) and -- (11m2)d,K, respectively. Wr; ahall close this section with a few varied comment8, First we recall that, although we have not illustrated it here, the possibility exists of redefining sources and fields by linear transformations which change the detailed ~ t m c t u m of the field equatians and, therefore, of the Lapange. functions. Then it is noted that all discurnion has been concerlled with the vacuum smplitude (O+~O_)~. The shift of atbntion to other transformation functions k convittyed by a cfiange of boundary conditions in the action principk. Let us be mplicit &bout the time cycle transformation function (o-/o-)'-~'+. Here the action separates into two snalogous terms, with opposite signs, that are associated wilh the two smsw of time development:

K,

The accompanying boundary conditions require negative frequencies for X+ and positive frequencies for X - at times before the sources come into oprstion, and the equality of X+ and X- sfter the sources have sbut down. The latter has the following consequence. While the inkgrations in (3-5.91) can 'be regarded m exbnded over d l space-time, there is eomplele cancellation of those re$ons that me aubmquerrd to the functioning of the sources. Accordingity, the inbgr~tion, domain can be made ~lenni-infinite,bounded by any space-like sudace that has only source-free space in its future* The final comment is concerned 4 t h the rdationship b t w e m the Lagrangtl funetlong of massive and massless particies. X t is known that, as m 4 0, the states of a padiele with spin s efe~ompom i n b those of vafious h e i i ~ i t k &g, ~ , &(s - l), . . . . Thus the desc~ptionof all m ~ s i e padiclw ~s with helicitiw of magnitude ss, in ideger gteps, ahould be contained in the a~eountof a rnamive particle .ivith spin 8, "T"hiahas received some discumion in krms of sources. We m n t to t r a e the mrraponding field decomposition in two importsnt examples. Let the vector field and souree of a unit spin particle ivith m= m be expremd by where the latter is 1Eq. (2-3.44). Then the action (3-5.18, 19) becomes

798

Chap, 3

Fields

In the limit m + Q, this action separaks into two pafis:

which, as shall, be discussed in greahr detail laCer, deseribs the photon, and

rtlferring to a massless gatPticb of zero spin (helieitd. Notice that the Lagr&nge funetion of the spinless particle comes entirely from the mms term of the original Lagrange function, and would have been overlooked had one merely get m = O in. (3-5.19). The other example is spin 2, where we express the tensor field and its source by = h,.

--

(2-lt2/m)(a,A.

+ d.A,) + 6 - " 2 [ ( 2 / m 2 ) d p a4-~

and a,TF9 = m 2 - ' l Z ~ & ,

@p&],

alrJ@= m ( 3 1 ' Z ~- 2 - " 2 ~ ) ,

(3-5.96)

(3-5.97)

the latkr being Eq. (2-4.21). These structures are such that

When (3-5.96) is inserbd into the actiorz (3-5.30,31), and the limit nz 0 performed, three independent parts are obtained, Two of them restate the unit and zero hellicity actions, Eqs. (3-5.94) and (3-5.95). The third one is

As we shall also discuss later in more detail, it is the (or a) gravifon action expression. This time the photon Lagrange function emerges completely from the mass term of the spin 2 particle, as indicated by

but the scalar field action has contributions from b t h parts af the original Lagrange function. Xt is an interesting unification of the actions repre~enting massless particles of various helieities to connect them with one action expression for a massive particle, Also impXied are the relationships between different spins necessary to amive ad a common description for a given helieify.

lnvarianee transformations and fluxss.

Ghsrga

199

3-4 IMVARIAFJCE TRANSFORMATIONS AMD FLUXES. CHARGE

The vacuum amplitude (o+/o-)' is unaltered by a rigid translation or rotation of the sources; for charged particles it remiins unchanged by B univ m d phase transformation of the wurces, Physical information is obtained through the relative tmnsformation of diflferend pa&s of the source distfibutisn. Relakive tramfation gives information about energy-momentum, relative rotation about angufar momentum, and relative phase displacement .tc?acbesabout charge. When the source distribution is causally arranged, with the dbjoint pieces treated a%units, one acquires knowledge of int-egral physical qu%ntitiesb t a l enerm, total charge. At the next stage one considers transformatiam that vary arbitrarily in space-tinrle, thereby supplying more locdized data about the various physical quantities. Fields are the instrument far eonve~ringthese data. To illustrate these remarks, let US consider spinless charged gttrtieles of mass m, as desel.ibed by the action

where K ( z ) and +(z) are now two-component objects in an appropriate Euelidean charge space, Invariance under the constant phase transformation af the source, K ( z ) = eiY9K(z), (3-6.2) follows from the existence of the compensating field transformation since all the Euclidean products in (3-6.1) are unchanged by s common rotation. Next, led p become an arbitrary funetion of position. For simpfielty we consider an infinitesimal phase transformatisn and write this generali~ationof (3-6.2) as

The msociated action variation is

But again we can introduce a compensating field transformation, which leaves K& and $2 unaltered, and fails to keep W invariant only beeause space-time derivatives now act upon $ @ ( X ) : Thus, through this method of caIculatian it is found that

2

Fields

Chep. 3

where j@(.) = a"+(.)ig#(.).

The eonsiskney of the two evaluations implies, with the aid of an irrtepation by pards, that (&G, 10) gPjp(z)= (z)iqK(z). This is verified directly, on using the field equations. When the righbfrand side iis zero, which is true in source-frw regions, we recogni~ethe local stakment of a con~rv&tion law. If the charge matrix Q is diagonrtliged and ~ompfexsources intrdtreed, the action expression. and %heproprties of jfi became

+

and

jfi = z"(a"4*4 - 4*a"@),

d p j f i= i(@*K- K*+),

(3-6.12)

where

th& #* is naC the complex conjug%teof 9. S t a b m e n t ~anaEoto all thew apply to the vacuum time eyele action with agproprbte afge braic sims in, 6t.F" Lo indicah fhe sense of time flow. Using this more general, framework, we now re-examine the causal situation with K = K Z ITz, where the phme of K 2 i~ changed by a eo&ant and that of XI i s held fixed. For the infixliksimd dransformstion b i n g consider&, we know that ~ t hthe , remind= PUB

+

This wei&ted average of the charge values rewmbXes an. expectation value. I n d e d it is one, if we consihr the time cyele function, with the phase of K,+, displaced, a f k r which $he two sourea are identifid with K,

this is an infinitesimal version of f he Xefbhand side of Eq, (2-2.123), for example. T o apply (3-6.8) to this c a w l situation, we Bepar&&.Kl and K 2by a space-like surface, which is otherwise! arbitrav, and then make 69(5) vanbh in the futwe of this su~aec:and be the conrJtant 6p in its past. The step function deriva&ive!,

Invari~ncetransformation, and ffuxes,

3-6

-a,

GP(%),

Charge

2@9

confines integration to the surface and

which is the identification

f n the time cyele degmbtion, inkgrations are exbnded up to s spwelikle @udaee,which foilows the working of the sources. Then the choice Su?-(x) == 0, So?"+($) = &pgives

which identifies

dgIr;iIr=

(Q):*

In the latter situation jP(z)is computed from the real or mutually conjugatpl mtarded fields, and is a real function. f t is evident that j"(z) provides a space-time account of t h distribution ~ and flow of charge-it is the charge Aux veclor or current vector, We sh&llevaluate it for a single-pahicle state. On referring to Eqs. (3-1.79-81), it is seen that the fields in the region between emission source K Z and absorption Bource .Kl, W* eiakd with a positively chsrged particle of momentum p, are

and if only this ex~itationoccurs, The source factors identify the emission and abwrption seLs [compare Eq, (3-6,19)1; the cument associakd with the particle is ZpP clup. We can confirm this by verifying that the total eharge is unity. But first it must b recsg nizled that the uniform value of the current is an idealization, which applies in the interior of the parfiele beam but fails ss one nears the [email protected] coume, the momentum is not s p c i f i d with arbitrary precision, as in (3-6.22), but within a cell of small but finite dimensions, having invariant measure bw,. Thrta, the correct description is given by

To compute the tots1 charge one can integrate the eharge density jyx) over all

202

Fields

Chap. 3

space a t a given time: Q(lp+) =

-

/(dx)lLap d% ~ X P ( ~X)P

Similarly, for a negatively charged particle, 4*(x)

(do,) '12e'pzi~zp-,

4(x) = (dwp)112e-ipzi~* IP-,

(3-6.26)

and jW(x)= -2pp dwp(iK~p-)(iK2p-),

(3-6.27)

with an analogous verification that the total charge of the particle is -1. The retarded fields of the time cycle description that are associated with a given momentum, and positive or negative charge, are [cf. Eq. (3-1.84)J

respectively, with their complex conjugates. The corresponding real currents are then given by jr(x) = &2pr (3-6.30) This is the contribution to the current expectation value attributed to a particular momentum and charge, being the current per particle multiplied by the expected number of particles of the given type emitted by the source. Incidentally, the consideration of a single momentum should not obscure the presence of interference terms, in these quadratic field structures, when several particles of different momenta are present. Such interference terms disappear on performing the spatial integrations that evaluate the total charge. The non-conservation equations, (3-6.10) or (3-6.12), connect the particles and charges observed after the operation of a source to the activity in the interior of the source. Let us test this in the physical circumstance of the time cycle description, with K(-, = K(+, = K. Integration over a region that contains the source gives

where the surface integral refers to any space-like surface that is subsequent to the source region. On any surface that precedes the source, the retarded fields and the current vanish. The explicit form of the right-hand side is

lnvslriancs transformations and ftuxes;,

3-6

(I/i)[bmt.(z

--.

X')

- i?rrCt.(zL z)]

- z" = aft)(z -- 2') -- a(-'(% -

";=-

(I/i)[b,,t.(z

Charge

203

--.

(3-6.33)

and

~vhirthis the expected rcsult. We have not, yet remarked on the ambiguity in the current vector that is defined in general by (3--6.8). To a particular jP(ctr) vector can be added any exprcs"siort of the form d,tnp"(x),tvherc nzp" is an antisymmetrical tensor, for

This supplement to the charge density, V n(x), ~vheren k = m'*, adds a twodimcnsianal surface ixrtegral to the charge associated with a three-dimensional volume: (3-6.36) Thc calculation of tatnl cllarge is not affeckd, therefore, nor is the value of the flux vector assigned to a uniform ~ituationsince this is also fixed by total charge corisidcrstiorrs. Wlxy can one not ignore the ambiguity and just accept the currexrt expression that is naturally associated with %beLagzlrtge funetion? One reason is that alternative Lsgrange functions can produce: diaerent currents. This is illustrated by the unit spin, situation. Tlre seco~ld-orderL ~ r a n g efunetiorr (3-5.19) and the first-order Lagrange

fune tion (3-5.28) imply, respecdively,

and

In the abrsense of the source M,,, these current expressions are equivzttent. But, when we use the Lsgrange function ($5.231, there results

where Q1piq+" is indeed an arttisymrnetrieal tensor. The alternative current expression is sowwhat simpler, since d,rpY vanishes outside sour= regions, and

the tofizl charge can be caleuXated as

Let us apply this to the region betweert tbe %M-o causally sep&raM souree8 .l JS, !, where the fidd is [Eq. (3-3.3)]

The eigenvecfor propedies and the narnzaiigation C* @pxqe,pxq

=1

enBure that the contribution to jp of a speeifie particle state is The flux per padicte, 2pp hp, imlearfy of univemal appliwvdity. Aa we have mentioned, it is fixed by Lhe normalization condition, in the manner m d e prec k by (3-6.25). One might think that the arnbiwiw of current exprmions is an! asp& of second-order Lapange funclions, wit)l their ogtisns in arrandng $WO cferivatives, &nd would disskppear if firs&arder Lavange functions m r e adopted. T o dispel this iXlusian it suffices ta wmidtjr pin 2 charged padicXes, where two 6mbordr;r fwms art;?avaiXlzble. From the Lagrane function (3-5,s) we obtain,

with the latter form applicable in soureefrtze: regions, whik (3-5.41) gives

The application of the phase transformation procedure to the spin produce^

(%$.m,51)

action

jp(,) = &~.(z)y@r~@(z),

(%6*47)

a,jP(z) = +(z)r0i9?(z),

($8.48)

bgekher wi& which is also s consequence of the field equations. The field in the interval

Invariance transformations and fliuxas.

23-43

between the causally separated sources +(X)

and

= C [@p.p(~)itllp.p P@@

+

Chargs

266

is [Eq. (3-2.14)]

+pop(~)*itl:prgl,

where, it is recalled, = (2m d~~)"~u,,,e'~',

*

0

UpuqY UP@Q =

1,

The contribution to 3'"" ttsscrcinted with a single-particle s t a b is

which identifies the ffux per particle:

This expected result expresses the evaluation

whieh ha8 %befotlot~ingderivation, bawd an the normalisation of IEq* (3-6.W) fef, Eq. (2-6.97)j:

O

=

u;u:7°(rp

= -2pp

+ m,7')

g,,,

+2m~~~,r~r~u,.,.

(3-6.84)

As an afternative to computing charges by surface integration of jp, one can consider the volume integrtl of %jP, extended over the region occupied by qg. The field asociated \~\-ith the latLer gouree does not contribub l o &his calculation,

since the mLisymmetrieaX m a t ~ xQ; removes the mateh between the antieommutativity of the sources and the antisymmetry of the kernel ?'G+(% - X'). According1y,

which exhibits the charges p = & l assigned to the sin&@-particlestabs. Although this interpretation is quiLs etear, it may be helm1 Lo give EL mar@formal discussion, b ~ ~ upm e d tfre analogue of the rdsttian (%ti.lS), or

Chap. 3

Ta andyze the lefbhand side : (3-6.58) we have only Its note %h&

and the charge vdues are identified as

I L has been entpha~sizdthat the arbitmsinw in m~iminget charge flux vector is not just the po~sibilityof adding any p3,mpVkrm, butt is inberent in the existence of akrnative Lagrango funetion dacr+t.ions of the given physied 8ystRm. This is Lme of .spin EM well. A third-rank spinor that is antisymrnetrical in a, pair of indices give8 a spin 4 description. The current deduced from (s5.73) is

C

+C

jr = +#~(j ~ g ) q +

+L.Y+(Y:

- T;Y&~,B~

a<@

+

3+[231~~:&~231f

Outside all the sauree~the auxilietry fields ?C/raal v a ~ s hand

+

g

.

(3-6.61)

An appeapriate expremian for the third-rank spinor is ~vritten.as

It is anthymmetriesf in tI and Sz, and the propesties (&6.62), m ilfu~tratedby

with matrix notation regtored for the third spinor index, are satisfied since

frr^(l/i)a, -l-

= 0.

(3-6.65)

The current that is defived from the first Lerm of (3-6.61) by inserting (3-6.63) is

tnvariancs transformations and f Iuxes,

3-6

Charge

207

With the aid of the identity

m,, =

-- -2 -l 9~@"'.~'k 3 2m 2

The o;,term is the only such structure that does not involve coordinate dcrivn-. tives. Ana;Xogous but more elaborate comparisons can be medc between alterneLive descriptions for particles with spins 8, 5, , . . . The technique of vsriabfe phase transformation. hag been used t o give a more detailed space-time dewription for the average charge distribution, It also supplies such information about charge fiuctualions, We shall illustrate this for spinless particles, confining the discussion to the simplest measure of ffuctuations. Consider, then, the time cycle vacuum amplitude with the sources

K,+,(z) = e " " ~(X > t

K(-I(Z~== K(%)?

(%6,70)

which is

An infiniteeimal v a ~ a t i o nof the phme constant

u, gives

This equation, with cp = 0, has been discussed Now let us differentiate once, before setting u, = 0,with the consequence that

where

is a charge fluctuation flux vector, In, evaluating this vector from

j?+,(z) == P4t-r-I (X)~~~P(+I[X), (3-6.76) one of the 4(+, fields is taken a t (P == 0, and beeornes + r e t , ( ~ ) . For the other we

Eq, (3-1.19), appropri&teto the vwuurn initial &ak, and obt&in (S6.77) with

In k r w of the~efield8 we have which urn the charge prop&y = 1.

It is helpful Go decompm +(g) into r e d and im&nary p&&: whem, weordjing to the mlation the maf,compnent is

In the circumstances to whieh (3-6.79) refers, which are made explieit by the apprance af the funetion eret,(g)# the advan& field vani~hw. And the contfibution ts 4 of $+fat. caned#, XeaGxlg the re81 fom

Since the lstter is a solution of the homogeneous field egustion, one finds that A oalculstion of the total eharge fiuetustion can be performed by integrating over the muree : (h) (dzf)~(z)h'+'(z-- z f ) ~ ( z 8= ) C K , , I ~ . (3-6.87) P@

The latter is the expected total number of particles emitted by the source, and thia fluctuation formula

((Q - (~))')f = (N+ f ~

-)f

(3-6.88)

is contained in the more general statement (2-2.123). In effect, jt,,,, is a partiele flux vwtor.

Invariance transformations and fiuxfslo.

3-7

Mechanical propertisrrr

205)

INVARIANCE TRAMSFORMIATIIQNS ARID FLUXES. MECWAMICAL PRBPERTliES

The wLion for spinless particles is invariant under zc rigid source translation,

R(%)= K(%+ X).

(3-7.1)

This is expressed, in (3-Gel), by the existence of the compensating field kransformation (3-7.2) * ( X ) === $(a: -I- X ) . The infinitesimal versions of these transfomations are

The following are the proposed eneralizations of tberse expressions when 6XY becomes an arbitrary function of position, 6 z v ( ( z ,

The distinction between these farms is necessary do maintain the invariance of the sourec-t t e r n under the eonnpensating sauree and fieid vftriiations:

The response of the Lagrange function to the field variation of (3-7.4) is

only the first term on the right would appear for tz rigid translation. An equive bnQgresentation is 62 == d,(6zp&) - tPya,&X,, (3-7.7) where tPw(z) = aP4(z)av+(s) gpve (+(z)) = $"(X). (3-7 -8)

+

Let us also xlote the relation

mrhere the fins1 form involves the use af the field equation. The chsne;e induced in the action by the source variation of (3-7.4) is computed alternatively ss

270

FSelbs

Chap. 3

and The compsrisoa of the two versiorls implies t;hat

whicl~can be verified directly. This is the local stahment of a vwtorial conservation law when the right-hand side is zero, whieb is true in soume-frm ~giona, In the causal circumstance indicated by K = K Z K Z ,B rigid infinitesimd displacement of K 2 induces

+

( ~ + l o - )=~ C (O+I

{E)

where

j /o-)~' +C(o+iIa))Ri[lt i b X * P , j ( { n J / ~ - ) ~ ~(3-7.13) ,

P,( which can be writbn

(4) = C Pp@,,, P(I

(3-7.14)

Similarly, in the time cycle situation the displacement of K(+, and its subsequent identification with Kt-t = K gives

To compare (3-7.15) \vith (3-7.111, we separate K t and K z by a space-like surface and let 6%" vanish in the future of this surface and be the eonstant &X' in its past, That &ves (3-7.17) which i~the identification

The analogous equation of the time cycle description is

where ESY(z), kvhi~his computed from the real retarded fields, is also red. The distribution and Aow of energy a~ldmomentum is described by t"(z). It is the enerm-momentum flux vector, or stress tensor. Let us evaflxak it for the state of a single particle, chosen to be neutral, for simplioity. In the region betgveen the causally separated sources, the field that is associated with at particle of mometrtum p is

3-7

lnvarian~atransformations rand fiuxaa.

Mechanical preperties

211

Xf only this excitation is considered,

which obeys the conservation law i),tM" =; 0. In any consideration involving time or spsee averages, where the corresponding components of p" set $he scaIe, the oscillatory terms of (3-7.21) can be ignored, The firs%tc3mm exhibits the anticipated ftactars: the representation of the emission and abmrption acts by iK2,, iK:,, the particle flux faetor 2pp dw,, and the measure of the quantity being transported-the energy-momenturn vector p". In the analogous d i e cussion using retarded fields, we have

The disappearance of all interference terms through integration can be vePified, in the causal sifuation for example, by using Eq. (3-7.12) to obtain

where, in the region occupied by the emission source Kzl

The first parL of Lhe: field, associated with KZibelf, does not corrlribute:

since the gradient of the symmetrical function &+(X -- z F )is tan antisymmetrical function. Accordingly, with a slight rearrangement we get (dz)(dz')iK l (s)(l/ i ) a ' ~ ' (~X ' -- zt)iK2(z')

which exhibits %bephysical property p-hat is carried by individual particles between the initial emig~ionand the final abssqtion acts. Perhaps we should ktho note that

in1 )KliKtpi~zp(inl

=

@+I i% + 1,)

+

i l)({% l,]

Io-)~', f 3-7.28)

which shows how (3-7.18) is used to produce the general enerm-momentum

252

Fields

Chap. 3

evatjtutltion (for neutral particles)

The analogue of (3-1.27) for the time cycle description is the expetation value

Enerw-mame~tumAuctu%tionscan aXm be given spacetime localization, in for charge, but we shall not discuss the details here. Spinlem pta&icles have Ptn al6emative characterization in the arcfian (3-5.15), whiah scalar and veedar fields and source^. The derivdives of the field variation (3-7.4), a(d,Qt) azY+v(afiqb)-4- (a@@)d, cfix', (3-7.31) provide a model. for the response of vector fief& to coordinab-dependent dispfacements: b#,(z) m ~ ~ v ( ~ ) ~ ,4,-+ B ( ~ ) &"(g)* (S7.32) The eamegponding muree variation is that required to maintain the invaritance of i ( d z ) ~ f i +namely ~, Notice Ghat the distinction between the two forms die~ppearsfor a rigid rotation, where (3, 62, = ---a, = SW&@. (~7.a) The reBpome of the Lsrsnge funetion

is

e = --+Pad + ace#. -

(~7.35)

616 = Zixvdpd:-#- (p+'- dic"dYQt- cfiYa'tb)ah 6% =.= a,(b~"&)- tc""d,

with pp

vaV+ +

---- f14V+

= tPp,

(3-7.36)

(3-7.37)

%nd (3-7.38)

We aIm have a,pv -.. -KaY# J, ( a , ~ @ )3+ ~K ~ ( ~ F ay+p), + ~

(3-7.39)

which e m be given oLher forms with the aid of %hefield equstionfs. The only fieldaependent term on the rlghbhartd side can be w ~ t h ss n -av@, nultiplied by the effective source K - 8,KP. In the absence of the veetoriaf soK,, the two vensiom af the ~ t r e mbnsor coincide, The Lapange functions (3-5.19,28) for unit spin ps&ielm contain vector field@,and their derivativw in the cud combination. Using %heh t b r ss the,

3-7

Invariance transformations and fluxes.

Mechanicat properties

213

model for an antisymmetrical .tensor field GFv,we infer by dihrentiation of fS7.32) that

When tensor field sources are introduced, their infinikgimal variation is But fimt we consider =

-&cppc,.- *m2@+,, G.= a,& - a.+@,

(3-7.42)

which gives 6& = 6zv4,d: - GFrGvhd, 62, - m2+p$va, 8%. = iaV(6xVdG) - tpva, SzV

(3-7.43)

and (3-7.44)

Here 1''

= G@"'~ ==:

+ m'+@< + grv&

IF@,

(3-7.45)

and we note &hat

+ 46:

l = Gp'GPv-4= -m2$'+),.

(3-7.46)

The direct evaluation af the source vagation efifect i s

and the eompa~sorrs h a ~that ?~ apt@"= J,GMY ( a ) , J @ ) t ~ ~ , or, more immediately, a,tr""= ---Jl"av+, 4- ar"(JP9"),

+

where the t a t term does not eontributc:to the volume inkepakions %hatevaluate the total energy and momentum emitted by the source, The alternative Lagrange function (3-5,23) digem from (3-5.19) by the divergence of a vector [Eqs. (3-5.%, 2511, and Chat remains true of the variations 6C. Such additiond terms da not contribute to &W, But let us note that the relsfion (dx)dh[f (%)aFd z v ]

with the last b r m annulled by the re~trietian

implies3 arm arbitra~newin the stresrs k m r , which is indicaectid by Iheeombina;tion Thia arbitrarinem &appeam# however, if we in;sigt upon a propeAy whi~h,thus f%r,has emerg4 automaticaHy-the ~;ymmefryaf the bmor P". Xnt, order that tfre 8yntmetry prowrfy continue to apply in (3-7.52) without recourm to iliifferc?nti&lidentiliw, we dennand f ""(z) = f k"(z).

(3-7.59

But $he eombinatian of (3-7.51) ftnd (S7.33)has the folla~r?tg comequeDce:

Phy8ical im$icatians of the gymmetry of the s t r w tansor will be gowider4 later, in connection vv;rth.the di~cusgianaf anmlsr momentum. Now we turn to the fiapange function

and obseme that it, Gves the ~ t r m hnsor

1L remains tme, bowever, that

According $0 the field equtalion

the field-dqendent kerns QD the righaand side of (3-7.57) can be and expressed in terms of the effective vector source J, d'M,.. The specialisation MP, == 0 idenfifim fhe two vemi~nsof the stress t Another qum1ian &ouL uniwenttss presenb iksdf, The generdisation to arbitrary displacements could have emulahd more eiosefy rigid zota$ion kmhavior, ss tha;fc ($7.32) would h&veh n

+

3-7

lnvsrianea transformations and fluxes.

Mechani~sfgropeftias

218

with the associated vector source variation

Tadeed, the distinction between field and source variations disappears completely were we to adopt

How does this freedom to choose the form of the displacementinduced v%riatiorts r%Eectthe identification of the stress tensor, defined generally by

and presumably made unique by the requirement of symmetry? The various choices for 6 4 p differ by terms proportional to

which vanish for rigid transfatioxls and rotations. This tensor is a, memure of the dilation produeed by the displacement, and includes the scalar measure c3, 62". The same dilation tensor appears in (3-7.M), in consequence of the symmetry of the stress tensor. Consider, for definiteness, the effect of the additional field variation on the action expression (3-5.18, 19). Sinee

contains second derivatives of the coordinak dispIacemen.t;s,s partis1 integration is required fo attain the form of (3-7,64). This introduces second derivatives of the fields, and the field equations will be called upon to eliminate them. The initial choice of field variations was sueh as La obviate the need for any application of the field equations. The dilation dependent variations can be treated m a unit, and the stationary action principle invoked.

which shows thttt the stress tensor is changed, by the addition of

This is also what is requird in order to maintain consistency with the dimet

25(3

Fisjtds

Chap, 3

evduatisn aft the zaetion v a ~ s t i o n :

We conclude that sitress temom do have tt degree of arbitraniaess eomesponding to the frwdom in sssigning the effect of general coardinadt? displmements, but t;hisl arbitrarines~is confined to the inkrior of the ~fsources,In the soureefrm reeons where the &=ss tensor controls the flux of: energy and anamenturn iL would %em to be unique, wcor&ng h %hemlm we have laid d a m for ilts evalaation. Rmrrangement that require the use of the field equatbw c m d s o be sccomplished by a different choice of field varialion. We iftuslrab %hisin the spin O situation. with 6
+

which gives

Tfiis stress bnsor is

+

tPp= ap$av

-- +a2(+2)I.

(8-7 '74)

The field equation, asse&s that an$ the ori$ttnd stress tenmr reappears, with an add4 brm. that vanish= outside sources. But the direct w e of (3-7.74) yields

which, in source-free regions, is the result obtained in (s7.9) through the use of the field equations. The? question of uniqueness intrudes agrtin in this exrample. The rearrange ment that eonneets (3-7.72) and (3-7.73) mighf have been haadled digereatly : This wmld eh&&&

the identifies%ionof %he@itresstensor by t h added ~ brm

Not@that this gymmetrical z~tre~s hnsor eon&;t.ibutionhas a divergence %ha$

3-7

tnvarlnnce transfarmstionar and fiuxslr.

Mgchaniesl proplttties

,237

Fu~hermore,it malres no contribution to the t o t d energy and momeiltum, in vidue of the inbeation theorem (1-4, f l),

This is true for angubr mamntum also. The d d e d term i s p r e ~ e n t din the form of (3-7.52) by choosing fh ' = &[B"3p +hydfl-- 28'ya"+2. (3-7.81)

+

Observe that this expression is symmetrical in p and v, but does not have the antisymmetv that is stakd in (3-7.51)- The annulment of the last krm in (3-7.50) comes about, instend, through the differential identity (it is (3-7.79) again) akarfk" 0. (3-7,52)

The rejection of stress knsor terms that invalve digerential iderttities is thus an essentid aspect of the computation rules. But we art? noiv going to sese that stress tensors are quite analogous to charge flux vectors, Arty current vector 'j can be replaced by jw a,mp",with arbitrary &ntis~;mmt?kicaf ntF"". Qmmight ag;ree ts rejeet such additional terms in studying a given Lstgritnp function, but the existence of different Lagrange funetions for the same system, leading to current expressions that differ in just this wayl shows that the arbitrariness is intrinsic. The arbitrariness in symmetrical stress tensors i s expressed by the possibilify of repfacing tp"(x) \\-ith

+

where and the symmetry restriction nZ~v""IX

assures that

mm"",x' .+ rtzx",l^"

8

a , a , a , ~ a ~=~ ~o.~

It also guarantees that,

and it will be verified later that the total angular momentum is equalXy unaflected by the additional term. In the simple example provided by (%Ten), As in the discussion of currents, the exisknee of two sets af firsborder field equations for spin 2 particles provides s valuable proving vound for unique RWS questions,

2l8

Fidds

Chap. 3

The tensor variation structure (3-7.40) is compatible with both antisymmetry and symmetry of the tensor. Accordingly, the symmetrical tensor field of spin 2 particles can be assigned the displacement variation

Two significant derivative combinations of this tensor are given in (3-5.37) and (3-5.44). For the first, antisymmetrical combination we deduce

while the symmetrical structure obeys

Notice that second derivatives of the coordinate displacements appear here. First, we consider the action expression (3-5.33), simplified by setting the thirdrank tensor source equal to zero, so that GxrV = ax+Kr

and

-aApx

+

d: = -;tGhflYG~r, +GAGx- *m2(+pv4p,- 4').

(3-7.92) (3-7.93)

Infinitesimal coordinate displacements induce 6s = a,(6zVd:) - trv'a, 6s'

where

- ~ " ~ a 62,, ~ a ,

(3-7.94)

~ P V=' p '

- GFK'G',~+ 4@"'GKvx- GNG' - Gx(GX" + G'") + 2m24'"" - 2m24@" + gP'd: and F ~ P V=

-

+ +'G,vv.

(3-7.95) (3-7.96)

To extract the stress tensor we use the following identity:

A partial integration in the action variation shows that 1''

= 1'"'

- ax[FA"'+ F"" - F""],,,,,

(3-7.98)

where ( p ) indicates symmetrization in the indices p and v. According to the evaluation

lnvclrian~etransformations and fluxas.

3-7

NIs~hsnlcalpropertim

219

this gtress tensor must obey

When we confine our aLtention tr, sauree-free regions w considerable simplification. occurs since all vector and scalar field combinations rranigh in these circumsdances:

and further reduetion c m be acconnplished with the aid of the field equations ahck" = m2+pp,

Thus,

axc;pAp .= 0.

+ f G""C~'~

= m2$pK+vK

ah(Gh'*+'.)

(3-7.102) (3-7.103)

gives from which we obtain

to = --m2@'+,.. In discussing the alternative action expression (3-5.40), we shall p c e e d directly to source-free eondi.tions by omitting all scalar and vector fields in the tagrange function, This i s jmtified, despite the varistion that is going to Be perform&, since all such fields occur in pairs, one faetor of whieh continues to vanish after the infinitesimal coordinak displacement has been etppfieid. Then, with (3-7.106) &'X a p + v ~ 4- ~ V + & X ---- d ~ + p . , we have (3-7.107) e = ~H~'%H,.,- +mzyp6&. and "LI;:

+

+

1% = --+(H@~'H.~' H P h % ~ ')) +@'a: 2dh(~'"+..

-- H X " + ~ ~ ) ~ ~ ~ l p (3-7.108)

in whieh we have also uwd the field equation

The comparison of the two stress tensors noiv econfirms that

220

Chap, 3

Flsida

satisfies the symmety requirements laid dawn for t h i t~e m r . A ~ h o r k defiv* r dion of the rmuft is b on direct eonsideration of the diEerence in the L* grange funcliorrs, (3-7.1 12) CH - C@= $a""'a,+.k. Concerning the expre~sionfar the displacement induced &an&@of soure@ and field, we note that the availability of dditional b m ~ that involve the general dilation Gensor (3-7.635) is strongly spindependent, if dditionai coordinate derivatives are esefrewc?d, Matrices &ha$a ~ I tupon objects carrying pin s arc! limitd Lo the e o n s h ~ t i o nof teaorr~with rank _I 2s. Af least unit q i n is requird in order to r e d i ~ esmotld-rank Lensora, and only the gealar conThe spin O example traction, of the dilatbn bnsor is av&la;ble ta pins O and is illustraw in 63-7-71). We no\v discurn apin +. On referring ta Eq. (2-5.11), whieh describs Lhe regponsr?,of sny m u m to coordinah transformwtions or the equivalent rigid trandaLion and rotation of the sourn, \ve are led, for the spin 19 source 7 f ( ~ )to, the generalisalion

+,

The spin Lerm only contains 3, 6%" - d, 6q,. There i s no symmetrical hnsor %hater-tn be deviged from matrices, other tbwn g&" multiplid by the unit matrix. A specific mui%ipleof d, &X' appe~m,analogous to the seatar source reBponss! of (3--7,4). The corr~spondingfield va;ri~tion,~vhicbis &sign4 to leave intact $(h)?ro+fis (3-7.1 14) &\d.(%) == &'(~)ar$(~) iQo"@+(z)&8~,(2),

+

The spirr. 4 Lagrange funetion has the following irnmediab responm to the variation (3-7.1142,

But the symxrretv of the second de~viativepicks out the matrix comGnalion and, since the Farthermore,

are symmetrical matrices, the last term of 642 vanishes.

[rk,+opv] =i(ghF -~Bh~7r), ~

and we get 66: == a,(lEa;'C)

- tPVd,

tvith

(3-7,118)

(3-E .f 19)

+

F' = 1'. = ++ro$[rp(l/i)ap $- rp(l/i)a@]+ fve.

(3-7.120)

The associated sealsr i s

t = -m4+r0$ - fVro+,

(3-7.121)

3-7

lnvarlancs trrrnsformationa and fluxes.

lVIschanicr@ proper?les

since the field equation implies that

and this stress hnsor must obey the divergene@equation as one can verify directly. The various terms of 6A? show explicitly how the funetion l e d s to the cancellation.of the rotational 8csllar nature of the Lagra~gt?_ sfrueture, leaving the dilationd part and thereby producillg a symmetrical stress .tensor, In the re@on b t ~ v e e ntwo causally septtrated sourees, and qz, the field cllahd with a particle of specifid momentum, spin, and charge is fEq*34.49)f (3-7.124) $(z) = $ p . p ( z ) i ? ~ p ~ p4- $ ~ ~ q ( ~ ) * i q : p . ~ .

Thr?eorre~pondingstrem b n m r contribution is given by

aecoding to (3-6.53). The expecked enere-momentum flux per particle is evident here. When pafiicles of spins Q and l are described by spinors of the second mnk, it is natura! to follow the example af spin and write the displacement induced fieid v s ~ a t i o nm 8lt. = &X%,$ i#(a"f &$;")$a, 6zp. (3-7 126) Its eKeet an the Lagrange function

+

+

+

ia quib andogous to the spin 4 situation, apart from the 6zy tern, for now [u = rfr:] 6 8 = azPa,e f ++r&i (?:av r:ap)J.a,h, (3-7.128) - ++r+(r:+~:' r k i ~ : ' ) ~ aaz.. ~a,

+

+

The identity (8-7.97) prduees ithe symmetriea2 stress hnsor

to kvhieh the null terms involving (?@ckY) can be added. This is useful for the matrix dr4tnseripdion of (3-7.129),

where the double commutator can be replaced by the double antieommutator without agecting the value of this term. The relar;ion between this and earlier stress ~F~EISOX" foms will be e x a ~ m d , wing the gero spin example. Inserting the m a ~ field x (3-5.83) one finds

where the (z~lpropx"iatf3 Lagrange funetion, given by (3-S.%), has been replac4 by the Lagange function of Eq. (3-5.16) and the necessary divergence krm, Observe how the derivatives of the vector field cancel, leaving which agrees with (3-"1.8) and (3-7.37) in regions where the veetor sowee vanishes, The generalization of the field variation (3-7.126) to an arbitrav multispinor is

which can 8180 be applied to the vmious auxiliary fields #raBlt .. . . source-free space all auxiliary fields vanish, and this property is not aBectc3d by arbitrary displacements since the transformation law (3--7.133) refers only to the 6eld under considerrttion. Accordingly, anly the first term of %heLagrange funcLion (3-5.78) coxrtributes, and the resulk obtained for the stress tenmr is an, immediate generalizatian of (3-7.X29), but wifh 6: = 0 befits the sourceless circumstances: J/iaB)lavt

Xrct,

We shall elaborate fibis for n = 3, using the spinor, antisymmetrieal; in a pair of indices, that gives an alternative spin 4 description. The eal~ulationr e sembles the, one already performed far the current, and the re~u1Lis

where

tg = +qr"+(rp(l/i)a' + rp(l/i)ap]q

(3-7.1 36)

is the stress tensor of the simple sginor or Dirae equation for source-free conditions. The additional terms can be exhibikd in the form of Eq. (S7.831, with

Invariance transformetions end flux=.

3-7

NIechanfoet properties

223

The relative rotation, of sourcm supplies anwlar momentum information, To d i m u ~this, consider the displaeernent

6 d t ( z b-dow(z),

62' = dwA'(z)z*,

(3-7.138)

the aceomp~ayintgdilation The latter v&&%, of eourmJ when Gu~,(s)ia a constant, dea~ribixlga rigid rohtion, The implied actian variation is where jhhr

= Z~th'

--

%'tkpe

The dhmative evalualion refers directly to the regpom of the souree, In order to deal uniformly with all spins ws? use the multispinor de~cription,for which

Comparison of the two computations givw

This re8uIt is not independe~tof the divergence ewation for the ntress bngor,

since ghjx"~

ghaxtkv

-- xpaxthfi + pp - p,

(3-7.145)

It is hem that we recognize the significance of the symmetry prope-rty possmd by the stress tensor. The quantity jkv, which i s formed merely by taking first moments of the sGrf?ss knsor, is eonsewd in ~ource-freeregion@, As the m&&tional analogue of P', jh" evidently provides a spsce-time sceount of the distribution and flux of angular momentum, in s fourdimensional mme. As in eadier discussions, the rigid rotation of aource q 2 relative to thecau~allydetached source is introduced by letting 6c~,,(z)vanish in the future of an inhrmdiate spa-like surfam, after being the congtant 4, in its pasLa This gives (3-7. f 46) AR analogous e-qustion applie~ta the time cycle demriptim, wbwe an@has the

impler physical inbmreta%ionin h r m ol an evetation value

Surface? inbgration is replaed by v ~ l u n eintegration, 6 t h the aid of (3-7,143),

%hecombination of orbib1 and spin 8 n ~ l a momenta, r is e~denChere, The eig~vaienfm~dth other fornnulatiaas?;is illwtraM, for fhe vwbrial sourceand field of unit spin p&ides, by [cf. Eq. (3-7.49)j

Wer ~ h dnow l confirm that the evaluation of total ~ 1 ~ @ momentum ar is anafleeM by the mbitrariness of the @%re88 &mar. AL~e~mpanying the redefinition

The verifieatioxr that the additional ~ c a x l d - d e r i v i ekrm gives no eontfibu&ionto l&&Pfi'is identical to that of Eq. (3-7.87)--only the oyclic symmetry prapr,rLy (3-7.85) is invokect. The lafter also impbw this slakanent of antisynnme%qin X and a: [ m h ~ , a ~ m b . a r ~+ m a ~ . h v l = o8 (3-1.152)

_

_

from which fdlows the vanishing of the s u d ~ eintevd for the Xwd term of (3-.7.151). Ps&icle s t a w %ha% are labled by three4imensioztrtl angular momexttum quisb~tumnurnbr~,ar~therthan by Xinew momentum, have been exhibited for spirms 0, *, 1. T b i r use i~ suEciently similar to the charge and enerw-momentum &aewiom that we Eskdl not enbr into detail8 here, In ontrwk do $he fidd translations and rotations of phygieal inkre&, $da$iorrr~ would =em da be a d y a device thaL assists in the identification of merw-momentum fluers. There is, however, a 8ubset of these tran~formations that plays a more physical role in (the spc;ie,ialcireumtance of 18 is the ~ o u oZ p hatropic dilations that is characteris& by These conditions are vemy. re~krietive.We fimd nob the seafar relation

The &vergenee of $he kawr equ~tio~1 then gives

tnvsrlance transformationsand fluxes,

3-7

and

et

Mschanlcai propcrrtiba

226

fu&her divergence asserlks thad

But even more is obtained by applying the operator a' to (3-7.153), namely Thus &p(x) is Einnihd La a finear funetion of the ~oordinahg, The corresponding form of 6xp, apart from infinite~imddran~Xa?;tiong %ndrotations, is the quadratic=function Tagether with tran~lationsand rotations, them fransfarmations form s group of 15 parsmeter&. It has the structure of the rotation group in 4 2 dimensions, in the senw that the homogeneous Larentg group is the rotation coup in 3 I, dimensions, Perhaps the quiekest way to recogni~ethis is through the introduction of homogeneous coordinahs,

+

-+

that are defined in the five-dimensional space of the null 'sphere'

The ten-parameter space-time translation-rotation group now appeam er*s the subgaup of homogeneous transformations on: the null sphere that leaves q15 y~ invsrian t :

+

the?! one-ptzrannebr &a transfomations, uniform scale change8 in ~spaeetime, are the "rotations' and the transformations parametrized by Cibv hold g5 -

fixed,

The quadratic form of (3-7.161) also admits refieetiow, including 116 -+ --g&, which hezs the following effect upon the xp coordintaks:

an inwersion in the origin. A sequence of two inversions, first p e r f o r d at the: origin anci then at the point with (coordinaks &bp,produces the infinibgimd tramformation of (3-7.159) with &a= 0.

226

Fietdzii

Chap, 3

The isotropic diltllions are known as conformal transformalions, Their physiestl relevanw emerges on considering the part of the Lagrange function response t o displacements that is given by which, singles out the racalar- t as the significant quantity. Inspection, of the examples with spins 0,#, 1, 2, shows that the scalar E, evaluated a t source-free points and with m -+ Q, either vanishes or is a second-derivative structure of the form implied by the arbitrariness of the stress tensor, Since d l physically interesting possibilities are included in these examples, we shall forego the luxury of a general proof, Rut what is invofved should be clear. When X/'VL is infinik there is no standard sf length in the action. If the scale of all coordinates is changed uniformly, the form of the action can be maintairled by a corresponding scde change of sources and fields that is determined by the number of derivatives in the Lagrace function when the minimum number of fields is used, This is illustrated for spins O gnd. % by Considered in. infirritesimal farm these settling lakvs specify a definite multiple of 4, Gzr-an example is (3-7.71) far scalar fields. The invariance of the action, stating a local property of the Lsgrange function, requires that the scalar t vanish everywhere or, more generally, be the divergence of a vector. As one can verify in the simpler examples, this vector employs the gradient vector for its construction and (3-7.168) emerges ss a generally vaIid statement for a, suitable choice of field variations. Then, since the general $ @ ( X ) , Eq. (3-7.158), has vanishing seeond derivatives, and the action is invariant for the whole 15-parameter group that ineoflporates confomsl transformations, As aIm-ays, invariance of the action impties conserved physical quantities with space-lime distributions and Auxes, The procedure is standard; the constants 6a and lib, are replaeed with arbitrary coordinate-dependent functions. We shall assume, for simplicity, that t = 0, If, instead, (3-7.168) is applicable, additional terms appear in the various Awes but nothing basic is altered. The response of the action to the generalizations of the conformal transformations is

Tha slectromagnstic field,

3-8

with C@ = tpyz,,

C"

Magnetic charge

= 1fik(2~%2' - ~"Yz').

227

(3-7.173)

The tensor c"^"is not symmetrical:

-

C ~ v

==

-22hjhpu,

and the implied scalar is c = @pp8v

2zhek.

JR source-free regions, the Ioeal conservation laws are

and the existence of conserved total quantities is indicated, for Lhe time cycle description, by (3-7.177) The physical eontetlt of these conservation statements will be discussed in the corltext of the most familiar massless particle. 3-43 THE ELECTRQMAGI\1ETIC FIELD.

MAGNETIC CHARGE

Although frequent. reference has been made t a the m --+0 limit for unit spin particles, it is important to give ark independent discussion of the field associated with the massfess, unit helicity particle-the photon, The slarting point is Eq, (2-3,45), written as

I n defining a field A,(z) tlzrougl.1 the effect of a test; source 6Jp(z),

(dz) GJP( X ) A, (X),

(3-8.2)

strict account must be tnkezt of the source restriction, which demands that

one shouId not identify the eoefieiexlts of 6Jp(ts). The correct conclusion, is that they differ by any expression that leads to a vanishing integral in consequence of the restrickion (3-5.3). The general form of such an expression is

Chap. 3

and therefore (3-8.6) The aspect of A,(x) that is governed by the arbiCrary scalar function h(s) is p k k d out by forming the divergewe of (3-8.6). This gives a.kV(z) = a2x(z),

(3-8.7)

and the application of the differential operator --aZ to (3-8.6) then provides

us with the mcond-or-der difflerential equation We reeogniee Eq. (3-3.6), with m == 0. Since the arbitwriness af X(z) persistfs in this &Rerentis1 field equation, the latkr musk be! unsffectd by any rdefinition of A, in the form A,(z)

+

A,(%) 3- dpX(z)t

(3-8.9)

which is known ss s gauge trsnsfarmation (mare frequently, as a guagc? (sic] transformation), This gauge invariance is emphmimd by ~vritingthe field equation in the equivalent form [earnpare Eqs, (3-3.7,8)]

since %heddition of ra- gradiend term to a vector does not alkr the: curl of the vector. Thst the divergeneetess nature of J P is built into the field equations is 8ho emphasizled, for 8,Jr = d,iZ,PV E 0, (3-8.12) owing Co the sntisymmetry of F@". The curl construction of F,, is given, another form in the differentid equa;tions

The tensor dud to F,, ia &fined by fEq. (2-3.60)] 'pY = +e@,gh~Ek,

%?herethe totally sntisymnretrietal knsor of the f o u ~ hrank is normalizted to c0123

- +l.

(3-8.15)

Using this coneept, we express the differentid equations (9-8, f 3) as d y *F@'(%)= 0.

(3-8.16)

The pair of equations, (S8.10)and (3-8-16), me MwxweII's equations for the Lensor of ejectromagnetic field strengths, F,,. We recall the identifications of

3-43

The stearamagnstio field,

Magnetic-charge

229

The alkmative evaluations of W ( J )thftL l 4 to the action expression. are

The eleetrom8gnetie aetlon, fareshadow& in (3-5.94), is wi%hthe Lwaxrge function

The Lagrange function is explicitly gauge invariant, and so is the action beclzuse of the &iflerentialconsemation property of J@. This time we have bemn with a,

c o n ~ w a t i o nlaw and inferred an invariance OE the action. In =king expressions for the murce and field vadsltians that are &soci&bd with a r b i b eoordinak ~ displacements, id is natural to mabintain fhe conserve %ionprope&y of J P and the gauge inrrafittn~eof FP,. The vecbr souree tramformation law (3-7.33,47) has the required characteristics since iassures the continued vanishing of d,Jfl; the amoeiated knsor transformation (3-7.M),

&FP.=

+

6 ~ ~ a h F , . Fk.a,

+ F,,9,

6zk,

(3-8.23)

involve8 only gauge invariant quantities, And the vector field prescription which w&sbased initially on the tramformation prapedies of the psdient of s malar function, is maintained under a gauge traasform8tion. The direet evlzlluatisn of the imducd &@&ion variation is

while %heLagmnge function regponse is

23Q

Chap. 3

Fields

with and The comparison of the two evaluations j v e s

Same explicit stress tensor expremions are

+

to' = &(E2 H'),

tok

(E X HIr.

(3-8.30)

A direct approach to the eonfofmal conservation. laws will naw be m d e . Multiplication of (3-8.29) by z, gives (I = 0) d,cp = J,FP"z,,

and similarly J , F ~ ~ ( ~ Z "-~ g"z')

= fd,tgk)(2z%' E

a,eE"V,

(3-8.3 f )

-- g"z2) (3-8.32)

The nstum of the eamfiponding integrd mnsewrttian b w s mogt closely rewmbles t h ~ oft

which, being an explicit function of time, is a statement about how something moves; it is the eentroid of the e ~ e r w distribution in this instance, The strongest asmrtiorr of this type that is implied by conformal invtadancl: is ~ontainedin

Thus, witb the weighting factor provided by the enerw density, the werage value of x2 varies quadratically in time, with unit coefficient of (~'1%.The inkrpretation in Germs of the motioa of the particles &at carry the energy ig clear: photons move a t the speed of light. The coefficient of z0 and the constant term supply information abouC the initial. correlation between position and velocity and tbe initial average value of X'. This view of i(dx)eO is consistent with its ~ignificaneein terms of the momentum distribution:

3-8

The electromagnetic field.

Magnetic charge

23t

The field strengths F,, and the vector potential A, are placed on the same footing in the following action principle:

where = -%Fpv(apAp- aVA,)

+ $p%,,

(3-8.38)

is explicitly gauge invariant. The field equations now read

avFNv = P,

dpAp

or a V ~ p= v

where *JP =

p ,

-a,,

and *MMvis the tensor dual to N"'. and

- a,A,

= F,,

+ M,,,

a, * p v = *JP,

*MPv,

8,

*JP E

0,

(3-8.39) (3-8.40) (3-8.41)

The stress tensor is symmetrical,

avrv= (JP+ a A ~ , , ) ~ p+vJ,M,~

+ *J, *M".

(3-8.43)

On setting Mpv = O and identifying the field strength tensor with the curl of the vector potential, the previous results are recovered. If the photon source function Jp(z) has the interpretation of an electric current, according to the first set of the Maxwetl equations (3-8.40), is *JP($), as realized in (3-8.41), a magnetic current? The answer is negative. It is consistent with this, but hardly decisive, that the total vdue of the apparent magnetic charge is zero,

provided MN"as the kind of spatial localizability that attaches to the source concept. The essential remark is that, through a redefinition of the field strength, the magnetic current is transformed into an equivalent electric current. Indeed, the equations (3-8.40, 41) are also given by

which contains the effective electric current already exhibited in (3-8.43). But this short-lived possibility does raise a fundamental question concerning the existence of real magnetic charge, distributed and flowing in a manner that, explicitly or in context, differs from (3-8.41). To study this question, we go back to the beginning, to the source. Is it possible to distinguish two fundamentally different kinds of photon sources? But the two kinds must also be closeIy related, for the structure of the Maxwell

232

Fialds

Chap. 3

eqwtions is rehined under the substitution

or, more generally with arbitrav angle p,

JP *JP

+ *JC^sin p, ---JPsin p + *JP p,

-+ J P cos p

COS

+ *FPr@inp, *F@' -+ -Pu sin p + *F@*cos p. F*'

4

FP'

COS p

(3-8,471

The consib3tency of the field strength sub&tutions involve8 the r e p t i t i o ~ p r o p e ~ yof the dual, **F~Y = -?-JwY, (S8.48) All this brinm to mind the discussion of Smtion 2-3, It vvm recopiad there thaG n~Lhiftgint~nsieis albred if aX1 photon. p o i a ~ ~ a t i ovwbm n are rotaM though the angle */g, thus repising the initial polarisation veotor set e,h by ( p @= Ipl) This suaests that the de~ireddisLinetion and relation b"Ewn two frintds of soumes is realized if the memum of their eaecfiveness in emitting a given phuh~t, 1abeIed pk, udililieg e , ~for one kind of sourn and *e,x for the other. Thzrt is indicated by (red pala~zationvectors are u ~ e dto reduce the number of @tars in our ctyc;s) $,X = (d@,)"2[@p~ $(p) -4-*@,h *J(p)]. (S8.W) Ths equivarfcsnce of the desefiptions that are eonnecM by the @@wee tram formation of (3-8.47) then expreBses the freadant to rotate bath s y s b m of pols~sationvectors -tXlroughthe common angle p. Again we consider a c a u ~ dsiluakion, brtf now with compnmt sour- JC;, V1; and J t , V;. The coupling betmen the exnission and &~orptionsowem that a single photon, mediwks is conveyed by

The intrinsic equivdence of the two mts of polzlrizadiun v e e b r ~is contained in the dyadic relation ancl the eommponding brmrs of (3-8.51) can. h &ven h v a h n t fom in the kno-run way. But what of the couplillg btlCween different kin& of sourem? We fin& note %hatthe summation in

3-43

Tha ~Iectromsgneticfield,

Magnetic charge

233

can be extended to include the third unit vector prardlel to p, thereby introducing the unit dyadie:

The coupling illustrated by JI(-p) X 'Jz(p) pipQ is very three-dimensional in. appearance, Nevedheless, in the physicd circumtarrees under examination, tbis is a Lorents scalar. That can be dirtj~tlyverified, It is mom rewarding, however, to w r i k this term in an explicitly covariant form. We begin by remarking that (causal subsc~ptsare o m i t w )

where f,(p) ha8 onXy a time component such that ~ P % ( P ) = 1. (3-8.56) The decisive observation is that (3-8.55) remains true in its fourdimensional form with any vector f,(p) that obeys under the eausal conditions that mquire the photon enerw-momentum relation Also relev&nt, but holding without regard to eausal arrangemen&, are the current conservation statements The fottowing identity, which is valid for arbiLr%rypp, is b w d on these conservation laws :

Of the two k r m s on the righbhand side, the second vanishes when gP is a photon mamentum ohying (3-8.582, and the first is indepnden-d of the gpcific choiee af vecbr &(p) that obeys the restriction (3-8.57). This is in, fact the proof of covarianee for cmsaf eireurnstanees, But same explanation is caitled for, A elarss of functions Ghat obey (3-8.57) is given by where n, is an arbitrary constant vector, ff n, points along the time axis, fa2 example, we h w e the situation of Eq, (3-8.56). That eharae.t*rtrizatioaof f,(p) is not eavariant; after a Lorentz, transformation is perfomed, n, will have nonvanishing ~patialcompsnent~,although it is still a, timelike vector. 1%ias here that the arbitrariness of n, enters, for we can replace the time-like vector by ons with only a kennporab component. It is &rough sueh coupling of the ehoice of n,

234

Chap. 3

Fields

to $be choice of coordinate sysbm that any discrimination among coordinate systems is avoided and covarimce achieved. The spaee-time transcription of (3-8.57) is

If, ss in (%8.fil),p(3s - z') is prapa&ianal to a constant vector n", the digerential equation is eRectiwly onc? in no $ 0,we get

et

single vrtriable, For the gituation of (3-8-56), with only

aojO(z- g') = a(xa

- X@')

&(X

- X)).

(~8.65)

The soluGion is not unique. Two alternative solutions that correspond to retarded and advane& bounday conditions are where

represents the Heaviside step function (the capital of the Greek letter is H, as the eapitrtl of i5 i s D, in the Chalcidian alphabet). Another choice aligns the vector n" tvith the third spatid axis, for example, Then and dternativt: soIutions are fa(2

- 2')

d ( ~ ' - z")

6(g1

- 2:)

6(za - zj)[?(za - X & ) ,

-?(%g

--

za)l

(3-8.69)

We specifieagly note the equally weighted linear combination in, which

More gener~lly,it is compalible with the diRerential e q ~ n * b fS8.62) n to imp%hesymmetry restfiction -F(z1 - 2) = fV(z- d ) . (S8.72) The four-dimensional replacement in (3-8.55) will be used Co perform the spiace-time ex%rapola%io.rs of dhc: murm oouplings inferrd under o

Tha ~Is~tromagnstic field.

3-8

Magnetic charge

235

stances, If one of the f" functions in (3-8.66) were adopted, an, additional causal elernenL, which i~ arbitrary and physically irrelevan%,would be injected into the description, ]in contrast, the kind of funetion illustrated in. (3-8.69) is bmporatly inert;, and its arbitrary aspecls are confined to spatial directions. Since causa1i"ty is a fundamental widing principle, we reject the use of funetions such m those in. (3-8.66). Without being commitkd to the specific examples of (3-8.69, '?Q), we do insist that f"(x - X') have a, spsce-like direction and be loealiaed in its time-like coordinate excursions. The desired space-time exLrapoltzLrion is glxven by

Xn verifying that this properly represents the initial carnal situation, we encounter the Fourier transforms and

whieh are involved in reproducing the last two terms of (3-8.51)' The latkr are interchanged by the substitution: JP

(P)

+

*J,(p),

*JP(p)-+ -$#(p),

(3-8.77)

To test W for this symmetry property it is convenient to introduce four-dimensional momentum xlotatiart :

The effect, on the last term, of the substitution (3-8.771, combined with + --p, and p ++ X, is

p,

and invariance requires that -~P(-P)

==

;F;(yz).

I n a eauaal s3ituatian, however, fp(p)oeeur8 anly in Lhe earnbination i p y p ( p )= 1, and this additions1 synrmetv praprty is nok Bud if the wlatiae htween the two source f e y p i s tit3 be maintain4 generally, &hecondikion, (3--8.80), which is dso (3-8,72), must be i m p d . 1%foUow8 fhat W(J V) pre=mm it6 form under the general murce tran~formationof (3-8.47). Two kinds of h s t sourcm define two Ends of fields:

.

which are, inThere are alm two independent kinds d gauge arbitrarin~m~ corporsM in the fie!d exprewiona

(dz') (dze')D+(z- z')f '(z'

- z") '( 3 : ~ .(g") -- ak'~,(z''1)

+ 4, *h($), where @@""as been used to form dud tensors. The following identity should be aobd [it i ~ the 3 eonGenf of Eq. (3*-8.13)J

Obmme alsa thellt, for mampfe,

3-43

The slactromagnetic fisfd.

Magnetic ehsrga

where the Iwt form involves the inhrehange of the indiees h: and gauge inv&nisrttfields

V,

237

The fwa

then obey dvF""(x)

JP($),

3, *Fp"(%)

*J'(x),

(3-8.89)

But only if *Fp"(;e)is the dual of F""(%) can we, procbim these to be the general f o m of Maxwellk eequa;ttions, with electric and magnetic eurrent~, A direct proeedurr; for &is purpose is to evaluate the curls of the two vechr patentials A,($), *R,(s) and compare the results in (3-8,88), Here is another identity that is valid for any antisymetrieal hnsor Q@,,

+

a, *G,& a, *G&,+ ax *Q@,= - % Y x x a , ~ a .

(3-8.90)

1xr consequence,

with a similar expression. involving *A,(s), and the use of the differentbl equations gives

The necessary dual relationship is exbibikd here. Notice that the gauge invariant field stren&ha are also independent of the arbitrary veetor $,' It is e~identthat thme kasors obey Maxwell's equations, The converBe is also %me; the solution of the MaxwelX equations with outgoing wave boundav eonditiom is just (8-8.93). To verify this the identity (3-8.90) is applied, hthe form

238

Chap. 3

Fields

which. produtees

-a2Fpv= a , ~ , --

3

,

-~*(a,~ *J, -- a. *JJ.

(3-8.96)

The dwired ~olutioaiis that stakd, with its dual, in Eq. (8-8.93). Apast from the characteristic freedom of ga;uge tr~nsforrnalions,the fm vector potentiebb ean be exhibikd in kms of the field slren@hs. Fir&, let ua sbmrve that

(3-8.97)

h= the Foarier transform

The conmquenf v~nishingof a,(%) is exploited to derive from Eq. (3-8.m) tba&

which also uses the diWFerential equation obeyed by f"(z - zf). Thus,

and a gauge tr~nsformationan A,(%) change8 X(%) appropsatety to mainldn. these relations. We da not mean Lo suggest, by the wsy, &at the? X(%) of Eqe~, (3-3.83) and (3-8.1W) am the same function. A common, symbol is used since bot;h funefians embody fhe eharacferislic (arbitrar;lness af the veceor potential* By introducing the compensating gauge transformation, the X(%) of Eq. (3-8.1QO) can always be reduced to zero. The rmult is et, p%rticuIsrgauge in which

and we note: %ha% the gauge conditions (3-8.102,104) are not independent statemenb, but are implied by the constructions of Eqs, (3-8.101) and (3-8.103).

3-9

CHARGE C1UANTIZIATlQN. MASS RIORMAILIZATION

Preparatory to exhibiting various action expressions, we note some integral identifie~that ineorporah the field equations. Thus, from the Maxwell equac dion8 fS8.89) we infer %ha&

while the equation8 of (3-8.88) lead to

Xn 8rriving at the last expressions the following property of the dual is used: which ~ltlsoimplies that

' P *FIIYm --F"E""F,,. The qua&atic W(J *J) expression eiln be presented as

or in various equivalent foms, ineluding

The latdtjlr passea the action propedy. But one must etppreciak the context, describing the independent field variables. In (3-9.Q, for emmpte, the fieMs A,, F,, are subject to independen%variation, while the a;ymbol *A, stands far the functionat of the field s6renGt.l. Gensor stakd in (3-8.103),

*

(dz')fp(z - z') *F~.(Z').

This is verified by performing %heindicated operations, whieh e;ive

(3-9.8)

2

Chap, 3

Fields

Proeeding from the dual La Che last equa;tian, the =and equationis, a, *F,,(z) = *P(z~,

mf of M%xwellt~ (3-s.xo>

is generahd by diRerentiation. The con~tructionof A,(z) M l m s M in Eqs. (3-8.99,100). The use of (3-9.7) is anslogous, with *A, snd *FpF,, as independent fields while A, i s defined as a functional of the *F,, by (3-8+101),

The ~pgymme%ry in-vot~edlin e q l o y i q either A, or *A, as i~dependentfields is overcome with yet a fhird aetion expresgi~n:

which is explicitly invariant under the rotational transformstion of (5-8.47), and usea, $he field stren@hs EMindependent variables. The equation produced by the fstalionav slction principle can be presenhd as wing momentum rJpaGe for coxnpaetn~?ss, where

Then, since K@($)is divergeneeless, PVK'(P)

we learn, successively, that

= 6,

~P(P)K*(P) =0 and

K"d

= 0,

which also uses the positiveness of -f@(p)f,(p) that expreases the choice of np

and %herebyof ip(p)in (3-8.61) m a pace-like vwbr. A ;sfimilartreatment af %hedual Lo Eq. (3-9. 13) suppliw

and both sets of Manuell's equations have been derived in s symmetrical way from the i4~;ftian expres~ion(3-9.12). By this time the bypodhetical alert reder of li~tiessddeatian, hencefadh aefanycnicdly known as Harold, can no longer reatrain h exchawe ensue^.

Chargs quantiration. Mass normalization

3-9

243

H.: You showed in the previous section that the apparent magnetic ~fZ&rg@ given in (3-8.41) could be transformed aws~y. It was intirnahd that a different kind of mapetic current would be forthcoming. Yet the action pdnciple of ($9.6) and the Geld equations (3-9.9) etre identieral in form to (3-8.57) and (3-8.391, with and indeed

-

How then can you claim that true magnetic charge is rrow k i n g disc S.: Mistake me no$, goad Sagredo, er, Haratd. The fundion, (3-9.a) does differ-in context-from the source function of (3-8.41), far it lateks that depw of facalizability which is characteristic of sources. Consider, for example, the choice of $,(s - z') with only the spatial companenf

giving the nonvanishing tensor component

Unlike the spatially limited magnetie charge distribution *J0(z),*Mo3(z)becomes independent of after one ha8 passed through the charge dist~bution, moving po;;jitivety along the third axis, This limiting value is

and the sudace integral

whieh need not be zero, That is in contrast with *he null value of (s8.441, which ~vasbmed on the sptstial loedizttbility of "dd,,fz), Had we umd the add fC^ function of (3-8."i"), the explicit form of *Kfo3(z)would be digerenl, bud xlat the value of the surbce i h g r a 1 that produces the totd magnetic charge. Ths, it is through the special properties of the class off functions that we make the transition from mere semblance to the redity of magnetic charge. At the setme time this poses a fundamental problem since the detailed description would seem to depend upon the arbitrary choice of the f function, for whieh there is no physical basis. Surmounting that formidable difficulty is the task to which we now rtddress ourselves, Let us introduce into (3-8-81), which is the diEerential statement of the dependelice of W on the source functions, those expressions for bJp(z)and 6 *JP(%)

242

Chap, 3

Fields

that convey the efft;clof an arbitrary coordinate-dependent displacement:

up(%) = ~,(~z~(z)J@(x)) - Jv(~)a,azP(z) =

-a,[

~x~(z)J~ ( szV(z) z ) JP(Z)]

(g9, 26)

and 6 "J@(z)= -a,f&rc""(s)*JP(.)

- 6zV(.) *JP(%)].

(3-9.27)

The conservation requirement8 (3-8.82) are identically satisfied, This insertion gives

dlternative expressions for the two terms sre exhibited in writing

Two elementary statemnls concernjng explicit f p dependence emerge from these

forms. If electric and magnetk currents are proportional with a wziversd constant, the f p term vft~fi~hes, as it should since tbis ia a rotated vemion of pure

electric charge; when electric currents are causally separabd from magnetic: eurrents the fB term vanishes, weording to the restriction s n the etas8 of fC" functions that confines it to space-like vectors connecting points in space-like relistion. It is the situation of electric and magnetic charge coexisting with different space-time distributions that poses the problem of nonphysical P dependence. To make this very explicit, supposef""(z - z') is chosen a-s in (3--8.701, a spatid veetar of fixed direction with its sawort, its nonvani~hingdomain, confined to a fine of that; direction. Those points in the two source distributions that can be conmcted by this line canlfibub to 6W, When the direction of the line is varied continuously, bW and W itself also vary, continuously, thereby denying to W any physical meaning. Is Lhis the death h e l i of mslgnetic ehargt?? No. Them is a subt;le possibility concealed bere, I t depends upon the precise fact that not W but expfiWjis the physically significant quantity. If, in altering the direction of f p continuously, W were indeed to change, but change discontinuously-by multipIes of 2~-the exponential would remain unaltered and the mathennatied arbitrariness of p should be wri%houtphysical consequence. This is impossible, of course, when, as is assumed above, the sources are eantimuously distribrtkd objects, Instead, they must h w e a panuJar atmcture, gving values of the p

inbgral that differ by finite amounts according a&rs the f@ line does or does not penetrate the kernels of that structure. And, since the magnitude of the integral it3 ~ l s measured o by the product of electric and magnetic charge, this combination cannot be arbitrav but must be re~trictedto certain discrete vdues. Theae are? remarksfile conclusions-charge is eomplebly locdized and q u a n t i ~ din magnitude, The sweeping nature of such inferences should b-tt, emphssi~ed. We are encountering =strictions on the structure of photon sources that are required for the consi~~tency of a theory of electric and magnetic charges. Sourees are introduced as idealiaations of realistic physical mechanisms, idealizsllions thaL dispense with individual characteristics but respct all general laws. Xn uncovering fundamental restrictions on sources, we are revealing general laws af nature, Sueh was the argument when the divergenceless nature of the vector pftoton soufee, demanded by the null photon mass, was interpreted as the rtssertion of s general eonsewation law, that of electric charge, A realization of electric and magnetic currents in hrms of the motion of point charges is given by

(instead of "8, syrnbola such as g are aho used hut we wish to emphasize? the symmetry between electric and msgncsLic quantilies). The eau~almotion of the points is conveyed by the restrictions

The conservation. properties hold individually, according to the ealeufation

since the point % p ( & ) is infinitely remote from zp a t the teminalis of the integration, The evident identification of the e, and *e, as charges attached Lcl the individual moving points is consistent with the evaluation of the total charges, ars in do, dsP(s)6(1: -- ~ ( 8 ) )= e,, (3-9.33) where the inkgation sweeps the whole four-dimensional domain with dcr, dzp(s) acting abs volume element. We cttnnot sirnply insert theae expressions into W(J V ) ,however, The latter was devised for continuously distributed sources and should not be applied to s collection of point charges without reexamination of the physieab sipificance

244

Chap. 3

Fields

of W . But it is useful, and serves as an intermediate stage in the development, to modify the known results in a manner that is without effect for continuously distributed sources but makes the consideration of point charges mathemdieally meaningful. This is achieved by introducing an arbitrarily small space-like vector Ap and constructing the provisional action

+ * J ~ ( x*A.(% ) *U - 3FpV(z> (apAv(~* X ) - avAp(z * X ) ) + ~ F p v ( ~ ) F pfv X)], (~

W ( X )= / ( ~ x ) v ( x ) A , ( x f X )

(3-9.34) where the appearance of f X signifies the procedure of equal averaging for expressions containing + X p and -Xp. This action continues to be stationary for field variations about the solutions of the Mamell equations:

which uses the possibility of performing a displacement to transfer fX' from one field factor to the other. To evaluate W(A) we use

* + *Jp(x)*A,,(z * X)].

W(X) = & / ( ~ X ) [ J ~ ( X ) A . X( X )

(3-9.36)

The point charge construction of the currents gives where

is symmetrical in a and b, and

Wo(X)= #(c:

+ *e.')/dr ds'

D+ (%.(S)

- zo(d) * X).

-

(3-939)

The mathematical existence problem which the X device is designed to overcome is concentrated in Wo(X). I n the neighborhood of 8 - 8' 0, D+ would be singular without the addition of the space-like K to its argument. This

difficulty is re8tficM to the real part of B+, however, (cf. Eq. (2-l.M)], as contrast& with Thus -

cos px

h= nsturesl ulpper frequency limits if the motion of the particle: ia without di* (S9.42). To csntixluity, and the limit X' -+ 0 can, be introduced directly i ~ t Q discuss w,(X) = Re W@(&)

far suficiently smdl kC"and s comesponbingly eloiose to a', it S ~ G to~ consider S uniform motion. Let W use the wst frame, for simplicity, and identify ds wifb. dz: in thst frame of reference, while choosing AB to be a spsti%lvvector. Then,

Doe8 wa(X) have a pfiysical sipificance"2t does nod. This quantity is woGi~kdwith a single poinl; eh8rgf: or partide. 8inse the pa&icles that cornprim a source have prescribed motions they are being idealized as very m m i v ~ parlicles, which are uninflueneed by the effe~tsthey praduee. The desc~ption of their indi~duafmechanical propertie8 lsgically precedes the discussion of inler~ctions, The nature of this description can be infemed from the rwultp, concerning stress h n ~ o rand s their vdues in single-particle states: P"= 2 h , p p p ' . As we have? expldnd, &is is a simgXificatian valid in. the intt3rior of a barn whesc? fhe variation of momentum rand the associated firtih spatial exkensian can km neglwbd. To reinshh these, we identify pp with the p a d i e d of a @me function ~zndintroduce s variable weight funetion, tp"(2) = p(z)aYrp(~)d"p(s);

the m-

rest~ction, dpcparp $ m2 = O

recalls the momentum sipifieance of

aFa. N o b that

and the foeal rneehanicd consemation laws are satisfied by the consemation of padicle A l u , ar(~d)"v)Q. (3-9.48) This interpretation &o supplies the value sf the intepal : (3-9.49) Within this picture of prescribed motion it is cornistent to take

Indeed, the consewation law is satisfied,

6(2

-- S(&)) = 0,

(3-9.51)

and (3-9.52)

I n transferring these results to the connection bekvveerz action and stress Lensor,

one must not forget the meaning of 6s,(s). It arose m a generalization, of the ~ g i displtaeements d given ta soureefri, which were inbnded to simulak $he &hplacement of a referexlee aurface and are therefore in the opposite mnse. Thus, when tranglating h t o the motion of point particles a minus sign must be s&ed:

It is now necessary to generalize the identification of ds with dzo, in the rest frame, to the invariant proper time definitian -(ds)2 = dZY dzy.

Its consequence for a vafiedt molian, --ds 6ds = dzVdsz,,

eonvert;s (3-9.54) into

(S9.55)

3-9

Chrrgs quantixation,

Mass normatization

247

and supgies the action expression for a single particle, labeled a, performing a preserihd motion, (3-9.58) The phenomenological orientation of source theory has the folfotving eorollary, Physical parameters identified under restrickd physical circumstances do not change their meaning when a wider elsss of phenomena is eonsidered. The mass parameter m, is determined from the response of the particle to weak, slowly varying, prescribed forces as in beam deflection experiments. When eleclrornagxtetic interactions among several particles are considered, this parameter i s not assigned a different value. It has already been fixed, normalized, by experiment. Thug the single-particle term (3-9,44) mast not be added to (3-9.58), thereby changing the value of m,. There is no question here of assign? ing some fraction of the total mass to an electromagnetic origin, What is a t issue is the consistency bet\veen the various levels of dyrlamicat description through kvkich one passes in the course of the evolution of the theory. The prescribed forces of the most elementary level become assigned to the motion of particles a t the next stage, but in neither one is there any reference to individual particle structure and the phenoxnenoEagicaf parameter m, must be common to both. The eonclusiorl is that the real zt*,(X) krms, which contribute neither to the vacuum persistence probability nor to the couplings among sources, must h struck out. Here, then, is the action to be assaciakd rr-ith a point charge realization of photon sources: W = Lim [W(X) -h-90

w,(X)]. Q

Consider again the eRect of a source displacement, now pictured through the motion of point charges, We use (3-9,28), but tvitl-i attention to the X displacement and the minus sign required to translak cfizY(;lr) irfto 82:(5):

x f,(z.(s)

- I&(s~) f X)

dZbr'S"

---

ds"

-C 4W,(X) a

The antisymmetrical product of two vector displacernexlts defines a two-dimensional element of area, 4xf dz; - 6~:: dx: = dcz", (3-9.61) and the antisymmetrical product of three displacementa produces a thmedimensional volume element, or the equivalent directed sudace eXement for the coordinates zz - z,; d *C:' dzb, = d ~ t b . (3-9.62)

Chap, 3

The corresponding pre~entationof (3-9.W)

4 d@:'[e,F,,(z.

-C @a tab

- *e,es)

f X)

S 'e, *F,,(z.

de:afN(x.

- ~b

f X)] f X)

- C 6wa(X) a

1

&"+d

is no Ionger limikd to infinilesimal displaeemenfs; the integrhion~extend over the gt?ometrical domains defined by the initial and find p b i c l e trajectofies, Given the various three-dimmsiond 8z;zdwes thezt occur in (&9.63), all the individual fIr intepals can be made to vanish by 8pprop~stechoice of the $" support, which n d not be restricted to straip;ht lines, For any other election of j'p that Il;lves nonvanishing values to one or more of the inkaals, those valuw must be confin& to irtbgral multiples of 27r. Consider a pair of pafiieles a and b, f'ar which the three dimensional surface cr fhat i8 t r a e d out by X: - XI: is eEectivefy displacctd by &Xp. We desimatti! these surfmes by @(&h) and wfite the condition guarankeing physieal uniquenem

where n is an integer. In order to ensure thabonphysieal etennmb do not intervene during the limiting process ?P--+ 0, we demand that this hold for aImosG d l X@. The scale of p iis fixed by the diEerenLiaI equation (3-8-62), or the equhalent integral statement (3-9 "65) referring Lo 8ny surface that encloses the origin. The diserekne8s required by (3-9,64) implies thak the suppod offp on any such sadace i~ ca~lfin~d. ko a f i ~ k number of points. And, in virtue of the qnnnnetry properLy (3-8,72),

fhat number must be an even integer, 2v. We may visualize $his number of filaments drawn out from the o ~ $ nin a way $had assims to e a ~ hfilament its image in the origin, Let the contribution to the sudam integral (3-9.65) th& i s ~~aciaLt?d with an individual paint a,a! .- 1, , , . , 2 ~ be , designs64 r, so that

The basic 8ilua;dian far (3-9.M) is that @ ( X ) , for example, incfudes a single point a,while @(-X) contains no support; point of p, Then

and the addition of sueh, sxpresaiong repmsents any other possibility. Xn. par-

G h s r ~ quentization. s

3-9

Mass normalixat4on

249

tieulnr, the summation over all a = 1, . . . , 2v gives

or, making explicit that the paints of support oeeur in pairs with equa! values of r, and n,,

we get the charge quantization condition

Xote that the kveight faetom r, take the rational form

If all 2v points are equivalent, r, = 1/ ( 2 ~ 1 and , the integer nab is an inkgral multiple of v, The simplest possibility, v = 1, is illustrated in the $" funetion of (3-8.70). With the suceess in removing the arbitrary aspects of 6W through the re~o11;nition that only exp[iWf is significant, we can present (3-9.63) effectively as

4 dfl~'[eaFL".(~. f X) -k *e, *F,,(x.

f X))

-

&W.(&) a

This might seem to pose anotiier problem, hosvever, Although WE: retain the symbol $W, it is no longer the change of a quantity W and the question of uniqueness arises. Consider rz corltinuous deformation of the trajectories that finally retur~isthem to the initial eonfigun-ttion, thereby defining a surface exlelotjing s three-dimensiond voIume. As the covariant generafizatiorr of the three-dimensional reletioxl

(3-9.75)

and similarly

4 do" *F,,

=

--

(3-9.76)

The net change of W an completing this circuit is, therefore,

indicates the three-dimensional volume, associated with padicfe a, tvhere @,(&X) ~vhicahis subjected to tire alternative space-like displacemtfnts &XL". The integrals

250

Chap, 3

Fields

of (3-9.77) record the amounts of electric and magnetic charge within the various volumes, Here the basic situation occurs when particle b lies within the volume @@(X), for example, but is outside of a,(-X). The associated contribution to d W is *(e, *eb - *e,eb), a multiple of 27r according to (3-9.69). This aammation of the single-vafuedness of expfz'Wfwas inevitable; it was only of some interest to see how the charge quantization condition brought it about. The charge quantization demanded by magnetic charge provides a most satisfying explanation for one of the more striking empirical regularities in nature. Uespik the widest variation in ather proper-t,ies possessed by partieles, the magnitude of the unit of pure eleetric charge is universaf. It is measured by the fine structure constant a == e2/4a 1=: 1/137.036. (3-9.75)

If we Msume that the smallest magnetie charge magnitude, "eo, eorresponds t o the smallest; integer in (3-9-72), the latter becomes

and

This is very large indeed, being the equivalent of the electrie charge 2(f 37)e. However, one might think, if only for a moment, that this great asyntnniely could be apparent since there is the freedom to redefine; dl eleetrie and magnetic charges by the rotation of (3-8.47) :

+

eh = e, cos rp *G, sin p, 'eh = -e, sin p f *e. cos p. (3-9.81) Of course, there are invariants of this rotation in the two-dimensional charge space, including 6 * e t , e, *eh - *e,eb, (3-9-82)

+

which correspond geometrically to lengths and angles Between two-dimensional vectors. 14fso relevant is the inequality (e, *eb - *e,eb12

2 (ez + *ez)(ei

+

Now consider the following invariant slatemend. For all known p~;t"ticles, (ei f * e , 2 ) / 4 ~is small compared to unity. Comparisoll of the inequality (3-9.83) with the charge qusntization condition (3-9.71) then shows that the integers n a b must all be zero. The corresponding points with coordinates e,, *e, are confined to a single line, which thus acquires an sbsoluk significance. It is conventional to identify that line with the axis of pure e1eet;rie charge. The complete l-eduction of the line to equally spaced points demands the existence of a. second class of particles for which (e.2 *e:)/4r is large compsred to unity. Among such particles there is no necessity for an. absolute charge line although,

+

Charge qusnfization. Mass normalization

3-9

261

if the integers of the charge quantixation condition assume only moderate values, the charge points will clusbr near a, line, which is the conventional axis of pure magnetic eharge.. It is remarkable that we have been led to the existence of two types of charged particles that are characterieed internally by relatively weak and reIadively strong forces, for this corresponds to the empirical distinction between leptons snd hadrons, respectively. Certainly hadrons-mesons and baryonsare? not magnetically charged particles, nor do their interactions possess a strength as great as (3-9.80). Rather, we view them as magrretica1l.y neutral. eornposites of particles that carry both eleet~icand magnetic charges, with the observed strong interactions of hadrons emergir~gas residuals of the considerably stronger magnetic forees, lvhich thus far have successfully prevented the experimental recognitioxr of free magnetic charge, I t is essential far this explanation that a magnetically neutral composite appear as an ordinary electrical particle, If we have a group of particles wit11 charges e,, "c, such that

the comparison with a reference pttdicle of charges eo, *eo gives l

X

-(e, *eo -- *gago) = 2 4%- a

E n.~

(3-9.85)

a

This is the required charge relation, The automatic appearance of conventional electrical behavior for a magnetically neutral composite is significant because the individual electric charges on parficles that carry both. ty pes of chargwdual charged particles-can assume uncanventional values, We make the specific assumption that the smallest magnetic charge, "eo, is found on a dual charged particle with accompanying electric charge eo f O [the value of eo \vas irrelevant in (3-9.85)f. For any other set of dual charges, e;, "eh, refererice tn the unit of pure electric charge s h o w that *e(; is a multiple of *eo, *eh =: *%l (3-9.88) md the application of the charge quantization condition to the p8ir of d u d charged particles gives

T h i ~exhibits eo and e W independent units in 1% two-dimensional lattice that produces all possible electric ehargea. Since m nnefilsures magnetic eharge, in units of *eo,we &gabrmognize that a, magnetically neutrd eamposik is m t ~ c b d ta t: m a charge unit. It also follows thak electric charge digemnms, for a eommon value of magnetic charge, me confind to multiples of e. The discurnion of electrical pa&icles and of dual ehargd gadicles ntafurally ~ u ~ e the s bcon~ideratianof purely magnetic partier@. The unit of pum magnegic charge, *e, must be an inkgral multiple of the sma1lesL magnetic @barge, ;ELS in (s9.88). We write this s p i f i c relation in terms of an inhger N, The ~bnalopeof (3-9.79)1 connecting the unit, of pure eleetric charge with the ~mPtflmtm ~ p e t i echarge, is the following connection between the unit of pure mwnetic charge and Ithe smaliiegt elwtFic charge:

From our various tzssunrpdions, which are grounded in the symmetry. hlwws electric and magnetic charge, we have inferred $hat the charge units on 8 d u d char@ padicle are the same fracLion, 1/N, of the uniks of pure e l e c t ~ cand magnetic c h a ~ e . Among the possibilities, 2, 3, . . . , which value hw nature elected for the integer N ? But fir& we musk digess $0 discuss the relation b t s v a n the tati is ties of eompwitc3:petrticlm and their constituents. One approach uses the spin-statiskics wnneetiam. A compasitc?b m e d of an odd number of particles with inbger 4 spin (F.D. 8%&ti~tiesf h= a regultant spin anguIar momentum thatt is also inbger "f- &. This eamposite particle o b y s F.D. sta%istics,I f there are an even n u m b r of eonstituent particle8 with inbger 3 spin, the compsite p inbgral spin and is a B.E. particle, It is as though a F.D, (B.E.) particle @&fie@ a, nninus (plus) sip and these ;9iws are mdtiplied to give the statistics of s composite stm~ture. This is more than s mnemonic, far the $us and d n u s signs identify the dgebmie propedies of the individual m u m 8 &at are m m pfict-d to produce the eEeetive sourge of the composite sysbm. Now, as we have mentioned, there are two varieties of hdrong; mesons, which are B.E. parkiclm, and baryons, w h i ~ h&reXi".f). particles, If bLh dyps of hadrons are to km con~ t m c t e dm mametically neutral eornposites of dual ehargd pa&ieles, the lattctr eannot all be BB. particles, The simplest msumptioa is that they are all F.D. partides; ~n wen number of such constituentss p d w m a B.E. particle, an d d number builds a F.D. particle. Cm the dud charged particles exhibit only one 8trength of magnetic charg~? N o b %hatboth sign8 of the magaetic charge, linked to sign ehange~in e l m t ~ e

+

+

charge, will occur. This is the antiparticle concept, with both charge^ involved in order to maintain the structure of the two @etaof Maxwell's equaliom, which. ha-ve the field stren@h bnsor in common. If the only values of magnetic ebarge are fl/N) "e and -(l/N) "e, they must be combined to produce a neutral composite, and sueh p a h of F.D, d u d charged padicles are B.E. particles; bavons cannot be manufactured in this way. Eence there must be at lemt two digerent eharge magnitudes, According to the magnetic analowe of the elmtrie lattice cowtruelion (%9.90), the mametie charges on dual charged padicles with the same electrie charge must differ by a multiple of *e, the unit of pure magnetic charge. It would seem to be a rertsonable charachrieation of dual charged pa~iclesto describe them as carving charges that artit smaller in magnitude than the uniLs of pure charge. If that is granted, just two values of magnetic eharge are admitted, With a conventiond sign choice, they are --(X/N) *e and [CN - l )f N ] *e. The possibje values of electric cha~grjwe analogoug: -(l/N)e and [ ( N ---- 1)/Nje. Either electric charge can. be assigned to either efioiee of magnetic charge, giving four dual charge combinations, although them may be duplieatioxls of these assignmenb. In ezddiLion to neutralizing a magnetic charge by its negative, which builds a maon, we can now balance .the mapetic charge f ( N - IL)/RT] *.a against N - l units of the magnetic eharge -( l / N ) *e. This is a composite of N F,D, particle@, and N = 2,3, . . must be odd if the resuit is fo be a F.D. baryon. The simples-1; possibility, which we adopt, i s N -. 3, Thus, bsryons are v i w d ~ZSe o x n p ~ ~ i h of three entiLies that bear the magnetic charges, in. *e units, af 3, ---*, -9, We learn, incidexllally, from *e = 3 *ea, that

.

It remains undeeidd whether the two magnetic charges of

-*

refer to duplicates of the same particle, or to BiRerend particles with s common value of magnetic charge. To this we can only offer the observation that, withoul reference to antiparticles, the magnetic charge average over all distinct dual charged particles will not be zero in the first possibility, but does vanish in the second one wbercr eharge -g has twiee the multiplicity of charge 8. We accept the situation of greater synrmetq, and extend it to electric eharge as well. Thus, whether we speak of electric charge in units of e or msgneLic & a r e in unik of "e, there are three options with values g, -9, -4, I t is natural to regard these nine possibilities as differ& slates of a fundzbrnerztal dual eharged particle. To emphasize its basic dyadic eharwter in regard to charge, this pa&icIe is called the dyon, Although the hypothetical picture of magnetic charge ills the bmis of hadroaie behavior is still quite incomplete, we haye alredy far outrun. our ability to test it, particularly rsince et quantitat;ive phenommologiical analysis of the properties ol hadrons is not yet before us3. We must turn away from these heady s p e d a -

P54

Fields

Chap, 3

tiom and bgin the study of ordinaq elwtgeal pa&ides in dynsmied eontexte. Hawever, Harold finds ffimwIf compelle-d h eomment.

W.: You were quite pemuaeive wncerniw the imporlanm of svoi&ng spculakive wumgtioxls about the stmctwe of partiefw, and yet you have just e n k ~ i n ad very bold spe~ula$ioxlinded. Is %hisnot incomishnt?

S.: The final goal of rt phenomexlollogieaf theoq is fo mtabllish contact wigh an underlying fundamental,Lheov. My injuxlction was agaiwk the conhsion of phenomexlolo@cal theory with fundabmental bhwv. The organiaatio~and $hearet,icaf simplification of ex~fimentaldab should not involve impEici"c stmaturd assumptians. But, quite? independen%$ of Lhst develiopment, o m may de;vim speealative candidztks for evmtual contact with %hephenomenolagt ~ a$heory, l fifltim&te~uccessssboulld be spedttd through the la@cal aeparstion. of these two phwes. &'IQ

PRIMITIVE ELECTRQMAGMETIC INTERACTIONS AND SOURCE MODELS

The comemed nature of the photan eleed~esource JP(%) sets the patbrn for any realkation of ~ u e hsource8 by an deetfie current metar wociated with ~pecifictype of particle. The electric currenf-a that we h&vealredy comidered for v a ~ o u sspin choices fail ta meet this standard since they are conserved only oubido soume rM0n.s. Let us rope& that discussion for spinlem pahicfes, using %heslightly diRerent procedure thiat is b on (;fie ae%ionexpremion

The eortsideration. of infrinitessimd, variable p h a s tramfornations of the sources:

and of the compensating field tsansfarmst;ions

with

The eompafism of the two evaluations implies that Natjea .that we have written q eveqwbere, replacing the ehtsrge matrix q of &heear&ertreatment, in order to memure charge in %hephygieaf unit e,

3-1 0

Electromagnatie interactions-

s ~ u r models ~e

265

The observation that jp is not conserved in the inkrior of sourees means only that the physi~~kl d-~?~~ription begins with the creation of the charge-bea~xlg particle and ignores the pre-exisbnee of that, charge? if nod Lhrs particle, in the aouree. We must find a way to insert f he fact that charge is transmi-t;ted, nod created. I t will be seen that this requires the indroduetion of an, electromagnetic model of sources, which is simplified to the point of retaining only the charge con~rvationproperty, but still has arbitrary elements. One prace-dure e r e ~ t m a conserved electric current by smputt.ttingthe nonconserved part, in a wrty %ha$ retains the ari@nal current in the regions that are cmally separated from the emission and absorption acts, where the current is eonsemed. This is accomplished by the construction

where d&fF(rz:- X') = &(s - X')

defines a, no$ unfamiliar class of functions. When the support of f@(z- 2') is restricted to space-like intervals, the subtracted term in (3-10-8) vani~hesad any time for which the sources are esusaly inoperative, To keep uniformity of treatment between jCI(z) and JP(z),we shall relate the canwrved vector, now designated J:a,,, (X), to an arbitrary vectorial. function J p ( z )by

The vector potential Ap(z) must multiply the total current, in the action expression. That can be rearranged to give

in which A',(z) = A,(z) - a,

(dzk>fp(z - xj) A ,(x",

(3-10.12)

and where, for convenience, we have accepted the symmetry restriction.

which ha8 no apparent pbyslcaf sigrrifiesnee here. Note that the construction. of A:(%) from A,(s) is a gauge transformation, such that the new vector potential i a charaekriaed by (3-10.14) (dzt)f"(z - z f )A:(.') = 0. This is a unique characterization, for, if the general gauge transformation

26B

Fields

Chap. 3

is designed to make X,(%) satisfy (3-10.14), we get

which produces thc?construction of Eq. (3-10.12). When, two digereat kinds of pa&iclw lkre @ansideredundm physical conditions af noninter&ation,the vmuum gmplitudm are multiplied and the actiom atxtded, Thus, for noninteracti~gphotons and spinless pa&icle;s,

An interation. Ibetwczen photons and charged pet.rticles is introduced by r e placing J&,, with the total current. We call this inforaetion primitive beeaustz it is not the final gtatement of d l inhraedioms, but rather charackrizes a, first elementary stage, which implie8 and is supplemenbd by further, increasingly elaborate levels of descniption. Pmeisely in wb;af sonso it is the first of a s e ~ e of s dynarnlcal s k p s will be discussed later. The action expression th& e h m l e r k a this first stage is

where we have chosen to incarparate into the Lagsange funetion %heinteraction k r m jr (x)A:(z) = +'(z)i@q+(z)A,(z). (3-10.29) Although the Lagrange function here employs the vector pokntial of a specific gauge, it is a g ~ u g einvariant combination that remains unchanged under the unified gauge and phase transformation

This is a con~quemwof replacing a,+, with transfolemrttion behavior by %hega,uge eovdsnL combination

(a, - ieqA,(z))+(z)

--t

ege'"'" (a, - i e p ~(z))+(z). ,

(5-10.~)

The field, equations deduaed from the sLsLionay wtion principle by varying @ and rf, are, re~petiv?rfy,

where the gauge covariant wmbination stays intact since the sign rever~alOf the derivative on partial integatian is matehed by the antisymmetry of the charge matrix g. In performing the variation of A: we must not violate the gauge restriction on the veetar pokntial,

Thus, the correct conelusion from

where ?(S) is arbitrary as far ss the action principle is concerned, But tha divergence of this equation gives a,dP(x)

+-a,ji"(z)

(3-10.27)

= ?(X),

and .tve recognize the MaxweEl equation Ta connect the use of f@(z- z') in defining s specific gauge with the concept

of electromagneticsource modele, we perform the fdlowillg phase transformation on 4 and cap, without the accompanying gauge transformation: +P(z)

+ @-ie@Acz)

4

e-ieqA(zj

S

c35P f d ?

(3- 10‘B)

where and A,(rt;) is the veefor potential in an a r b i t r a ~gauge. 'This transformation does two things. I t replaces A: in b: by ~ : ( z )4- apA(z) = A,(%),

(3- 10.31)

which is the inverse of the gauge transformation (3-10.12), and the transfer81 of the uncornpensakd phase factor to the saurecs replaces them by ,n(x)=eie~b'z)~(2), K , A ( ~ ) = ~ ~ ~ ~ (~ z( +) ) K (3-10.32) With the introduction of the arbitrary veetor ptential A,(z), we return ta the uut; of Jta,,(z), The additional label will be omilted, however, for one can understand from the cantext %.hether JP($) is sn arbitrary vector, since the vector potential is limit4 to a particular gauge, sr is a conserved vector, since the vector potential admits gauge transformations. The new sction expression is

The gauge invariance of the Lagrange function is now matched by that of all

the souret, terms, since A, -+ A,

+-

A(.)

and

KA

+

eiegh(l)

4X ),

induces

+ A(z)

(z),

"4- &(X) K: (z) -t eiegh'"' K A, )

(3-10.34) (3-10.35)

While the charged partide field equations that are implied by the action (3-10.33) continue to be given by (3-10.23) with the sources K", K:, the eleetromagnetic field equation. presents ab different aspect. In contrast with the action of Eq, (3-10,XS), &A, is arbiLrae md the charged padiele sources are furrctionds of the vector pakntial. The implieation of the latter fiaet is indictzkd by ( d z ) + ( z ) ~ ~ (= z ) (dz)4 (z)iepxA( X ) &A(2)

(3-10.36)

Thus we ROW get $,PP(z) = JP(%) 4-jP(lz)

-

(dz8)fp(z- z') [4(zt)iep~" (X') f &(z')ieq~: (%')l. (3-10-37)

1%iss just the Maxwellt equa;tion of (3-10.28), since but this time we have m8de explicit a, contribulion to the electric cument that is associated directly with the ehargd pareiele source, Cowider Lhs fallowing fictitious source problem: A point charge e moves uniformly with four-vector velocity %l^, until at a given location., which we aidapt as $he o ~ g nit, g;oes out of exisknee. Whad is the description of the phobns emitbd or absorbed by this act"i'he curred vector is given by

it obeys the nonconfiewaition equation

where &his@f, function is time-like,

3-1 0

EIectromagnetic interactiona-

source models

259

and has the momentum representation [cf. Eq. (3-8,61)] if' ( p ) = -npi

dse-i'pn - nE"/pn,

pn 3C5 0.

(3-10.44)

Recall the description of the emission and absorption of an arbitrary number of particles, here photons, by a given source distribution, J B ( z ) . The factor in the vacuum amplitude that couples J P to the creation and detection sources, J ; and J:, respectively, is

[/

exp i (dx)(dx')J';( X ) D+($ - zt)J,(z)

+ i/ (dx)(dz')J p ( x )D+(x - x ' ) J ~(X')] ,

where A.,(x) combines the field associated with J $ and the initial photons with that having analogous reference to the final photons. Xn view of the causzal. arrangement of sources, wherever A,(x) is of interest in (3-10.45), it is a solution of the source-free Maxwell equations or, in momentum space,

If we insert the current of (3-10.42) into (3-10.45) it becomes

But, observe that

which shows the equivdence, for the purpose of evaluating (3-10.471, of the time-like jp function with the space-Iike

The latter is also an odd function of p without restriction, unlike (3-10.44) which mirrors the asymmetry of the coordinate function in (3-10.43). We recognize in (3-10.47) precisely the exponential factor that is associated with a single charged particle emission act, as in

/ (dz')4(x')lCA

(X') =

(dx')+(X') exp

[- iep/

( d z ) f p(X

- X') A, (X)] K ( z r ) ,

where x' serves as the reference point at which charge eq disappears in the source and emerges on the particle of interest. The members of the class of fC"functions given in (3-10.49) differ only in the choice of the tirne-like unit vector nC",which represents the motion of the

280

Fields

Chap. 3

charge in the source model. When fhe coordinate system identifies with the time axis, f (p) has only spatial components that are independent of p@, and where

There is one choice off@that avoids the reference to an external unit veetor by devising the latter from relevant physical. paramekm. It requires an exknsion. of the structure sf f @ fa include akebraic funetions of derivatives that act upon the source funetion K(2). We indicate this repfacement in (3-10.50) and describe its meaning by writing

(d2)fC1(z- S', P)A,(s) K ( P ) . (3-10.55) When K f P ) reprwentx the emission ar absorption of particle^, the timelike repIa@ apart from a scale faetor. Thia gives vector P@

where the Imt form is the analowe of (%10.44), one that is equiv81ent for the ea;tleulation of phofan processes* The discussion of spinless particlm is pa&ieufarly. simple. A rsysbnr without( intrinsic angulm momentum ewn anEy exhibit scalar properties in ita rmt frame. In the electromagnetic conkxd this permits manopole momeat-ch forbids multiple moments, More generftfly, a particle of spin s, in its mg&frame, @an possess multipole momenk ta the rnrtximunl order 28, That is, a spin 3 pa&iele can have arbitrary dipale moments; a, particle of u ~ spin t can have arbitrary dipole and quadrupale msmenk; and so forlh. A sufi~ientlygeneral eument expre~ionfor spin $ is

This way of writing the eoeBeient in the term thwt hras the form d , d ' antieipab~ the _identification of g as the wrclmsgnetie ratio, the? magnetic moment in tbe unit &e/2m relative do the spin angular momentum fEq. (1-2.4)). Th8t be-

comes clearer on. wing the i;dlen.tity (3-6.67), applicable in source-free regions, to remite (3-1 0.57) ara

The magnetic moment of a system i%

which here hcomes

ma&ng explicit the roles of orbital angulztr momentum, spin. a n p l a r momentum, and the g faetor. The dipole moments permitt& to a spin -& pafticle inelude an electric dipole moment. It would supplemen&the second term of (SXQ.53)by 9t gimilap exprwion of arbitrary coefficient that u ~ e the s d u d spin tensor

No such progerty has yet been detected, however. Since .the second te-m of the cument is identictitlly divergenceless, we still have [a, factor of e is i m h d campared to Eq. (3-6-48)] (3-X 0.62) a,jr(z) = ~ . ( z ) ~ ~(g). ieq~ The currend (f.E-10.57)is ineorporai;ted in the fot10~ngslcti~nexprwion, an8logous do (3- 10,Is),

g(+,A,) = -&Fp'F,,

- ++ro[rB(--$a, - epA,)

4-

m]$

(3-10.63)

4- +F" 2m (-&g - l)$+~%.&* The omitted electric dipole interaction term is sirniitar to the last one, with either of the andisynrmetrieal tensorrs replaced, by ita drxgl. The Lap8nge funcC.ion is invariant under the gauge transformation

and this propedy san be utilized, as in fhe spin O discug~ion,ta m m v e referenm to a specific gauge whirs introducing an eleckrom~elicmodel for the particle 8ource : (z) = eicpb(i)q(~). (3- 10.65)

2#2

Chap. 3

fields

The fi& equations impXied by the action (3-10.63) are

ther witb the Maxwell equation8 employing the appropriate ~onmmQ?d ewrenk. The appe8ranw of the gauge @ovarian$derivative aations (%$Q. 18) and (3-10.63) h completely general. f f hgween the twa ways in whieh d e c t r i ~cutmen& b v e k n introdurnd. The f i r ~ t one cowiders tche imfinih~imalresponm to a variable phme transformation. For s typieal p&icle field ~ ( z this ) ig

where the hetor of e in (3-X0.a) provT_des the ntppropriab elwtroma~etic memure far the cutmeat. This Enematied definition ia not unique. The d y n a ~ ad definition of electric current imitate8 the role of the phobrt wwee. In. padiculsr, the reBponse of the action Lo the field variation BA, d, 8X is .E.;

Thua, the identity of the two eoncepls is impo~edby imisfing that the aetion b invariant under the unifid gaugephae tr&nsfomationwith The replawment of desvatives on chargebeafing field8 by gauge e~vmiant derivatives &coontpli~hes.t;hia for the whole poup of gauge drawfomna%ions, Pvhich ia Ablian in struetm. And the possibility of adding independently gaup hvas~bntbrms, M in (3-10,63), conveys the a r b i t r q aapeab of %heIcinennatieaI t dt?fini$ian. It is generally believed &a;t them is something padieularly and a;a%uralabout the ctjtectromametie coupling produed by wing ody the gauge covariant aubgtitution, and there is gmfh in thb. But i-d magt not be forgotbn that ttlkrnative de~criptionse G ~ for t the same spin vdue, and by haowing a common procedure we rsr~veat diEeren%electronrametic prope&iw. Thus, the third-rank apinor dmariplion of @pin+,b m d on the L a g ~ a g efun* tion ($5.73) with gaum cova~antderi ,&vm the eument of Eq. (3-6.6X), &p& from the f m b r of e, and the orrdi% g value, m eora~nedin (W.68,69), is 8. If the very striking nem+quality, +g S 1, that is abwpvd for the electron and $he muon has atay single moral, it ia the apmiall relevance of &hesimple Dh&cspinor equer,Cionfor the description of them p&ielw.

To illugtrate the direct use of a gauge i n v a ~ a nLavarrge t function far introdueing primitive electromagnelie inkraetions, we shall discuss charged partielw of unit spin. Such a L a p a w e function, generdized from (3-5.28), is

which finally us= an abbreviation. for the gauge covari~atderivative, Notice that we have devised two independently gauge invariant 6errns. Tha a r b i t r a ~coefficient8 a and b will be related to mabgnefie moment and deetGe quadrupole moment. VVe shall not consider Che two additional couplings produeed by replacing FE^" with its dual. They would describe e l e c t ~ cdipole and magnetic quadrupole moments. The parti~Xefield equations derived from the action principle are

D,$, - By+, -- C,, - (blmZ)(~,kieqCA, -- F.hieqC\,)

= M,,, DvGpY m2+@- aFpiep.+, = JP, (3-10.73)

+

&ndthe electric cument vector, in source-free regions, is

Lf we are ixrtereskd in the intrinsic electramagnetie properties of the parkiele, and not; those induced by the electromagnetic field, it, S U E t~ G ~implify ~~ (3-10.n) with the aid of the uncoupled particle field equations: ths lrtst of which, i8 an innpodant but not independent statement, This giva

and the implied coupling with an elechnzagnelic potential in ccsmpbtely soume-

fm regions is eonvqed by &)f A,(ar+'ieq+v)

The identity

- ft.

---.

a -t- b)iF,p(sbPz"eq+')

+ (blm2)ah~,.(aE^dieg4v~1. (ap+%ieq$') = (dp#'iep$" )+ ap(+'ieq+')

(3-10.77)

(3-10.78)

haws also that the field ~Lrengthderivative in (3-10.77) should ba symmetriad in the indices X and v.

2

Chap, 3

Fields

For a slowly moving partich, of charge f e, the three field components dominate, and are eonvenienay combined in the vector 9. The spin matrix vector s is represented by the rotation

We use the spin matrices to present this speeiztlizatioxl. of (3-10.77) as

where the d y a d i ~"ErEis symmetrlrzed, and we have dao picked out the term^ that deseribe the propagating particle in a crausd arrangement. WiLh Lke coupling of the scalar potential A' to the charge fe serving as s reminder of the nomaligation, the linear coupling of the spin vector to the magnetic field identiifies the g value : g== 1 . - a + b , (3-10.81) while the quadratic spin term @;iveg the quadrupole momexrt Q, in the unit, (&e)/m2,as & = 2b. (S10.82) The idividual results obtained for g values when oaly the gauge covariant derivative is used (s = fr, g = 2, 8; s 1, g = I), are given unifQrmXgby the geneml nnulti~pinarLagrenge function (s5.78). The current %ha% the latker implies in source and field-free @paceis =I.

The vanishing of all auxiliary fields under such circumstances, as expregmd by

hplies the set of field equations

Hence, the rearrangement used for spin that compose (3-I0.83), giving

She@the padiek: spin vector is S

=

ean be applied to each of the n t e r m

*C@,,

Extended soureas.

Soft photons

265

the g value is immediately identified as and all other multipole moments are %em, Note that the actud spin value eaters only through the inequality s C: i n , and where the equality sign applies to totally synnmet~calspinors, Incidentally, a very similar unified treatment applies to all gpixtar-symmetricaE tensor fields used to describe integer spin values, As one can recognise from the examples of spin 8 and l,# Lagrange function^, Eqs. (3-5.55) and (S-5.581, the gauge cov%riand electromagnetic interaction implies a current vector Lhat, in source and field-free space, is

++

The sarne spin. 3 rearrangement, combined with. projection of o on the total spin and the observation that (r3, for example, is unity when SS === e5, give8 directly ys == 1. (3-10.91) 3-11

EXTENDED SOURCES,

SOFT PHOTONS

Complementary to the pt-inciple of space-time uniformity is a principle of uniformity for phenomena that differ only in the values of energy-momentum that are engaged. The source concept was inkoduced as an ideali~ationof collisi~ns in. which precisely the right balantte of enerw-momentum or, invrtriantly expressed, msss is transferred to create a specific particle, But the sarne laws of physics are operative when less mass, or more mass, is transferred. Long ago, in Section 2-43, we used an extrapolation to quasi-statie source distribulions, which are incapable of emitting particles, in order ta connect the properties of photons with the Coulomb-Amp&ri%nlaws of charge and current interactions. Perhaps in our recent preoccupation with the very familiar equations of Mmwell, we may have forgotkn the initial logical ba&s far that contact. And now, through our concern tt-ilh the electric currents that are assoeiztted with the rtpwation of charged particle sources, we are moving in the opposib direction. The physical situation is quik simple. The creation of a ebarged pvticXe generally involves the transfer af that charge from other particles having different states of motion. Accelerhed charges rsdiate. Hence, unless precise eontrol is exercised over the energy-momentum balance, the charged particle has t% nanzem probability of being accompanied by photons. If we were to take too narrow ail view of the source concept and decline to extend it to this mrxltipztrticle emission act;, we would divorce the dynamical, significance of ehacrge from its kinematiesl aspects.

26Q

Fields

Chap, 3

The emission or absorption of photons is not a foedized process. The photon that accompanies the creation of a charged particle cannd be zts~igned $0 the agency of that particle, nor to the charge8 in the Borne, but involve@ intedererrce between both effects, This is implicit in the dditive congtruction of the electric current from con-t;ributionsof the pfcrtiicles and the source, It i s im%ructiveto examine such phenomena in some detail, We bedn by evaluating the probability amplitude for the ernig~ionof one photon of momentum k p and potafiztztian X, accompanying the creatian of one spinless particle of rnomentunn p" and charge fe. The physieal conLext Ghat underlies Lhe use of the primitive interaction to compute this probability amplitude is that, between creation and detection, particle and photon propagate under conditions of nsninteraetion, Aeeordingly, it is useful to review the description of that situation when the two particles are produced by independent sources. This is contained in the vacuum amplitude (3-1 1-1) ( O + ~ O - ) . ' ~ = (O+\O-) J ( ~ + j ~ - ) K the k r m involvimg one emission and one absorption source of each kind (we place K" = O im. these considerations),

where we have used p as an aI.ternative to zl" for assistance in diskinwishing between the two kinds of particles. (And let us hope that no confusion. results from speafring of particle, in the singular, when we mean charged particle.) The application of the primitive inbraetion will retain the noninteraction GO& b x t but replace the independent sources Jg((), K 2 ( z ) by a joint soume, J$(6)K2(5) which we now exhibit. The restriction to the single action of a photon detection source can be intr* dueect. by considering (3-1 1.4) Qr (3-11.5)

for one csn identify $be probe source 6J" with JT. Since the field A,(4) is to be evslunkd for $,(E) = Q, it is given by

apart from an irrelevant gauge term. The process in which m are interested involves the @%us&coupling of three sources: J"; KK1, snd &. Tbe emission source K2 is u s 4 to inject into the system the mornmtunn P p that, is redized m two particles, P" = F-+-k@, (3- l l.7) where

Thus

This sowce is aperating in the extended sense, and we shall urn the designation 'extended source' to distinguish its mode of action from that of KIPwhich detects the partide by absorbing mass m. A souree uLili~edin that way, performing only its initial mission, is a 'simple source.' Now, the current of Eq. (3-11.6) is a quadratic functional af the particle source and therefore ~ontainsa porkion fiz(atthat is bilinear in K 1 and K 2 . ~ h igives i a factor on the righehand side of (3-1 1.5) thaL a'Iredy has; the required three sources, All ather te different processes than the one af intertlst, whkk is displayed m

The relevant c u m n t stmcture, obtained front Eqs. (3-10.37) and (3--IQ.@), with K@== 0,is

The omission of %nother f erm involving K l ( f "ieq&z (t") expresses the caustll ) related to iits source by tzrrangement, The field + 2 ( ~ i~ or, in momentum space,

+

The fact that P2 m 2 # O [Eq. (3-11.9)) means that the field 4z(z) has no propagation efiaraete~sdics,and is localised in the neighborhood of the source Kz(z), Thus the field cba(z) will have no overlrtp with a sufieiently remote detee-tdion. source Kl(x), which is the assumed causal situation. The term 'virtual particle' is used to extrapolate ordinary particle concepts to such sihations where the energy-momentum balance is not suitable to the creEtlion of a 'real' particle. With our new terminology we can characterize the content

268

Chap. 3

Fields

of (3-11.11, 12) by saying that the extended source may emit a virtual particle which quickly is transformed or decays into a real particle and a (real) photon, or it may emit both final particles in one act, although the photon originates a t a different point than the particle. The precise meaning of these phrases is conveyed, on comparing (3-11.1 1, 12) with (3-11.3), by

where the first derivative refers to the X' coordinates. An equivalent momentum version, which also introduces (3-11.14), is

In this form it is easy to verify the conservation property

left.

k * ~ z ~ ( W n ( P ) = 0, which is valid for p2

(3-11.17)

+ m2 = 0 and arbitrary k2:

An important simplification appears when one considers "soft" photons, those for which energy and momentum are negligibly small compared to the values associated with the particle. Then (3-11.16) can be written as

in which we have also introduced the form (3-10.44) for j,(k). The interpretation is clear. From the viewpoint of the soft photon, the charge eq has made an instantaneous transition from uniform motion with velocity n, to uniform motion with velocity p,/m. This is expressed by the photon emission source

which is the transform of the conserved electric current

Notice how the two contributions, one associated with the particle source, the other with the particle, are fitted together in an equivalent photon source. This is an illustration of the self-consistency that is demanded of the source concept. The source is introduced as an idealization of realistic dynamical processes.

3-? 1

Extsndd raurcssr.

Soft photons

2@

The dynsmieal theory that ia erected on this foundation must, under appropriate m~t~cfiong, validate iLs ~tartingp i n t , Thus we learn, not s u ~ ~ s i w l %ha$ y, the aimpie photon murce dewnption becomes wficable to w realistie syr~tenn when there i8 ne@igible re~.(tionassociated d t h the rsmimian or abmrption procem* We should a h recognize the phpical significance of the cavarianf f , funetion &ven in (3-10.56), which we now \$?ribas

where the Ifitkr version refers to soft photons. The eEwtive phof;on source vanishes; the ehotrge hw not changed vejoeity and doe8 not r d i a h , This i~ the most natural Csf 80ur~emdels, in which the ernittd particle dekrminm ing the velocity of the charge in, the murce and thereby supp the a~comptlny radiation. That mppre~ionis not limiM to mft photons, however, I f we imrf the unapproximated version of ifp(k, P) in (3-11.16), it becomes (kg = 0):

The prabability amplitude for the emission of the two particltts labX1ed kX, pq requires, beyond (3-11.23), the additional factors (dwk)'I2 and (dw,)li2, together with the explicit slection of charge h e ttnd the photon pofari%a&ionX- The? latter is produced by scalar multiplication with the polarisstion vector et:, and There is anothm point that ean be illustrakd by the eBective wurce (3-11 1.16). Equiv~lentta a pmicle source mde1 charaeterizr4 by fH(k) is the

mignmenf of elect~calpmpertim only ta the padiele, cornbind with the use of v m b r paknLiEEX~in a apeeifie gauge such that [Eq. (3-1Q.14)] The veebr pfentirtl thsf represents the emitM pfiobn i s proportional to the polarisation vector cif, and the gauge condition (3-11.25) demands that (3- 1X . 26) f r ( k > & ~= 0, which 8~pPfemnts(3-11.24). Thus, with the Ghoice af I,(&) that is display& in (3-1 0.49) we have n,dr = 0, (3-1 1.27)

and this becomes c$& = O in the appropriate coordinate frame. The significant obwmwtion i8 that, an mdtiplying (3-1 l . 16) by one of t h w plarizr&tionvmbrs,

270

Chap, 3

Fields

Xt is pomible to remove the limitation to single photon emission, at lemb wktln atkntion is confined to soft photona* Since there is still only one papticle deketion source, we change tttczties and use

in which 6K(z) -4 K l ( x ) and +(z) is related to the aowee Kz(s)by the field eqtltttiarrs

-(a, - iepA,(z))@@(z)+ m2+(g) = K;'(%),

(a,

- ieA,(z))@(zf

= +&(z).

(3-1I.W)

The elimination af &, gives the ~econd-orderdifferentid equation

Since both p;a&iele sources already appear in (&X1.29), the clws of proeeams wile wish -t;o aelect are exhibited by

where the notation emphasizes the dependence of .titre parti~fefield +z(z) upon the veetar pokntial tf?(l)that represents the emitted photons in mlation ts their deteetion souree JVfE Let us fir& recovw the known ~inglephoton, result; in this new way. For this we need the part of @i1(x) that is linear in the vector potential. The field equation (3- l 11.3 I) retains just that amaunt of informa"cion when; it is simplifie$ to ).

We get wh& is requird in (3-1 1.32) by multtiplying this field equation by Qil ( z )and integrating :

Tfie first term on the right represents thtl rdiationless enzission. of the particle, and the geeond one reproduces (3-11.1 l). The nth hrna of the power series expansion of @ $ ( X ) in A'(t) describes %photon emission processes. If we agree ta consider only soft photons, all such processes can be combined in. t l ~C O R R P ~ L G ~ farnub which, as we would now expect, is equivale-nt to w photon sotlrcjr? dmeriptian.

Extended saurces,

3-f1

Soft phatone

273

The dihrential equation (3-1 1.31) is formally solved by

(dzt)A$(z,X') exp

(dglfF(z"

--)A,([) Kz(z"

),

(3-1 X ,352

where the Green's function A$(%, z') obey8

[-- (a - i

e p (2)) ~

+ m2]A:(z, X')

= 6(2

-- z').

(3-11.36)

We introduce the following transformation : LZ$(Z? 2')

(3-1 1.37)

= exp

in which the integration path is a straight fine canneeding x and x@,,as paramet;~zedby

This tr%nsformationinduces a gauge transformation on A,, replacing it with

and giving the new Green" function equation The identity

produces the gauge invariant caastrucdion

This vector pokntial has two ather ~ignificantpropertiers, fn regions; far from the eleetrornagnetie sour-ce J v( t),

and, general1y, (Z

- z')PA;(z)

= 0.

Hence, if we were to begin a construction of &$(x, z') ss a power series in A:, reprwenting photon fields far from their d e b e t i ~ nSource, the initial hrm

272

FIelzls

Chap. 3

would be obtained from

(-aa

+ m )A+ (z, 2

A"

X')

= 6(2 - zt)

+ 2epAL

(l/i)apb+(z

(3)

-- z') +

*

.

(3- 11-45)

But &+(z - z'), being an invariant funetion, depends only upon (z- z ' ) , and its gradient is a multiple of the veetor (z -- . ' ) z We learn that h$(%, z ') has no term linear in AL. is More can be mid; if the field strengths %retreahd as harnogenmus, appmpriab to goft photons, of negligible momenta. Then

which implies the tran~laLiona1invariance of the Green's function, A$(z, z') = h$(%

- zf),

and the digerential equation (3-1 1.36) becomes

The anll;ujtar momentum structure of the linear field streneh, trsrm assures it;s commutstivity with a2; it also eommutea with the qusdnttic combination of coordinate8: [tF@r(g,a,- %,a,), ~ z ~ F :=~ Pz ~F] ~ ~ S ~ = (3-1 1.50) since = F,.F""~, (3-1 13 1 ) is an. antisymmetrical function of p and v. AI1 this, and the rotationd invariance of 6(2), shows that the differential equation. (3-11.48) can be sinnplifid Lo

We shall not stop now to solve the above equation, I t suffices to know that ~ T (--z X') is an even function of field strengths, for this means that the field depndence of bhe latter funetion can be neglected relative to its partner in (3-1 1-37), since, earnpared La veetor potentids, field strengths contain an additions1 photon momentum factor. Introducing the8e soft photon simplifications, wrearrive ttt

The straight lim inkgral that occurs here bgins a t %bandmoves, in 8 dimtion ned by the vector (z - g':")@%ward , an effeeCiv~!lyinfinikly dista;nd point,

Extandad sourosar. Soft photons

3-1 f

273

~incethe photon emieion processes are localized near the extended s o m e K%, And, if the eoupling betwwn the padicle sources iis to be appreciable, the g* me$~caldisplacement (;e - x')" muat coincide cIowly in direction with that of &hemomentum vee-t;or of the exchanged parkiele. Accordi~gly,

&Eer~only in~anwquenlidlyf ram (3-l X .21), through %heexplicit appearance of z k the transition p i n t ;it is umd implicitly as the origin in (3-1 L.21), since the variation of 2' over K z is not signifie8nt in the soft pfroton conkxt, This is %he antieipakd source dweription of multi-soft; phobn emimion proc Notice that the eRecl;ive photan source c h a r ~ k r i z e fshe probability amplitudes for dditiond phobn emission, relative ta that of the rdia%ionlemprocess, which is s u p l i d by the ~ignificzllnceof K 2when if acts a simple p a ~ i e l eemie sion s0urc-e. The time has wnne Lo face up to a eharaeterisfcic feature of soft photons. With ra continual diminution of the e n e r a migned to a soft photon in tt given experimental amangement, one e~entuallyreaches a, point where it is no longer pmible to decide whether the phohn has or hm not been emitM. X tl is a somewhat complementary spsm-time obsemalion that, with increasing wavelength, one eventually loms the pomibilily of isohling the soft photon emission procm sineft the diswsition of sumoundixlg matbr hecomes relevant, Thus, more than %heu ~ u aarnount l of det&ilconcerning the aperinnendal arrangement is mquird. This is emphasized by using $he photon sour= (3-11.55) eompub the Bverage number of phohms enniCM along with a given parl,icle. That number is

To we the emenee of the situation, id suaces to eonrsider ra slowly moving particle,

m

+ fmv2,

p = m*.,

/v

(3-1Z'$7)

and a coordinste system in which nphw only a time oomponent, no = 1. Then the eompoxlents of the vecbr combination in (3-11.56) are, approximately,

6x1writing

where dQ is the solid an&e within, which the phohn moves, we get

This photon eaerw intepal doe8 not exist matbematica;lly, divergng both at the upper and lower limits. But clearly there are physieal r e s t ~ e t i o nrtf~ bath en&, When one reaches energies sL which the photon eeaws t;o be soft, the evaluation ( S l 1 . W ) no longer applies, and a lower limit is 'by the minimum delectable photon enerw of the experimentd amangement. Onee upon a time, the m~themsticaldivergence at zero energy was taken literally, and this soft photon phenomenon boame known as the 'infrared catastrophe.' As s. eatastrophe, it nzhs rathw low on the scde. Consider %he&Berence that is impfied in the vdue of ( N ) , depending upon whether the softmt photon considered has s wavelength of visible light, --10-' cm, or has a wavelength comparable to the nominal radius of the universe, cm. Since v2 < 1, that difference is

If the radius of the universe is reglmed by a typical laboratory. dimension, this difference drops to -10'"". To ilfwtrak the discussion of spin rraluecs other than gero, we shdf consider spin 9, using the Lapange funetion of (3-10.63). The current of (3-11.1 1) is repIaced by

+e

j?,(t) = @1(t)r07'&2(0 Z;;t (+g - ~)a*l$l(~l~~@"*&2fE)1

where [?(l/.;)la

+ dtbz(z>= a%(%)

is ~olved,in momentum space, by

The comparison with the exchange of one particle and one photon under noninteraction conditions,

supplies. the effective two-padiele source that represents the emission. of the

3-1 1

Extended sourc~s. Soft photons

276

extended particle source :

The momentum space equivalent is

5

)v

p

1

eff.

+ Ge (h- l)gp~k.q]4, ( P ) - f '(k)ieqv,(P)

= [rpeq

(3-1 1.67)

Using the latter form, we observe that

-rP

m eff.

and, on writing rk = r P

- l ] eqn2( P )

+ m - ( ~ +pm),

we get P

eff.

(3-1 1.68)

=-(YP+~)

m

-YP

eqs2( P )

(3-1 1.70)

But this is to be used in the context of Eq. (3-11.65) where the field t,bl(x) represents particles far from their detection source, and the Dirac differential operator in (3-11.71) produces the required null result. Alternatively, we can use the momentum form (3-11.70) and recall that (X)

where

=

E irl:,.,(2m

dup)" 2 ~ - ' p z U *P ~ ~ P

(3-1 1.72)

P"!? U~.,YO(Y~

+ m ) = 0.

(3-1 1.73)

Let us also note the photon analogue of (3-11.72),

A: ( E ) =

U

(dwk)"

&X, *

(3- l 1.74)

kX

with since both factors are useful in producing a simplification of (3-11.67). A relevant algebraic property is r"(m - r P ) = ?'(m - r p ) = 2p~' flvik,

+

- YpYk

+ [ ( r p+ m)rP+ P],

(3-11.76)

where both terms in the square bracket can be omitted for our purposes. Simi-

6

Fields

XarXy, we note &hat

-o@'ik;,rk = (rprk since"k

+ k@)rk

[kfi~kj,

s = ~

(3-1 1.77)

QO,and

where noneonlfibuting k m s have been isolated in brackets. The result is

where one can alm use the substitution

1%is evideat %h&,in the Emit of sof* photsns, there is an ef-Teedivephobn source which is identicsl with the one encounkred for zero spin. This i s ta, be expc?cLr?d, for every spin value. The suecwsive multipole moment efXecb involve ixlerettsing powers of the photon momentum, and all become negligible compsred to the charge aeeeleration rsdiabtion for suficientfy soft, photons, But the particular choice of f@(k;) fhst removes the acceleration radiation no longer suppresses photon emission completely, since the spin-dependent effects of magnetie dipole moment8 remain in (3-1 1.79), and no ~pmializationof g can, annul bofh term^. We have illustrated the e x k n d d source concept in the eonbxt of emission. 1%can all be repeated w i m the exkaded; source acb h absorb ab padicle and a, photon. But the= inverse procet3ses are also d a t e d by the TCP operation, concerning which nothing bm been said reeenfly. The eEw&of the Eutllide&n basd coordinste transformstion on sources and fields is given by

and, far spin

5, q(rr?)=rsv(z),

The field4epeadent gource

$(Zc)==Y&#(z)*

3-1 2

fntsrsctian skefston,

Scattering cross sactfons

277

has %hesitme trtansfomstion behwvior zlis ~ ( x if) which finally provides a physical basis for the synnmetq propedy that, thus far, hw been adopted for convenience. The puirely elcetrom%meticpart of the action retainis its form undw this tran~formwtion,

while the particle contribution, including the interacfion brm, reverses ~ i p :

But the compIc?& statement of the W P opration ineludes the revergal of sll factam. The anticommutativity of the sources and fields msociaM with the spin +,F", D, particle provides the addition& minus siw needed fo produce the anticipratd invariance of the action under the TCP transfammation. The TCP operation inverts the causal order, and inbrchanges emission and abgarption processes. On applying the transformation. to (3- l l .M), one quickly ve~fiesthat the whole ~truetureis mainlaind, and it is therefore only neceBsary to change the eeusal labels. The same rem~rkapplies to the momerrturn version (3-1 l .67), of course-, except that we folIo~vthe practice of r a v e ~ i n g the sips of all momenta when absorption proeews are being deseribd, which the transformation automatieaIIy supplies. What has been shown in the spin 4 framework is of general validity, 3-?2 tMTERACTLtOll5 SKELETON,

SCA-ERINQ

CROSS SECTIONS

A @ven primitive interaction implies a fet of coupled field equations. Here is the example of the photon and the charged spin plicity, with $g = 1:

3 particle, writkn* for sim-

In view of the nonlinearity of thia system, the construction of the fields in Cerm of the sources wil be given by doubly infinite poxyermfie5, That is atss the

nature of the action tvhrtn the fields are eliminated snd W is expremed as a f u n e tionsl of the sources, The successive terms af this series, W,,, with n particle and v phohn saurces, represent increasingly colnplieated physical praces~es whieh are thus mbo~t~ledged to occur, but will not be given &heirfinal dweription rztt t h i ~first level of dynamical evolution. That is the meaning of an inbraetion ~keteton. At later s&agesof the dynamiml development, proceses already present in skeletal form are provided with more complete descriptions, and mme additional processes are recognized. I n thiss ~mtion,we propose to carry. %hedigeu~sionof the simplest terms in the interaction skeleton to the p i n t of displaying their observational implications. There are t~\-oasymmetrical ways t;o eliminate the fields. In the fimt, one introduces the formal solution of the parti~lefield equation: ( d s f ) @ $ ( z ,z f ) q A(X'),

[r(--G-

(z) )

+ m]@$($,X') = 6(2 -- S'),

(3- 521.2)

which gives the partial action expression

The stationary requirement an vasi%tionsof P1, reemem the Mamvell equation of (3-12.11, Gcth #(S) given by (3-12.2)wa highly nonlinear equw;tiorr for the metar potential. One can still exercise the option of removing A, from the particle source by sdopting the special gauge of the A: potentisls. The latkr procedure is psrtieuldy mconnmended rf we hUow the econd course and eliminafe the vwtor p t e n t i d , replseing it with

where j&ne,(z)

jP(z) -

(dz"lf""(Z

- x")aj"(~~)

(3-12.5)

and the gBuge condition betermina X(z) as

Another way sf p r e n t i n g this pohntial is [JPis now an arbitrary vetor] ( d z ' ) ~ / + (z zt),,[J'(z')

+ jP(z')],

(S12.7)

where, writ&& in msmentum spsee h r convenience, D/+(k),, = (g,. =

(@@P

-- i k ~ s ( k ) ) @ ' ~ D +(g*. ( k ) - fk(k)ik.) - ikJ,(k) -- f,(k)ik. -- Skdh(klfk(k))D+(k)

(3-12.81

31-12

tntersction skeleton.

Scattering eross sections

279

i s the Green" function of the second-order Nfax%vellequation that sati~fiegthe

gauge condition fp(k)~:fk),.

= 0.

(3- 12.9)

The second pta&ial action exprwsion ean be written as

or in the equivalent form that uses the nonconserved currents and ~ $ ( z- z)',. The nonlinear field equation for J/ that is derivd from this tzetion is that of (3-12,1), ~ t A,h replwed by (3-12.4) or (3-12.7). Wbieh of them asymmetric hrms it is mos&convenient to consider depends upan the process of intermt. Suppose, for example, that no photons arts in evidence. Then one ean e t J P = 0 in (3-12.10) and marnine the nonlinear prope&ies of the pa&iele fiftld, If the causal situation is such that interaetion~ aeew far from the padicle emission, and dekction saurees, which is part of the amangemat of s s e ~ t b r i n geqefiment, the p Wmi in jtoa8.-causally tied to the sourcecan be i ~ o r e d .The inkrsetiot-t.tern of (3-12.10) contains few particle field8 and therctby ett least four sowee factors, When we consider prmesws that involve only four sources, as in particleparticle scattering, fhe stationary aspecCs of the =Lion principle permit us to identi.fy $ with the field

XR omiLLing further k r m ~of equation,

8

mare cornpleb solution of the egeetive field

which are a t Xeast cubic in the source, what is thereby lacking in W fim no let35 tf-trtn six powers of the source since firstorder effects of the field change are: annulled Lhrough the stationaw action prope&y, Thus we have ideIldifieib

where jr(,) = IC(~)~~?@~.~PJ.(Z)

and JI(1;) is the field given in (3-12.11). Analogous r e u l b hold for any other spin value. With spinless p&rticlr~?s, far example,

2

Chag 3

Fidhs

and (dz')A+(%

--

X')

K(%').

(3-12.16)

at involve only t~vop&&iclesources but any n u m b r of pftatoxl demfibd most cmvenienlfy by the action (S12.3). The s& fionary action principle wrxnits the identification of A, with the field of the photon. mure@6, the o ion of jf', which is a t Xewt qadratia in %hepadicle t h m four paticle murem, so-, chanGng thorn b r m in ~ W thort contain no Thw the whde m6 of 8keleLd inbrraction hrms is ¥ by

The refe~eneeto the vector pokntid in the p&icle sowee has been dropped, d t h the undersLanding that (3-12.17) vvill be applied ta pmes3ses in which ~ ( 2is ) umd as a ~imple pafiicle source, all partielephohn interaatiom occunlng far from any of the sowees, To eixhlbit the individud W%,,we must expand in power series the A@dependence of G$(%, S') and extract the term containing v vmtor pobntials. For this p u r p e it is useful ta r e h k the Grwn% function equatiorr of (S12.2) (--ria+ m)@$(z,z8)= 6(2 - g') e P ~ ~ ( z ) ~ $ ( z , z '(3-12.19) ), which is e o n v e ~ dCX7 an i n % v ~ equfttion I by the formal mlution

+

G:(.,

2')

=

C+(% - z')

$-

(dt)C+(z

- 8 e q r(6)~ ~ $ (S'). t,

(3-12.20)

The d m i r d p m r series expansion can now be constructed by suece~ives u b ~ ~ t i h t i ainn this equsfion. Such manipulations are facilihtd, however, by sdopting a matrix notation in h i e h the e~ordin&les s and x' join the &scmk spinor and charge indice8 as continuous row and column labels. Thw, we transcrih (3- 12.m) into (3-12,21) G$ = C , c + ~ ~ ~ A G $

+

and wrik %hefomal solutian of this m a t ~ xequation a% A compact statement of the expansion is, therefore,

to (3-12.17) and mite out the suceemive W2,. In doing this one recognises &hateach parficle source is mullipli~edby a prop~gationfmction

Interaction rikarlaton. Scattering cross seetions

3-9 2

G+ to form the field

281

+ of (3-12.11) :

(dz)(d~')#(~}r ' e g r ~(s)G+ (z

- z ' ) e q A~ (2' )J.(z') ,

(dz)(dz') ( d z " ) $ ( z } ~ ' e ~($1 ~G+(z ~i X @+(X"

(3-az.24)

- z')eq~A(2')

X " ) ~ Q Y A (Z")#(Z"}.

The spin O analogue of Eq, (3-12.17) is

where the? Green" function difjterenttidequation (3-1 1.36) is presenkd ss

The equivalent inbgral equation is of the following symbolic appearance,

which fiw the formal solution b$ = [l = A+

-- A + ( ~ ~ +( ~A pA) -- e Z A 2 ) ] - ' ~ +

+ ~ + ( e q ( p A+ A p ) -- e2A2)&+

+ h + ( e p ( ~+~Ap)

--

+

e 2 ~ Z ) ~ + ( e P ( pAAp )

--

+- .

e2A2)4+

g

*

(3-1 2.28)

The successive powers of A@are not p w ~ n k dwite4 so neatly as with spin The first two terms of the series W z , ara

3.

in which, it has been expedient to retain the symbol

which means th& the careful orctering of factors can be ignord if the vector potential has a vanishing four-dimensional divergence, as is the sjitualion Eor (Slt2.18). Bokntirtls having this property am said to be in. the Lorcsmtz gauge.

The imnndiak applications of the interwtion skelefon for which we have b e n preparing refer to scatbring processes. Let us fherefore review the general eonneetion between the source description and Lhe tr~nsitianprobabilitiw that describe the @fleetsof inkractions among partictfts. The causetf ~ikuationis this. Emission sources, generally referring to different kin& of psrtioies, act to produce a multipartiele state sf parti~fesin a physicdly noninbracling condition, owing to their initial apatiaf separation. Afkr zt sufficient Gme lapse, some of these padiclm approach each ather, inhraef, and then separate to be eventudly mnihilabd along with their noninkracting earnpanions by suitable detection sources. The causd tknalysis of the arrangement is @yen by

where the individud probability ampIi%udes({%l 1 (R" ) desefibe the traxfaitions induced by the psdicle inkraetions, and

represent the nonintersating muldipartiele ~tates. The Eakhr &relabled by the numbers of parlicle~in the wrious single-paAicle mdes, d the prodwts also range over all the differen%kinds of ptzrtiele~,of bo$h statistics. Aa a geaerating function of the probability amplitudes, (3-1 2.32) ia more u~efullypresexlbd in this v e ~ i o n ~

since the vaAous power8 of the emimion and detection sozlrces Bewe to direetlig identify initid and finat stake^, The vaeuum prob8blliLy ampfitude is detemined by the action (3-12.35)

in which we have specifically exhibikd in symbolic f o m the qusdratie ~trueture thst represents noninteracting psrtielef;. All relevant types of particles are includrsd, so &at S is being used as a supersource. By r e w i n g fram both sides of (t2-12.34) the expwssion thaL Mers to noninbracting psrtieles we arrive at [exp(iW' (S1~8%)) - I]

Interaction skeleton* Scrrttering cross sections

3-1 2

where

tF""(Sl,Sz)=Wf(Sr+Sz)-Wf(Sx)-Mcl"(S2).

(3-1Z.37)

The factor ~ X ~ [ ~ J S ~ represents ? G S ~ ] the exchange of those particles that happen not to interact. And higher powers in the expansion of ezspfz'Wt]iindiertlt; the possibility of repating independently in disjoint space-time regions all configurations of interacting particles. Thus, the irfeducible interaction pmceslsrjs, those that do not contain ndditiond noninteraeting parlicles and cannot be analysed into two or more disconxreekd processes, are obtaind from

Invariance8 of the action imply selection mles for the tramidion probabili$ies, Ri@d translations or constant p h a e transfomations of all souxces, far example, which do not change Wy(Sx,S2), must leave the righthand side of (3/-12.38) unaltered. The emission and absorption s o m e prducts are multiplied by reeipro~alphase canstrtnts, re1a;ted to momentum and charge in these examples. The individual transitiolrz. probabifitim must vanish if the phase constants do not cancel, expressing the neeessaw eonservatisn of momentum or eharge in the interaction process. The fwtor that imposes momentum conservation,

will emerge fmm a space-time integration over the inksactition ~ g i o n .We make this csxplicit by writing

thereby defining the elements of the transition matrix. T. Thc inbgral is no&a four-dimensional delta function since the integration domain is not infinite, T o sppreciate this we must recall that the precise specification of individual momexlta used here is an idealization that holds well ~vithina partiele barn, but faits near the bounda~es. Where the initial and final beams overlap to &vct esusail definition to the inkeraction region, (3-12.40) is applieabfe, and limiting the integration to that finib volume is sn ~pproxirnateway of recopizing %he realities of the situation, It is probability that is physically significant, and we are actually concerned with (dz) exp [i

X

(nh - tl,)p.z]

(dz)(dzf) exp [i

(nb

- n.)p.(z

- X')]

The 5 integration ean now be identified as a delta function, and the X intepal memures the Wtal inbraetion volume V, within the uaeedainfiea attached to the bounday liayers. The proporttionality of the transition probabili-ty to the volume of the four-dimensional inhraction region ixldicabs that the impadant quantity is the cwffieient of propontionality, the transition prab~bilityper unit faw-dimensional volume, or, per unit time in a unit three-dimensional volume. This ratio is which suppliw the physical interpretation of the transition matrix. Led us b & n the svcific discussions of skeletal inhracdiona with the scatkring af spinless pa;rtictes, as describd by (3-12.13, 15, f 6). The field #(x) is requird in the inderaction region, which is eausaHy intermdiah bcl.t~sreenthe e ~ s s i o nssoumt? K2(z) and the dekction source K l ( x ) . The b t a l field is the @uperpositionof pads related to fLfie8e wurees, *(X)

= cBl(x)

where

+ +zCzZt

(dz')~'-'(g

(3-1 2 . 4 )

- zf)KE(z'), (3-12.45)

and the particular forms of A+($ -- z') disclose the caussl situation, The prscessj we %reeoneerned with invdves the action of two emis~ion,sources and two absowtion sourca. Thus, when (3-1 2.13) is considered, with the c

we m u ~ retain t only tham eontribufiom having the required overall characteristic,, &s conveyed by the causaX indices. Those term8 are

811 others having tso many or h a few emission or detection indices. In earfier discussions of chargd spinless particle8 we have worked with complex sources- But experien~ewith spin 4, for example, has shown the gnzahr mnvenience of retaining red multiconnpontend source8 and making the apprw pr"i8te complex projections for specific charge valum* Henceforth we shall write

where .the &WO complex ch%rgeeigenvectars are =1

-

*

(D-

= 2-"'(1,

i).

(3-12.49)

3-1 2

Interaction skeleton. Seattatring cross ssetions

2815

Thew vectors have the properties of orthonormality,

snd, relative to the eharge matrix

they obey pP p;

= 'p;eqr,

pp,.

= ~'@p.*

The eReet of complex conjugation is given by

Using this notation, we present the fief& of (3-12'45) as

where +Pg (z)=

( d o p )' l 2(pqei~'

is the field associated with the specific particle Iabeled pqr, which enters the inbr&&ion re$on after its creation by the source K%,,. Sinnilarly, +P,(z)* is the h f d of the particle labeled pq which, after leaving the interaction region, is annihilated by the detection source K:,. The charge structure of the various partid currents that compo~e(3-12.46) is of importance. In j$,(z), far example, the charge frtctar assoeiakd bvitlh two incident particles of charges qhand g" 'is

+

As we sec, it vxnishes unless p' p" -- 0; anb zero c h a r p is brou&t into the interaction region. A similar restriction to opposite charges applies to jt;,(x). When we consider fl2(x), the charge factor associated ~ r i t hcharge p" enkring the interaction region and eharge qf leaving it is

and the necessary equality of p' and p" implies thet no eharge accumulates in the interaction region. These am different ways of satisfying charge eanstirvation in the scattering process. The seeond term of Eq. (3-12.47) does not contribute to the sewttedng of particles with like ch~rgeand m examine that process first.

The form of the current j";z(z) is

in whi& we can recognize the cument, q 2 f h,, %h$ B tsingle undegected pwtide. When only contributions from incid~ntpardicles of $he =me ahrsrge are ret&inc?d,

where

exhibiLs the momentum form of the plnofon pmpagcttion funcl;ion, L)+(k) == (k2)-l, snd produces the space-time integration that enforces energy-momentum clon8ervation. In pieking out the d e ~ k e dT matrix ekemen%we m u ~ take t inb acoount tha;t %hesowee f a ~ b r gidentifying a particular pair of incident p a i e l w , iKZ,,,iKz,hp, and a particular pair of scathred particles, iKf,,,iK~,;,, can each be produced in two ways comeapnding %athe s y n n m e t ~of thme pmduefrJ3 in p,, p; snd in p,, p:. Thus the transition matrix element will have those symmefries, which is a statement of B. E, statigtics, The mstsix elemenk is

L explicitly symmetried in p%, p: and d s a has the mquird pl, p: csymmetq &meovemjl momentum conservation implie8 th&t W&&

The exprimental memure of the eEectivenetss of s given ~ c % t % e faet, i ~ ga o h m & in beam amtzngennents, is sxt arcs or cmss section, It exp r e e s the at which the dwigna;ted process ocaum per unit tinne and p r uniid spatial volume, mlg%iveta the incident padiele flux and the den~ityof the scatbrer~,in. the u m d 8iLuation af fixed brget, The controllable fwbm refehng Lo the initid particle8 can be dven s general foPm fhitt permits fhtt cro8s smtiorr eoneept b be applie?d to aolliding b e a m as well as ~ h % i o m v targets. Le% be the particle flax vmtors of two asuch barn. An invahnt

3-1 2

interaction skeleton. Seatterrlnft cross sections

287

measure: of their relative ffux is suggested by the requirement that it mwt vanissfr when the vectors are: proportional, and the beams run with the same velocity. This definition, is 2 2 11% F ~ ( g a s b ) -t (3-1 2.64) which does produce a real positive quantity since the Aux vectors are time-Xike. If we write this out in terms of particle density so and particle velocity v = #/so, the flux definition hcomes

and. ~vhenone of the beams is a stationary target (vb = 0) it reducm to the magnitude of the incident flux multiplied by the target density. Sinee the padicle flux associated with a single particle in a, small mamenfurur cell is SI( = 2pr dwpr (3-12.67) the version, in ~vhiehwe shall apply (3-12.64) is

which introduces the masses of the particles, Other f o m ~can be used, particul~rlyone involving the total mass M, the invarisnG measure of %het o t d moxnenturn, = -(pn-tpb) = m: -i- Zpapa, (3-12.69) namely F = dw. d&b2[MZ-- (m, rnb)']li'[M' -- (m. -- rnb)'Izi2. (3-12.70)

+

The following ratio, probability of a transition per uni-t;four-dimensional volume [(3-It 2.43)j divided by invariant flux f (3-1 2,"i")], defines w, differential crass aeetion. 1%is digerential since the final pahieles are gpecified within small ranges of momenta, ets Ennited by momentum conservstion. 1nLel~;rations over these diRerential elements supply variow diflerential erass sections of lmser degrees of specification, le~dingfinally to a total cross section, althoagh the latter mw,y not misf if very slight deflections e a r q a dispraporltionwtf? weight. We shall uw the symbol da" generally for all diBerential eross sections, relying on the explicitly stakd differentials La indicate its precise nature. Energy-momenturn conservation in a two-particle scalteGxlirrg prmess fixes $he energies of the scatbred particleg and leaves frw only two pa,ram&ers that give the direction of the line alorrg which both pa&icleg move, in the rest frame of the total. mamentum. We may as well inkgrate immediw,Le?lyover the distributions af those variables that mmme precise vsltlw. Let us consider sny pair

where the m o d vemian refers to the rest frsme of P, in urbich P' == M, The magaitude of the relstive momentum

P

S

Pa

-B&

(3- 2 2.72)

is @ven by 1 - [M' --2M

(m.

+ m b ) 2 ] " 2 [ ~-2 (m. -

(3-12.73)

h carqixlig out the enerm inbsa;tioxl thst selectis this value one must d k

where dft is the element of wlid angle for the relative momenturn. The immdiale resuit is

1 1 3Sa2 M a P f 2 - (ma

X-.--""-.

+

mb)'~'''[~~

- (m. -- ma)'] l"

da, (3-12.75)

which rdtrees the &fierentid aspeet to the angles that 8pesif.y %hedirectian of her ee m We nob that the same squareroot kinematics1 mf a c b r ~aceur in the final 8tah ixlbpation (3-12.75) and in, %h(?: incident flux (S12.70). Thew relatively camplicaM factors will e~ncelfor a purely elstic seatbring proew where initial and find perticks are the same. The tramition xnat~xelement (3-12.62) provides a eimple application of $he cross section. definitiorr, giving dirwtly

In the latter form, B is the defiection anlgle, and the full equivalence of the anglers

3-1 2

lntarraetion skeleton. Scattering cross sectlons

289

The W - B apremes the indistinguishability of the B. E. padicles, rduction is wrformed in the rest frame by noting that each of the four partiefe energiea equids +M and this gives, for example,

8 and

Of psrfcieular interest are the vePy high and very low energy limits:

Note that at sn~a;llseatkrictg angles the latter reduees $0 the Rutherford differential cross metion for the scwtkring of dislin~ishablepwfticles,

is the relative kinetic energy of the particles. When padiele~of opposite charge scatter, they are distinaishable by th& chrcrgw rand only one kind of term emerges from the analowe of (g12.M) thaf has the source factors replaced by iK:,,,iK:,i -,and iKPPPQiKIP; An BddiLiond minus ~ i g nis also needed. Qf eaurcse, each process a p p s r s twice owing to the combined wmntetry: plc--"pi, p2++ p;, q -+ --q. But now the second k r m of (3-12.47) comes into play, widh -@.

where mch current contributes two eqllia;l terms Lo a given process, correspondiag to the symmetries expressed by p1 C-' p:, q -+ -q and p a t ) p;, q -+ --p. The implied %ransitionmatrix element is

290

Fields

Chap. 3

Notice the simple connection between the matrix elements (3-12.62) and (3-12.83); they are interchanged by either of the substitutions

Correspondences of this type between different transitions have become b o w n ss crossing relations. Their origin is not far to seek, Emission and absorption proceljses are united in the field +(z). The formal substitution pp -+ -pB interchanges the? physical e.ETects that identify emission and absovtion acts. And the numerical eharaclerizations provided by the individual fields (~,,(z) respond appropriately : "bp,(.> #p-e(a)*(3- 12.85) Given the transition matrix element for one process, the substitution generates another oxre in which an initial particle of properties p, q is replaced by a final particle of properties p, --g, or conversely. Of course, this must be done a t both ends of the reaction if one i s to retain s s c a t k ~ n gprocess, When particles of opposib charge are present, the outcome cam, be a synnnnet~of s, given matrix element, as illustrated by the invariance of (3-12,83) under either of the substitutions P P $6 --1->2* (3- f 2-86)

Note that we are considering individual applications of a transformation that, used wholesale, is the TCP operation. The square bracket factor of (3- 12.83) has the foliowing evaluation : 2-

M2

- 2m2 1 M" - 4m2 sin2 (Bf2) I -

4m2

p

11/12

cos 8

The second term, (4m2/M2) cos 8, is relatively negligible both at high energies and at low ener$es. This provides a simple eonne~tionbetween the cross wetions far unlike and like eharges, one thsd becomes accurate asymptotically a t bath extremes of the mass scale and constitutes a reasonable interpolation between these limits: h"

COS

The scattering of photons by spinless charged particles is contained in (3-12.29), dong with other processes. The part we w a d is extracbd by writing, as in (3-12.441, (3-12.89) (a(s) @I(%)""I"cbz(x)t E

3-1 2

Interaction skeleton.

together with its photon analogue A'(4 = At (X)

where

Scattering cross stmtions

+ A$(z),

291

(3-12.90)

/ A$(z) = i (dz')f'~'+'(z - X')J2.(zf), /

A:(z) = i (d~')~''D'-~(s - z')JlV(z'),

(3-12.91)

and then retaining those terms that have one photon and one particle emission source along with one photon and one particle detection source. They are

w12

-

/(dz) (dzl)rl( z ) [ e q 2 ~( x~ ) A + ( ~ - zf)eq2~~2(z')

+ eq2pA~(z)A+(x- z1)eq2pA (x')l#z(z') I

- /(dx)4l(x)2e2~i(x)A2(z)02(z),

(3-12.92)

in which we have adopted the simplification that. expresses the use of the Lorents gauge for the vector potential. Let us recall that

Of the two terms that do not refer to polarization vectors, one vanishes because the source is divergenceless and the other, a gradient in coordinate space, can be removed by a gauge transformation. Thus, it is in a special class of gauges that we write where A$x(x) = (dud and they are Lorentz gauges, since

112

p

ikz

ek~e

,

Now select the coefficient of iJk*,X,iK,*,,and iJk2~,iKp2,(we have finally omitted causal labels on the sources since they are abundantly evident in the other indices). The resulting space-time integrals produce the Fourier transforms that convey the momentum specification of the scattering process. Here is an example: P

w h i ~ haltf~oU S ~ Sthe fact that the differential operators pp rand p" a& dirmtly upon momentum eigenfunctions* The transitian matrix element is obfaimd aa I r , ~ , l p , , ) = (dur, d ~ pdut, , do,,)"22e24:~,~,.e~,~,,

(3-12.98)

which uws the kinematical simplifieiztions

Other aspects of the kinematics are these, 7'he totat momentum is F

?E=

P I -4-

kl

E

pz 4-

h2,

(3-12.101)

and therfsfare -plkl

= -p2k2 = 4(B12 - m2)

while -plkz = - p 2 k z =

$(MZ- m2) + ktkz.

(3-112.102) (3-12.103)

Invariant expremions for the particle energies in the center of mass system, the rest frame of P,are

Written in terms of the center of mass sezltteltxing angle 8, we also have

We should also nok tbft crossing symmetries exhibited by the transition matrix element (3-X2,98,99). Sinee the sign sf q is irrelevant, there is invariance, under the substilution spcseificalfy of h,, for which the equdity of klplwith k2pz, and of k,pz with Iz2pf,is decisive. Concerning tbe photons, the use of linear polarizations with Feal potari~atioa wetars implies the transfomation

3-1 2

tnti;sraction ~ksleton, S~attsringcross setlens

;193

end the transition matrix element sfiwld be invariant under the interchangtt. This induces the exchange of p and v in Vp,, which indeed 8how6 the requir& invariance. The tensor br,, also has the fallowing imporlant proprlies:

They bring about the neeemary con~rvationof the egwtive soure@ for the emission of the final photon and the absorption of the incident one:

The summation of the transition probability over bath polagsations of &he scattered photon ean now be performed with the aid of (3-12-93),

If the ineidctnt photon beam i~3unpoIarizd, both polarizations appearing with: equd probability, the nceessary average can aIso be performed by me&m of (S12.93) :

A straightforward algebraic reduction gives

Let us again emphaize the relative 8impXicity of the kinematical factam in the cross m~tionfor elastic scatkring, even for pslrtieles of uneqml mas. The ratio of the inhgral in (3-12.Z) to the invari8nt Aux of (3-12*70) producw &hefaetar

which sugplie~the unit for a, diRerential seatkring cross swtion, Then, ailnee

2eZ = ~ K O Iwe , get directly

- m2

M %+ m2 -I- cos B M2 - m2

p

(3-12.117)

l + ~ z + r ncas ~B

vvhich u w the center of mms wwthring angle evaluation for plkz. The dif"Terentiwl and totd crass wctions for the extreme mergetic limits are

the latter Ling the Thornson cross sections. The conserved nature of the effective sources thaL emit the find photon and tabsosb the initial photon implies the gauge invsrianee of %hetrazlgitian prab* bilities, This perIxlits oae Lo exploit whakver simplifications cm be iarkaducftd by spwial gauge cheiees. The question of gauge in eonneclion wiLh the polerriza~ tion vectors is implicit in the choice of E@ which, in some coordinate frame, haa3 ita spatid component8 reversed dative to those of kC. This k e ~ p ~ ~ s l e d wi%hthe aid of a unit time-like vecbr [email protected] and tho polarizalion vectors have the two arthoganality properti~,

which are incorporated in the summation

We now return to (3-12.98,99) and observe that the identification of either py/m or p$/m produces fhe simplification

R+# with

One can wrify directly that the same resuft for the surnm&tioaand average over polarisations is obtained in this way. Applied to the final pkotons, (3-12.121) ghes

Interaction skeleton. Scattering czross tstlclionas

3-1 2

298

and, then

If we sef

12

= p t/rn or p2/m and insert the relation

we regain (3-12.113, 115). The $auges we have just described are particularjy useful when the particle is at rest initially, or finally, Another choice of the vector lz ia P / M , and this is most convenient in. the eenter of mass frame, f n ail these examples the coordinak e ~ npwith the time axis, Let us u8e the cenkr of ma;iss system is ~ h ~tos identify description lo study the polarizafion dependence of the scattering cross section. The trajectories of the incident and scattered particles define a plane, We first choose linear polarization vectors that are either perpendicular la the plane, or lie in it at right angles to the appropriate phston momentum. The differential w s s sections for the various polarization as~ignmrsntscan be read off from (3-12.98,99), m gimplified by the special choiw of fbe several vectors, including the eenter of mws momentum relation

O = p, 4- kl = P 2

+

kg,

which gives

The cross section. vanishes if one polarization vector lie8 in. the scatkring plancl white the other is perpendicular to the plane, When bath veetars are perpendicular to the scattering plane,

and when they are both in the scattering plane,

--

M2

m2 + + cos 6 m2

- m"

.l+ ~ ~

cos B $ m 2

Fields;

Chap. 3

The average of the Lwo, appropriate to an initially unpolarized barn, is (3-12.117), which gives the latbr stmcture greater physical meaning, 1%isr interesting to o k m e that photons s e a t k r d through the angIe d e k r ~ x l e dby

mid be complebly polarized perpendicular to the scattering plane. The diBerenti,iaf erosss wction~referring to circular pla~azztionor helici$y sr%aksean be produced from the linear polarizalian results. The eireular polafiea%ionvecbrs are linear eombinzttiom of thaw p8rallel and pepndicular 4x1 the seatbring planet relatively shifM in phw by =&W [(2-3.29)]. Since the complex conjugate po1ariza;t;ion vector repremnb the outefoing photon, the probability amplitude with the same helieity initially and finally equals half the sum of the two linear polarization amplitudes, and, thst for opporsite helieitiea~is half the digerenee. The diff'erential cross sectism corresponding to no change in h&ici.ty, or to a helicity reversal, are therefore, respectively,

The geometrical facton thst appear here, cos' +B and sin' 48, are familiar ss probabilities, for unit anwXiar momentum with magnetic quantum number +l in tl given direetion, that a measurement m d e in a direction at the relative angie 8 will yidd magnetic quantum numbem +l and -1, re~pectively, 'Shere ia also a dynamical wei&ting fpbctor that is unity at low energies, M rrr m, and suppresses helieity changes a t very high energies. The tots1 differential cross seetion, vvhiGh is independenl of the initial heiicity, is the gum of the partial erass geetions in (3-12.131) :

It is equivalent h (3-12.117). At the scattering angle determined by tan $6

==

Mint,

(3-12.1133)

the two partial cross sections are equai, leading to zero average angulsr mamentum in the dimetion of the scatter& photon. This ie, of eourBe, the same as the one given in (3-12.130) at which the scattered photon is linearly polarised. Other processes involving two particles and two photons are contained in WzZ. When we SeXect terms with tvva partielet e ~ m i o soureas n and two photon deketion s m e s we are considering pa&icle-a~%iparticle andhilati~ni n h t;uro

3-7 2

lntsraction sksteton.

Seatasring cross sections

2337

photo-; two photon emimion sourees bgether with tmvo pafiicle deteictisa ~ ~ u r e indicah es the inverm process, the crestion of a padieleantigartick! pair through collision of two phobns, The h t k r , for example, is deacl.ibed by

and we extract the coefficient of iK,',,iK:;

_, snd iJxgk,iJi;~;

to get

where

Notice that the symmetry klXz m 8 the kinemadical reIatiom

k ; ~ ;is explicit, as is plp ++ p; --q when one

--plkz = -pikg = +M[+M -- (*M% - n t 2 ] l J 2cos B], -pgk$ = - p f k z = +M[*M

+ ( * M ~- m2)"'

cos 81,

(3-12. f 37)

where 8 is the ande htween particle and photon, relative mamenta. The location of the: tbrmhold for the reaetian rat M = 2m i8 app8rent in the square root that gives the mapitude of the cenhr of mass particle momen$um, a11 parfiele and photon ener@esb i n g equal to $M. Using red plariealion vectors, the matfix elemeat (3-12,135,136) is obtained from (%12.98,99) by the craasing tr&mformrti.t;iorr p2 -+ --p$, klXl -+ -kbA&. (3-12.138) Since the final pa&icles naw differ from the initial ones, the ratio of the kinemadicrtf square root facdam appear8 mpfieilly in the differential cross secs assipments, first using tke linear %ion. We shall give it for v a ~ o u polarizst;lom po1ar;igatiaxzs that are parallel or wrpendieular to the plane of the remtion:

the cross section vanishes when the two polarization vectors are st right angles.

Provided the threshold energy is exceeded by at least the factor 2'j2, M 2 > gmZ, there is an angle at which the diEerential crass section for parallel polahgations vanishes,

The cross ~ection,appropriate to unpalarizd photons, an average of the four psmibilitie~,is *

(3-12.141)

The behaviar near the threshold, and a t high ener$es, is given by

The transition amplitudes for circularly polarized phobns are a g ~ simple n linear combinations of those referring to linear polanaaliom. The photons are opposikfy direct& in the ceder of mms frame and as~imingthem the same helieity, for example, means %hatthe photons have oppo~ikeanwlar momentum doag the common line of motion. f t follows that $he tr&mitian amplitude for equal hellieities is half the sum of the twa linear palariaation amplitudes, and that for opposik heIieiGies is half their differenee:

The dominant reaction thus shifts horn equal helicifies near the thre~bsldtZ) opposite helieilies at vew high ener~es,The crossing poixlL aceurg at

3-1 2

Interaction skelston, Scattering cross se~tionar S 1 9

wlrieh is a restatement of (%12,1.10). Note th& the geomc;tricr;tlfactors, l and ( g ) sin' 8, also refer to angular momentum. The first affirms the equivaienee of all directions in s state of zero ane;ular momentum, and the second gives the probsbility, for angular momentum quantum number 2, that connecb mal$netic quantum number f2 in one direction with mapetie qu8ntum sera in another direction inclined at the angle B. The transition matrix element for the inverse process of particleantiga&icle annihilation into Lwo photons has the same appearance as (3-12.135, B@), with the causal lsbels reversed and complex conjugate polarization vectors eubstituted. Since the roles of initial and final padicles have h e n reversed, the kinematicat syuare root factor becomes inverted, but all dse is the same. With helicity labeling of the final photans,

and again the predominant helicity relationship of the photons change8 in going from low to high enerdes, I n the annihila.t;ionaf slow particles, M c=i gm, there i s no relative angula~momentum for the photons to carry away and equal bdicities for the oppositely moving photons dominates. A t very high energies the photons sustain the mrttximum. angular mmentum along their common line of motion. Nevertheless both: differential cross sections are isotropic, and the tatal eross sections are

The variation of the cross section with inverse relative s p e d v, when. v f<1, means only that the rate of the annihilation process per unit volume is pmportional to the product of the beam densities. The eomputzlLion of the tatd eroes section for reactions in which the final state contains two identical particles, such as the B. E, phatons in this mnihilation process, needs one note of caution. I n summing individually over the find tstaLes of both particles, which is here tlke summation over heficities and the integration over all dirrsctions of motion, every physically distilnet slate of the Lwo padicles is counted twice. That trap has been avoided in stating the cross swtions of 63-12.1463).

3-33 SPIN

-& PROCESSES

Let W b @ n with tbe scatkhnf~of rspin particle8 that have like charges, As withod reference to specific in the @pia0 discussion, the relevant part of charge v%lue~, it3

but s d y the first &mmapplies to pwr2;ieles of tbe sarne charge, with

The form abtrallned for W4@, %ndogousts (3-12*M), is

The q fabl, which is common .t;a df spixrors, It% ben, omi_l;l&. Note t2b0 that the. id%idorder of the tatally antieom.muta%ivemure@f a e b r ~has h e n r e amwd wiLbout the inkwention of minus signs:

The tmmition matrk element will be defined relative to %heor&r of murce~ &at appeam explicitly in (3-13.5). f3ut am must, of course, tske into account $hat a pa&icular p d u c t of dehtion sources occurs twice in Lhe s u m a t i o n with B rel~%iw minw ~ i powing , to the F. D, antisymmetq under the e x e h a ~ e plc8 ++ p:@:. A similar remark sppfies to the emission sources and the permuW o n p,~g ++ p&;. Thus, the required matrix element, whieh shows the antithat are charack~gticof F. X). a&&@, is

Even when one is not interested in specific spin values in the inifiai and finaf states, perhaps the simplest general procedure is to evaluate the higerentiaf cross sections for the various helicie assignments, That is already suggeskd by the photon pola;riz;ation eonsidergfions of the preceding metion, urbere the oudcome of poIarization summations and averages required some algebraic reduction to attain the result that was produced directly By considering the various helicities (or Xinear polarizations). This simplification lvas particularly m a r k d in high energy photon scattering where the helicity strongly preferred not to change, The same tendency appears in the present situation, which we may as well call electron-electron scattering since that is the outstanding realization. The general construction of the spinor up,warj giwn in (2-6.90) as

where the v, sse v @eigenvectors with eigenvalue 3-1. When they are chosen to be eigenve~torsof U pllp/ as well, identibing cr with the helicity, we get

where the latter is the high energy limit in which helicity becomes linked to the f n the cenbr of mass frame where X aX pa&icle energies equal eigenvalue of +M, consider the following high energy evaluations:

since irs has no diagonal matrix elements in the 7 ' = +I subspace, and, using the relation, TOY

= iY5a,

(3-13.1 11)

We see that (~~,,,r~r'u,,,) vanishes if ol = --@a. The helicity does not change in these products a t high energy because rar' commutes ~vithvs. Accepting the restriction ol= a2,a: = g;, we find that the product a g pearing in the first term of (3-13.7) is

There is a basic identity, expressing the completeness of the four mateces, I, a, which is presented in Eq, (2-5.58). When the general lmratri~esX and Y are

Chap, 3

appropriately chosen as dyadie products, it tells ua that

which is antisymmetrical in the indices crz, G; and in g,,g;. But a word of eau$ion about notation is cdled for here. Although we have M-rittenvoz, 8&y, one must not forget the implicit reference to the direction of the momentum p2 slfong which the spin is projected to give the component a2,m a t we have just referred to as antisymmetry in g 2 , a; is, properly speaking, antisymmetry in ~ 2 and ~ p 2h i . The combinittion (3-13.14) does not vanish when the helicities crz and cr; are equal,. We consider fimt the situation of equal, initial helicities. Then precisely the combination that is evaluated by the identity (3-1 3.14) appears in (3-13.13) and, owing to the antisymmetry just mentioned; the two terms of ($13.7) are combined into one, with the factor

To evaluate the spinor products that appear in (3-3.14) we choose p2 = --p; as reference direetion-%he x-axis-and express the choice of equd belicities, or apposite spin projections along this direction, by standard t ~ v ocomponent

The spinm v,, and u,;are in the same relation, but ~vithrespect to the direction af p, == --pi, wrhieh is rotated by the sngb 8 about the y-axis, for example. That is expressed by

which, as a combination of matrix elements, is also the determinant of the unimodular rotation matrix. Since $he various factors of 2m and df cancel in the matrix element, leaving 2eZ,the result i s immediate :

3-7 3

Spin

3 procssses

303

For the situation of opposite initial helicities, the combination Lhat appears in (3-3-18), apart from a minus sign, is

The two contributions of (3-13.7) are now wsocia-ted with different final states, which do not interfere in dihrential cross sections, The helicity labels are, . respectively, cl = crz, c; = o; = --gz and ox = a; = --c%,a: = ~ 2 AlternaLiveIy expressed, unit angular momentum along the initial direction of motion can lead Lo either of the magnetic quantum numbers, +l or -1, along Lhe common direction taken by the scattered particles, Leaving aside the multiplier of 2, the factors contributed by (8-13.20) in the two situations are

(y:e(l a1

= G;,

i2)isrvU+) ( v : e ( 1 / 2 ) i @ @ ~

v+) = cosZ*8,

= Cr2:

*

(v-e

(1/2)illa

%+)

(3-13.21)

* (1/2)iBo. (U-e @v+) = sin2 48,

if one is careful in translating the heiieity specifications into spin projections along the two felevant directions. The diRcrential eross section produced by adding the noninterfering eontribu%ionsis

The trigonometric factor in square brackets can also be tvritten as

The dieerential cross seetion appropriate to initially unpolttrjzd electrons (one unpolarized beam will do) is the equally weighted average af the more specific cross sections given in (S13.111)) and (3-13.22, 23). ThaL is

which has two remarkable katures. It is s perfect square, as though only tt single process contributed, and it is identical with the spin O high energy differential cross section of (3-12."7). This interesting equivalence of different spin values is restricted to very high energies, That is most evident in the scattering of low energy particles, M = 2%.

Then the spinors g,, essentially reduce to the rot= $1 eigenveetorsv.. Helicity t degcription, the referral of $1 spins fo a common. ceases to be the m o ~ u&ul direction in space taking its place. fnded, with that chaiee, and the individual scattering praeosmss t ~ k pl&ce e withouL ehsnge of mrwnetic quantum numhr. In this low m e r limit, ~ spin and orbitd motion are dynsmially i n d e ~ n d e n t in , contrmt ta very high enerw conditions where they are tightly linkd. When the initial, and therefore the final, mapetie quantum numbers are equal, both te af (SL3.7) cantfibuk, with rever~edsign; for ogpo~iteinitiaX mapetie quantum numbers one or the other of the two t e r w is effective depending upon the msignment of oppalsite mgnetic quarrtum numbem in the find state. The spin-averagd differential cross section, is, themfore,

The latter f o m comesponds to the alternative of averaging over the three symmetrical spin states, vvfiieh are anbisyrnmetrieaf in spratid coordinaks, and the single anfisymmetricstl spin state, which is symmetrical in spatial coordinab~. This F. D, re~ulLdiBFers, of coure, from %ha& of (3-12.79), where only the symmetrical spatial combination appeam. States clmsifid by the total spin of the particles are useful at high e n e r ~ ~ ~ r too, provided the reft3rt;n;ce direction, of the magnetic qu%ntumnumber dif?Fers for the initial and the final parLicles in con,fornnity with the altered direction of motion, For unit anwlar momentum there is also a mp~ration.of the stabs with unif magnitude af the magnetic quantum number from the one of zero ma6;netic quantum number, reminiscent of that for a unit spin particle ita m m vanishes, eEeetively a high energy limit. We have already noted that the transitions of the initial unit magnetic quantum state tstke it only into fins1 states of magnelic quantum number f l. indeed the weight factors, cos' f B and sin' that apgc?;ar in (3-13.22) are just the probabilities for unit anwlar momentum %;hat; connee.t;magnetic quantum number +l in. the initial direction with +l and --X in the final direction. T h v occurred within a pho-d-onconkxt in Eq. (3-12.131). The unit spin state of aero magnetic q a n t u m number is the symmeticstf. eombination of the two ways that realize equal helicity, B == B' = &l, The aero spin state is the c o r r ~ p n d i n gantisymmetrical combination. Since the helieitim are maintained in individual seataring aeb, and reversing the sic of all heliciticts is without effect, there are no trrtnsitiom between sGzbtes of diEerent spin and

a@,

the differential cross section (3-13.19) applies to either spin state af zero magnetic quanhm number. Not mueh more effort is required to obtain scattering Gross sections for arbitrary energy, using the helicity classification of states. Now helicity changes in scattering do occur, as exhibikd in. the general evaluation of

However, the vector structure

still requires equality of the helicities. We shall merely list relative corrtribudions of the various processes tE.lat appear additively in the spin-averaged differential cross section, They are classified by initial and fins1 magnetic quantum numbers that refer to earresponding directions of mof ion, and aeeoPding to ~vhetherbelicity trr\.nsitions have occurred, O + 0,no:

0 -+ 0, yes:

1P

0 --+& l , yes: X

+ 1,

both-:

1 "4 - 1 , both:

For the last two processes, the constant --4 is the contribution asso~ialkedwidh helicity changes, Adding these terms and supplying the appropriate factor gives

which interpolates between the high and low energy famh (3-13-24) and (3-13.26) respectively. While resembling the zero spin result in (3-12.77), it differs in detail, except tat high energies.

3

Chap, 3

Fletds

To discws electron-positron scattering we return ta &13.1) and cornider both .terms, with

where the two equal contributions Ghat refer La a spcifie pair of appsidely charged parLicles h w e already bwn ~0lle~Cf3d.The transition matrix eLemenf that is defined by %hecoefficient of

4-

(U:;.;

-97

Q r *

Y

@Pl@lP)(%p..g'I7

(p1

+~

0 ~ f i ~ p ; . ;-P)

$ 1 ~

The: crossing re1a;tion btween this matrix element; and the one for like charge scaLkring again follow fmm the unification of e ~ m i o and n abmrption prw in %hefield $(X), as convey4 by the formal substitutions When ~lppliedts (3- .13.7), the h~ubstitutions

produee (3-13.34), with the additional &nu@ sign coming from the reamange men& rrecaBaq to realize (3-13.33), the stand&rd mul%iplicationorder of the sources [i' is omitted here] :

since the two soufees asaociatd with charge ---g must revem Iheir relative pasition.

3-1 3

Spin

processes

3W

Ia order ta treat both terms of (3-13.34) in the same way, we use the relation of (2-6,134), up* -'I = i o ~ ~ u ~ . ~ ~ (3-13.38) which gives

where the now mahhing charge labels have been omitted. The effect of the additional r5 factor is ilfustrated by

The first statement depends upon the opposite motion of the two particles in Lhe cenkr of mass frtzme, There is an irreconcilable conflict btween the numerical fector, demanding the equality of the helicities --@l and o:, and the spinor product, which requims equal magnetic quantum numbers and therefore opposite values of the helicities --g2 and er;. The situation is snalogous to that for a spin 1 padiele, where the time component of the vector field rranishea in the rest frame. As we recogmizr?.from (3-13,.i10), high energy collisions with the same initial helicities (a2=. g;) are dominaLed by the first term of (3-13.34), which diRers from its andague in like particle seateering only in sign; therefore [(3-13-29)]

The high energy evaluation of (3-13.39), in the psincipal circumstsnee a2=. ---c~.;, = I-bj, is

6 1

according to (3-13.B) and the nuU property noted in (3-13.40). The firs* b r m of (3-23.34) contributes only to the process ~viiitbcpl = crz, and, reedling that

(PI-4- p : ) g = -AfZ,

(3-13.43)

Chap. 3

R-eget

The average of (3-13.41) and (3-13.44), the diEerential cross mction for unp o l a ~ m dpadicles, is

This, Loo, is identical with the high energy spin O differential erom section. As in (3- 12.88), the simple f &etor cos4 +B relates high energy electron-positron scatkring to high energy electron-electron scattering, Owing to the disparity of the denorninatom in. (3-13.34) a t low en.er@e~, only the first brm is significant for d4 2%and

A general fomula, incorporakixlg both limiting cross mctions, (3-13.45) and (3-13.4@),can be derived, S d t h electron-electron seatte~ng,by eonside~ng all the helicity transition8 that are powible a t inbrme&mte energieg, Bud the crossing relations make it unnecesB%ryto do tEs, The connection% between h&vidud tran~itionampli%udesatso apply Lo %hehelicity sum of the Squares of t h m amplitudw. Thus one can be@n with the electran-electron. scatbring result exhibikd in (3- 113.30) and derive the requird electron-positron form. The crossling tr%asfomationpi t., -pi implies or, Lrans1ate-d into the g8rameters M and 8,

retains its meaning. As a quick illu~trationof the procedure leg us make t h a e substitutions in the bigh e n e r a limit of electron-electron scattering, in&G&& by

= cos'

4s

Spin 4 procaraes

3-1 3

%H

The relation between. the two high enerm differential erass sectiom h- bcomc? quite transparent. When the substitutions (3-13.48) are introduced in (3-13.30), the general dectromi-electron. differential cross section for unpolarisd barns, the result i s the eorre8pondin.g electron-positron differential eross section (the kinematics1 factor 1/M2does not take part in these transformations, of course):

Although written in a slightly different way Chat exhibits the dominant, low energy and high energy behaviar, the fimt, square bracket, term is idenbiezlbl with the spin 0 diEerentia1 eros9 section for scatfclering of opposite charges. The latkr was only stated implicitly in (3-12.87), one half of which is the entry in the square bracket of (3-13.51). The two additional terms in this equation %re relatively negligible at both low and high energies, but they can be of quantitative significance at intermediate energies, In order ta illustralcr the scatter;ing of dikrent kinds tlE charged partictes, we shall also consider the interatetion Between at gpin O partisle and a spin g particle, The appropriate electric current vector is the sum of tbow associated with the two types of particles 8xf.d the interaction term in is

We can write the transition matrix element directly:

where all primed yuankities refer to the spin O par$icle. This will also extend to the masses, m and mf9auf the spin i$ and spin O particle, respectively, A, simplification. can be introduced with the aid of the h t a l momentum,

P

==

PI

4-~ "$32l3-P$,

(3-13.54)

for ?(pH

df-5) = 2 W -

-

snd

(3- 13,55)

+

u ; ~ ~ , Y ' Yf( ~p6)up,,, ~ = ~~;,.,Y'(YP m ) ~ ~ 2 @ 2 = -2u,*,.,(~

-- my0)up2.2,

(3- 13-56)

in ~vf-rlchthe last form refers La the center of mass frame. The introduction of

310

Fields

the heIicity consGruetion (3-13.0) gives

lvhere the kinematics of the situation, state that the eleetron enerw and momentum magnitude, whieb remain unalkred by the collision in the eenbr of m m frame, are

At high energies, where the: individual p%&iclemass- are negXi@bXe, the electron he:licit;y is maintained in scatlefing,

and

M

dc n2 cos2 4, --------. din sin4 48

>> m, m':

When, a t low energies, the electron. spin is referred to a fix& direetion no change in magnetic quantum number aecurs on sealeering, and

The general result, s u m m d over final spins and averaged over the initial spins (the latter process is unneeessaq here) is

It is interesting to consider the two limiting siGuatisns in which the mass of one particle becomes v e q l&rge,not only compared to the other m s g , but Ls the total eaergy of the second particle, ff m' iis that l a r e m%,it is m m eonvenient to introduce df

- m'

+

-4

0,

(3- 13.63)

where the electron mew in the center of rnam fiame i e indistinguiehable from the enerw in the coordinate system where the h e ~ v ypadicle remaiiols zr;t re&.

The limiting process givm spin 9:

The analogous limit in whieh it is the spin .5; particle Chat has become v e v heavy has the same form as (3-X3.M), kvithout the trigonometric faetor in the numerator : spin 0:

&a

fbo now superfZuous prime on m has been. omitted. The two diflerential cross sections have been identified with the moving particle, the very massive one acting only a%a stationary charge. The possibility of applying t.t, gource dt3;scrip tion to this circumstance will be developed in the next section. But first let us examine some processes involving photons and spin -& pa&ides, as eonLained in the expregsion (3-12.a) for Wa2. Electron-positron annihilation into ttwa photons is described by (dz)(dz')~., (z)roeqr~ I ( X ) G+(z - zr)eq"/A1(z')J.z (X'),

and the coefficient of i ~h l ~; ; ~ h;iq,;,; ;

-,iq

(3-13.66)

gives

The B. E, symmetry in kih; is evident, and the F. D. antisymmetry in p ~ @ z pp;@; , --p can be verified (recall that rr = --~~r,r@) with the aid of the kixlennatical relations

JVe shall find it more coxrvenient to tvrite the dyrramical factor of (s13.67) as

In the center of mass flrarne the energies of all electrons and photons equal

*M,and --2p2kl = *Jf[dl

-- ( A f 2 - 4m2)'12 cos 81,

-2p2k; = $fii[Jf

+ (MZ - 4m2)'iZcos 61.

(3-13.70)

Particularly simple is the annihilation of sloif--lymoving particles, df c=t 2m, for ~vhieh(3-13.69) &comes

Multiplied by 2m, this reduces to

(e2/m)@'v%,.[c- e*,

i.

(?Or

= irs@)

ee'*]a- kv, = 2e2igf6,.8ee

X

et* . klm, (3-1 3.72)

\%-herethe unneeded causal Xabels have been dropped. Only terms with an even number of Ys fachrs survive here, and tl-e have used the fact that e X ef must be directed along k. The equality of the helieities staks that the two magnetic quantum numbers are opposite, in the antisymmetrical way impIied by Lhe faetor G'. Accordingly, only the singlet state of zero htef spin can, undergo twophoton annihilation, for slo\vly moving particles, The corresponding aero sngutar momentum state of the two photons, a linear combination of the two equai helicity states, is identified by the perpendicular polarization vectors of the %\Wphotons, When we recall that

the differential cross section per unit solid angle for a given pair of perpendicularly $arized photuns, with the particles in the singlet state, is oMained as (uz/m (l/v). To compute the total annihilation cross section for unpolarized particles we must supply the follo~vingadditional factors: 2, for the number of polarizations available to one photon, the other polarizatisn b i n $ fixed by the requirement of perpendicularity; 4, the Aatistical -tveigbf of the singlet sts;te; 2w, the total solid angle aecessibte to either photon \vi%houtduplication of the

fins1 states. This gives

which is half the analogous spin O annihilation cross weetion, I d is inkresting to observe that the folXotving effective interaction term,

will directly produce the traxlsition matrix element expressed in (3-13.72), ~ v h e ~ r evaluated for the same low energy collision. Its space-time locality, in contrast with the nanlaeal structure of (3-13.@6),is a specific reffectioxl of those limited energetic circumstarsees h hi eh prevent any more detailed space-time characterization of the process. I n tz high energy evafuaticln of (3-13.69) the mass ns ixl the rlumerat~r~vould be neglected. We shall see that this is justified if one excludes very small forw:itrd or beckward scattering angles, sin 6 p> m-/df. (8-13.77) Then, since the resultixtg matrix commutes with ? g t the helicities are maintained, --g; == cr2, and annihilation occurs only in urlit magnetic quantum number states. One must be careful not to confuse the latter statement, tv1.tich refers to the spinors a,;,; and upZCz,, with the properties of the spinors U;; and U,,,, where the magnetic quantum numbers are opposite, since the helicities -a; and 6 2 are equal. Written in s simplified notation and multiplied by 2m/e2, the high energy version of (3-13.69) is (--cf =r a):

-.;

-,

f L is convenient to use the photon msmentum k as the spill reference directionthe z-axis. Then the arthogonal particle spirkors describing magnetic quantum numbers of &4 in the p direction can be ~vrittenas v, = v+ cos fiB

+ v-

sill $4,

U?.. = --D:

sill 18

+ u*

cos i 8 .

(3-13.79)

The photon polarizzltiorr vectors appear in the combinrttioxls

tvl~iclrhave the effect of rnisirkg and Iotvering particle magnetic quantum num-

314

Chap. 3

Fields

bers by unity; *(g.

+ i.,)a-

= v+,

v:*(@,

- i@@, = V;,

4

-+

=,

v:+(@,

+ iq,) = v*,

(3-1 3.81)

all other combinations being zero. Thus wet can ewily work out the values af (3-1 3.78) for any choice of photon helicities. With X = --Xt = +l, for example, tvs get

-- 2cot 38, (3-13.82) and similarly

t,r;Izile null results are obtained far equal helicities, h = X L & l . As in the high energy annihilation photons carry only the mmimum spin 0 discu~~ion, angular momentum, & 2, along their line of mation. Again there i s an elementary interpretation for the geometrical. factors of (3-f3.82,&3) which appear ixr transition probabilities ns sin2 B cos4 48 and sin2 B sin4 +B. These are the spin 2 probabilities that connect rnagneti~quantum number +l in a given directioxl wilh magnetic quantum numbers +2 and - 2 in another direetion at, the relative angle 8. The transition amplitude factor i/sin2 B also appears for spin 0, in conjunction with the geometrical factor sin2 B, \vhich produces an isotropic differential cross seetion. No\\-, hoi~ever,the spin averaged differential cross section is

and this alteration in angular distribution is attributable entirely to the change of the initial state from zero to unit mbignedie quantum number, It is the singularity of this diEerentx%tl.cross seetion a t angles B = 0 and n that denies universal validity in angle to the high energy evaluatim. These singularities are spurious, and trace back to the failure of the higEz energy approximation

Spin 4 processes

3-1 3

at @

==

Slti

O and r , re~pectively,A suEciently more %cur& version i~

which is d w wfuX in the form of the product

making explicit the origin of Lht: angle reestriction. (3-13."1"1) If ls-auld m m thaL one had only to replwe (3-13.M) 1~ifE.1

leading to the total annihilation cross sect;ion.

and this is correct. But there is more here than mels the eye. When the improvemends of (gL3.86)are: introduced in the denominators of (3-13.78) and thereby in (3-13.82,83), the resal$ is

which. is not the same as (3-13.88). Xn fmt, something h= b e n onziLM and that, is the eonlributictn of the m-kmms in. (3-13.69), which ara not negli@ble for sin B -- m / M , Them terms mtieommuk with YB; the helilici%yraver= ar B; = cr2, and only initial sLraLes of zero magnetic quantum n u m b r are significant. Multiplied by 2m/e2, this contribution to (3-13.69) is (g' = o)

Since this nneGbanis~nris unimportant under high enerw con.diti~m,except for small v d u e of sin 8, we need not distinguish Between %bedireeLions associated ~vithfhe vectors &p and &k. A particular choice of spinom is

and only photons with the same helicidy can be emitted:

That give8 the follo~vingsupplement to the spin averaged differential cross ~ection,

and its addition to (3-13.90) effeekivefy produces (3-13.88). Node th8L the differential cross section for forkvard and baekrvard emission comes entirely from this last process, The value of that cross section per unit solid angle, $(a2/m2),differs only by a factor of 2 from the result of the low energy calculation, tvhen. the kinernatical factor 2/v is replaced by its high energy value of unity [Eq, (3-1 3,73)]. The general evaluation of (3-13.69) involves little that has not, been encountered af;high energies, apart from the frevenl appearance of the parameter

The helicity eonstmction of the spinors in (3-13.69), dogether with the factor 3m/ez, gives it as

Here is a list indicating the various possibilities:

Spin 4 processes

317

The only transition not considered in the high energy discussion is tbe one with zero initial magnetic quantum and fins1 magnetic quantum number of &2. 11; has the anticipated geometrical factor, sin' 8. The immediately obtained form of the differential cross section for unpolarized particles is

Explicit here is the contribution of the only pfoeesses that oecur at sin 8 = 0, thoslc with zero initial and find magnetie quantum numbers:

2

I a2 (sin B = 0 ) = - - (I f 4 rn2~

~~h

+

They are also the snIy ones that survive at low energ;)r. It is the funetion 1 that produces the variation by a factor of 2 in proceeding from Iou. energy ( K = 0) to high energy (K = 1). Another presentation of the differential cross section is

The laat term can be neglected Ett high energi:es, a d we recover (3-13.88). The total annihilation cross section is evaluated as

which reduces to the limiting forms (3-13.74) and (3-13.89) in the appropriate circumtanw. Apart frclm changing the kinemaka1 faetor K'"-" inta K, the same digerential cross section (3-13.101) rtpplies to the inverse process of electrcm-positron creation in the collision of two photons, Factor8 associated with the summaGiarr

M8

Fields

Chap. 3

over final potariaations md the averaging af initial onw do not change ~inee both particles, electron and photon, have two possible po1aril;stions. But there is a dtifference in the evrtfuation af the tokal er0863 metioa, far electron and positron are distinct particles and the full solid angle of 43r applies. This gives the tab1 pair ere&ion cross scletion

The differential crass section for electron-photon scattefing can. be derivd from the electron-po~itronabnnihibtion differential cross seetion by means of Lhe crossing substitutions ~ "-z PI, k+ -kz, (3-13.108) The corresl)onding LransEormations on the parameters M and 8 an: indicateci by the combinations

from which we de~vcr-

In the l i d of high energies t h e ~ eeomespandenees ~ i w l i f yto

If we apply them to the first version of (3-13.84), we get (the kinematics1 factor l / M Z is not involved)

mat Is the sig~ficnurceof the d n u s &m?

Spin prmesm

3-3 3

Consider the individual transition m&trix elements, which are mdtiplm of They change from red to i m e n a r y values in wpm fo cot +B and tan %hesubstitutions (3-.13,108). Sin@ it is the absoluk square of the mtk e l e m a s that @ve probeibifitiw, the additional factors of i are bout efllw1, bat i2 rz=: -1 makeg a ~puriowappearance when the c r o ~ ~ i nmbstitutions g am applied directly to Lhe difperential crom section. The gener&liLy of this @gee%, fos c r o ~ i n gsub~titutionsinvolving a single spin 3 parliele, can be with the aid of a technique that we have nat u e d t h w far-the evalaatian of pola~zstionmnmations of transition probEEf3ilitim by meam of the 8pinar

The crossing substitution on spinors, a,,

+-B

u:

_,, which is effectively produced

by the aegative complex conjtxgsltc?of (3-13.1 l@), gives

whereas the formal repfacement of p with --p ia (3-13.110) pro due^ %heneg* five af this rwdL. We did not eneounhr this phenomenon in retbting elmtronelectron sctatb~ngto electron-positron scattering since two spin. # ~ubstitutiom are used there. The high enerm limit of the efttctron-photon diEerenti8l crow sc?ction. far unp1sriz;ed particle8 is, therefore,

where %he%WOferms corre~pondCo callisions with equal and with apposik s i w of the iaiLid helicitiea, respectively; the dectron and phobn helicitim are mainbined in scatkrlng. The apparent s i r r m h ~ t yat 8 = W is removed if we use (13-13.88) and the high e n e r u commponderzee

The result can be present& as

and %hecome~padingtotd cross sation is

To get the electron-photon diffemntial cross seietion a t arbitrary enerrS;c;s we make- tha appropriate subsahtbns in (3-13.101) and remove the Gnexnatiaaf,

2

Chap, 3

Fields

factor 1 / ainee ~ naw inifial and final particles are identical. Thia give8 directly

The Imf tQ?rmdoes not cantribute EtL hi& enerdcts, where (3-13, P 14) is reg&ined, nor a%low energies where fhe Thornson cross ~ ) e e t i emerges, ~n

The total cross section for electron-photon sedteringt: is

3-14

SOURCES AS SCAnERERS

The photon wurees that appear with inereming powers in Lhe interactioxls 'I;tTzl, ]FV221.. . can also be umd in the exbnded gense fo give an idwEz;ed description of charged particles. As we have already sugge~%ed, this Bimplifid treatment is appropride for a particle that is sufieiently heavy fo be uniaftueneed by its ecattering partner. Conaider then s point charge of stre~@hZe "Ghat ia stationed a t the orilyin,

for which the potentials are

Beginning with spinle~aparticles and the inkrackion W 2I we have, rc3premnting a scatkring proww,

When dealing with an immobile matterer, et11 reference to momentum camervaLion ie last, but enerw tzon~mettiongurviviE?~.The defirrition af the transition matfix i ~ai eome~pondindysimplified version of the general definition (5-12.40), mta-ining only the time in@& factor, and (3-12.43) similarly degenerak~to a statement of the tramition pmbabilily p r unit time that digplays an the ri&&hsnd side a single fwtor of 2z and the one delta function thati establishe~ enerm consenrslion. In the premnt situation, then, the tr&mition m a t ~ xifs

giving the transition probability per unit time ess

A diflerentid cm88 ~ c t i o nin angle is pmduet?d(on dividing this by the incidenk particle Bux, 21p( do,,, and integrating over the well-defined final energy, p; = p '; p:

322

Fields

Chap 3

QP

which does indwd agree with (3-13.65), apart from the us@of af the stationary scatterer. The similar consideration for spin begins with

m the charge

+

wbich is expressed by 1,,u,,)

= 2m(dw,, d ~ , , ) " ~

and the transitim probrabiility per uniL time is

When heli~itystates am used,

where %hefactor slin f i e second entry reproduces the algebraic s i p s that are r?xhibi;t;ed in (3-13.17). For either ehoice of a2 the summation over (zl dvet3; the differential eraas section

as contained in (3-13.M). It is quite clear, in (3-14,13), that the electron retasixlg i b helicity a t high energy while the spin remains inert in spaee ait low energy. Xf IFzx describes scattering by the fixed char@, what do TVz2,WZ3,. . .repre%at? Consider, for example,

Only the field $ ( X ) refers to propagating particles and thert3fore Wgz dso describes an electron scatkring process, as do all the &her Wz,, Thus, the expansion in powers of the static potential A@is no longer a classification into sue~essivelymore complicated prooesses, but represents successive latpprlroxirn* tions ta the complete treatment of the rnation of the padiele in the Coulomb

field of the point source. The inkraction slreletan here acquires more substance, and thereby indicates one aspect of the dynamical scheme that is generally lacking a t the first dynamical level, namely, the? possibility for unlimited repetition of a particular interaction mechanism. Sinee all po\vers of A' contribute to the scattering proeess one should like t o avoid that power series expansion and work direetly with the appropriate , or G$(x, 2'). Unfortunately, the ability to solve Green's function, ~ $ ( z X') the Green" function equations in a reasonably closed form is restrickd to the nonrelativistic limit, in the physically interesting situation of a point source and the Coulomb potential. The latter has a simple connection with the diflerential equation (3-11.36) for A$, whieh here assumes the three-dimensional form

when one introduces the time Fouricr transform in this time translationally invariant situation:

The transition to the nonrelativisti~limit is conveyed by $he altered meaning of energy,

(p0)2-m2d2mE,

pa--+m,

(3-1 4.18)

and the term quadratic in the scalar potential is neglected. It is this omission that produces the essential difference between Lhe two regimes, whieh other~rise are connected by the reciprocal correlation of (3-14.18). T h u ~if, we begin with the nonrelativistic farm of the differential cross section and i~ztrodueethe inverse of thc substitutions (:3-14.18), we get

in agreemexlt -with (3-14.9). This is, furthermore, the exaet consequence af (3-14, X), with the (A O) term struck out, since the nonrela.Civistic solution has the special property that tlll higher polvers of A', or Z, lead only to a multiplicetive phase that disappears on forming the transition probability. Accordingly, the first sig~lificantdeviation from (3-14.9) sriwl.;from tlte last, quadratic term in, W Z 2 ,Eq. (3-22.29). It produces the following modification in the transition matrix: 6(1,,,1T/1,,,)

= (dw,, dw,,)'/2(~a)z

l~herc 1 ( d x ) exgfi(p2 - pl) . x] ----

[XI"

==;

417

(dx) exp[z"(pg - PI) xl

3

Fietds

Chap 3

The earreetion. to the transformation matrix is indicftkd by the substitution

which chaagm the digerential cross =%ion far pin O m g ~ t k into ~q

The fmfar Zip implies that the comec%iondiminishes the cross smtion for pa&i~lm of like eharge and in~remwit for the serttbfing of agpossite charge#. f n dealing only once with the egmt of the quadratic inkr~etianbrm and ignoring %he phase f a e m tbti reprment the consquenee of r e p a b d linear interwtisns, we have obtained only the first term of a rnultipli~ativepower eEieb4 in Zaq. Gs thi~3first tern displetys, there must a l ~ be o at leaat one faetor of %hepaft;icfe @p&, &nmthe corrmtion i8 8 dfttivi~lrtiephenomenon. T b comegpanding discussion of spin. scatteriag proc&s mmewhat mom indhctly since both the deaird relettivisfie correction; and the repetition of the effectiva nonrelativistic inbraetion am combined in TTzzl Eq. (3-14.15). Taken as i t ~ & @ n%be d ~Xathr , implies the following %r%mition matfix modification,

rvhere the time ink?gra%ianhm i n t r d u o d the Fourier trawfom of the electron Green's funetion, which. appsr8 as

In the nonrelativistic limit, r@--+ 1, 7'7 = i y 5 s is negleokd, p' --P m, snd @@l2 - m' + 2mE. This indieate8 the stmeture of the terms that sre to be rqarded sfs already included in the phme faebr, The nonreiati~sticprewnce of 2m in the Green's funetion implies that zpOis its reletivistic counterpart. Indeed, the Adition of the relations

supplies the replacement

--

*(PI%f P?),

(3-14.n) witbin the context of the spin m a t ~ xelement u a d irr (3-14.24). Aceardingly, the mtuaf earneelion contained in (3-t4.N) is (but see below) +

PO

where

The symmetry of this integral in p1 and p2 idicates Lhat V is direct& along p1 -$- pz and we- therefore write:

with

Before diseuming this integd, let us obsente that, when hefieity staks are used,

which meam that the earreetion is confixlied to tramitions in, whieh the helicity does not change. NOW the last b r m of (3-14.311, with %heeomfsnt, fachr +(pI -- p2)%,can be identified ae altering the phase associated with helicitypresehng trawitions. Put another w%y, this term is ima&n%qand, to %he accuracy with which we are wol;king, it does not inkrfere Bvith the prheipal ne:glwbd, &long with the eont~butisiantto the scattering matrix taad can imrtginav parts of the other term, The remaining real hrms of X, the only significant ones for the cross section modification. we

and the two equal inkgals illu8trated by

where the return t;o eoardixlah space fiss b e n advsntagww, Also utilizt=d is the thre-dime?r;tsian.almomenturn i n b ~ a l

(dp)

ed((p@j2-m71j9zt

X

* (3-14.35) 444 The mdifiesltim in the transition amplitude for scattering without helicity change isJ conveyed by

( 2 ~ p2 ) + ~ mn -- (p')a

cos 98

-4-

ah2

*8

1 sxn2 46

- is

1/2

cos

sin *@(l- sin l@)

and the corresponding differential cm88 section for unpotaxlised padicfes is

sin *@(l- sin 38) In this resuit, anid in, the structure of the coordinate intqral of (3-14.34), we recognize s mechanism thst is common to spin O and spin $. The sina c o r n %iaah m ia speeifict to spin %.Another way of writing the last, rsquwre bra~kee, faclor emphmizes the msoeiatiorm of the correction krm with helicity-prwrving $ramitions :

Before con6ixrGng the dkeu8sion, W@ shsfi evaluste the imaginaq pad of 8, wjbeh played no rob in the crass section calculation. On r e c o ~ z i n g&at

infegrates the vector n over the unit sphere. The unit vectsrs nt,na g p c i f y the; directions of pl, p ~ .The inkgra-f e m b ~ r . ~ d t e n

+

where f (al nZ) and f (Bl - nz) are perpendicular vecton of length cos f B and sin .$8, reapeetively, Basing a choice of spfie~calcoordinaLes on .them @vea

cos 98 (ess *B - p) sin%)Bcos2 p

1

2;; dp (1 - F eos 4@)z - (1 -

ge

dp 1 -- p cos #B

cos +e - p

I -- 21log . sin 88

(3-14.43)

Thus, the complete walualion of S is

The complex structure of S, expressing a relative phme shift be;Lwmn kelicity-presewing and hdicity-ch@n&?;irag transitiorrs, hrts a physical inalplica6ian that cali best be appreeia$ed by relinquishing the helicity desoription. With an. unspecifid choice of the U@ spinors, we pre~c3nL$he transition m a t k ;ast (IPI"IPITIIP1'2'1)

MU.,, (3-14.45)

= 2m(d@~l d@P*)fi2

where

M = = f+2'go"-nl

Xng

and

The computation of the btttl differential cross srtction for an arbitrzlry idtid spin invofve~

Accordingly, if f/g ifs a conrplm number, there ia an explicit dependan@@ on the initial spin, proGdd it has a nonvanhhing expectation value in the dirwtion pwn&c61ar to the plane of scattering. A state of definite helicity dws not

have that properLy; it require8 a linear combination of the helicity stabs, b i p r o c a l to the dependence of the dtiEe:rential eram section on the i ~ t i a l pin Is the appearance of plssizatian in the parfieleg far an inithlly unplsriaed beam. At a given scattering angle, the average final spin is

where v is the unit vmgor normal ta the s3catb~ngplane,

ng X

B%

= in Clv,

and

In the special situstions exprwsed by f == =trig gin 6,the wla&ation is complete: p = &l, Notice that the game polw~~etti.on p&r&mekrexprwBw the relative de~nde-nmof the scatbring differential cram =&ion on the i ~ t i spin: d This effect can be demomtraM experimentetEly by a dauble s e a t k ~ n g&mangemen&,6 t h the fim-t,a3cafbAng act pola~eingthe padielw and %hemeand one defecting &at pllitri~ation. If m dmipate them m a and b, the insadion of the polarisation produmd by the fimt Mwfion into the cross section for the seeond scatkfing the reXB;Live factor

The obmrvationatt s i p that neither p, nor pb is zero thus eomm from a dependenee of the final inbn~ityupon the mlative orientrttion of the two aatfefing plsnes. In pa&iculsr, if thf? two phnm are the same geometrically but deflectiaw in opgosik seases am cornpar&, .= &v,, the ratio of the inbwity for &flee%ion8in the 8atme seme to -tlaat for the opposilti! s e w is r &tan one when the individad 8eatkfing a & ~are iiientjeal, p, == pb. The preference for sucemive deflection in the same m m will be b i d with any choice of individual seatte~ngaaglm if, W in the pment di~ussion,the pohfisation paramettt3.r hadJ a definife! sign at aU angfes:

This

it3

3-3 4

Ssurees as scatterers

329

We turn now to examples of the class of phenomena in which both simple and extended photon sources are involved. These are ehared particle interation8 witb fixed charges, in which phoGons are emitted or absorbed. The simplest illustrations are contained in W22, They nre s i d e photon emimion during sealt;ering in a Coulomb field and the creation of a pair of charged psrtieles by a photon passing through a Coulomb field. The relevant part of ITzz for a spin O particle that emits a photon during a collision ~vitha mwsive pa&icle of charge Ze is

where A$(%)indicates the vector potential associRted with the chwge Ze. W~iing the form of the latter that is stated in (3-14.2), the transition matrix element appears as

where

and npis the unit time-like vector that has the single component no = 1 in the rest frame of the charge 2%. Enerw conservation takes the form iL is used in verifying that fb eEectivc: pkobn emission source is conserved, for this is the algebraic pmperty First let us observe the simplifieations that appear far soft photons, where the photon momentum k is negligible compared to p1 - p2 and = P.: This @ves

whieh is the trntnsitioo matrix element for scattering in the Coulomb Eidd, multiptied by the probabiliLy amplitude for photon emission by the source that represents the instantaneous trangition of the eharge ep from velocity p z l m to velocity pl/m. This conforms vvith expetation. We should remark, h o r n @ , fsr future reference, that the connplek negbet of the pboton meehaaied proper-

536

Fields

Chap. 3

ties a t sufficiently low frequencies is justified for finite particle deflection angles, but require8 m r e careful eonrsideration when the deflecdian angle is very s m d . Glomly rebted fo the soft photon situation, but &skinet fram it, is the low enerw or nonrelativistic Iimit. Here, the photon momentum is negligible but any frastion of the initial kimtie enera can be r a d i a w as the photon energy

Using the gauge eE,h, = 0, the trsnsition matrix element simplifies to

and &hediEerential Gross section for specified polariaetLion, emisgion dimctions, and photon enerw is (ulineee~~ary labels are omitted)

The successive operations of summing over polarizations and then integrating over photon emission dirwdions duce it to

and the fufther integration over alt pa.pticlc?se&tering angles gves a cross ection fsr the photon energy distribution:

It is also int;eratiag to consider the digerential emss per unit solid angle dQ l;fitzL

is i n h g r a t d over JI photon energies, from the minimum debetable erreray This is

&Einto the maximum energy aet by the initial kinetic energy T = pg/2m.

?\?hiehuses the inbgration variable

The i n b g a l can be evalusltd in general, moat simpXy by f t z c b ~ n gthe denominetor into 1 - z, 1 $ z, 1 -- xe", 1 - X@-'@, but we shall only present

the reeulL for the circumstance k$i,
(3- f 4-69)

where

- (n - 9)tan 98 This is not v~tlidfor arbitrarily small angles, however. XR ~ o n t r a with t the apparent singutari%ya t B = 0, (Sl4.67) there yields

The apptichility of (3-14.70) requires that 8 >> kki,/T. Still another c a v a t must be mentioned. As in the discurnion of nonradliative wabtering, Wattis only %hefirs%of an infinik series of proeesseb that contribute to the emission af a photon during deflection by the Coubmb field, But unlike elastic ~cft.Lb~ng in a Coulomb field, these ndditional processes do alter the ems8 section, particularly at low emclrdrss and large Z. We do nat intend to go into this matter bere, however, Let us return to the transition matrix element (3-14.57) and no& the fallowing expression for a digerential eross seetian that atill refers La the dedailed energy distribution :

where fhe ~urnnnabionover the polarization of the emitkd phoGon is given by

Here is another, invariant m-a,y to write the digerentiaf eross seelion:

although wc? have not troubled to iatroduee the invariant equivalent of the initial pa&icle momentum. The four-clirnensionaI delta function slaks that

382

Fields

Chap, 3

and, in the rest frame of ?P,which is the coordinate sysCern of physical inbrest,

+

b(nkz) &(P! 4- k! -- p:), k: = (PI kl -- p,)2, (3-14.76) which regsin~(3-14.72). But the expression (G14.74) has a sugestive eharaete?r, for process= rmenbling psrtiele-ptrobn 8esthring are being considered. Of course, the incident photons are virtual, since kg > 0. Nevertheleas, this point of view haa prmlical advantwgw a t high ener@es. Viewed in a wifable coar&natcr;system, a, major fra,cdion of the digerential cross section can h evaluaM in terms of the propedies of r e d pholons. fn the physical coofdinak system, the incidcnt pa&iele is considerd to move along the %-axiswith velocity et ru 4-1, so that How think of the coordinate system in. which the particle is at rest inifiablly and

the- charge 2% moves dong the z-axis with. vdocity

--D.

In &is frame the vector

n@has the components [(O, z,y, z)]

The requirement nk2 = 0,which asserts tbe static nature of the field in the pbysicd, or &attached, wordinate syslern, becomes in the particle rest frame

and therefore kg = k$

+ kgl[(l - v2)/u2],

where kg = kK

+ k:,.

Thus, in circumstances for which

and k i is sufficiently small, it would seem that the virtual photons dould be approxirnaM by red ones. There is one app~retltdifficulty, hawever, Playing the role of pol8riaation rrecbr for the incident photon, is the vecbr np,which is indeed such that nlcz = 0. But we should etxgecb that the pola~zl~fion vmtor of a red photon is, or can be cho~en,without time component or component along the propagation direction, which i s here the negative z-axis. This suggesGs performing whrat should be a gauge tran~formation: (3-14.83) n' -+ nC - (?/kg)%, w h i ~ his comtrucbd bhave vanishing time component in the pa&iele rest frame. The z or loxl@tudinal component of the new vec-tor is then

and, provided

r k ~ l k> ;> I/T,

the transformed veetor w i U be predominrtntiy the multiple -?kT/k; of the tramverse; unit vector kT/kr, which acts as the ineidrtnt phokon p o h r l h o n vector, But all this is contingent on. the magnitude of the addidiond term introduced by the transformation (3-14.83)1 which is p r o p o r t i d to

Now, k z ~ g=

kip,

- *ki,

kzpl = klp2

+ +ki

(3-14.87)

and therefore

whieh indicates that the substitution of real photons for the viPtu~t1pElotOns will be justified if suitable upper limits are placed on kg 2;: k:. A suggestion of the magnitude of this upper limit is obtained by comparing, in $he gauge ef, &,pz = 0, the pa&icle r e ~ frame t values

and

namely

k~

< m.

(3-14.91) We shag eonfine the discurnion to the diEerential crag@section that $ive8 the mew spectrum of the d t L t n l photons in the Z eoordinete system. Since we are ROW f i d y establi~hdin two &Rerent eoodinate systems moving rehtive to eaeh other at pmctieally the spectd of light, a few notational distinctions are needed. The Z h m e photon e n e r a will be denoted by

K = --nkl= r(k!

+ vkl,) r: rk!(l

-- eos 8),

(3-14.92) where B is tbe photon scattering mgfe in the paPticfe frame, The kinematics of tbe phaton s c a t k r i q process in that refewnce frame, as derivd from 0 = (p,

+ kz -- k112 + 'm

= -2m(k$

-- k:)

f 2kyk$(1

- cos 8),

(3-14.93)

334

Chap. 3

Fialds

is expressed by k? =

k:

1

(3-14.94)

+ (k$/m)(l - cos B)

Some derived relations involving K and the Z frame particle energies

E2 = rm,

E, = E2 - K

(3-14.95)

are -K= E2

(k!/m)(l - cos B) 1 (k:/m)(l - cos B)

+

% =k! g P E1

(3-14.96)

and

The latter shows that the incident photon energies k: that can produce a scattered photon of energy K (two different coordinate systems are used here) will be restricted by

Also useful is the differential relation

E2 dK k: - - = - sin B dB. El El m Considered in the rest frame of the incident particle, the differential cross section for photon-particle scattering is

where the factor in the denominator arises from division by the photon flux multiplied by the particle density, 2k: dok22mdo,,. The polarization summation and average is

f

C le:,r, . et2~,12= 1 F [l - (%er,r,)2)

X I As

= f (l

+ cos2 B),

(3-14.101)

which also follows from (3-12.124), with n = p2/m. The final momentum integration can be performed with the aid of the kinematical relation

and

which eould also be produeed by transformation from (3-12.117), the center of mass expressian. for the digerenlial craras swtion. The form in which we shall use this digerential cross section is obtained from the relations of (3-14.96, 97, 99) EM 2 dK 1 [mkido] = 2ra (3-14.lM) E% 2

-

The real photon contribution to the differential eroas section (3-14.74) is

or, with (dlez) = r dk$ dk2,dki,

If we change the seale of the incident photon energy in the follo~\ingmanner,

so that z range8 from X to m, this r e d s

Acoording to the restriction (3-14.85), the kg integml should be stopped a t a louw limit that is s fraction of ( k g / ~=) ~( m 2 K ~ / 2 E 1 E 2 )one 2 , that ia large compared with l/rZ, say l/?, but a negligible error i s introduced by extending the inkgral down to zero. The value of the integrerl, un&r the condjifions

and &hen

The suggestion implicit in (3-14.91), h, S m, is that no significant interactions oceur for larger values of the Cr~nsvemmomenhm, We propose to e x a ~ n Chip, e que8ti:on. A quick indication of the quan$itaLively eorrect resullt is obtained i f one merely aempk %hatthe eEee.t;irre replacement for &, ia indepndent of' K . Then it sufiees Lo conrzpgre (3-14.112) with the Merenth1 crass swtion Bppr* p ~ a l eto soft photons. This diwwsion tsktls place entirely in the &attached physical coordinate system. The paliafization summtioo, in $he digerential e-mm swtion derived frsm (3-24.61) i s

This expres~ioncon$ains the only reference ta the direction of Lhe emitttld pholon, md we shall integrate over all solid angles. Removing the factor l / K Z ,that iategal is

where R is now the uniL propagation vwtor of $he photon, and high enerw, @oft photon simplificatiom have not yet been irtfrducd. We first obgeme t h ~ t

m denominabrs, it is ub3ef~ltO To inhgrak the km containing G

(EE-J E2 --- E )

Sources as seattrrrarsl

3-1 4

337

as one can verify immediakly. Then we have

The v integration can also be performed, sf course, bud it is preferable to leave if as id stands. The soft photon differential cross section is

Lvhich still needs to be integrated over the deflection angle 8, But now we must recall the M-arningthat the soft photon sirnplifications need to be qualified for very small angles. I n contrztst to the singularity of (s14.118) at fl = 0, the minimum value attained by (pl k l - p2)2, ~vhichoccurs for scattering and emission in the fortvard directiorr, is

+

It ~vouldbe more accurate to replace (pl -

by

and we recognize a characteristic aspect of (3-14.1 09). f f is unnecessary to incorporate these refinements, hawever, They atre the content of Lhe real phooton compubtion, The soft photon evaluation need only be applied at angles such that

rn2~

2E sin H@>> -t 2_E;"lE2

C3-14,121)

[shere (3-1 4. X 18) can be used without correction* Ernployirlg the variable y

"=:

(Elm)sin $B,

(3- 14.122)

338

Chap. 3

fieldra

we begin the integration at a conservative upper limit to the real photon disc u ~ ~ i okmaiX n , << m, and thus Ymin

r==

(kmax12m) K< 1.

This gives

(3-14.124)

where the y integral simplifies to I

dv(l

+ v2) log f. -X

---

u2

and 16 z2a3dK -

(3-14. f 26) 3 m2 K Whexl the real photon contribution (3-14.1 12) is considered under soft photon conditions, E l --. El2, the addigion of the two part^ precisely cancels fog (mp,,,) H, and the inference is that, $enemlly,

d@

--m

virtual

+-

The inference is comeet, as we shall verify by repeating the virtual photon calculation without using soft photon simplifieations. But Haraid interrupts,

H. : f have not forgotten that you decided to omit all reference to historical nlatters, but your use of il, parametric device for combining denomintztors prompts me to ask about a smafl historical point. The technique of introducing parameters to unite denominator products into a single denominator is invariably ascribed h Feynman in the literature, 1%it not true, however, that the usual intent of that device, to replace space-time integrations by invariant parametric integrafs, \vas earlier exploited by you in a related exponential version, and that the elementary identity combining two denominators, (3-14.1 16) in fwt, appears quite explicitly in ra paper of yours, published in the same issue that contains Feynman's contribution? X. :Yes. In the physical coordinak system, at high energy, radiation processes occur predominantly near the forward or longitudinal direetion. We express this through a dceompasition into longitudinptl and transverse components, sts illus-

whi& ase d t t e n in terms of Ihe dransverse momeRfunn 8uppfie;d by the *B tionav charge (3-14. 12921 = h 4- PIT, as --klp,=(Ea/zK)oz, - - ~ I ~ I = ( E ~ / ~ I ) ( E ~(3-14.1301 /~K)~~, h

7

The virtual photon contribution to the differential cmss section that is produced by (3-14.72,73) is ~.---"---..%.=--

virtual

r2 E2 K (S14.f 32)

U B ~ ebpprop~~k R~ variable tra~glalions,we hwe

and, carnbining denominators with the aid of the v psrannehr,

The iatraduetion of &hetvariable y = kT/2m

Flelds

Chap, 3

which doe8 inded differ from (3-14.124) only by the faehr E1/g2that i~ nededt. Co combine properly with the general real photon contribution (3-14.112) and prduce (3-14.127). To give sn analogous discussion for spin 1 psrtioles requires, first, the explicit f o m of the electron-photon diRerentital cross wction in the r a t frame of the incident efeetron, That is waitstble Lo us through transformafion of (3-13.1 l?), %hee n k r of nnw e r a s section, but, there is some i n k r s t in a direct derivation. The transition matrix element is

whieh urns purely spatial polarization vectors and a simplified aodation. The m a t ~ xfaetor in square brackerts reduces to

where the latter exploits the fact that uz i s an eigenvector of ?%with eigenvalue -4-1, and introduces the aotafion nl, z far the unit propagation vwfmirs of the photons. C o n ~ i d e e ~real g polarization vecGors for simplicitcy, the tr%nsition nnad~xelement becomes (d%, d o k , do,,dwt,)1122e2u:[-el

+

e2

+

+?,(g

e p nl

X

e2

+a*e2aeal X el))uz. (S14.139)

term and the spinor8 are omitbd we ge6 the comespondirtg spin O If the expremion. The summation of Lhe tr%nsi.tionprobability over final spins can be pe$armed with the aid of (3-13.11Q), giving the pino or faehr

where the kinematics of the mXIision dekrmine

The matrix prduet is reducd by omitting a11 krms that eontain a rli fa~tOr, since uz is a eigenvector, or s a faetor, the latter expressing the averaging over aft initial spins, A quite sho& edcufation then gives the foffo~4ngfar

X [l

- cos B(el

=

e2)'

+ el

e2n1°ezag et g

- el X

-

ez alet X

ozJ,

(3-14.142)

where II~

(S14.143)

= cos 8,

The appamnt dependence of the ~ m n d term on the pa)$sri%stionvectors disappwm on. invoking the identity

C@Z

X

X h3 *

ez) m21 = (et X e2)% cos @

E@l

=

-- el X e2 * nlel X et nz --el . e2n1* e2aS elf (&l4.144)

and we g@$

replacing (et e2)' in the spin O cross section. When summed and averaged over photon polarizstions the differential cmss section that appears in plme of (Sli4.103)3s

For the purpom o f evaluating the pbobn ennission emss section, this is wdm

acwrding to Lhe ~ecandrelatiun af (3-14.W) ~ n the d spin O rwult (3-14.104). The e o m ~ p n d i n gmo&fication of (3-14.109, 1f 1) is

mK -

log 2BiEt

The virtual photon contribution i n h r r d frbm enerw appearance :

fi2hsts Lhe fobwing high

whieh is to h aurnmed over the photon polakaation and the find electran spin, and averaged over the initid spin. As in the dimurnion of pfxoton-elechn s~afhringtat small angles, transitions with electron heliciw ehsngw %res i p fiea ant, The cafeulatian can. h prformed dvtitnbgeau~lyby methods tha$ have dresldy k n illustraled, using photon helicify states and the photon e ~ s s i o ndirection for reference, and expreming the efeetmn helieity states wilh fhe aid of suitable rota;tion mal;rices. We shall only give the re~u1GsBere, which are classified with respect to helieity change:

For soft photons, helicity ehsnges are fehtively negligible and the spin O &metare is reproduced. The sXighLIy different integral assacigM wi.t;h KeXieity ebanges is 1

clu

dv(1 -- vZ) log (1

The individual, eram section~sare da virtual, no

virtual, yea

--

v2)

which add La & virtual

This virtual phohn part and the real fiaton contribution of (3-14.148) cambine: ta give the final high ernerw f a m of the diRerentia1 cross =%ion for phobn e ~ m i a r by r an elecfran defleetd in a Coulomb field,

The proeegs that converts a pfrofon into a pair af oppositely charged para croming tiele~,in the neighbarhood of a tafstianary charge, is m l ~ by transfomations to the reiaction jurzlt eonsiderd. T h m %ramforrnatiorrsare In order to recan~truetthe absoXuk squard transition matrix element, we take the known differential, era88 section for phohn emis8ian, d ~ ~ refer~ng ~ ~ j ~ ~ . CQ definitr: spins and pola~zations,and f a m where the second high eneru version also emphmizes that we arc?inlerf;sM only in the energy specification of the particles. Under fhe croming transformation, dwk,. The differential the kinematieal factor do,, dok,h,, beoomes dw,, bp; er088 =etion refeming to an inciaent photon beant woufd require division by 2K dwr,, K = kg, and thus ap8l.t from the spurious minus sign tkat accompanies this formsl substilution, with spin 8 particles. When cross sectians involving summatioas over final helieitiea and averages over initial ones are used, appropriate correc%ionsmust be made far the different weight faclom. With spin 3 partieia this if9 '1;18f required, in thew reactions involving one initid pa&iclc?:and fwo final particles, ainee b t h elwtron and photon hsve t ~ r ohelicity staks. For spin O padiolm, howwer, the photon emission crass section, summed aver the phobn p~1arizaLion, contains an additionztl faetor of 2 relative ta the photon sbmq%ianer088 section, where photon polarisations are averaged. The implied pair produetion cross sections are 8 z~ ~ QE ~ E ~ spin 6: cFo = 3 m2 K 2 K (3-14.159)

1abls are omit$&, tomther with charge indices, since the paditioning of the phabn enere,

K =E

+ Eft

(3-14.

XW)

doe8 not de*nd u p n the ~peifiecharge wimmentts.

In u8ing exbndd photon. sources 4x1 reprmnt heavy eh&-& padielw, tbe the pdnf t h ~ new t kinds of p&&iele~am eneounkrd. They are idealixd vemione; of cornpiLe s y ~ t e m ,S i n ~ eh y b o g e ~ ea b m ~ are the mast familiar examplr?,they will be bmed H-pafiielee. We first conerider a sbtic mume d i ~ t ~ b u t iJ"(x), a n and the p h ~ t i a fAl@(x)in Bame convenient p g e . The time tramlwtiona1 iinva~ance of Green" funetiom, A+ for emmple, is conveyed by

where

The B. E. slynnme;try

af

this zero spin Green's function

&a

in which rnat~xtramposition is applied to the eharge irrdice~. Eigemfuxlctions, solutions of the homogeneous Grwn" funetion equa&ion,aw~ktin eonstmcfing the G m n % function. They are Xabld by enerm and charge values, supple mentd by. odher qumtnturn numbrs which air@usually relahd to a n e l m momenturn. The homogenmus equation and ifs complex canjugab are

Notice that the joint sign reversal of p" and p' interohanges the forms of the two differential op.etra;tors. Hence the eigexlfumtions can be so chomrr that where a'* iindieaw a rf3X~hdSt?t af quantum rrumbrg, If G' iaeludes a magnetic quantum number, for example, am refern to %henega$ive of that quantum numbw. If i~ pomible to chwse the eiigenfuncfiona in a way tha&identifiw at* with a', While not usuauy convenient for individual problem that Gbaiae simpXifiw gener8f disewion~. To avoid canfugion, we &all a h undem%and that p" is s positive quantity unless there is a specific indication otheraise.

Another consequence of the equation pair (3-15.4) is the formal i n h p a l relation (dx)4,op,t,f(x)

* (p0' $ p'" --

2 e q f '~( X ) )

+,eoppopf ( X )

= 0, (3-1 5.6)

~~rhich is incorporated in the statement af arthonormdity : (dx)

+pop,r,p(x)

'(p" + p"' - 2epfA@(X))+ , o ~ ~ , ~ ~ ,=~ ~ ( x ) E

~

~

o

~

?

~

(3-15.7) f n the &Beme of the staLie source, this properQ is obeyed by the Imotvn eigen-

funcfGions mmciatd with small momentum cells,

according to the orthonomality of the % and the clarification of the spatial nmmslization given in Eqs. (3-6.26,23). Here the momentum veebr p f l a y s the role of the quantum numbers a, which aXso specify the energy value. There is an analogue of (3-15.6) in which the bamogeneaus equation obeyed by is replaced b y the inhomogeneous Green" function equation,

+ , o * ~ , t , ~ f ( ~ )

supposed to be isolated, Sufficiently near a particular energy eigenvatue however slightly, fmm all: the others, the Green's function is dominakd by the eomwponding eigenfuncdions, and (3-15.9) implies that,

The analogous behavior in the neighbarhood of --p"7 demanded by (3-15.3), is W

: &+(X, X',

P@) --

*+p@#qfcar ( X ' ) C +p@tgr@p(x) pot +

q"O"

(3-15.11)

A representation of the Green's function that ia valid near any part of the physical enerw speclrum, or its negative, is given by

in vvhick we have dso exhibited the appropriatr? urn of the parameter e -+ +O in order to satisfy the time baundrtyy. condition--pasitive (negative) hquenciesl

~

,

346

Chap, 3

Fistda

for positive (negative) time differerrccs. This is verified directly:

Wheu the eigexrfuactioxrs (3-15.8) are inserted in (3-15,12), the 'Emojvn form of the free particle propagation function is recovered. If \re nolv allow to assume both positive and negative values, the Green" funetion can be premnted more compactly as

~vl~ere ( f ) signals the extended meaning of

and

Not to be confused with the infinitesimal parameter e is c(p''), stating the algebraic sign of p". Another \\.ay of writing this function, in ~vhichthe scalc of c -+ +O hass been changed, is

The time-dependexlt versioxl of (3-15. f 5) is

We urc intcjtrested, in this section, only in tltst podion of the energy sptlcLrum which is inaceemible to a free particle: < m, Such s$atf?scan exist, localized in the neighborhood of the source, if &hereis force of attraction btween the particle and source, of sufficient strength and range, Tn the familittr situation of the long-rang& Coulomb interitctiotr between oppo~ihlysigned charges, no minimum strength is required, %ndan unIimidd number of sueh bound s t ~ k s

exists. These are the H-particles, Wllat are the emissiorr arrd sbsoqtion sources for H-particles? The insertiorr of the Green's funetion (3-15.18) into the souree coupling t e r n

TXre time dependent quantitim

are sources associated rvith the particular H-particle label4 by pat, p', with of appearing as an %dditionalindex snalogous Lo pin. Thsf the= sourcw mfer only to time conveys the immobjlity of the very massiw W-particles, The repeated operatjon of these sources will inject any number of parLIeles info bound statea. 8ince no a~countis being 8ven of the inter&cfctionsamong the particles, we ~haffbe concerned only with the propertie8 of rr single particle, bound to the murect anid forming an H-padiele. Nti3verthelesa, id i s desirable to verify that probabili6-y requiremenk are satisfied in the dynsmieally simpfifid many-pa&icle situation. The usual consideration of tc causal arrangement of emission and absorptioxl source8 leads h

and the eausaf arrangement restricts the energy summation in (3-15.23) to the physical, positive values. The mulLiparLicle Aabs produced by the caueat analysis of the vacuum amplitude have the usual canstmction in krms of gauree producls, and probability normalilration implies that

The direct verification of this property employs the relatioll

For spin $ particles, the transform Green" ffunctian intmdtrcd by ivriting

b~nathe F. D.. symrnetq [r"G,(xf,

X,

-pO)lT = -yOG+(x,

(3-15.28)

X',

and o h y s

Eigenfunctions are defind by %hehomogeneous equstiom

The tmroequations are related by the Hemitisn chsracter of the matrices If, imkad, atkndion is paid ta the i m a d n a ~naGtxre of the nntktriees Y", we ider the eigenfunctian connection (for suitable choices of a') The eigenfunctions are normttEiz;edin accordanet:with the inbgal property.

earnbind with the symmetry requirement of (%15.28), leads Lo the Greenp@ function construction (pa' > 0) :

OX", more

compactly,

~ ~ o e =( - ~~ - ~~ " l )(

- ~ ~ )

=

1 p@'(1 - ie)

-

*

The r?;xpEciltime depndenee of the Green's function irj given by

(3-15.37)

in which

- so')= - G - ~ O ~ ( Z O ' - xO)

~~o.(zO

= n(xO- zO')v(po')i~b$? (zO- zO') - *(X'' - X ~ ) ~ ( - ~ ~ ' (X') ~ A 2"). ~ ~ O ~(3-15.39)

-

Time dependent H-particle sources are defined by

1( d ~ ) $ ~ o( x~)~* ~ Oa *t ( x x ~ )

s p ~ , q , a (xO) t =

-qta'(x0)*, and

With a causal arrangement of sources, the vacuum amplitude becomes

(0+10-)*= (O+IO-)TL exp

[

P

,i q & o , q t a ~ i v ~ p ~(0 ~+1s0~)' a"t ]

(3-15.42)

qa

where we have introduced

The completeness of the multiparticle states, which have the usual source product representation, implies that (p0' > 0)

Direct calculation from (3-15.41), with the aid of the relation

gives

since, in accordance with F. D. statistics,

Now, let the static source that represents a heavy charged particle be supplemented by a simple photon source. The terms in FV that contain one such

3M)

Fields

Chap. 3

photon source deseribe processes in tvhicE.1,through transitions between difierent H-particles, a single photon ifs emitted or absorbd. Using two photon sources, we describe transitions that result in two photons b i n g ernitted or absorbed, rand also transitions in which 8 photon is scattered with, or kvithottt, an secompanying H-particle transition. And so on. I t is convenient to use the characterigstion af electromagnetic prwesses in which all interactions refer directly .to the ehaqed particles, with the electromagnetic m d e l of the partieIe source transfornled into a gauge re~trictionoxr the veetor pokntial. We recall the space-like ehaice [Eq. (3- 10.49)f

The static source defines a coordinate syskm in which nc"can be chown to have only s Lime component. Then fc"(k) h= only spatial components, \vhich are proportional to the vector k, and the gduge condition reads:

This gauge i8 called the r&distiongauge, since the proprty of transvemalidy to the momentum vector is characteristie of the polarization vechrs asmeiated witit photons. Xt has atso, but less appropriately, been termed the Coulomb gauge. As i s most evident from the three-dimensional form of the srscorrd-order MaxweX1. differential equations,

tile scalar potential A0(z) in the rdiation gauge i s necessarily given by the instantaneous Coulomb potential of the charge distributiort,

But the coxlverse is not true. If it is required that AQ(z)shell be the instantaneous Coulomb potential, presumably the intent of a Coulomb gauge, the inference is that the time derivative of "C" A(z) must vanish, No restriction is thereby placed on any static compnent of the veetor potential, A(z). I t is $he radiation gauge la, ~vhicfithe sta;tie potential AP(x) refer^, The vector potential A(x) can be used to represent the field of nuclear magnelic dipale moments, leading to the hyperfine structure af H-prtrticles. Xn the follotving, however, attention will be confined to the static chmge density and its scalar potential. This avoids notatiorlial conflicts with the pokntials that are associated with the aimple photori sources. The latkr are only needed far from %heemission or detection sources, There, they reduce to the vechr poten-

ia the transverse or divergenceless part of J(z) and, indad,

The solution of (3-15.52) in regions that am musally inbmediatk? between emission and debction sources is, of course,

A(t.1 =

C IAkX(z)iJzth kX

with

ALh(x)=

( d ~ kliZet*eiL' )

4-

,

i~:th~kk(z)'l,

(3-1 5.55)

V.Akh(z)=O.

(3-15.56)

We now distinguish A+(%, X?), which contains the static scalar potential A0(x),from A$($, 2'). The latter also describes the effect of the vector potential A(x) that represents photons. The differential equation for the Green's function &$(X, X') can be presented as (p = --$V) (--

(ia, - ecl~'(x))2- $ m' - i e ] ~ : ( z ,X') = 6(2 - z') + (2epp A (z) -- @'A(%)')A$($,

2')

(3-15.57)

The use of the Green's function A+(%,z') converts this into an integral equation,

which can be solved by an iteratian procedure. AIL this is entirely analagous to the discussion of (12-12.27) and as there, the suceesslve interactian krm, W%,,WZ1,. are most ~ ~ m p ~ eexpws~ed tly with the aid of the particle field,

..

(3-25.59) Thus,

and so forth. The particle field +(X) is related to H-particle sources by

At a time tfr& is eausally inbrmdiab htween the actions of emission md absorption sources, this becomes (p" > 0)

Aewrdingly, the transition m a t ~ xdemtjnt far a pmcm in which the H-p&&iele labled po"af' transforms into the H-particle denoted by p@'o' (charge specificstians &reomittftd 8ince only one siw of charge, gay qF8d f be bound to the 8t&tio murce), vvitht the tr: ion of a photan, kk, is (dx)dp@ta. (xltp . e:he-R'x#p@erorfi (X). (&XS*M)

The transition probability per unit time is given by

8ince enr?ra cammation precisely specifie~the photon enerm, > = - PO',

(3-15.66)

there is no n e d tA, mfer to the enera distribution, We @hall e%w out the pola~z;ationsummation and the inteqa;tion over alf photon emimion directiow, using, for simpfici(;y.of illwtrstion, the nonreI~ttivisdic~i%a&tiaxl.Eere the phofon momentum, but not i b enerw, it3 negligibje, and $he p a ~ i c l eeigenfunctions are relrttd ta the con~entionallynomaE~ednonrela&iviaticwavefunctiom (X) by t pot = m E': (3-25.67) cftPalaf(X> 2 czm) 1 f z $&#a#Wt +*E"t

+

in virtue of the normali~ationcondition (3-15.7). Using ~ t s n d ~matrix rd element notatian, this dves

Since polaizatian v ~ b xealiae m oxlly two of the three o&honoms'l unit vecbrs, the palarisafion gumm~tionand integration aver emkion directions produce the f~ljowingexpression for the tr%nsitionprobability per unit time;

which also wea the m a t ~ connr;.ctian x between the parkicle veloeify and posiifion vectors, This is the probability per unit time for spoa$antsous pbsfon emimion

in a transition between the specified H-particle states and, as such, is Einstein's A-coefieient. A less gpecific A-caeEcient that refers only to energy is summed over a' tand averaged over a"". When a photon is incident on an H-particle in the state p'"a", a transition to the state po'a' can occur if

The transition matrix element is

ttrhich implies the transition probability per unit time:

After integration over the sharply selected photon energy, this can be expressed as the Einstein B-eoeEeient, which relates the transition probability per unit time to the photon energy density per unit angular frequency range: k0(2k0 dwr/dka). Averaging over the incident photon polarization and direction of motion @yes the nonrefati-visticexpression

The simplicity of the ratio, for a given pair of H-padieles,

is a reminder that, apart from the kinmadittal factors involved in the definitions, the transition probabilities for single photon emission and absorption, are inkrchanged by the photon crossing transformation, k p 4 -kp, The emission and tllbsorption rates are equal when the definitions refer to single photons of definite polarization. And, as we learned long ago in the simpler coxrtexl of a probe source, if n photans of the appropriate frequency are present initially, the absorption rate is multiplied by n and the emission rate by n 1. The latter represents the combination of stimulated and spontaneous ernimion processes. The analogous discussion for spin $- particles begins with the Greenpg function difierential equation

+

and the equivalent inkgral equation

The succe~aivephoton interaction krm8 arc: exhibibd, with the aid of the padicle field (3-1 5.77) as; [compare Eq, (%12,26)f

sad so on, The particle fi&

eotn be expre~sdin brms of H-p&&icleB Q U ~ C I ~ B ,

and, at a time internedide between %heoperations of emimiaa and rzb~orption, gaurces, is given. by $(X,

s') =

C

' [$p@larG*(~)e-ipa

0

iqtp~'a8ap 4-

*e

itl:p0~pfa8#p0tqfar(~)

1.

(3-1 5.86)

pQfqglo6

The tran~itionmatrix element for single phohn emission is

One can eombine the eigenfuncdion digerendid equ4ltions in %hc:mamer af (3-6.67) to produce the inbgral identity

In a non~lafivistielimit, where the

Jtpofa<~) m

ra,the last tern, containing the matrix TOY

approximab eigenveebm of

= i Y a ~ is , relatively negligible. It

would be inconsistent to retain the W * k X e* contribution while repla~ing @-a.z by unity, d a r e it multiplies p * e*. The next term of an expansion i s

and one mcopiaes the orbital contribution to the magnetic moment, wbiGh ad& Ga the @pinmagnetic mantent in the mannrsr rc:presenM by g = 2. If we neglecl thia mametic dipale radiatim, and the rdated eXeet~equadwpofe rdiation, which is tke other b r m on the right-hand side aE (3-15.83), them rmains only the rd4ttion of the electric dipXe moment, eq'x. This i s radiation associded vvith aceebmbd charges, and is indepndent of spin. Inded, witEr the sirnpiifieations irnpIjed by retaining only the firf;t two te

and replacing spin 4 eigenfunctions by nonrelativistic Brave functions, in &ccordarrce with the normalization (3-15.33), we have

This coincides with the correspondixlg spin 0 limit, (3-15'68). A similar consideration, relai.Led by the photon crossing transformation, applies to the tabssrptian process, We shdl discuss orlily in the context of photon scatbring. Using the spin 0 structure (3-15,6t), we insert the causal field decomposition

and isolate the terms of interest:

The general transition matrix element is

where the dyadic V has the components

i112~12&+(~, xl,

- kg)p;e-"l

"'j~,o~~,~~(x'f

(3-15.88)

The A+ symbols are the transform Green's function A+(x, assig~ledone of the values p"+k~=p"'+k~,

po'--k$=p'"-k~.

X',

p'),

with 'p (3-15.89)

Far simplicity we consider only the namrelativistic limit, where the photon momenta are neglected, as are the terms in &+(X, X', )'p having denominators p'f $ p' N 2m, in contrast with the denominrttors''p -'p = E' - E. This gives (using a slightly simplified notation)

3W

Fields

Ghsp. 3

If we restrict our attention to elastic scattering (E' = E", ky = k i = ko), the msult is la digerential cross section far deflection of the phobn into the solid angle dSZ :

from .inrhich less spwific erass sections are obtained by summing over aband 8veraGng over a''? by summing over XI and avr?r&ng over X2, and by i n b u ~ t i n g over all solid angles, At ghobn enerrgies %hatare Xsrp in comparison with H-pa~iclt,binding e n e r ~ e only ~ , the 1mt tern of (3-15.91) survive^ and, with a' = a", one recognizehs the Thornmn cross section, which describe8 the scaftr;et;riagof (on a, rekL i ~ t i eseale) low enerw photons by a free particle of charge rf=eand mass m, Another limiting conneckion with Thom~onscattering should appear a t very low frequencie~,smsll in eompaeson with the mergy inkrvals beI;wwadigerenf H-grarticlea. m e n a, photon of essentially zero frequency is sca,&&redelmtically by a, part4eular B-padicle, the dynarnid connectians \;vith other W-pasticleg are not in evidmee, and the seatlering shoutd be descr;ibetd: by the Thornson fornuLa appropriak to the H-par$icXc?icharge and masts. Sinee the latter hafj been ideafi~edas infinite, the elsstic scattering erass ection should vani~has k@ -+ 0. This implies a set of relations, known as sum rules, which can be va~ouslygreesented. The immediate form implied by (3-15.91) is

Here is another:

(3- f 5.92)

and yef a third, intermediate f m , obtained by repfacing only one of Lhe momentum. matrix elemen* fetctorar by the corresponding coordinate matrix elemertt, (Bfa'lpklEa)-- -..-*E: -- E")(E"c4'(zkfEa), (EalpkfEkf')== &(E - Ef)(Ealza: is

(3-15.95)

The 1st version shows the mathematical origin of the Bum rules; they are matrix elements of the cammutation relation

The elementary ori@n of the sum rules does not detract fiorn their gignificanee its conditions of consistency for the phenomenological particle des~rip%ionof eomposik systems. That is emphasized by removing the idealization of infinite mass to obtain the necessary result involving the charge (Z" - I)@ and mass M of the H-particle, viewed as a, composite of the two particles with charge and mass assignment8 $ivm by -e, m (electron) and Ze, &g - m (nucleus). Tfre gcattering amplitude fhat appears in (S15.91) describes the pme e s ~ in e ~which the electron absorbs the incident photon and emits the seatbred photcllm. To this will now be added the reprwentatian of the processes in which the nucleus alone perEorms these acts, and of those in which both par-tieles are involved. Altfiough we have not developed the relevant general famalism, the necessary modifications here are quih clear. The ma;trix product terms of (3-15.91) describe two successive interactions with the electric current, to which both particles now make conlributians:

where is the rdative momentum in the center of m s s frame. In addition, there is a eoxltributian in which the scattering takes place in, one wt that is associahd with an individual particle; it is extended by

I n carrying out the reduction of Ghe matrix product, the relation betuveen relative momentum and relative velocity is now given by the reduced mass, m(M -- m ) / J f . Removing the factor of e2, we find that what replaces the amplitude of (3-15.91) for s, realistic H-particle as ka --t O is, apart from the gola~zittionvector product,

Tbis is just what is demanded by the phenomenological H-particle description. Let us idroduee these realistic modifications in (3-15.91), while retaining sn arbitrary value for k'. The sum mle (3-15.92), together with another sum rule fhat expresses the null value of the earnmutator Exk, Q], ~ a be n used to

rewrite the er088 ~ e ~ f i oassnl

Here, -ed is the internal eleetric dipole moment of the system in which df position ve~torsrefer do the cenhr of mass veetor

which relaks d to the relative position vector

In this version, the fsw frequency behavior characteristic of the H-particle is explicit, while the disclosure of the constituents a t high frequencies is assured by the sum ruEe.es, which here produce the earnbination

If is the amplitudes for individual scaLLhring by the two particles that are add&, and not their cross sections, since this simplified treatment neglects the photon, monnenbrn and thereby amumes that the photon wavelenglh is large eompard t;o the particle separation. That restriction, is easiXy removed by in~ertingthe relative phase factore and, with increasing frequency, the coherence between the two scattering amplitudes disappeam, It is also possible to derive (3-15.101) directly, by using s digerent gauge whieh is specifiedty adapted to the long wavelength regime.. If the electric field of the photons is homogeneous over the inkrior of the H-pahielc! and procems involving the magnetic field are negligible, s suitable choice of potentiwfs is A'(X,Z@)= -x*E(R,zO),

A(x,zO)=O,

(3-15.2M)

where

E(R, zO)=

( d ~ "2ik'[ekh i) exp(& R kX

- i~:t&e:&exp(-ik

*

R

-ik'z')i~~~~

+ ikOzOf].

(%I 5. 107)

The seafar wkndial ~ouplesto the c h 8 ~ density, ~e

Transitionis btwwn diBerenL H-padi~1-warf3: e x c i w by the inhm&i d i p ~ l ~ moment derm, ernd this contdbution Lo photon scattering repraduees the sumanalion k r m of (3-15.101). The exkmaE digole nnomen6 (%5 - J)aR affects only the motioa sf the given H-pa&icle. The scatkring amplikde $hat it prais gi-ven by the diagonal matrix element of the a p r a h r

The reference to the rest frame, the state of zero momentum, redurn8 %histO

which eompleks the derivation of (3-15.1011) since there is no joint effect of the Bihrent kinds af dipole mannenb, The analogue of (3-15.86) for a spin 3 part;icfe is

and the dyadie thaL replaces (3-15.88) h= %hecomponents (dx) (dx')#,o~.~(x)*~'[r~e-''~

"G+(x,

X',''p

+k : ) ~

te"2*''

Again, we only consider the nonrelativistic limit. But this time the k r m in p@ = 2m cannot be neglected. the GreenP8funetion with denominabtors We shrall need the explicit sbtemenl of completeness for the eigenfunetiam. It em be: inferred by eamparilrg Lhe high enerp;y limit of the Gseen's funetion

+

@+(X, X', P O ) ,

Lim (-rap'~+(x,X', pO--toc

= 6(x

- X'),

(3-15.113)

3

Chap. 3

fietds

with the constructian (3-15.35), which gives

We exploit this relation by writing the Green's function as

D+(x,

X',

p') =

+,o~,~,~

(x)$pot,r.pfn')*ro

,ka"

+

and then introducing the nonrelativistic simplification''p 'p z 2m. The is negligible. But the last term of (3-15.115) becomes correction to (pof - p')-' -(2m)-'rO &(X - X'), and this supplies the following addition to 2Vkl:

it is the Thomson term. Introducing the nonrelativistic equivalence of Y@T .t;o plm, the complete structure of the spin, O rmufd [Eq. (3-15.W)] is realjlfi~id, as one would expect. The situation is simpler when the gauge (3-15.106) is used. No matrices appear and the nonreIativistie reduetion of -t;he eigenfunctions to wwe funciC,isns can be performed directly, with the justifiable neglect; of the 1/2m term in the Green" function, The immediate result is (3-13.101) (withoud the 1/34 term, of course, since we have been using the souree description of the charge Ze). We notv have before us some simple physical situatioons in which the incornpfeteness of the skeletal description of phoeon interactions bwome~evident. Spontaneous emission is described as proceeding at a con~tantrak, even though. the initial H-particle supply woufd be exhzrcuste-d after a su&eient lsp,se of time. Under conditions of exact 'resonance,' k' Et = E, the photon scattering cross section is predicted to be; infinite, whieh is ailways unacceptable m the angwer to a physical question. In the next section we shall identify the sign nifieant phenomena thaL are omitted in the skefetal de~eription,and remedy these difficulties.

+

3-16 INSTABILITY AND MULTIPARTICLE EXCHAFUBE

Although the need to describe unstable gerticfes as naturally m stable ones i s one of the motivatians in devising the theory of sourcw, %heH-pmticle pro-

Instability and multiparticle exchange

3-1 6

361

vides our first encounter with unsthle particles. The distinction between. stable and unstable particlm is a matter of time sede. Within suitably restrickd time intervab, the mechanism producing particle instability is ineffective and the stable partiele description is applicable, provided, of course, that enough time is still available for the accurate dekrmination of the characteristic particle properties. Ot;herwige, no single-particle description is mestningfut. The Hpa&icles supply examples of sLPLble and unstable parlicles, The particle of minimum enerw is st&le. Those of greater energy are capable of emitting one or more photons, thereby transforming themselves eventually into the tllbsolukly stable variety. The initial description of H-paPficles wunned their stability, and i s applicab-le over a rerstricM time scale, The descrip%ionis false for very long time inhrvals because it mserts that weak H-particle sources emit and absorb single H-particles that propagate unaltered between these acts, But, given enough time, an unstable H-particle will transform itself into another H-particle and a photon. These two particles are also c%pableof recombining to form a single H-particle, Thus, a description of the coupFing b&ween weak, causally arranged H-particle sources that does not refer to the real exisknee af two or more particles propagating between them is physically incompbte. I t is the inclusion of such multipart;icXe excbantgeis between sources and the consideration of some of the physical consctquenees that wit1 occupy us in this section. The first task is the identification of effective sources for the emission and the absorption of an H-particle and a photon, This is analogous to the discussion of Section 3-1 1. The description of s noninteracting photon and H-padicle ia given by (using the apin 0 example)

Comparison with the vacuum amplitude term describing single photon emission, as contained in (3-15.W), gives 2==

eff.

8 ( ~

(3-16.2)

and the same form applies, with appropriate causal labels, Co the a b s o ~ t i o nof $I photon and an E-particle. Since this eRective photon souree is meant Lo bg/ multiplied by a vector potential in the radiation gauge, its appearance is simplified in comparison with the structure of (3-1 1.15). On replacing J";Zt ) K z( X ) and J2r(Et)K2(~" iin (S16.1) by these egective combinations, we obtain a desc~ptionof the causal coupling between H-particle ssourcw that is merfiabd

by the exchange of an H-particle and a photon, under physical conditiom of noninteraction. But, to be consistent with the use of the radiation gauge, we must first ehange the ten~orthat eouptes the vechr photon soumes, in relation ta the exchange of a gart;icular photon,

The polarization veetor summation is a spatial dyrtdic,

which extracts the tramverse parts of the mulfiplying currents. Thus, the coupling krnn in the vacuum amplitude is

An understanding of the causal situation is r e q u i d before the necessary eontml can be exerci~ed. As we see in. Eqs. (3-16.2) and (%16.5), the distribution of tbe two-partick emission and abgorption sourees is charscterized by the &lds +a(%') and (iil(z), Suppose the E-partide emission mume supplies the correct enerw to create a partide that is capable of spontaneous tmnsition to otlker H-pa&icles of lesser energy, with aeeoqanying photon emission. Thier process occurs a t a s k d y rate throughout the subsrfquent history of the particle; it has no csffeetive locdiz;ation in time. Therefore, in order to exert a causal, tempore1 contrd over the aet of two-particle emission and the subfequent absorption, we must use the H-particle sources in the exbnded sense. Then the Bources emit and absorb virtual H-particfw, which cannot exist far from their sources and, consequently, are transmut?ed into or are produced by a real photon and a real H-particle near these sources. This is what gives us the &iliity to influence vvhere (when) the acts take plaee. The quantitlative equivalent of these remarks is contained in the exprmsion for the field,

which has no propagation chafaeteristics if the sources K,op,t.t(Po) vanish for = As indicated in this formula, we shell use the symbol P' to denote fhe e n e r a that is injected by an extended H-partiek? soume, converted i n b a real EX-particle and tt, real photan, ~ n dfinally absorbed by an =.beadd 3% particle dehction csclurce. Through the use of exterrded sourceg, %hen,we emure th& the fields and have supports Lhat are eausa2ly relaLE?c;I. This pernits us to we the causal

PO

fnstebility and multiparticle exchange

3-1 6

363

forms of the propagation funetions in (3-16.5),

where, as indicated, only positive values of''P appear. The resulting form of the vacuum amplitude coupling term is

with

~vhichare the elements af a positive Hermitian matrix, The additional coupling between K-particle sources csn be expressed ss a modification of the propagation funetion ~ , o . ( z @ - E''), ~rhichnow beeomes %I matrix, def ned generally by

where

5,,,,., pa#.u..(~O -)'2

= &-porraFr,

-pofar(~o' - z').

(3-16.11)

The emission and absorption sources of the vacuum amplitude term (3-16.8) occur in the combination

- zO'), We recognize in the central factor the prop~ga4ionfunction evaluated under the causal restriction z0 > zO'. The space-time extrapolation

W

Fields

Chap. 3

of this structure is performed under the guidance of the symmetry property (3-16.11). I t suffices to define r,otap, po..a.t (P') for negative values of P':

r-pae,anre-pa~a~(-PO) rpa~a~,pa~~a~~(PO)p (3-16.13) and the symmetry propedy is then satisfied by

The effective limitations on P' must also be removed if this propagation function is to be meaningful for arbitrary sources. The P' integral can be defined to simulate the initial consideration of extended sources, by excludi~ljlgneighborIf this is done symmetrically about these hoods of the values p'' and p"'. values and then the limit of arbitrarily small excluded in&xlr8!% considered, with a speeial provision for''p = p'",

the result is to use the principal value of the singular P' integral. I n contrast to other reeips $hat m~igncompbx values fo sinwlar intepala, this proeduro h- the mtisfactory feature of preserving $he essential wociation of eonnplex numhrs with the propagation function d p e ( ~ ' - z"). For S more expli~ittmt of these extrapolations, we examine how &hesimple propagation function is modified, by choosing p'ra' = p""af', and considering x0 > zol,g?@" 3:

The physjcally inkresting regime be&= afker a time Iapm of many periods, p"(z' -- zO')>> 1. Then the integral is dominated by the immediate neighborand one can introduce a simplificstion by hood of the singularity a t PO = replacing r (Pa)with

The principal value i n t e p ~is l egectively computed as

3-1 6

Instability and rnultipctrtl~ll~~ exehangs

366

The result is which intrduces an amplitude that diminishes in time, without alhring the time varying phase. This is in acmrd with the phenoxnenolo8cal viewpoint of source theov. The H-particle energies that have been t?~~eur&tely. identifid over the finite time intervals r,a...(zo - xO') << 1 do not ehange their values when the time scale is enfarged. We are not eoncernd here with e x a m i ~ n g how the theoretical understanding of the energy spectrum changes EH we move to another level of dynamical description. For our pre~en1purposes the numhrs ''p are given, whether by theory or by experiment is immaterial. The unit value of the absolute square of exp[-ipO'(zO - X@')] represents the cedainty with wlnieh stable particle will be found in the same e n e r o state afkr any lapse of ttime, The square of the amplitude fmtor in (sI6,fZ.O) scribtls the changing probability &at an undable EX-parLicle (r,ol,f > 0) shall still exid after the time interval x' - x" -. f, There is an initial deercse, at a r a k dven by r,o,ai.But this result become8 unsatisfacbv at larger time values. The persistence probability of the Hpadicle, according tx, (3-16.21), rewhes Eero at s finite time; it then inereme8 and eventually becomes larger than unit;y. The probability formulas is evidently linnihd in. physieal applicability to smdl values of r,ap,lt. Wlka~Lis still missing in the physieal %count is this: We b g a n with an ex6ended H-part;icfe source emitting a virtual H-particle that quicHy trane formed into ta real E-particle and ab photon. This situation endured until both p8rtiafes reached the neighborhood of %heexknded detection. souretr;where they recombined to f o m a virtual E-particle th& is absorbed, But, @ven enough time, the recombination do form a vir-lual H-pargide can occur far from detaction sources with this excitation rapidly deesmgosing back into real, part;ieles, The cycle can h repeated mfl~nytimes before the virtual H-pafiicle is findly absorbed by the debction. source, Qthewiw expressed, the fields appearing in the coupling term (S16.5) originate, nod only directly in the ssurcw, but also indirectly through other, efleetive sources which sre assoeiakd with the virtual H-particle8 that form far from the sources Ghrough the propagation of resl particles, The qualitative description. in the last sentence is @ven a quanlitaiive meaning by the following integral equation for the field &,...(zO) :

where %hem t r i x funadion TX describes the mechanism whereby, for the tmf dime, zt virtuaf H-pztdiele goes through the cycle of transforming into a real

%M

Chap, 3

Fields

R-particle and phulon, then back into a (no6 necessttrily the same) vi&uf H-particle that is detect& by the pro& source used to define the field. The exciting fidd that appear8 in fhe inhgral expression surxznrg~zesthe @fleetof %heinitial source excitation and of the u n l i ~ t e drepetiLions of Lhe~erevemible conversions and is, therefore, considering all pDfa'together, the very field that is being construeled. This point of view is similar to a xnuXLipfe setzt%!ring analysis in terms of the last eoXXision. Xf this integral equation were t a be solvd By iterafion, we would indeed be conside~ngmcceg~ivelymore; elabora;ts repetifions of the same bmis grocesa, The compa~sonwith fhe h a m descriptian aE ane such action then identifies the matrix n. Tkis coqarison is ffl.cilitaM by wfiting.

w h m the modified pmp6t.gation funetion obeys the inbgral equettion

X &p:az

,,a ..a. (20,--

zO').

(8-16.24)

The identification of (3-16.14) with %hefimt two t e r m of %heihrative solution of (3- 16.%), @ves, using transfom propagation functions,

The corfegpanding farm of the intttgral equation (3-18.24) can be presented as

Al%bou& these are ra%hergeneral equ&ions we shall prsduee only an appsoxinrrak solution that is aipplieabfe ordinary circumstances, as indicactrtd by the specialization, (3- 16.27) rp~tofiapo..~t(~') 6.~.~rpa~(Po).

Such staftjmenfs exprw the. rotational i n v ~ ~ a o cofe isolackd systerna, when the a' are identified ss angular momentum quantum numbers. Only equal mer@e~are consider& in (3-16.27) since attention is abo resdll.ic-(lddo the d o d n a n t elements of the propagation matrix:

3-1 6

Instability and multipartick exchange

367

The resulting simplified equation is

which is consistent with the symmetry

-

This symmetry is maintained when c(p0')e(P0)is replaced by unity, as is justified by the predominance of the contributions for PO p''. The inference that the integral is only of interest for p0 p' would seem to be contradicted by the factor (p0' which vanishes strongly under just these circumstances. To see which tendency prevails, we approximate l',o0 (Po) by

and consider the integral

which uses a complex equivalent of the principal value of integrals, according to [Eq.(2-1-62)]

The integral is evaluated by closing the contour a t infinity in either half-plane, as is convenient, with the result

The imaginary term in (3-16.34) is more directly inferred by writing (3-16.32) ss

Thus, the structure that appears in (3-16.29) is

368

Firstds

Chap. 3

doe8 illdeed s u p p u s the reill part of the integrsl, but the faetor not its ima@naw part. i f ?,et Z 0, the finite imagina~yterm maintains the sign of the infinitesimal imaginary quantity, -ipO'c, and the latter is superfluous in the resulting approximate equation:

The implied time behavior is

eoxlsistent ~viththe symnnet~y

The timedependent connglex phase factor continues to identify the enerw p@' > 0, but the variable smplitude 1 - fr,o.l, 1 = z0 -- zO' > 0, haa been replaced by exp(--- +"u,o#t). This is quite satishtory sinw the implid prabeEbility, exp( -?,at t ) , never exceeds unity and decreases monotonicslly to zero with inorewing time. The expnenLiaX, function generalizes the linear deerewe of probability over short time intern&, extending it from the initial instant to arbitrary later times, according h

where r At << 1. Xonrmlativistic .~lpproximationscan be introduced in (3-1 6.92, and tvc arrive at a simptt?explicit formufa for r,ol [earnpare (3-15.6ti)j,

This statement is xlat restriekd ta the nonrelativistic limit, of courrj-e, for it c?quates the initial rate a-t kvbieh the probability of persistence deereases to the cometsponding r a k a t ~vhichtransitions are made ta H-padicles of Iobver enerm. It is inkresting ta verify ChsG this bdanee of probability persi~ts%tarbitr&ry timtits. We first consider the most elementary siluation, an un~tatsleH-pa&icle, designated X I , trl-zich e m only radiate down La the stable H-particle of lotvest energy, 1,zbele-d1, indicated by

3-1 6

tnstsbitity and multfparZlelaaxehanga

M9

The probability t l ~ a tH-particle 11 still exists at at time t after its creation is exp(-rnl) < 1. Is t h h 10s~of prabfLbili@ cornpensaw by fhe probabilify %h& H-paAic1e I exists, accompanied by ra, photon? 9"s evduah the latker prob8bility we must extend the formula for Wzl, Eq. (3-Is.@), .tto the new esilu&ian of urntable particles, From the \riekvp.point of H-partcicles thirs formula de~erib8 t ~ v astages of emissions and absorptions. After the creation of the initial Eparticle it propagates until the moment that the phobn is e m i t t d when iG ee to exist. At that imtant the final H-padicle is created and is eventudfy de&l;ed, Evidently, we must now US@ the modified propega%ionfunction to ~ p r w n t these voyage8 bet1t;een emission and absavtiom acts. Sourm are i n t m d u d do dmeribre what is common to aIE emimion and abmvtion mwha~smrsof a gwticular type, and the corresponding propagation function is of general applieabiliky, f ndividud realistic mechanisms will a180 have specific features fha%n d a In the present essentially additional characterization. This we s b l l d i ~ e u later. no~lrelativisticsituation, Itowever, the description of the mmhani~mfor Hparticle transmutation. and photon emimisn requires no gisignifielznt eomwtion and the introduction of modified propagation. funetions is quite su&cient for our purpose@-= we shall we. apply with the What has just been said is that (%fS.fiO) eontinucjtss changed meaning of field given by (3-16.23) and the simplificaticion (3-16.28). In the situation being considered, the; camally I%beledH-parficle fidd +2(z) that appears in (3-1 6.43)

is given by [we use the nonrelativistic approximation (3-16.67) but retain the relativistic origin of energy]

The time t2 is R fiducial poillt \vithin the source K 2II(z"), and correspondingly tlxe definition of H-particle emission source appeam as

The use! of a reference point that is interior to the source rather than arbitrarilJ; &oen is always posgible and can be useful in identifying the xnwhanical propep ties of states. It hcorrseg mnndataw in demribing unstable pa&icles.

376

Ctrrrp, 3

Fields

The coupti~rgof emissioxl and afssorptiorr sources for H-particlc is given by

tz

ga~rticularunstable

tvfrere, illustrated by particle 11,

and t l is a. reference time that locates the source K l ( z O ) . The fa~toriastionof (3-X(i.46) clearly separabs the three st;ages of emission, propagation for the interval t = t x - t2, and absorption. The possibility sf describing unstable particles as stable over shod time intervals must certainly apply to $he large&% values assumed by z0 - t I and t z - zO', which are limited to displacement8 within the sourem, Thus the decay factors in the H-particle sour= definitions (3-26.45) and (3-16.47) can be omitted, and these sources play the same role as with stable particles. In l;his way, %hen,the weakening af the coupling (s16.46) tvith incremirrg time int;ervaE between the sources, owing to the instability of the particle, is specifically associated with the process of prop%gation only. The same kind of description is used for particle I, even thougl~?I = 0, and the probability amplitude deduced from (3-16.43) is

The initial and fixlai times are now explicit in the specification of stateg, although only t ==: t - t is significant?,as we have emphasized by using E l as the reference time for the photon fielid. The time integration is evaluated as

tltld the traxtsition probabilitjr, summed aver photon polarization8 and emission directions, but still differential in the photon energy, becomes 711

k0 dko 1 -- 2e-"'2""L cos (kO -- k f I r ) t + em'"' , %2 (kO - h.! l d Z f ( * 7 1 1 ) ~

(3-16,50)

3-1 6

where

lnstobillty and multlpartlcta sx~hange 371

k f 11 = E11 - El

and, of eoursc?, ~ K= I

$akf rrl (IlplmlII)I 2.

The total probability is produced by carrying out the k' irltegration. That is approximated under the assumption of \~\.eakinstability, r 1 1 << k f 11, by replacing k v l f 11 w~ithunity and evaluating the integrals as

together with the speeializwtion to t

-=

0. In this j v v \ye obtailt tllc probnbifit~

of finding H-particle I afkr time t as

5vhicE-i is the required value. The spectral distributiox~of ttre emitkd photon is also exhibited on evaluating (3-16.50) at a time rrlt >> 1, such that the radiativ~ transition has certainly occurred, The result,

is the familiar Lorentzisn shape th%tidentifies the decay constant 711,the reciprocd of the mean lifetime, tvith. the tvidth of the spectral line at half-maimurn. This is the shape of a spectral line emitted in a transition to the stable W-particle. But what if the final H-particle is also unstable? Now eonsider n third H-psrtiele 111, ~vhiehcan only decay into X I , with the subsequent transmutation of the latter into the stable variety 1. In this situation two photons are emitted and we must use W z zto describe the process. There are tutoanalogous terms in the relevant probability amplitude which are related by the B. E. symmetry of the phobns. But, apart from the special circumstance kf"lr 2rl h:& 1x1, only one of them terms is appreciable depending upon tvhich of the photons has its frequency near kp 11 while the other frequency is close to kfI 111. Thus, it suffices to regard the photons as distinguishable through fheir frequerlcies and use only one of these terms. The probability amplitude for the rvhole process is

tf"l1ere the time propagation functions detail the sueeessivc enusul nets of tllc drama, In writing this expression, we have proeeedd ss thaugh the H-padicles of types II and III were unique, although additional indices a11, a111are necessaw. T h a e detdls can be inse*d and do not d e e t the resulb, under the physical circumstances indicated in (3-16.27). The z0 time integretioll is thc one already performed, with zO' supplying the latver limit instead of tn. The Lime inkgrd faetor of (S16.26) is, therefore,

The next integration poses no dificultiies, but we shall be confed to ev81uak it only for such large 1, r l ~ >t> 1 , r111t >> 1 , that it represents the completed process of cascading decay. Only the exponential containing Er contributes, in contrast ~rittrtthe one containing Ell, to $ive the value

The implied transition prsbability Chat refcrs only Lo the spectrail distribution of the photons is

+

El of The successive emissions are not independent. I t is the energy k' the p h o h snd pa&icie i n b which I1 dec%ys,rather than the energy E ~ Ithat , determines the spectral distribution of kO'. When one integrates over kO', the result is just (3-16,55), h i e h means that W-partide 11 is certainly produced at wme time by decay from Iff, after whi& the previous disemion applies. The sns\\*erto the question concerning the spe~traldistribution of the ptrobn radiakd in a Cransitian bett-veen unstable H-partidea is obtained by. intiegrating over k'. It is instructive to write this integral as (E = ka f E I )

which describes an energy-conmwing radiative %r&nsitian.b t w e n two enerw didcibutions of Xlorent~lian~ h a that p have the widths 711 and rgrr, respectively* Aocarding to an elemenl%vcontour integ~alwaluatioa, Lhe ~ s u l t i n gspectral

3-1 6

Instability and multiparticIsexchsnge

373

wiLh a width given by the sum af the individual H-parLicle widths. This conclusion is parfieufarly transparent if one recognise8 that the double enerw iatepal of (3-16.W) is equivslenl to s single time irtkgral:

It ~-outdbe hard aot to suspect the existence of another approach that is capable of producing this formula directly. We shdl find it, not surprisindy, in the time cycle description. But, first, 1eL us give an. analogous discussion of photon scattering, in order to verify that the unphysicat infinite ttross section a t exact resonance has been. removed by the explicit recognition of H-parLicle instability, Elmtic wscattering by the stable E-pa&icle L will, be comidered, Then it ~u&ee1;3tro introduce modified E-particlis propagation functions in (3-15.88), which will be used only in the aonrelativistie limit and in the gauge of (3-15.lM). The sigrmifiesnf change is the ineroduction in (3-15.101) (\v@ ignore the l / n l term) of the substitution while E -- (E1 - k') remains unaltered. To understand this it is necessary to be somewhat more general than (3-16.37), where r,ot (P') is considered only for PO = We return to (3-16.29) and proceed as in (3-16.351, but with rPo. (P') retained, and get

-

showing the general form of the imaginary term. This distinction is unnecessary near resonance, p' -- p'', or EX k' E, but it is needed far from resonance conditions. Otherwise we should have, incorrectly, added an im~tginarykrxn to E -- (E1 -- k'), where%? rE(E1 - k') 0, (3- 16.615)

+

sinee no photon emission can occur if the Lots1 energy is less than E x . H-particle If becomes strongly excited -when

Under the% circumstances the dominant contribution ta the differentid cross sectian of (3-15.101) is

This differential cross section far specified polarizations is replactea by the b t a l G ~ S mction S on summing over final poltzrizations and directions and averaging aver the initial palwization (and direction). Recalling Ghat

together with the orthsgonality stated in (3-16.27), we find that

whem g11 is the mulLiplieity of parti~leXI, the number of different values m8urrtd by axr. The fwm of the cross section, a t exaet resonance,

is typicd of any rwonant scatbring process. The bmic resananf crass seetion is 4.1rg2, h m 4rr/(kf x1)2, which is multiplied by the number of resonant states, grx, and divided by the multiplicity of the initid particles. Tfist is jusf the fmtor of 2, referdng to the two phobn pofsri~stions,sinee H-particle I has bmn amurn& to be unique. The promise to exhibik another and, more direct derivation of (3-16.62) will be fulfilled, even to the point of generalizing this formulst so that it, refers to any pair of urntable H-padicles, which are capable of dwaying in other sequences khan XI1 + 11 X. Here is the statement of the mare general problem. The arbidra~ unstable H-particle f I1 is creELCed near time zero. I t can decay to a particulrzr unstable H-particle I f as we11 W in other ways, and these secondary unstable particles continue the eabsedc?until the stable particle I is reach&. Wh& is the differential probability for finding B photon of frequency ko 11 k f r 111, wiLhout reference to the ather photons of different frequency &hat are also emitbd? For s spwified polarization, that probability is expressed by

which msumes 8 time in,terval long exlough to have the probability attain. its final value. Let us supply two additional hetom, Gtkp the pmbability amplitude for detecting the photon kX, and -i& it8 ,complex conjugate. This produces

tnstebiiity and mulripsrti~leexchange

376

a quantiLy th%tcan be presented as

Apparing here &resuceesaive s t s s s of a time cycle, in which two anafogom photon sourcw act, ooe on the fortvard time path and the 0 t h on LXle refum path. Thus, we are now inkrested in fhe time cycle gener~liaationaf Wzz. T o use a, cornistent nonrelativistic dmeription, one should subtrmt m from the f rtltquencies in the propagation function (3- 16.38). This reduce8 the positive frequencies to nonrelativistic energies, for z0 > xof, but converts the negative frequencies to values 3 -- 2m, for '2 < g@'. The latter produce negligible contributions to time inhgraXs, and the nonrelativistic version of (3-18.38) is, accordingly,

When the g a u p (3-15.106,107) is used, the stmcture of z"W22,w~t'(cenin a simplified matrix notafion, becomes

The transition to the time cycle is made after time t2. Time t1 is now emounhred on the return path, which is certainly 'later' than tz, and q(tl -- i2) is replac4 by unity. Also, time t is reached 'efLer9irne tr and q(t - t l ) mu& be replaced by q(tl - t). Since both t and tI refer to the return time path, there is no sign change in the integral. The proper treatment of the r Lerrns is fixed by %he physical necessity of maintaining the damping, the weakening of the ~oupling ~ C inereasing h dime interval, All this gives the subsfitution:

With the H-particle saurew operating in the vicinity of 1 = 0,

3

Chap. 3

Fields

supplies the required time eyele quantity, and the coefficient of ( - i ~1:)($Kr ~ in the f o m HI-) J~-~s"(+~ = e2 dt dCa exp[2'EIIx~ - pzIrttl

We must still extraet the ooeEcient of

snd of --gkk from E(t2) and

E(tZ),re~pmtively. According to (3-15.1 07), Lhct: fir~kof thme is and i b coxnplex conjugate, evaluaLed a t 11, applies on the rever8e Gime path. k0 czs k f i 111 picks out the contribution from the specific Ef-pafiieXe XI, the de~iredpdability, as it is dedueed from (%16.77), is Since the restriction

With dk0/2?r removed, the factor in front of the double time integral, summed over polarizatiarrs, is the A-eoeseient for the dmay XXI -+XI, Ia the h k r m k of a more uniform no%%tion we: shailt now denote it by 711 ux. T o aimplify W ..time irrbgrals we introduce new v t t ~ a b l e ~ : which mnges from -m t~ QD, and t<, the snadier of the two %inn@# whiah vahw from O to m, Then, %hetransformations

d have ~ e p r d ~ ~ c x f When XII can only radiab to If, 7x1 1x1 = "Frrr, a ~ we (3-16.62). Mart3 generally, the probability of emitting any hquency in the neighborhood of k f 111 ~ is (TI1 I I ~ / Y I I ~ < ) 1, ~ecarding tO

Instability end multfparti~leexchange

3-1 5

377

and this expresses the competition between the specified transition and all1 othem that 311 can undergo, The sum af these fractions over a11 decay mode8 of If1 is equal to unity. The time cycle extension of W z z also gives a direct derivation of the remnance scattering emss sction (3-llieCi9), or, rather, ifs generaliszttion in which I1 beeomes an W-particle that can decay in ways other than down to the gtable particle 1, and rr becomes the corresponding total. cross swltion. A photon is incident on X, and eventuizlly one again finds I, aecompzznic;d by one or more photons. The total probability for these phenomena, \vith s given interaction time, is

When the initial particXes are introduced by appropriate sources, two of each kind, this beeomes a Lime cycle vacuum amplitude, deseribd by iW2z. The result is obtained from (3-16.77) by replacing XI1 with the stable I, and using the field of an incoming photon insted of (&X6,78) :

The integrand depends only upon the time variabb 1 =. l z -- t l , and the integration over t< is identified with the duration of the inter~etion. The total erom section is found by dividing the photon flux 2k0 dwr into the transition probability per unit; time, On recognizing thaf e2(k!11)'

a

/(IIajx * e 1)i2 =

T ~ I ~ YXI, I

we get the dominant contribution to the cross section, for k'

(3-16*88)

kf 11, as

,P

we regain (S16.69). The additional factor, Tr II/rIx C X, When 71 11 == evidently represents the diminished ability to excitc! I1 directly from I, w h i ~ h is the reciprocal aspect of the fractional probabiIiLy for reaching I directly from fE in decay. From this point of view, the elmtic scattering cross section should

be obtained from the total cross section by multipfyirrg the lathr tYith an additional rx II/rrrfachr. That is indeed the result dducrsd from (2-16.67) whm the quantity mntaind in (3-16.68) is $;iven. its pneral interpretation W the partial width r 11 :

The above discussion is ixreompfek since no mention has been made of the ddi.tionsl t e r n in IFzz that is dewnded by the cmsaing symmetry of tbe photons. It is produced by reversing the sign of k'. This tRrm is certainly nonresonant. But, more important is the appeesance of the initial energy as EI - k a ; the value that should be a~signedto the damping constant of H-pa&icle I1 is not rI1 but %em, as in (3-16.65). Then the resuXting dime inlegrd gives 6(kf 1- k 4 0. All the developmends of this aeetiotoxl have used the exampie of spinless particle8 that are bound to form El[-particlw. A similar treatment for spin p a ~ i c l e swould run in exact paraIXet, with occasisn~iinserLians ar deletion8 of c(p') factors, for example, to represent the changed statistics. The nonrelativistic results are identical. The natural instability of H-particles has direckd attention to the necessity of considering multigartiele exchanges, in addition to single-particlepropaetion. It is a complementary aspect of $he principle of spaee-time uiformity that couplings identified through the examination, of red proeesms eontinue to be meaningful when. appfid Lo virGual processes, This says thad multiptzrtiele exchanges are significant, although the energy CO produce sevemX red par-tietes m6y not be available. Thw, &henext stage of d y x l a ~ c a levolution. is %be ~ystematicgeneraliz;ation of all single-partick exchanges b e t w ~ nsources to those involving two particllcls, including their unlimited repetition. Before embarking on. this masive progfam, however, wre shall give a relatively b ~ e disf cug~ionof the ~avitationalversion of ~ u e hconeepk as primitive interactions and gauge invariance.

+

+

3-17

THE GRAVITATIONAL FIELD

The field wsociahd ~ t massless h particles of ihelicity &2 has not yet been given an independent discwsion, We refer back to (2-4.241, but use the mechanical mea8uxl.e of Tp,,according to (2-4.33) :

The symmetrical tensor field h,,(%)is defined by

Tha gravitationat field

3-$7

379

subject to the source restriction

a, 6 ~ ~ ~ (=x0.) The corresponding arbitrariness in the identifieatiion of h,,(z) is exhibited in

Contraction of indices in the tensor gives (3-1 7.5)

and therefore

The introduction of the source restriction, through the divergence of this equation isslate?a the a~pectof the field h,,(z) that is governed by the arbitrary (z) vector, (3- 17.7) a,(hpv(x) - ~ ~ " h ( z= ) )a2ty(z). Returning to (3-1 7,4), we deduce

which is the second-order difirential field equation,

It is also the form of the equation obtained by placing m = Q in Eq. (3-3.19). GaxrLractixlg the indices in (3-t7.9), or converting (3-17.5) to a diEerentiaX equation, implies -a2h(z) a p a v h p Y (= ~ )-*KT(z), (3-X7.9a)

-+

from which we derive another version of the differential field equations,

equation. it is (3-3,17), with m = 0. The structure of the left-band side of t h i ~ is such thst its divergence is identically zero. The vanishing divergence of the source tensor now appeam a8 an algebraic consequence of the field ~ U ~ Z ~ ~ Q X ~ F

380

Chap, 3

Flalds

Since the arbitrariness of the vector E,(z) is still maintained in these field equations, they are unaffected by a redefinition of the field h,,(%) having the form

which is a gravidationd gauge transformation. The following definition, analopus to (3-3.23),

with its eonsequenee F,($) = r,kh(x) = a,h(~),

pravides a fimt-ar8er form of the field eqiuations [(3--3.2-2), wiLh m = Oj

The gauge i n v ~ i a n c eof the Iefbhsnd side of (3-17.14) is not realized through the invariance of r,,~,but rather

and

Note, however, that thege gwgr?, transformation rmpsnses do not invdve first derivatives of Lfre Ex fz). Another ~ y a k mof first-order differential equations fit is (3-3.20, 211, with m = Oj is praducd by the definitions @~XE(%) = - w V X ~ (= ~ ) aph~W(z)- dlh~@(je)

(3- 17.17)

and oy(z) = wPhh(~) = aph(z) - dhhyh(s),

(3-17.18)

name1y, ah@,.k(z) - a.w,(z) = K(T~.(S)- + ~ , , T ( Z ) ) .

(3-17.19)

The reBponse of thme fields to gabuge transformations i s gven by

+

@ph~(~)@~xP(z)ak(ap~r(~) - ~P&(x))

and @p

(4

+ @p

(z)4- a p a k t k ( ~ - )d2& ( X ) .

(3-17.2131) (3-17.21)

It is observed that Lbe divergence af the vector field m,($) is gauge invariant, A comparison of the form of the divergenee inferred from (3-17.18) with (3-17.9a) shows that d,wg(x) = +KT(%), (3-17.22) which is dso the eantraction of (3-17.19), sinee

3-1 7

The gravitational field

389

An arlalwus w e of (3-17.14), however, introduces ta. new vector field, &I"(z)= I"",x(x)= 2dVhv~(~)axh(x),

(3-17.24)

such that rx(z) - XT'(X) = ~wx(z).

(3-17.25)

This is s contraction of the tensor relations

Another connwtion between the two third-rank tensors, which implies this one, is

The sueeessive stages involved in producing an action expression Lo represent the first-order field equations (3-17.12, 24) are

where the last version introduces the relation dhhpr(x)= d ( ~ ~ * ~ ( x~ ) ~ ~ ~ ( z ) ) ,

+

(3- 17.21))

i(rP(z)i- 'r(z)),

(3-17.30)

i t s contraction

dyhpv(z)

E

and (3-117.13). The action is (3-17.31)

with

Apart from a, divergence krm, this Lapange funetion is the analogue of (3-5.411, with m -. 0. For simplicity, we cfo not include a source for the third-rank tensor field, in contrast with (3-5.40). The stationary requirement for va~ations of hp', or h""" ---- +@@"h, recavers (3-17.14)) and variaLions of F,,x, &er rearrangement~indieakd by the structure of (3-3.45), reproduce (3-17.12). The Laf~;r&nge funetion is not gaum invariant,

382

Chap. 3

Fields

but the aetion is invariant. If we relinquish the use of pas an hdepndent variable and Xet it be befind by (3-17.12), an appropriab Lapange function is ~ . e ( h )= f(r"'Yr,.h

- Arrh)e

( ~ 1 7 . ~ )

When writfen out as a qudrtttic funetion of the fint defivativm of h,,(z), this fi-ange funetion differ@from that of Eq. (3-5.99) only in the 1-6 germ: which itlustrat@ the freedom to add divergence terms, This athmative has d r e d y been n o M in Eqs. (3-5.31,32). The similar development that i s bmed on the fimborder diEerentiaX equsGions (3-17.17, 19) stafts with

~vherethe second version u& the substitutions

The Lagange furretian is obtained rzs

Apart from di-vergenee terms, it is (3-5.34) wifh m = O. Again, the muree coupled to the third-rank tensor will not be u s d . The vafiatiaion of hp" - 3 f " h reproduces (3-1 7.19) and thlikt of oh,, yields (3--11.17) reanangements that; are indieat4 by %hestructure of (3-5.38). The response of this Lapange furzetian to gauge transformations is

which msures the inva~anceof the tbction. Whm wh,, Io~wi k independenf sfatus and is defined by (3-17.f7), the Lavange function clan be chomn M

which is the quadratic function of the fir& defivakives of h,,(tt) that is pmdued by averadng the two alternatives of (3-17.35) :

3-1 7

Ths gravitational field

383

The stress tensor Fy(s) has been given a kinematical definition, m~hichis not unique, through the response to infinihsimal coordinate deformations,

It aequires a dynamical defiaition by imitating the role of the paviton source Tpp(2),This is inacated, in the re~ponsc:of an action expression Lo infiniksintal gauge transformations, by adding t'" to !Py,

The two concqts are identified by requiring invariance of the action under the unified gauge-coordinate transformation,

Thus, infinitesimal coordinah transformations induce the infinitesimal gauge transformations 6h,,(z) = i (a, 6zv(x) t a* sz,(z)), 6I",,x(z)= d,3, Qixx(xZ

+

The use of the total stmss tensor !Py P",as the fwtor of h,, in the mtim, is the introduction of a primitive interaction. Same modification of tB"is needed since it is not conserved inside pa&iek, aaurces, and a gravitational model of particle mmes must; be introduced. But let us dekr the biscugsion of that question and proceed with the development, whieh is modeled so ctosdy on the eleclromagnetic one, in order to reach Lbe point of divergence between the two very different physical sy~tems. Consider %heexample of spinless pa&icles, using the simplesl stress Lensor form, Eq. (3-7.81, As in the electromagnetic anetlogue, the eoupling term &,P"will be combined with the particle Lagrange function, -to form

At this stage in the elwlromagnetic discussion, faeilidahd by the usc? of a sfightly different Lagrange function, the gauge eovsrisnt derivative a, - ieqA, appear&, and one vefified invariance under the whole Abelisn uaup of gauge transformations. The related gravilalional situation is only partly produced by the suh~litutionkqA, -+ h,,ay; there is also a pavitaLiond eoupling that does not refer to derivatives, But, much more significant is what underlies the replacement of the single matrix czq by the four diaerential operator8 (I/.k)d,.

384

Chap. 3

Fields

The general coordinate transformation group is non-Abelian, as indicated by and the extension of invariance under infinitesimal transformations to cover the whole group is not trivial. It is instructive to examine just how infinitesimal coordinate transformation invariance comes about. The transformations associated with describe the scalar nature of the particle field, and give the induced gauge transformation of the graviton field,

+ *(a, ~ x , ( x )+ a, ~ x , c , ( x ) ) ,

= hpV(x)

including

E(T)

+ a, 6 # ( ~ ) .

=h(~)

(3-17.52) (3-17.53)

The invariance of the mass term in the action is stated by

= /(d5) ( l

which is satisfied if (dz)( l

+ ~ ( z ) ) + ( z ) ~ ,(3-17.54)

+ h(x)) = ( d z ) ( l + E(@).

(3-17.55)

For the infinitesimal transformation (3-17.50)) the transformation law of volume elements becomes (dz) = (dx) det (aF/azV) = (dx)(i - a, ~ x ' ( x ) ) ,

(3-17.56)

and it is required that

This can only mean that h(x) is restricted to be a very small quantity, permitting ha, 6 9 to be neglected as a second-order object. The situation is similar for the quadratic derivative term of the action. We first notice that isolates the factor 1 Then,

+ h that compensates the transformation behavior of (h). g"'(%)= Q"'

- 2hpv(x)

(3-17.60)

The gravitrrtionslfZeld

3-1 7

388

must transform appropriately to produce a watar combination: & 6 ; ( ~ ~ 8 " " ( ~ ) ~ vzzz ~ ga,@(.z)8"""(z)av&(g)t Sz)

(s17.61)

~ " ( 2= ) s""~)a.z@ahz@.

(3.- r 7.62)

or

This property eharacbrizes g p v ( z ) as s contravarjlant bnsor of the second rank under general coordinate transfornnationa For infinik~imaltransfomation~, that transformation.taw becomes which doe8 reduce to the first statement of (3-17-45), if one neglects seeond-order quantities by replzccixrg gg"(x) with g@', on the right-hand side, The ten~org,,(%), in~verseto gpv(z),

bfts the transformation law of a c o v a ~ a ntensor t of the swond rmk, The implied behavior of the dekrminant S(%) zzz

det gpv(z)

(3-17.66)

is

g(z) = g(z)[det ( a x @ / d ~ ' ) ) ~ ,

(3-X 7-67)

and therefore

(-@(z)) " 2 ( d ~ = ) ( - g ( ~ ) )l l Z ( d ~ ) .

(3- 17.68)

E t is consistent do regard this as the generalization of (&17,55) for, under the weak field conditions ia whieh the latter refers, and Aeeordingty, to enBure invariance of the action under arbitrary coordin&k transformations, the Lagrange ftrnckion of (3-17,48), which is appropriate to weak gravitationd fields, should be replaced by We must find a ximilar generalisation of the weak field fmm of the p a v i h tionat Lagrange funetion,

in kms of which i s (3-1x32) with dl reference to third-rank tensors s%a%d

386

Fields

One rec~gnigesin hp" - ggp"hsrt of the weak field evaluation

The missing constant term can be added in (3-17.72) since it changes the Lagrange function by a divergence. Then the strong field generalization is clearly indica;ted: (3-17.75) Z K (g(%), ~ ~ ( 4 ) (--II(x)) li2gpv(2)Rpu(2), with Rhv 8hrtv- aPr;h ~ ; ~ r ~k~r:h, $~ (3-1 7.76) E

+

This will indeed contribute an invariant action if yMvRp.is a scalar with respect to arbitrary coordinate transformations, The required covsriant knsor behavior of R,,(z) must emerge from the transformation law of the three-index symbol ) The latter should resemble a third-rank tensor but cannot be entirely of this nature, according to the ~veilk.field transformation of (3-17.45) which ~ontainssecond derivatives with respwt to coordinates. A suitable generalizatioxr is stated by r;,(i-);1.~" rig(~);i,~Pa,~g a,a,zk. (3-17.77)

+

This transformation Xatv is such that a coordinate covariant derivative of firstrank contrsvariant vectors can be defined:

v,vP(~) = (a. + r,(z))~v"(z = aa.VP(z)+ r:&(z)vk(z). (3-17.78) The matfix notation facilitates the consideration, af

[v,, v,lw"

where

=$X

V&,

(3-17.79)

(3- 17 .so) R,,"~ =. a,r:~- aVr:& 3- T:,T";~ - r:,r,Ph is indeed a fauxrth-rank tensor, which is antisymmet~ealin p and, v. We can now recognize the tensor character of

The nation of covariant derivative, identified with ordinary differentiation for sc~lars,is extended to firrsbrank covariant vectors by the requirement

and to arbitrary bnsors by generalizing the difierentiation rule for prodtteb. As an application, we note that 6~3,(3), which is any infinitesimal change of the r,",(z), does transform as tensor, and

&R,, = [a, ar;,

+ r",

= V& 8 ~ ; ~V@

- ar",ri. - arigrk.]- [a, art, - 8r",r:,] (3-1 7.84)

The eovariant derivative of g(s) is defined by the deterrninantal diflerentiatian fomu18, Vxs(z1 = g(z)gHV(s)Vxs&y(2) (3-17.85) = g(x)fvIz)P~~~~(2> - %x(z)f"xv(z)l = axa(x) - zg(z)f""x(x), 01"

vk(-S(~))

H2

=

ak( - @ ( X ) )

- (-g(%))

112riv(~).(3-17.86)

A. simple consequence is the divergence formdla, These results are used in applying the stationary action principle to varistions of P:, as it appears in g(g, F). The vanishing coefficient of 6Pi, in 6W states that [gppis gH"(z)j

v,[(-o) 1j2s"l - a i ~ , [ ( - ~ )

lizy'Y] =

0,

(3- 17.88)

which imp1ies

v~[(-~)'~= ~ ~0." ]

(3-57.89)

From the latter pmperty one d e ~ v e s ,successively, the vanishing of the covariant derivalives for g ( ~ ) g'"(~), , and g , , ( ~ ) . The last statement, leads to the explicit construction mfhieh is the strong field generalization of (3-17-12). The weak field vemion af (3-f7*90) appears in (3-17.29). As one can verify directly, the vani~hingcovariant derivative of g ( ~ )implies, according to (3-17,86),%h& which generalizes (3-1 7.13). This form ensures that Bp,, as defined in (3-17.76), is rz symmetrical tensor. The variation of gp"(z)in the pul;.eXy gravitational contribution to the action induee~

where

4, R,,

R = FvR,v, $gPvEp and we have used the deterninandal property X

m

(3- 1%94)

The Zlensor G,, obeys a differential identityt which is a consequence of the coardinatti?invariance of the gravitatisnd action term. We firrat note %heinfinitesimal response of ( X ) , analogous to (3- 17.63),

which generalizes the weak field e u g e transformation of (3-17.45). On ~ t i n g where we eonelude from the invariance of the action Ghat

The variation of ggvfz) in the matter part, of the action, defines a hnsor t,,(z) that generaIizes the stress tensor,

f nspectiorr of the Lagrange! function (3-17.71) show8 that

We slm note the generalization. of (3-7.91, in source-free regions.

When, the atalionary properky with mspect La 6 variations is invoked, the coordinate invaeance of this action k r m leads, as before, LE) a differentid slabemend, V,t"""(gz) = 0. (3-l"i"lQ3) Another form of this generalized local conmrvation law is

The field equation dedued by va;ti).ing g"" in the complete action

The gravitational Qield

3-1 7

389

it lis Eimtein" pgra;vitationat field equation The sfress tensor divergence eondition (3-17.103) appesm again, now as an identity demande-d by the ~tmeture of the p a ~ t a t i s n a field l equation. The replacement of spin O particles as the model of mavitating matbr by okher inkger spin, parYtielm is relatively straighithmard, A rather special but interesting example is provided by photons. The Lagranp function is innmediahly getneraliaed to realize invrariance under arbitrary caordinab tramsformatiom, while maintaining electromagneticgtauge inva~ance,by writing

Af points not occupied by electrorngra;gneficssurees, the implid field equations are

(3-17.1 IQ)

The stress demor deived by varying gPY(z)in. (3-17.1W) is where aX1 eontrava~antand covsrianL indiees are reIated by mean8 of the t~3n~0r gP,(2), Another instructive derivstioa can bet dven, By redefining FP' to absorb (-g)Ii2, which praduees the Lagrange function aU reference to g,, is ctoncentrsled in the Imt term. The stress bnsor (3-17.1 12) i s regained, divided by (-g) to conform -4th the d t e r d meaning af F""". BuL now we can see something very clearly: (--g)-"2~r.g.h i s homogenwu~of degree gero in the campanexrts of g,,(zj, which is to gay that the Lapange fmetion (3-27.113) is i n v a ~ a nunder t the transformation for arbitray X(%), unity, i~

The impEiesLlion, for an infiniksimal deviation of X(z) fmm (&)(-g)

112 v

P fi, SX = 0

(3-1 7,115)

or l(%)= gb,(z)t""(x)= 0,

(3-17.116)

which, is tme.. Id is e ~ c l e nthat t we me now considering a generalization of $he cadormaf transformE1Ciona that were o~@nallyinfroducd through the can-

3

Chap. 3

Fields

sider~tionof isotropic dilakions ((3-7.153)l. Incidentally, while the alternative Lagrange function. af (3-1 7.113) was helpful in recognizing eonformaf inv~rialrtce, one can also use (3-17.109), combining the eonfarmal transfomation (3-17.1 M) with the field transformations

ta attsin the invariance of C. As ure have noted eariier in, connection with (3-7.168), the kinematical: arbitrariness in stress tensors must be considered in tes"cing for conformal invariance. The arbitrariness can be placed in a dynamical, conbxt, akin to $he electromagnetic procedure illustrated in (3-10.63). Returning to the weak gr8vitational field situation, we examine the possibility of replacing any given stress tensor P" by [Eqs. (3-7.83, 84,8511 is symmetrical in p and v , in K and X, and obeys where mpYsKk

The addition, to the Lagrange function term tP" is, effectively, mr**"bKa&h,y = -$mp'e"QR,A where

-+

R , , , ~= aJlaxhxv ava,hxp-- a,axb, -- +a,h,~

(3-17.120) (3--17.~2~

is produced by using the eyelic property (3-1 "l.X 192,

Also contained here, in

is the requirement that m&**"'be symmetrical in the two pairs of indices, as already found in, pa~ieutarsi-t;uations[f 3-7.88), (3-7.11 l), (3-7.137)f. The four index object EpKVk Inas many symmetries. 1%is antisymm&rica;l in p and K, in v and X, and symmetrical in the two pairs, prc and v&. The sum of the three krms obtained by cyclic permutation, with one index held ked, equals zero. As the notation betrays, R,,,x is the weak fieXd version of the hasor derived from (3- 17.80):

We conclude from these resulk $hat posrsible dditiontzl brms in %heLagiznge funetion of matter have fhc? form

"'

where mNF9is a tensor, refening to the matter field and the gravitational fieid, that h@ the symrnetries previousl-y noted. An illustration for s p h OIgenerfziized from (3-7.88), is

For definibeness, the coacient is ehosen so that the new ~trmstensor, in the absenee of the gravihtiomtal field, .,8

=

a,$aR

- f g,v(a"

+ m2+2)- it(a,a"*2-- g,.

(3-17.127)

has the prope&y 1 -"c.

and vanishe8 for m = 0. When (3-17.126) is uged in the pme-ding equation we encounter R,K.ksps'h= R,.'hgK" E (3-17.129) and the modified spin O Lagrange function is

1%would be indere~tingto ve-fify that this system, with m = 0, is conformally iwari%nk,in the sense of (3-17.114) supplemented by an itppropria6e response for +(X). One sws $hat X(z) is eonstanl, To complede the test If is 8uBFieienl .t;o cornider an, infinitesimd variation of X(%) from uniiGy, 6X(z), Then,

is suitable, if

where

is to be computed from &l,v

= &kg,,*

1rrvolved here are = 6(3, log (-g)

'l2]

= 2gP bX

and

5/""vr;,

.=;

-gwap

&X,

(3- 17,135)

thus assuring the invariance of the action, for m = 0, under the ~ o u pof conformal transformations. Harold interjwts a question,

H. Your preoccupation with conformal trangformrationa in the context of what is eustonna~lycdted general relativity makes me suspc?cd that you, inkad to give a aource theory setting for some of the more recent atbnrpts t a enlarge the fmmework aE general relativity. I am thinking particulshriy of the ideas of P, Jordan and Brans-Dickc: (B-D), and of Diekek related eRort8 to e~labfish.s discrepancy between the residual perihelion, precession of Mercuv and the Einstein prediction, The B-D progosat is based on Mach'@ppriciple bvhich, while a very intriguing notion, is devoid of immediate obsewational. content. caxt one sugge~tp o ~ ~ i b i l i t of i e ~modifying the Einskin theory on somewhat more ph ysiealt gounds? S. That is indeed my intent;ion. h t uss begin, by asking wbedher, through some exhnsion of the theory, conformal ixlva~aneecould be made an. exact symmetry property. Celrtainly the msss term of (3-17.130) can be multiplied by cr(z)', where a(%)is a new scalar field that responds to eonformal transformations as Furthermore, the conforrnd response of the grravitadional Lwrange function (3-17.75) could also be compensated by multiplication with ~ ( z ) at ~ ,less* for constant X(%) which leaves R,, unchanged. And, when one recognizes %had f -g) 'Rc2 is part of the conformally invariant Lsgrange function (3- 17.130) [with m = Q and 9 --P a]the generatigafion h arbitrary. X(%) is clear, leading to the complete?earzformally invariant

where R = g@"R,,retains its meaning in terms of the g,, and their derivatives. It would seem thsf we have acquired rt new nnasslws particle of spin 0, repre ~ n h by d the scalar field ~ ( 2 ) .But something is amiss, In a weak field situsLion, with g(%) 1 -4- ~ ( z ) , (3-17. 140) the dominant f/;, Ldtrms in this Lagraxlgr: funetion are

The p derivative term has the wrong sign. And the source of the p field, proportional h R ~ i ? , vanishes according to (3-17.107). All this indicaks &at the p field does not descrirbe a physicd excitation. It can be transformed away, by introducing the conformal transformation \%rith(X(%)) I t 2 equal to to(z), which reduces the latkr to unity. Nevertheless, the eonfsrmal invariant version is valuable in pointing out a new direction, As we have b e n learning in high energy particle physics, nature does not always seleet what we, in our ignorance, lvould judge to be the most symmetrical and harmonious possibility. Perhaps the formal inva~anceunder conformal transformations is broken in suck ai. way that tz, ma8sless, zero spin particle does exist, IDespik the principle of noriloeality for mmsle-ss padicles, one cannot object to such a psrlticle on experimental grounds if it interacts with matter sufficiently more weakly than a ~ a v i t o n ,In order to realize this suggestion, we must add an additional contribution tro the Lagrange function that eaeictively reverses the sign of the ~r derivative t e r n and msigns it an arbitrary coeacient. That is not enough, however, for the tr field would still have no . can be done source; it is necessary to destroy the combination R $- ~ t This arbitrtzriily, but $he possibilities are illustrated by two elementaq stfkrnative procedures: remove the g2 faetor that multiplies m 2 ; remove the g2 factor that multiplies R. The firs%procedure gives a version of %hie?" B-D theory. The second one has the same practical csnsequenees, and seems somewhat simpler. Xt is described below. The modified Lagange function is

+-

+-

where a > O is a new empirical constant. The factor I a is introduced in order to retain the o ~ G n aphysical l significance of K * This Lagrange function leads to the f ~lowing f field equations :

where t,, is the total stress tensor, adding to the matter contribution that af the er field,

394

Fields

Chap. 3

The equatianv (3-11.143-145) are independent of the made1 used for matter, provided the matter part of the L%grarrgefunction has been made conformally invafisxrt by the local introduction of the a field, implying

The field equation (3-17.144) first appear8 as

which is reivritden in the sfstGed form by eliminating 1,. It is also obtained direetly by applying the sction principle to the conformal response of the noninvariank f i a ~ a n g efunction,

giving

-(-s)-"2a,[(-g)

CUK l~ZB@uau@l] = --@R -1. 14-a!

(3-17. 1.50)

The quickest to draw the practical consequences of the modified theory is by returning to the souree procedures of' Section 2-4, now supglemenkb by a k r m referring do spin O particles. The respective spin (helicity) 2 and spin O sources are appropriately normalized ss [ ~ / ( 1 a ) ]'j2Tp'and [ja~/(1 C X ) ] ' ~ ~ T where the latter is bmed on the field 4(02 - 1). This produces

+

1 wz=--

EL:

2 14-ar

fdz)(drt.')[PP(z)D,(z- z")T,,(z"

+

- *T(x)l>+(z - zt)T(zt)

-+ +ffT(s)l>+(a: - zt)T(zf)f,

(3-17.151)

and the inkractian enerm with a fixed body of mass M, replacing (2-4.36,37), is

Under the eireumstanees trr << to@,the Newtonisn potential energy is retained, along with the gravitational red shift. For light, with t k k = to', the deflection and the slowing of the spwd of light are reduced by the factor 1/(1 $ a). In diseus~ingperihefion precession, the klnetie e n e w correction factor L (22"/m) isr hanged it~to1 [(l --- a)/(b a)](2T/w), which gives

+

+

+

The gravitational f Sald

3-3 7

and the perihelion precession is reduced by the factor (1 - ia)/(l convenient presentation of the correction factors is light phenomena: perihelion precession:

+ a).

395

A

1 --- fa/(1 -$- a)]; l - * [ a / ( l a)].

+

At the time of writing, memurernents on the time delays in radar echoes from Venus have produced a result that is 0"8f 0.2 of that expected from the fensor, or Einstein, theory. T h i ~limits the parameter a to 0.1 & 0.2. Z t hss also been claimed that 8 percent of the Mercury perihelion precession cart be assiwed to a solar mass qundrupole moment, leaving 92 percent to be accounted for by the scalar-tensor ~odificationof the Eingtein theow, This @wes Until independent evidenee for the solar quadmpole moment is fartheaming, perhaps from continued obsewation af the asteroid Zcarus, the question whether a weaHy coupled, massles~, spin O particle exists must still be considered sub judice. Now that a scalar-tensor t;beory of gravitation has been devised without reference to Mach's princi;ple, perhaps a word about this cosmic speculaLian is in order, I t is a natural hypothesis from the soume theory viewpoint, for it asserts that the weak field decompositions, ideniify thc! Gdds of nearby sources----2h,,(z), p(x)-.-and Lhe fields of very distant sources-g,,, 1. To draw a qualitative inference fmm &is idea withoul having to use weak field approximrttians throughout the cosmos, we eonsider an averaged situation in whieh the 'mass of the universe,' M, and the 'radius of the universe,", provide the only scales for fields and sources. This is expressed by

where the funetions r,,, s, s,,, IP are all of order unity which is to be judged here on a logarithmic scale. Then, if we exhibit; only the scale f a t o r s on opposite sides of the two field equations (3-17.1.1-3, 1441, they read

The implications are a-

logarithmically eonsistenl with a

==

X,

0.6 X IQ-', and

398

Chap, 3

Fields

-

-

-

The latkr is a well-known empirical eonnection, between the eonvent;ianal ~ ~M 10" g 10" cm-', and the grsvitsorders of magnitude, R 1 0 em, - [Eq. ~ (2-4.40)]. ~ If this relation is viewed as chartionsl constant K l ~ cm2 aetessdi~of Mach%principle, it cannot be said of source t h w y that the situation. is qualitatively altered by the infroduction of the scalar field, for (3-17.lfi0) aho %ppliesto the pure Lensor theor-y. The suggestion tha%the value of a would be significantly restricted by a common solution of the two field equations is not qttib b r n e out on examining a mast e f e m e n t a ~model. To describ-r?it we use geometrical tanwage (for the first time) and char~eterizethe space as w homogeneous, isotsopie, threedimensionally flat space. This is one of the Ffiedmann models;

To be consistent with the purely time-dependent tensor field, the scalar field is also of that character, a(t), Xf it is assumed Lhgt the matder stress tensor has only the energy component, too = Pm, the field equations imply

where the dot cjtesignates time derivative. We shall be csnknt ta pick out a particular solution: = (t/T)"3,

~ ( i= ) (a/3)'j2 log (l/$@),

p, = 0,

(3-17.163)

where T indicabs the present era, and This solution describes the matter density as negXidbXe compared go the energy density contributed by the a field, whieh, evafuakd at the present era, is

where H is the Hubbfe expansion paramekr

- --

Here is an illustration of (3-17.160), with R T,M pT3. The currently ~ 8s~ ~ aeceptcd value of H -. 2.4 X 10-18 sec-' implies p 1.0 X 1 0 g/em3p s u n n i ~th& a is fairly small compared to unity, Pre~umablythe simplifying feature of the model, p,, = 0, means that the nnalkr density ip, at least an order

3-"t

The grevitationaf field

397

of mszgnitude less than this value of p, bvhich is not inconsistent with the observ* tional data. The only sensifive dependence on a oecurs in to, the time a t which Lhe laws of physics bmonne quafitatively similar to those now prevailing, in that c($)> 0, t >z to, To the e-xtent that there is evidence for the winbnanee of these laws over a significant fraction of the age of the univerm, a is correspondingly bounded from above. The nominal value a = 0.06 gives to ~ o - ~ T . The ease with which integer spin L s ~ a n g efunetions acquire general codoe8 not extend to psrtieles ordinette invltriance by suitably introducing of integer spin, T o appreciate the difference let us folloui. the earlier wwk fidd procedure, now using the apin 4 Lagrange function

-

++

and the stress tensor [(3-7,1=)iO)f

+

t,. = ~ + ~ O b [ ~ , ( l l i ) ar.(l/i)a,lJ. v

to form g($,h ) = g($)f t,,hp'

+

(3- 17.1438)

-- --(l $ h)f#r0[r,(grP -- hrp)(l/i)a, + m]+.

(3-17.169)

Appearing here is, not Q ~ ~ (CZ A%g&Y ). - 2hFy(x),

(3-17. 17a)

but something resembling the square root of this combination, The necessary generali~atioscan, be car~r;dout, hawever, if we distinwish the vector index in gc"" - hP"(z)that is asmciszted with the coordinab derivative d, from the vecbr indw &at is tied to the matrices 7,. To emphasize this we henceforth use fatin fetters to indicate a focal Minkourski coardindtte system, which will be ret;ained for the description of spin. Thrzt; notational diskinctian is used in writing the generalization. of (3-71.171) m g" (z) = efia(z)g.aePb( X ) = eB'(z) etfz), which, is maintained under general coordinate transformations if

(3- 17.172)

There is an. independent invariance under Xocal Lorentz tnznsformations: aPa(~)= lab(%)@p6(z) , where

l", (z)gab16d(z) = grd. With the definition

(4 = @PP ($14 (2) we deducd from (3- 17.172) that

Chap. 3

and (3-17,178) g,.(z) = e,"(s)g.be!(z) = e;(z)@~.fr) Regarding the fist form of the l a ~ equation % aa a, matfix prduet, vc-e infer the dekrminantal relation -B($) = (det (3-17.179) or (-g(z)) ' l 2 = det @,.(X) = &(g). (347.18C1;)

The indicatd provisional generalization of (3-17.16I)) is

The eammponding aetion is certainly inmriant under arbitrav c o d i n % & dmnsfclrmaptions i f the field $(z) is transformed W a, scdar, But we shdX also require inva~anceunder arbitrav Xocd Lorentg dr&mfornn& %ions,for wbick (3-17.181) ia irradequak. To ztpprwbte the physical significance of the lmt requirement, Iet US cansider the response of the matbr action Co va~atioxrsof e : ( ~:)

which defines tE(z). An infinihsimai local Loreats transformation is with Invariance wiLh rapeet to arbitraw transformalions of this tyw 'thus deman&s that tb"(z)= e@"(ztC(x) = tU"(z), (3-17.186)

which i s dso the symmetw p r a p e ~ y The t e m r that is required to be symmetrical is indeed the stress knsclr of matter. This fotlaw~on. ~ f i n g which repmduces the 8Gress knsor definition (3-7.1002,

The response of

+(S)

to the local LorerrLz Lransformation (3-17.174) is

The gravitational field

3-1 7

399

where L~TOL

= l"flb.

= ?02

(3-17.191)

It is the action of the coordinate derivatives on &(l($)) that disturbs the foeal invafiance of the Lagrange function. (93-17.181). A coordinab displace ment induces an infinitesimat. Lorentz transformation and an associated field tansformation : Eag(z 4- dz) = la,(x>[8i4- hcb(~)]p

L(l(z f dz)) = L ( l ( z ) ) [ lt &'d~.b(z)+@"I,

(3-17.192)

in which

dwab

==t

-doba

E

lCa dlcb,

(3-17-193)

and therefore

L-'d,L

=

+iF,a,lebgab.

(3-17.194)

I n order do compnsate this eRect the coordin&e derivative in the tagrange function is replaced by a, - t i ~ . , ~ @ " ~ , (3--~7.195) where ua,b(;c) behwes like a covariant vector with respect t o general coordinate fransfarmatians, and responds do Xoeal Lorentz transformations in such a manner that L-'(a, - ai@.,boa6)L = dg - aiwapboab. (3- 17.196)

The required transformation law is

A fundamental mixed t e n ~ a is r defined by the cornrnutrtLor

which has the character of an antisymmetricat Lensor of the seeond rank for general coordinate transformations, and of an antisymmetl.iea1 tensor of the second rank with respect to focal Lorenta transformations. A scalar in both senses is construct4 by e'"(z)ePb(z)Rpv.b(x) = R(z), (&17.200) and provides the basis for a gravitational Lagrange function:

I n the weak field limit, where the linguistic distinction of %heindices is removed and @"(X)c?i g@@- h p a ( % ) , (3-17.262)

Flslda

Chap, 3

this Lagange function rduces to (3-1?.38), ixpa& from s divergence km. The ~ p l i e a t i o nof the stationary action prinoiple to variations of U,,& in, a(e, CJ) E;ives 8.[e(e'@ePb-

- oo,,[e(&'ePb - @'epC)]- w$,[e(&@epe- dcePo)]= 0. (3-1 ?.m31

The following definition of

ii

quantity $h& is not a tensor,

enablm one to prrtsent this equation W aabc

where

+

@aeb

"-

A. = e-'a.(eeL)

@cab

"“k gbcha

- W.*

=

- @&BAG

-(nut

Q:,

-+

(3-17.m) (3- 17.m7)

meordintg t-o the dehrminsntal fornnuta On eantrm%ingb and c in (3-17.209 we get various muItipIm of ,X, implying that

from which X, = O ia %%ilk recovered. The East ref%tionis solved by it is the ~ t r o field ~ g generalizeztion of (3-1 7.17). Amth:er w a k Md propem, Eq. (3-17.n), is generalized by defining

vvhich syrnrnetry p r o ~ & y=presses the rc3latian of (3-17.210)* The additional definition (3-l"1.213) rip= e z e ; ~ . ~ . e ~=' F,X enables [email protected] w ~ t (3-17.212) e

EH

Mul%iplicatiom;af this equation by ePa, foflowed by symmetrisatioa in y and v , removes the ua&b krm and givm

3-1 7

The gravitational field

401

This is recognized aa the statement that the covariant derivative of gf"" vanish=, , : ' l with the quantities of (3-1 7.91), known usually a s Christoffel and identifies symbols. After this, it is abundantly clear %hat the two objects defined in. (3- 17.124) and (3- 17..f99) are connected by

where the correctness of the algebraic sign can be verified in the weak field limit* Thus, the two gravitational Lagange functions, $(g, l") and c(@,O) are idemticd. Returning do the spin $ Lagrange function (3-X7.181), we insert the eoordinate d e ~ v a t i v egeneralization stated in (3-17. 195) and obtain

where, i d should be noted, $he total antieommutativity of the field extracts the antisymmetrical part of the matrices u0r*obC.This removes the terms with a = b or a = c, trhieh are proportional to the symmetrical matrices roro. Then, since

we can write the Lagrange function as

+

+ m]$,

c(+,e) = - e i $ ~ ~ [ ~ ' e i ( l / i ) a , *oGiv,r,

(3-17.219)

where

*&

= aC*bcdWabe.

(3- 17.220)

The notation g($,e) might have been elaboraled as C(#, e, w). When this structure i8 added to the gravitational Lagrange function (3-17,201), in whieh eg and are used as independent variables, the W,,& dependence of g(+,e, W ) produces an additional term in (3- 17.203), ~vhich removes the symmetry property noted in (3-l"7.212). This is a natural form of the theory, But we shdl prefer, far simplicity only, to regard a a b e as defined by (3-17.21 1) and therefore not subject to redefirzition through the appearance of @,be in the matter Lagrange function. I n order to identify the stress tensor t,, directly, we consider the special variation 6eg = -$-ligPve,,, (3-17.221)

die21 is consistent with the construction of gP"ronrr the e,: and gives

A little cafeulalian sho\vc.s that

Chap, 3

and thus,

where the first two teirns can be united through the reintroduction of the spinar eovariant derivative (3-17.f 95). The sedar defived from this k n m r is

A. aimplifieation can be made through the we of the field equation implid by the Lagmnp funetion. But what is requird here can km obtain& marts directly by 8ppfying &heaction principle to Lke gadicular field vadation

The rmpontse of the mat;ter action term is

since no contributit;ion involving 8,6h appears, owing Lo Lhs symmetry of the matrices rave. The conclusion is that the Lagrsnge function vanishes (at points not occupid by pa&iele sources) and The lmt wnsiderakion is intimately relabd to the p o ~ b i l i l yof exhibiting s @onformallyinvafiant matter Lagrange funetion khrough the introduclion of the scdar field cr(s). Conformal tmnsfarmLions on the t e t r d of veehr fields G(z) appear as or, using infinitesimal &mnsformations,

The approprlak conformal bhavior of 9(2) is alwady @tat&in (3-17.22fi). It is such that %hereplacement of m by mrr in ($17.219) maces to produce a

The gravitational Pfafd

3-1 7

403

conformally invariant Lagrange funetion. The deriv~tionof the ~cralart thmugkr the tr-dependence of g($,e, a) according to (3-17.147) then @ves (S17.228) directly, with mcr substituted for m, The temptation to extend this diseusgion to arbitrary multispinor fields will be resisted. Instead, wc %urnto the long deferred topic of the pavita&ional model. of particle and gra.viton sources, I t is wodh appreciating why id ia thaf we have managed thus far u~ithoutexamining this question, The arena of gravitational phenomena is confined essentially ta ~tronomicalbodtim, which are beyond our exger"rxnentft1 control. Ftceordingly, urehave no overt use for ~ o u r e e ~ , lvhich give idealized expression to the experimenter's ability to manipulate the physical situation being studied, The graviton source concept has already fulfilled its primary mission by serving as the modet upon which the eoordinale invariant dynamical theory has been erected. NeverLheless, some remark8 are ewled for, &though, as the pmce$ing comment; indicales, they ean be limiM to the use of particle and gravihn sources under the weak field graviLaLiond conditions prevailing in terrestrial experiments. The following brief analysis wed not be applicable to experiments conducted on a. spaceship in d m orbit about a, rapidiy spinning neutron star. The first point a t issue has previously been raised far charged parliiicle sources. The generalized stress tensor consewation law, (3-17.103, 1M), ~411 fail inside particle sourees unless one recognizes the pre-exisknee of the enerw and momentum that is trzlnsferrd to the emitted particle, There is, however, no eIectromag;netic andogue to the graviton murce problem. Photons are electrical1y neutral, rvhereas gravitons carry enera-momentum which must dso be transferred rather &an created within the source. Just as an explicit A,dependence was introduced to provide the correet gauge transformation bhavior of charged pal-l;iele sources, these source problems can be viewed as the search for the explicit gPY-dependence$fiat will give the varisus BOUI"~C?Sthe eomeef response to general coordinate transformations. The simplest; example is a scalar source K(s), appea,~ngin the action through the term (dz)(-g(z)) L'Z~(z)@(z). (3-1 7.234) We must replace x"" by a functional of the g,,,zpfx, g), such thstt under a generd coordinate transformation xp(z,8)

for then

z==

~ ' C x g), t

(3-17.235)

shows the requird dynsannkal equivafeme of the field8 y,,(a), #(s) a d 51,,(2), +(X), Under weak field conditions we write

where the inva~ancepropedy (3-1 7.235), staged for infixritmimal f retnsforma%ions(2""= f l - &Y),requires that

This must hold as solution is

a,

consequence of the gawgcj translcrrmation (3-17.45). A

(dz"

where f"(z

- x')

)(dx")fIr

(X:- xylf" (X' - S'') 4, (S"),

(3-17.239) )is

one of the familiar elws of functions obeying

aufc1(z - X') =

&(X

- a').

(3-17.XO)

Tbe matter strms kngor that is rtow dc?rived, slakd in. the absence of %he pavitation field for simplicity, is the consewed objeek

(3-17.241)

where tC""is fhe tensor given in (3-7.8), which. is sueh Lhst

The possibility of ma,vil;a>rremission otscu~ngdirectly from the rrtafkr source is exbibiM, for single graviton rta&a%ion,by writing the coordinate i n v a ~ a nft o m of the murce term (3-17.234) m

Asmrning that the graviton detection sources do not overlap fhe K support re@on,one can use Lhe sourwfree, weak gr~viLtls1ionalfield equations (3-17.13,14) La derive

The gravitational f Deld

40Ei

This enabf es one do present (3-1x243) as

Alternativety, one might have b w n with the la& form, where the additional k r m mrvw tO remove the responw of + ( X ) to infinitesimal coordinate transformation~.Similar discussions can be given for any other type of matter source and fidd, wifh appmpriate &Lention to their transfornnation properties. The weak field form of the gravitorr source term in the action is

(d.) TRdnI"pCo> 4 (dx)TY(z)~lr.P(z) P --)

(3-1 7.246)

where d J I T ~ ( x= ) 0,

and a eonstanL is added to &rrive a t the second vemion of (3-17.246). The physical property to be represented is thaL the radiation of an ztdditional grsviton can accompany the working of a graviton bouree as well as a ma;tder source. The mathematical problem is the removal of the responsle of g,(z) to infinitegimal coodinale transfarmatians, apad fmm gradient hrrns-gsuge transfarmationls -which do not contribute in (3-17.246). If we use the symbol to indicak identiw apart from gadient term^, Che response of gr,, to infinitesimal coordinate tr~nsf ormations, Eq. (3- 17..96), is expregsed by Then, appf ying the weak field stakement (3- 17.45),

which gives the required generalization of (3-17.246) :

The source Pp(%) that is notv derived through variation of g,,(z) is

Through its dependence on the gravitatiortd field to the required accuracy, P" doe8 respend appropriately to infinitesimd coordinate transformalions, and o b y s the divergence equation (3-17.104). Finally, it should be said that, tzs

Fields

Chap, 3

in the plnoton situation, the consider4ttion of additional radiation h r n the Bources ean be avoided by &opting s n equivalent g&uge, The pavitational gauge eon&tion is (dz')fP(z - z')hiP(xp)= 0, (3-1 7.253) feding to %bevanishing of the X h terms. This volume closm with s short exchange bet~ieenEarold arid the author.

H. How Can it be the end of the bmkl You have haray hewn. There are any numbw of addifionail topics X should like to 8% developed from the viewthe revimers, point of source theory. And think of the field day you wiIl who usually prefer to list all the mbjeets not included in a volume r ~ t h e rthan discuss what it dws contain. S. Quite tme. But we have now reslelned the point of tr&nsifionLo the next dynamical fevd. And, since this volume is already- of ft re-mnable ~ize,and many of the ideas of source thmfy are in it, if" hardly fully developed and applied, it m m s better to put it before the public as the first valume of a s e ~ e s .Hopefully, the next volume will be prepsped in time to meet the pawing demand for more Souree Thwry.

WOW T 0 READ VOLUME I

The first volunle was described as a resewch document, and a textboak. Unfortunately, the beginning student was given no guidelines to tell him into whicfi category a particular section fell. hecordingly, here are some suggestions for a first encounter with source the or!^, and relativistic quantum mechanics, a) In Chapter I , omit Section 1-4. b) In Section 2-1, the derivation of the Lorentz transformation behavior of the

source function from that of states can be omitted, I t is sufficiently evident; from the form of Eq, (2-1.38), for example, that K ( x ) is a scalar function. c) Omit the muiti-particle generalizations of the vacuum amplitude in Section 2-2, They are of interest primarily in many-particle applications, which are not yet a t the center of attention, df Sec3tion 2-5 need be read only to appreciate the general linear transformation of sources and its relation to spin, tagether with the .possibility of campsing arbitrary spins from more elementary ones. e) The discussion in Section 2-6 that begins with Eq. (2-6.24) can be omitted by recognizing directly that (2-6.26) is the covariant generalization at the projection matrix 4 (1 ps), which selects a definite parity in the rest frame and, tbereby, the two components appropriate to spin 4, f) Omit the multi-particle generalizations in Section 2-7, g) I t is sufficient, In Section 2-8, t o read the discussion of spin $. 11) In Sections 3-1 and 3-2, omit multi-particle generalizations. i)Omit the discussion of spins 3 and in Section 3-25. j) Xn Section 3-4, restrict attention to multispinors of ranks 2 and 3. k) The spin liznitations already noted should be continued in Section 3-8, E) The rambling discussion about the arbitrariness of stress tensors that appears in Section 3-7 should only be skimmed, m) The lengthy account of magnetic charge and its conceivable relevance to hadronic behavior [Sections 3-8, 3-93 is optional, However, don't miss the debut of Harold on p. 2401 nor the remarks on mass normalization [p. 2471. n) Most of Section 3-17 is optional reading, particularly the discussions of broken conformal invariance, cosmology, and spin gravitational coupling.

+

Appendix

Finally, we add two minor comments about specific topics in Volume I. X . The discussion of Eqt . (1-1.44) does not make clear that cornmutativity of the two

displacement operators remains an alternative possibility (the numerical coefficient zero cannot be changed to unity by redefining the operators), 2. The f o m s of Lagrange functions that yield first o d e r differential equations were merely stated in the text. The genesis of thew expressions might be clarified by this illustration far spin Q. Begir*ningwith the second order fom [Eq. (&5.12)]

we introduce the independent veetor\field 4, by adding to 9 the term

The nature of the system is not changed thereby since, on extending the action vanishes, apart fram a pssible source grinGiple to $, we learn that QI, term. But, on adding (A-1) and (A-21, the squares of the first derivativs cancel, producing the Lagrange function (3-15.161, from which the fimt-order field equations fallow, This procedure is the analope of one for ordinary mechanics that begins wit h the quadratic Lagangian

where m is a mn-singular symmetrical matrix, and inlrducw the indegn$ent variables $ by adding

The sum of (A-3) and (8-41,

yields the equivalent first-order Hamiltonian description,

Index AeLiont 186, See aka Lerange function additivity of, for noninteracting particles, 256 for arbitrary spin, 191 coardixl~teinvariamt gravihn sourw term in, 405 discontinuous ehange of, and eharge quantigation, 242 electrom~gnetie,displacement charge of, 248 with field strengths as variables, 340 modificsticln for point charge@, %3-244

for point charges, 247 of ebctromqnetic field, 229, 235 for gravitoa, 198 of graviton field, 381 pavifon source brxn in, 485 helieity decomposition of in nr-tas~lesa limit, f 97 for interacting par-ticies, 282 ixzva~anceof, 283 under gauge and coordinab transformadionru, 383 under TCP, 277 partial, 278, 279 for particle with grewribt3-d motion, 24'7 for photon, 198 quantum unit of, 8 mduced form, for second-rank qinor, 192 mcond-order formulation for third-rank spinor, 194 for spin 0, X 87 with primitive electromagnetic interaction, 256 for 8pln B, 190 bawd on srjcond4rder quafion, 193

for spin 1, 188 for spin g, 191 for spin 2, 189 for spin 8, 191 for spin 3, X96 stationary, 187 of time cycle deaription, 197 for two kinds of photon sources, 239 Advanced Green" function, 148-149 for Dirac equation, l62 AmpBr-Ian inkraetion, 77 Analytic contiauation, 36 Analytieity, 35 Angular momentum : commutation relations in three and four dimensions, 88 exgeetation value, relation to flux. vector, 224 and souree rot&ions, 223 sgecification of stab@,53 Angular momentum aux vector, local conservation of, outside murces, 223 Angular momentum operskr, 8 Antiparticles and t3pace-time uliformity, 47

Antiperiodie Green" function, 163 Area, f WO-dimensional,247 Astronarnicd appfieadions of source: theory, 82 Auxiliary fields: for third-rank multispinnors, 179 for fourth-rank mul6iirrpimrs, 181 for fif th-rank muftiqinor~,183-184 for @pin$ parli&s, 117 for @pin3 particles, 171, 172 hid degcription, 84

409

410

index

Bargmann-Wigner equations, 264 Barryolxs, 251 magnetic model of, 253 Basis, 2 Beam of particles, 41, 58,57 Bhabha sclcbttering, See Scattering of spin -& particles with opposite?! charges Booster, 8 Bose-Einstein statistics, 55 Boson, 113 Boundary condition: for Gfe(?n88 function, 146 outgoing W&*, 153, 157 of periodicity, 150 retarded, 148 of vacuum time cycle, 148, 162 Bound slates, 346 Brrtns-Dieke, 392 Brernssdrahlung. See Photon emission in Coulomb scatteGng Broken conformal inva~aneetheory, 383

prscticsl consrtquences of, 394 Broken symmetry, 393

C, 49 Cauchy principal value, 46 Causal analysis of vacuum amplitude, 51-52, 109, 119 Causal control of H-particle transitions, 362 Causality, 313, 37 and unitarity, 61, 122 Causal structure of propagation functions, 42, 5& 559, l20 Center of mass, 35 G.G.8, syatem, iv Chalcidian slababet, 234 Charge, 47-48 accelerated, and rdistian, 265 conservation of, 255 in interaction, 285 distribution of, 77 dynamical and kinernatical arspeet;gs, 2 s eigenveetors of, 284 electric: %xisof, 250

conservation of, 72 twa-dimensional lattice far, 251-252 universality of unit of, 250 electric land magnetic, coexistence of, 242 exmctation V ~ U Z Irelation , ta flux. vector, 200 and fermions, f 13 leptonic, 125 Iocalisation of, 243 magnetic, 231 unit of, 250 of multiparticle states, 53 purely magnetic, unit of, 252 Charge fluctuatim ftux vector, 207 marge Buetuation~and p h m transformations, 207 Charge flux vector: local conservation of, outside sources, 200 for spin O particles, 200 for spin particles, 204 for spin 1 particles, 203 for spin 2 particles, 204 for third-rgnk spinors, 206 Cbsrge matrix, 48 Charge quantization, 239 condition for, 249 Charge reflection mratrix, 49 Charge symmetry, 49 Christoffel symbols, 401 Clifford-Dirac algebra-, 105 Coherence in scattering, 358 Collisions, 37 caussll control of, 36 Commutator, 2 Complernentarity, 38 of source descriptions, 39 Complettlness, 41, 53, 58, 59, 121, 158 of H-partieb states, 348 for particles of arbitrary spin, 137 of spin particle states, 110 of spinors, 319 Complex conjugation :of Fermi-Dirac sources, t 10 of sources, 131 Complex fields, 153 charge interpretation of, 1% of time cycle description, 154

Components of a vector, 2 Composite systems, 344 consistency of phenomenologieal description, 357 Compton scattering. See Scattering, of photoas by charged particles Conformal group, 225 and conmrvittion laws, 2226 for electromagnetic field, 230 represented by %otatians,2225 and stress scalar, 226 Conformal invariance of action for massless spin 0 particles, 39% Conformal transformations, 388-3W Gonmrved current, construction of, 255 Constructive principles: of S-matrix theory, 35 of source theory, 31 Contact terms, 144 and field digerential equations, If38 introduced by source redefinitions, 173 in multispinor description, f 77 for spin g particles, 176-177 for spin 3 particles, 171 Coordinrcte displacement, local Lorentl; transformation induced by, 399 Coordinak invariance, general, and particles of integer 4 spin, 397 Coordinate invariance, infinitesimal, of spin O Lagrange function, 384 Coordinate systems: in @ace-time, 7 transformations between, 332 Correction factors for light phenomena, and perihelion precession, 395 Coulomb gauge. See Rdiation gauge Coulomb inleraction, 77 Coulomb pobntial, 320-321 Coulomb scattering : connection with nonrelativistic limit, 323 and dynamical levels, 322-323 of spin O particles, 321 of spin particles, 322 Coulomb correction to, 324 phaw shift between helicity transitions in, 327 spin dependence of, 328 Govariant derivative: of contravariant vector, 386

+

of covariant veetor, 386 of g, P",g,", 337 of bnsors, 386 Critique of particle theories, 24 Crossed reactions and anslytieity, 36 Crossing relations, 33, 290 and A- and B-cwffieients, 353 and electron-photon scattering, 318 and electron-positron scattering, 308 and pair creation by a, photon in a Coulomb field, 343 spurious minus sign in, 318-31 9 Cross section : definition of, 286 differential, 287 elastic, near resonance of photon scattering by H-particle, 377-378 total: for arbitrary energy electronphoton scattering, 320 for electron-positron creation in two-photon collision, 31S for high-energy electron-photon scattering, 319 near resonance of photon scattering by H-particle, 374, 377 for slow electron-positron annihilation into two photons, 312-313 for spin O particfe pair annihilation into two photons, 299 for two-photon annihiiation of highenergy electror, and positron, 315 for two-photon annihilation of unpolarized electron and positron, 317 Current algebra, iii Current vector, 201. See also Charge flux vector ambiguity in, 203 associated with particle stste, 201 electric : for arbitrary multispinor, 2434 asmeisted with charged particle murces, 258 identity of kinematics1 and dynamical definitions, 262 for spin 4, 260 for spin I particles, 263 for spinor-tensor field, 265 of nonconsrved point charge, 258

412

Index

Decay constant and width of spectral line, 371 rlelta function, four-dimensional, 146 Determinant: e, relation to g, 398 and Fermi-Dirac statistics, If 8 g, transformation bhavior of, 385 Differential cross wetion: for arbitrary energy electran-photon scattering, 31+320 for arbitr%ryenergy spin 4 scatkring of unpdt-srized like ehtzrges, 305 for Compton scattering by electron a t rest, 341 for Cornpton scattering by spin O particle at rest, 335 for Coulomb scatte~ngof &pin0 particle, 321-322 Coulomb correction to, 324 for Coulomb scattering of spin -& parlicle, 322 Coufonnb correction to, 328 definition of, 287 for forward and backward emission in high-energy electron-positron annihilation, 316 for high-energy electron-photon scattering, 318 far high-energy electron-positron wattering, with equd helicities, 307 with opposib bhecities, 308 high-energy, for two-photon annihilation of urrpolariged electron and positron, 314 for high-energy pair creation by a photon in a Coulomb field, 343 for highenergy photon emission in Coulomb scattering, of spin O particle, 338 of spin 4 particle, 343 for high-energy spin $ scattering of like charges, with equal helicities, 302 with opposite helieities, 303 for high-energy spin $. scattering of unpolarized like ehares, 303

for high-energy urrpalarised electronpositron scatbring, 308 for low-energy spin 4 scattering of unpolrzrized like chetrges, 304 for low-energy unpolarised ebclronpositron scathring, 308 near resonance, for photczn scattering by H-particb, 373-3cr4 for photon emission in Coubmb scattering of nonrelativistic spin 0 particle, 330 for photon scatbring, by H-particies, 358

by realistie EX-partichs, 3% Rutherford, 289 for scatbring, of circularly polarizled photons by spin O charged particles, 286 of linearly polarized photons by spin 0 charged particles, 295 of spin 0 rtnd unpolarised spin 3 pmticles, 310 of win. Q particle by massive spin 9 psrticle, 31 t of anpolarized spin 4 particle by massive spin O particle, 31 X far &pinO particle pair creation, by cireularly polari~edphotons, 298 by Iineetriy polarized photons, 297 for spin O particle pair annihiltiltian into circularly polari~edphotons, 299 for spin 0 particles, of like charge, 288 relation btween like and unlike charges, 8 0 for f wo-photon annihilation of unpalarized electron and positron, 317 unit of, 296 far unpolarizred etectron-positron scattering at arbitrary energies, 309 for unpolariaed photon %tatbring by f pin O efiarged psrticfes, B 4 Dif3Ferential equations, firstorder :for electromagnetic field, 228 for srbitrary-rank multi~pinors,f 85

Index

for fourth-rank multispinors, 182 for fifth-rank multispinors, 184 for graviton field, 380 for heliciw 4 fields, 176 for spin O field, 187 for q i n field, 160 for spin l fields, f M,178, 188 for spin # fields, 173 for spin 2 fields, 166, 189 for sipin 8 fields, 177 for third-rank muldispinors, 179 DiBerential equations, fourth-order, for fourth-rank muldisginorfi, 182, DiBerential equations, %can$-order : for electromagnetic field, 228 gauge covftrianl for spin O field, for gravilon field, 379 for helicity 3 fields, momentum space transcription of, I69 for second- and first-rank spinors, 180 for spin 1 fields, 164 for spin 2 fields, f 66 momentum space transcription of, 167 for spin 3 fields, 172 for third-rank multi~inors,180 Differential equations, transformed by source redefinitions, f 74 Dilations, 215 isstropic, 224 Dipole xraoment :electric, 13 of spin g particle, 261 magnetic, 13 Birac equation, wage of fields in, 1611 Displscernent : of coordinate frame, 39 of muree, 39 Dispbcements: arbitrariness in, field responm do, 214 in causal situation^, 210 variable, 209 Double scattering and polarization, 328 E)wl charged particbs, 251 charge assignments of, 253 Dug1 field strength tensor, 228 Dual knsar, definition of, 17, 74

413

Dynamical evolutian: first level of, Z8 wcond stage of, 378 Dynsmics: in operator field &gory, 33 in S-matrix theory, 36 Dyon, 263 Effective local interaction. betwwn Maxwefl and Diratc Gelds, 313 EEeedive tweparticlc? source: for photon and spin 0 charged particle, 266 for photon and spin 5 charged particle, 276275 Eigenfunctions: related do nonrelativistie wavefunctions, 352 spin 0,orthonormality of, 345 spin 4, eomplehness of, 35S360 orthonormality of, 348 Einstein A-coefficient. See Transition probability, per unit time, for R-particle radiative transition Einskixx B-coescient, 353 Einsbin grsvitational field equation, 388-389

Einsteinian rebtivity group, composition properties of, 15 generator eommutrttian relations, 16 Einstein theory, obmrvationat tests of, 82 EEectrie charge, Xoeal conservation law, 28 EIectrie dipole moment, exltt-rnal, of H-pal-ticle, 359 inkrnal, of H-pafiicle, 358 Electromagnetic fieM, 227 commutation relations for, 26 Electramagnetic model of sources, 255 Eleetromagnetie source models, 257 EXeetron, 34 Electron+lectron scsltering. Bee Mplller scattering Electron-positron scattering. See Bhabha soatbring Energy :internal, 11 of quasi-static source distribution, 77,

82 Energy-monnentum : con~rvationof, 8I expectation value, relation to flux vector, 220

of multiparticle states, 53 prwxistence Bvikhin sources, 403 Energy-momentum eon~rvationand kinemstical integrals, 288 Eneru-momentum flux vectar, 209 lac& conmrvation of, out~idesources, 210 Enmgy operator, 8 q u a t i o n of motion, proper Lime, 23 Euclidesn Green's function, 146 Euclidean postulate, 44 and arbitrary spin, 189 and *in 3 particles, 111 Euclidean propagation function, 44 inequalities for, 45 Euclidean space, attached Ito Minkowgki space, 43 Exclusion prineipb, 109, 119 E w c t a t i o n values, fr13 Exbnded sources, 265 Exbriar algebra, 106 f@: cfsss of, 233

eovariant choiee of, 260 covariftnt, physical interpretation of, 269 and electromagnetic source models, 257 quation for, 233 quivalenee of electromagnetic warm model. and gauge intnz-rpretations, 269 equivalence of timelike and spacelike vectors, 259 snd gravitation81 gauge condition, 406 and gravilcttional source models, 404 and magnetic charge, 241 W noneonserved etrrrent veetar, 258 and scatbring experiments, 279 upp part of, 248 asymmetry restrickion on, 234 uwd in conmrved curreat consItruelion, 255 m d in gauge condition, 238 to eharacbriz;e gawe, 255 1Fermi-Dirac statistics, X03 Fermion, 113

Field, electromagnetic, 26 Field equrttiona. See alr~oBiSerenthl equeztions of broken eonformal invariance theary, 393 Dirw, with electromagnetic inbraetion, 262 nonlinear, 279 gravitational, 388-381) MaxweU, with chsrged particle current, 257 in grltvitational field, 388 nonlinear, and physical proeessa, B7-278 for spin 0 padiebs with prinaitivcs ellectromagnetie interaclion, 256 for spin I. particles with interactions, 263 Fields, 145 msoeiabd wit h individual emission, absorption act, 154 auxiliary, 171, 177, 180, 182, 183 causal evaluation of, 153 complex, X53 far spin. 3 particles, 15%160 of Lime cycle; description, 154 correlation funetions of, 30,32 electronnagnetic, 227 elimination of, 278 equd time commutation relations of, 32 I[",,x, of gravitons, 380 grsvitationd, 378 graviton, as wesk gravitational, 385 helieity decomposition of in massleas limit, 197 of helicity 8 prtrtieles, differentid equations for, 176 of helicity 3 psrticles, dif"ferclntia1 equations far, 169 H-particle, 351-352, 354 integral equation for, 365 Maxwel.1, conformal Lr%nsformationof, 390

multispinor, 1'7'7 WBhr, of ~ ~ B V ~ L O R S380 , operator, 24 r e d and imaginary, 2CZS

retarded, 147 of Dirac equation, 162 spin 0, for causal arrangement, 284 of spin 0 particles, 145 spin 9, causal expressions far, 159 response to local Lorentz transf ornnations, 398-399 of time eycle description, f 61 of spin particles, 157 differential equations for, 160 of spin 1 particles, 164 differential equations for, 164 of spin 8 psrticles, E72 difffsrential equations for, 173 of q i n 2 particles, 165 digerential equations for, l66 of spin g particles, 175 diEerentia1 equations for, 177 of ;spin 3 particles, 167 differential equations for, X72 tensor, of gravitons, 378-379 of time cyele description, 147, 152 of unstable H-particles, 369 Field strength Censor: d u d to, 228 electromagnetiic, 228 Fine stxrrcture constant, 250 Fluctuations, 64 Friedmann models, 396 g, for arbitrltry multispixlor description,

265 dependence an description, 262 for spin 3 particle, 280 for spin l particles, 264 Galiban relativity group, composition properties of, 8 generator commutation rehtions, 9 Gawe conditions, 238, 258 on MftxweXl Green" ffunction, 279 Gauge covariant derivative, 262 Oauw invariance, B8 grwitalional, 380 Gsuge transformations, 228 Abelian group of, 262 grtrvitstional, 380 Cwa kinds of, 2363 Generators, 3

Grassmann algebra, 106 Gravitational constant, 82 Gravitational deflection of light, 83 Gravitationd field, 378 conformal Lrsnsformation of, 389-390 e;, 397 e;, 307 g@",384385. g,,, 385

f l i p , 400 F;,, 385 l'$, construction of, 387 X",: transformation law of, 386 rak,

400 399 Gravitational red shift, 82 Gravitational slowing of light, 83 Graviton, 80 wtian for, 198 etdditional, ernitkd by graviton source, 405 emission from nnatbr source, 404 helicity states of, 81 a parable, 81 vacuum probability amplitude for, 80 Grsviton source concept, 403 Grftviton source problem, 403 Green" function, 146. See also Pragw8ticzn funet ion advanced, 148-149 of Dirae equa;tion, 162 antiperiodic, 165 associated Euelidean, f 46 boundary condition for, 146 for charged spin Q particle, 271 for charged spin g particle, 218 Dirac, causal expressions for, 158 of Dirae equation, 157 Birac, momentum integrals for, 158 of Laplaeek equation, 260 Maxwell, olzeying gauge condition, 278-279 modified periodic, 155 periodic, 149 Fourier series construction, af, 151 retarded, 148 of Dirac equation, 162 %&b,

spin 0,eigenfunction construction of, 344

spin 0, expanded in powers of vector potential, 281 spin 0, iterative constmction of, 351 spin -$,eigenfunction construction of, 348

spin $, exlpanded in powers of veetor potential, 280 spin 3, high energy limit of, 35s3fi0 spin ikratim construction of, 383-354

Group :Abdisn, of gBuge transformations, 262 AbeIian, of translations, 5 cornmutation relation8 of gener~brs,4 eommutation relations for three paramekra, 7 composition properlies, 5 conformal, 225 Euclidesn, connected pieees of, 49 finib dimension&lreali~ationof, 7 of Galilean transformations, 8 generators of, 3 Lorents, connected pieees of, 49 hrentz, finite repremntcation~of, 86 non-Abelian, of general coordinate transformations, 384 parameter space, 7 structure constantsp4 of unitary transformations, 3 Gyromagnetic ratio, See ako g of electron, 13 of muon, 13 Hadrons, 251 m magnetically neutral composites, 251 Harold: on history, 338 on magnetic chsrge, 24%241 on modifying Einstein" theory, 392 on source theory and reviewers, 406 on speculation, 2% Heaviside step function, 75, 234 Helicity, 20 eigenvectors of, for arbitrary spin, 136 for particles of inkger 4 spin, 132 fsr spin partiehs, 129

+

of neutfinos, 125 spin I states of, 69 spin 2 staks of, 79 Hermitisn adjoin%,1 Hermitisn g matrix reafigation, crihrian, for, 6 Hermitisn operator@,1 infinibsirnetl, 2 Homogeneous electromagnetic field, and spin O Crwn" function, 272 H-partides, 344 fields of, 351-352, 354 I' matrix, 3fi3. ixthgrlat equation for fields of, 365 modifid propag~tionfunction, a(i3 persistence probedbility, 3Ci8 photon scattering by, 355 Il matrix, 365366 propagation function, time limitstion on, 365 rdiative transitions btween, 350 skeletlllt irtterwtions with photons, 351,354 sources of, M?,349 unstable, fields of, 369 photon emission by, 368 time kbavior of propagation. function, 368 s s unstable particles, 360-361 virtud, 362 as effective wurees, 365 Hubble e w s n i o n parameter, 396 Ido, Zchir6, 81 Infinitesimal rotiztions, response to, 9 1n6niLesimaI Lrsnslations, response to, 10 Infra-red catastrophe, 273 Interaction skeleton, 277 Interaction volume, 283 f nvariance transformations, 199, 209 Invariant flux, 287 InvarianL momentum apace measure, 31 f rreducible processes, 283 Isotropic dilations, 224 Jacobi identity, 3 Jordan, P,, 392

Klein-Nisfiina formula. DiBerential cross mction, for Compton scatbring by electron a t rest Lsgrange function, See also Action arbitrariness of, 188 far srbitrary-rank spinar, 195 with broken confomal inva~ance,393 conformally invariant, of nnwive pin O partides and gravilatiana1 field, 392 of electromagnetic field, 229 gauge invariancto of, 256 gravitational, 385, 399 of grsviton field, 381 of gravitons, response to gauge t ransformations, 381-382 of interacting Dirac and Msxwell fields, 261 for massless spin O particles, eonformal transformation khabviar of, 391 of matter, arbitrariness in grsvitational field, 391 modified, af spin O particles in gravitational field, 391 of photons in gravitfttionai field, 389 for second-rank spinor, 191 for spin 0, 187 of spin O particles, in gravitational fieid, 385 with primitive electromagnetic interaction, 2565 with primitive gravitational interaction, 383 for spin *, 191 conformaXly invariant form, 402 in gravitatianaI field, 401 in weak gravitational field, 397 for spin l , 188 for spin 1 particles with primitive interaction, 263 for spin g, 191. for spin 2, 189 for spin g, X91 for spin 3, 190 for third-rank spinor, 194

Lafiace" equation, Green's function of, 260 and trttceless tensum, 93, 98 bgendtre" ppolynonnirtl, 93 hptonic charge, 125 hptoxls, 125,2-53 Light : gravit &lion&deflection of, 83 graviktional sfawing of, 83 XIight phenomena, in scalar-hnsor theory, 395

Linear momentum operabr, 8 healised exeitslions, propagatian characteristics of, 30 1;o~aEh r e n t ~transformations: requirement of invariance for, 398 response of spin field ta, 398-8W h a 1 Ninkowski coordinate system, 397 h n g wave length gauge, and photon, scattering by realistie H-pa&iefes, 358 b r e n t z gauge, 281 Lorentzian shape of speckrat line, 871 Lorentz Lransfomations: composifion of, 843, 10&101 linear respone to, 85 represctntatisn by matrix similarity transformations, 103 response ctf -spin nratriees to, 101 Lorentz transformations, infinitelml: ogertztor field theory construction for generators of, 24 redization of generators of, 19 response of electromagnetic field to, 27 response of single particle states to, 39 responm af saurce to, 89 responrxt of stale h, 89 responm of &tressdenwr to, 25 responm of vector ta, 89,85 respoxlst: to, I43 and the space reffeetiun matrix, 1OO and spin matrices, 86 Mach" principle, 395 hfagnetie charge, 15, 231 apparent srbitrariness in description, 241

Magnetic moment, See also Dipole moment, magnetic relation to angular moments for pin 5, 261 Magnetic quantum number of multiparticle state, 54 Mass, 11, 17 Masshss particles: of arbitrary helicity, 141 of arbitrary integer helicity, 9G97 of fnelicity O and 1, 144 of helicity *, 125 of helicity l , 72 of heticity %, 129 of heXieity 2, 80 of helicity 8, 175 of helicity 3, 99, 168 infinite spin limit, 21 af integer $ helieity, X33 nonlocality of, 20 nonlocalizlability principle, 22 position vector of, 20 Mass normali~atian,247 Ntaxwell" equations, 28, 228 with charged particle eument, 257 with electric and magnetic currents, 237 Mesons, 251 magnetic model of, 253 Metric tensor, 16 Mode, 119 M#Iler scsttering. See Scattering of spin g particles with equal charges Momentum cells, 201 Multiparticle states, 52, 68 parztnletrized mixture of, 149, 155 for particles of arbitrary spin, 13&137 of spit1 particles, 110, l18 Multiphoton emission, soft photons, 270 Multipole moments: electric and magnetic, X17 radiation by, 354 and spin, 280 Multispinor, 134 relation to tensors, 142 Multispinor fields, 177 of arbitrary rank, 184

+

of rank 2, 177 relation to knsors, 195 of rsnk 3, 178 of rsnk 4, 180 of rank 5, l82 symmetry and spin, 185 Muon, 34 Neutrinos, 125 helieity of, 20 Neutron star, 403 Newtonian interaction, 82 Nonunitary transformations, 87 Operator field theory,iii, 24 consistency of, 28 dynamics in, 33 particles in, 30 speeulstion in, 34 and strong interactions, 34 Orbit, equation of, 85 Orthonormal spin-angle funelions, 115 Ortfxonormal vector functions, 70

P, 50 Pair creation by a photon in a Coulomb field, 343 Pair creation by two photons for spin O particles, 286-297 Parameter : Hubble, 396 of scalar-tensor theory, observations concerning, 395 Parametric device for combining denominators, 336 Parity, 50 for arbitrary spin, 141 intrinsic, 1 17 orbital, 117 spaee-reflection, 20, 95 for spin particle angule~rmomentum states, 117 for spin % particles, 109 Particle fiux vector, 208 Particle occupation number, 52 average, 149, 155 of Fermi-Dirac system, 163 in Fermi-Dirsc statistics, 101)

a

Particles, 1 with arbitrary integer spin, 85 sources for, 92 causal source description of s beam of, 41-42 charged, two types of, 252: composite, 36 statistics of, 252 creation of, 37 critique of theories of, 24 detection of, 38 w i t h dual charges, 252 elementary, 11, 22 executing prescribed motions, 245 of integer -/- spin, X27 and general coordinate invariance, 397

inberetcling, 12, 23 in a macroscopic environment, f 2 massless, 19 of arbitrary integer spin, 96-97 of helieity g, 129 af helicity 3, 99 of integer $ helicity, 133 in operator field theory, 30 phexlomenolagical theory of, 37 reactions of, 32 real, 267 source of, 37 space-time description of exchange of, 54-55, 118 of spin 0, 38 elzarged, fields associated with, 285 of spin *, 99 of spin 1, 67 of spin 2, 78 stable, 31 unstable, 12, 32, 360-361 virtual, 2437 Pauli matrices, 99 Perihelion precession, 83 in. scalar-tensor theory, 395 Periodic Green" function, 1149 causal forms of, 150 Fourier series construction of, 151 modified, 155

+-

Yermanexzt, and Dose-Einstein statistics, 55 hrsistence probability of H-particles, 368 Phase shift between helieity transitions in Coulomb scatbring of spin g particle, 327 Phase transformations, 48 in causal situations, 200 and charge Auctuations, 207 variable, 199 Photon, 34, '72 action far, 198 emission and &mrplion, 266 helicity of, 20 heXieity states of, 93 minimum detectable energy, 274 polarigation vectors of, 73 single-particle states of, 72 soft, 268 average number emitted, 273 vacuum probability amplitude for, 12 Photon emission : in Coulomb scaltering, related t-a Cornptaiz scattering, 331 in Coulomb scattering of spin O psrticles, 329 in Coulomb; scattering of spin 4 psrtieles, 341 Photon propagation function, 77 Physical system, 28 Paint charges, electric and mwnetic, 249 Poisson distribution, 65 Polarization : and double scattering, 328 of spin particles in C?ouEomb scattering, 328 Polarisation vectors: arbitrsrine~sin, and gauge invariance, 294 for massless particIes, 98 for particles of arbitrary integer spin, 95 for photons, 73 rotation of, 76, 232 far spin 1 particles, 68, 69, 91 for spin 2 particles, 79 Position vector operator, 10, 17

Primitive electromagnetic interactions, 25-4 physieal context of, 266 P ~ m i t i v einteraction, gravitiationd, 383 Principal value integrds in modifid propasgatio~function, 3 M Probability :for e m c d e decrty of unstable H-particle, 372 for emitting prtrticbs, 65, 66 of H-particle deeay, 371 for persisten~eof the vacuum, 43,52-53 for ~ p c i f i cdecay of unstable H-particle, 374 Probability amplitude :for radiative decsy of unstable H-par$icla, 370 far two-pbobn deeay of unstable H-particle, 371 Probability amplitudes, generating funetion for, 282 Probe source, 55, 118 Fro~eetionmatrix: for helieity, 141 for helicity in relation to efrarge, 142 for inGeger -f- spin, 131 Projmtion, operal;ors, 30 for angular momentum, l14 Proieetion. tenssr :dy d i c eonstruetian of, 94

for inkger q i n , 94, 130 for massless particles, 97 dyadic eonstruetion of, 98 relation to kgendre's polyn~mia1,94 for spin 2 particles, 79 for spin 3 particles, 170 lh.)ropwstionfunetion. See Green'8 funetian asymptotic forms for, 46 causal structure of, 42-43, 57,51), 120 complex conjugate of, 59 ditrerential equation for, 145 fourdimmsional repremndation. of, 46 of H-partielea, spproximate conadruction of, 36G36.7 integral equation for, 366 nnsdified, 363 nonrelativistic form of, 375 mdf iparLic1e generalisation of, 56, 65

of spin O particles, 42-43 of spin partiefes, X06 for angular momentum staks, 316 far arbitr~rymdes, 120 invaiviag charge matrix, 123 multip-~le"lrticIe generalisation of, 120 timedependent, for spin 0, 346 far win *, 348-841) of unstable H-particlers, time behswisr of, 388 Propagator. See Propaie;atia:onfunction P r o p r lifetime, 32 Proper orthoehranous Lorent~poup, 16-17 QuadrupoEe moment, ebetric, for spin f particles, 264 Quantu:n degree of frwdom, p h w spwe f ranslation group, 5 &uesntum ehectrodynamies, 34 Quantum mechanics, 1, 7 dynamical variables of, 33 Rdiation gauge, 350 vector potential in, 350-351 Reactions, 33 twwpartiele, 35 3Reduced mass, 357 Relativity :Einsteinial~,13 Galileart, 7 &normdisstion, 32 Retarded fields, 147-348 of Dirsle equation, 162 Retarded Green's function, 1.48 for Dirac equation, 162 btations: and angular momentum, 22% in causal: situstions, 223 in charge space, 232 invarisnts of, 24% in Eueiidean charge space, 48 of sources, 54 Rutherford c r o s =etion, 28% Satl;redo, 241 Yalviati, 241 Scalar eonfarmal field, 392

Scalar-bne;or mobification of Kingbin theory. See Broken conformal invariance theory Scsle transformations and eonformsl group, 226 Scathring: elastic, of phohn by H-particb, 373 electron-photon, 318 multiple, lmf eallision snlsfysis of, 366 of photons by H-particles, 355 of photons by spin Q charged partiebs, 29&291

resonant, sf H-particle and time cycle description, 377 of spin O and spin 3 particles, 309 of spin 0 particles, 284 with equal charges, 281i-286 with opposide charges, 289 of spin particles, with eqml charges, 300 with opposik charges, 306 Thornson, 34,3fM1), 360 and H-particles, 356 Scattering c r o s seretions, 286 Seatbring procesms: csusal analysis of, 282 irreducible, 283 S(catte~ng)matrix, 35 Selection rubs, 283 SeE4onsistency of muree coneept, 268 Shell game, 50 Simple source, 267 Simultaneity, 15 Single parlicfe state, 38 Singlet spin funetion, W Skebtsl interaction :incompleteness of, 360

for particlepartiele scatbring, a;79 for particlephoton procems, 280 for spin O particle-photon procews, 28 1 for apin -3; parliel*photon proceBm8, 280-281 S matrix, 35 S-mat rix theory, iii constructive principle8 of, 35

dynamics in, 343 q@cufstionsin, 36 Soft photons, 268 Sources, 37 algebraic properties of, 135, 137 algebraic redefinitions of, 173 for arbitrary spin, response to TCP, 139 charged particle, containing vector potential, 257 complex, 46 for lspin particles, 159 complex conjugation of, 137 contmt (overlap) h r n s , 144 definition of, for masslew particles of arbitrary helieity, 142 for massless particles; of helicity 8, X 30 for ma~slessparticks of inkger 7t. 4 heliciw, 134 for neutrinos, f 26, for psrtieles of arbitrary i n e e r spin, 96 for particles of arbitrary spin, 136 for particles of integer 4 spin, 133 for spin 0 particles, 41 far spin O particles and antipecrtiebs, 47 for spin particbs, f 08, l f 3 far spin 1 particles, 68 for spin 3 partiele~,129 for spin 2 particles, 79 directiondity of, 50 dispfacement of, 39 egective, woeiated with virtual H-particles, 365 for photon and H-particle emimion and absowtion, 361 for two-particle procesws, 266, 276275 electromagnetic model of, 255, 267 extended, 265 for ernision and absorption, refakd by TCP, 2716 Fermi-Dirac, eampbx conjugation of, 110 gravitational m d e l of, 403

+

Index

of gravitons, and energy-momentum, 81 enechanicd measure of, 81 of H-particks, 346-347,349 extended, 362 multiparticle exchange htween, 5656, 118, 361 non-eonserva3;Cion of current in, 202 non-consftrvation of stress tensor in, 21 l for partieles of arbitrsry inkger spin, 92 far particles of integer 4 spin, 132 photon, coupling of different kinds, 232 two t y p s of, 231 for photon helicity staks, 73 of photons, and electric charge, 72 reality of, 40 relation between emission and abmwtian, 41 restricted, 72, 81, 97, 9.9, 125, 129, 133, 168, 173, 176, 233 and general physieal laws, 2.43 rotation of, 54, 223 as scatterers, 320 mlf+onsistency of concept of, 268 simple, 2437 spin O particle, containing grmitationd field, 403 of spin particles, 99 for angular momentum states, 116 coupled by single-particlc! excfianw, 1011, 115 definition of for arbitrary modes, 120 response to TGP, 111.1, 117 of spin X particles, 67 spin g esnstructisn of, $30 of spin # psrticfes, 127 eoupled by ~ingle-parti~lft exchange, 127 of spin 2 partieles, 78 pino or, of spin O and @in f particles, 142 of spin 3 particles, 106 strong, of q i n Q p~rticles,50 tensor, 92 and transition prabtbbiliLies, 282 of unstabb L3[-particlt3s, 369.

+

a

of unstable partiefes, use of reference pain%in, $69 weak, efFect on multipartide states, 65, 119 of spin O particles, 38 Source restriction: for heIicity m parLieles, 126 for helicitly .n mwsless partieie~, 133 for helieity 1 nnsless pa;r.ticles, 72,2B for hdicity 1 and 2 m ~ s l ~ psrticbs, 8s I68 for heficity If mltsslftss pafticles, 129, 173 for helieity 2 massless particbs, 81. for helicity 4 mw1ess partieles, 176 for helicity 3 mwless partieles, -99,f 68 for massless partiefes of arbitrary integer helieity, 97 Souree theory, phenomenatogicsl orientation of, 2497 phenomenologieakl viewpaint of, 365 Space reflection, 50 for arbitrary spin, 141 lineetr responm of Bource to, Bfi. matrix, for spin 4 particles, 100 Space-time, 7 Spsce-time refieetion as a Eucrideri~m transformation, 49 Space-time uniformity, 37, 42, 43, 51 and antiparticles, 47 and energy, 77 momentum complement of, 269 and photon helicity, 75 Spctral distribution :of phohn emithet by unstable H-particle, 371. of photon in transition btween. unstable B-padjcles, 372-373 of urtstable H-partielt? nlxdiation and time cych description, 374 Speculations: in operator field theory, 34 in S-matrix theory, 36 Spherical harmonies: ddition theorem of, 94 generating function of, 94 Spin, 10

++

Spin and statistics, 134 unified proof of connection btovwn, 138 Spin matrices: commutstion relations of, 86

for spin 4 particles, 99 symmetry and reality of, 86 Spinor, 106 antisymmetrical, 136 Spin-parity, 141 Spin 0 particles, 38 behavior of sources under Lorentz transformations, 40 charged, description by real sources, 284

deffnition of source, 41 fields of, 145 primitive electromagnetic interaction for, 256 propagation funetian of, 42-43 scattering, 284 of like chargesj 285-.286 of opposite cbsrges, 289 of photons by, 290-291 strsng sources for, 50 Spin matrices: as, 127 a",U , = 0, , . ,3, 101 a,,@ = 1 ,..., 4,112 eigenvectors of: u, 107 eigenvectors of: v, 106

.

T 5 , 104 Y", X03 F,, 103 rat, 133 a,,, 104 @&*E, 112

trace of, 107, 143 Spin particles, 99 and charge, 113, 123 definition of source, l OS fields of, 157 parity of angular momentum states, 117 parity of single-particle states, 109 propagation functions for, 106,136, f 20, 122, 162, X63

response of sources ta TCP, X 14, 117 scattering of like ch&rges,300

scattering of apposite chargw, 3W source of, 99 space refleetion matrix for, 100 symmetry properties of source coupling, 102 Spin 1 particles, fi7 definition of source, li8 fields af, X64 helicity states for, 69 polarization vectors for, 68, 6Q seeand-rank spinor des~riptionof, 177 source of, 67 Spin 8 particles, 127 definition of murce, 126) fields of, 172 helieity eigenvectors for, 129 source of, 227 Spin 2 particles, 78 definition of source, '79 fields of, L65 heftieity states for, 78 polarization vectors for, 79 source of, 78 Spin particles, fields of, 176 Spin 3 particles, fields of, 16'2 State vectors, 1 symmetry of, 33 Stationary tbction, principle of, 187 Statistics: Bose-Einstein, 33, 55 of composite particles, 2-52 Fermi-Dirae, 33, 99 and win, 134 Step function, Heaviside, 234 Stimulated emission, 56 Sdone, Max, 83 Stress scalar: and confomal transformations, 226 of electromagnetic field, 230 of photons, and conformal invcarlanee, 390

for q i n 0 particles, 209, 216 far spin particles, 220 far spin 1 psrticle~,213 for spin 2 particles, 219 Stress tensor, 24, 210. See alga Energymomentum Aux vector

arbitrariness of, 217, 390 and angular momentum, 1224 for spin O particles, 2x6 ~rbitrarinesswithin sowees, 216 for arbitrary multispinor, 222 wwciated with particle state, 210, 221 commutation relations with hrenrttl generators, 25 conserved, of spin Q field, 404 of ebctronnagnetic field, 26, 230 equal time eommutaficrn rel%tionsfor, 25

generalized conservation law in gravitational field, 388 identity of kinemzttieal m d dynamical definitions, 383 implied by gravitont source, 405 of massive particles, 245 of photons, in gravitatianal field, S 9 mssponse of to infinitesimal h r e n f z transformations, 25 and scalar, for spin 0 particles in. gravitational field, 388 for spin -$ pmtiele in graviteLtiona1 field, 401-402 far seeand-rank spinor, 221 for spin 0 particles, 212 for spin particles, m using third-r~nkmultispinor, 222 for spin. 1 particles, 213 for spin 2 particles, 218 symmetry of, 214 S%ronginteractions and operatar field theory, 34 Sum rules, 32, 356 m pherrornenological consiskncy eonditione, 357 Super rsource, 282 Swfaee, directpea element of, 247 I", f;O Tchebichef % polynomial, 98 T W ,50,54, $9, 113, 117,126,129,276, 2w defined for arbitrary spin, 139 Tertmr: eo~ltravariant,385 eovariant, 385

387 diftFerential identity for, 388 B,,k, 390 @p,,

R,,, 388

a,&,

399 R & ~ 386 *A~ ~trese,t,,, 388 symmetrical: t,,, 3E)8 t;, 398 tracelessr, 83 Tensor-spinor, 132 Tetrad vector field. See also Gn-tvitatio~tt~X field, et: conformal behavior of, 402 Thornson erorss wetian, %4, 320, 356 Tints cycle description, 62 fieids of, 147, 161 for parsmetrised multiparticle stabs, 1511, 1516

and photon spectrum of unrptabb ~ - ~ a r t i d374 e, and resonant seatwring, 8 7 Time eyele vacuum amplitude, 62,154, 161 for charged particles, 156 and expectation values, 6.8 generdisstion to multigar&ie!e sgate, 165 for spin 4 particles, 161 Time refleetion, 49 Time scale and unstable particbs, 381 T matrix element: for Coulomb seal"dring, of spin O psrticb, 821 of spin 4 particle, 322 for electron-positron mnhilation into two photons, 31 l for photon absorption by H-parfiele, 353 for photon ennisgisn in Gulamb scattering of spin 0 partieb, 329 for photon scattering, by H-particies, 355 by spin 3 pa&icle, 340 for radiative H-particfe transitian, M2, 3M

for scsttering af photons by pin Q charged partielw, B2 e r o ~ i n gsymmelries of, 292

for scattering of spin Q and spin for H-pnrtielcs, 347 for spilt 2 particles, 78 particles, 369 far spin 0 particle pair creation by two Traeuum, probabilrity =trnplitude : causal photons, 297 analysis of, 51-52, 55, 109, l f 9 complex conjugate of, 59 for spin O scattering, of like charges, decomposition in zero mass kinnit, 80 286 with differetlt kinds of particles, 256 of opposite charges, 289-290 Eucfideai~tra~lscriptionof, 45, 60, l 13 for spin 3 scattering, of like charges, for gravitons, 80 306 for integer spin in EucXidean space, 140 of opposite charges, 306 Total a~rgularmomentum: classific%tio~~ for i n t ~ g e r 4 s p i l l in Euclidean space, of sources and states, 70, t 14 140 quantum numbers, 71 for xnasslcss particles, of helicity 8, 129 of kelieity 3, 99 Total symmetry of zt multispinor, 134 of integer helicity, 133 Trace, 107 T(ransition) matrix, 35 multiparticle gelleralisatiot~of, 56, 58, definitioll of, 283, 321 1X9 physical interpretatiol~of, 284, 321 far neutrinos, Euclidettn tra~~scriptisn Transition probabilities in source of, 126 description, 282 for particles of arbitrary integer spin, Transition probability amplitudes, 56 95 for spin 3 particfes, I21 for particles of arbitrary spin, X35 Transition probability per unit time for for particles of integer --/- -& spin, 132 for photons, 72 H-particle radiative transition, 352 Triplet spin functions, 90 in refsttion to energy, 77 Two-photon ttnllihiliation : of high-energy far spin O partieIes, 42, 47, 51 electron and positron, 323 for spill $ particles, 106 of slow electron and positron, 312 Euclidercn trtlnscriptio~~ of, 113 for spin 1 particles, 67 for spin 8 psrlicles, X28 Unitarity, 35, 59 for spin 2 particles, 79-80 and causality, 61, 122 for two kinds of photon sources, 235 Unitary transformations, I Vacuum state, 16, 29, 38 group of, 3 invariance of, 41 infinitesimal, 2 Vector, response of to Lorentz sueeesslve, 2 transformations, 69 Universe :age of, 396-397 Vector potential: electromagnetic, 227 geometricai description of, 396 two kinds of, 236 mass and radius related by Mach's exhibited in terms of field strengths, 238 principle, 896 Vector-spinor, 127 mass of, 395 Virtual particle, 267 matter density in, 396-397

+-

+

model of, in scalar-tensor theory, 396 radius of, 395

Vacuum persistence probability, 43, 52-53, 68

Wavefunction for spilt 3 particIes, 159 Weir;s&cker-Wi11iamsmethad. See Photon emission in Coulomb scattering, related to Compton scattering