, and ~ = are
~ ,
of
variables
the covariation)
: (9) d/~t
In t h e p a r t i c u l a r cal
equations
=
~HI~
c a s e when
were
~
/~t
,
Imh
established
=
= 0 and
2
in
paper
-~HI~
.
h -2Reh = i ~
2
such c a n o n i -
[12]
R e f e r e n c e s I.
O.N.Holenhorst:
2.
V.Dodonov~
V. Man'ko,
J.Klauder:
in
3.
Phys.
~.Lenczevski, 4.
Rev.
~19
1669
EoKurmyshev:
Symmetry i n
Science,If,
Plenum P r e s s ,
N.Y.
eds.
A26
150
(1980).
B.Gruber
and
and L o n d o n , 1987.
F. H a a k e , M . W i l k e n s : i n P h o t o n s and Quantum F l u c t u a t i o n s . E . R . P i k e and H . W a l t h e r , ~ r i s t o l
5.
D. S t o l e r :
6.
D.A. T r i f o n o v :
7
H . Y . Y u e n : Phys.
8.
~.Nikolov,
9.
3.~eckers,
Phys.
E2-E1-79E
10.
(1979).
Phys. Lett.
I.Malkin~
Rev. DI 3217
Phys.
Lett.
A48 165
Rev. A I 3 2226
~.Trifonov:
and P h y l a d e l p h i a
(1970);
D11 3033
,
1990.
(1975).
( 1 9 7 4 ) ; A64 269
(1977).
(1976).
Commun. o f
JINR
(Dubna), E2-81--797,.
(1981). N.Debergh: J.
Math.
Phys.
V.Man'ko~ D . T r i f o n o v :
J.
30 1739
(1989).
Math. Phys.
14 576
(1973). 11.
E.Onofri:
J.
Math. Phys.
i2.
A.Rajagopal~ J.Marshal:
16 1087 Phys.
460
eds.
(1975).
Rev. A26 2977
(1982).
Correlated States of Q u a n t u m Chain O. V. Man'ko
Institute of Nuclear Research Academy of Science of USSR
The aim of the work is to discuss the integrals of the motion and correlated states properties of the t i m e - d e p e n d e n t discrete string consisting of q u a n t u m interacting parametric oscillators. Some solutions for the stationary string may be found in Ref. [1]. We follow to the procedure suggested for nonstationary chain of oscillators given in Ref. [2]. Let us consider a chain consisting of N harmonic oscillators. The distances between neighbours are equal to a. When the distance between neighbours approach a zero, and number N tends to infinity the chain turns into the string. All oscillators vibrate with the frequency D0 and linearly interact with the neighbouts, the frequency D0 depends on time. The Hamiltonian of this system is
1~
[P~ + m ~ = ( ~ ) ( q , ~ - q , ~ + t ) 2 + m f ~ o 2 ( ~ ) q ~ ] ,
(1)
m n=l
where q,~ is a shift from equilibrium point of an n - t h oscillator, p,~ is a m o m e n t u m of the oscillator, m is a mass of oscillators. The chain is closed, i.e.,
qi+N---- qi ,
i = 1, N .
In order to take into consideration the most simple case N must be an odd number
N:2p+] The equations of motion corresponding to Hamiltonian (i) are [I] /i,~ = / 2 2 ( ~ ) (q,~+l + q,~-I - 2q,~) - 2t2g (~)q,~ .
(2)
The case when the frequency D0 (g) is equal to zero has been considered in [2]. Let us go over to new variables which reduce the system of N interacting harmonic oscillators to a set of N free oscillators. The new coordinates are
Zs --
qm rs~=l 461
COS
-
-
,
XN = m=l
q,~ sin m=l
(3)
1, P .
S~
T h e new m o m e n t a are Pm
COS
pm
,
m=l
N
Pz~ = m=l
Pv,
Pm sin
=
,
m--1
T h e new variables oscillate i n d e p e n d e n t l y with the frequencies
(4) a n d satisfy the e q u a t i o n s of m o t i o n
~,+n,=(,)~,=o,
~+no~(t)~=o.
9,+n,=(~)v,=o,
I f one i n t r o d u c e s N - v e c t o r s q-=(ql,--.,qN), tt = (:rl,
...,
y l , -.-,
zp,
P = ( p l , . . . , PN)
y r , , ~rN) ,
and
P# = ( P = , , - - . , Pz~,, Py, . . . .
, P~p, P~t~ ) ,
(5) the t r a n s f o r m a t i o n (3) c a n be w r i t t e n in m a t r i x form q=S#,
p=S-lp#,
whereS=K'T
T h e m a t r i x K is as follows k k2
k2
.-.
(k~)2...
kN = 1
]
. , ,
K
_
v~
k = (k~):
...
1
, . .
1
where
k=exp
(i,~m):
1
.
462
...
1
-1 .
(6)
The N
x
N - m a t r i x K is matrix of a canonical transformation QH : K q .
This transformation is connected with the irreducible representations of the permutation subgroup of oscillators in the chain. This subgroup is the group of symmetry of equations of motion (2). This group consists of N elements
cN, c h , ... c~ = m , where CN is a rotation by angle equal 27r/N. Other elements are degrees of this rotation. The m a t r i x T is a matrix of transformation from complex variables Qn to real variables #. #:TQ
H
The matrix T is equal to T1 T : i T~T40
T=
'
wheIe 2p - dimensional matrices T ~ , T 2 , T a , T4 are E T s = T1 : - 0 ~0
T2 = - i T 4 =
-.-
Ol I0
1
I0 Ol
-..
0 0
,E=
"¢2 01 10
0 ...
0
0 0
-..
10 01
The Hamiltonian (1), rewritten in the new variables, as follows,
[ =
P~'
P~-"
mZ'~ (~) (=, + y, ) + --y-=,~j
(7)
The Hamiltonian (7) is the sum of the Hamiltonians of 2p (where p = - ~ ) independent oscillators with frequencies ~ , (t) (4) and one oscillator with frequency $9o (t) . The solutions for harmonic oscillators are well known. Using these solutions we can write some results in our case.
463
Let us construct "annihilation" operators for variable-frequency chain i
N
X,C~l=~Zcos
(2Nm)
(
(2~m)
(
--
m='l
i
N
K(~)=~si~
--
q~) ~(~)~-~'(~)Tqm) lspm ~(~)--fi--~(~)~ Isp,-n
,
'
wt----1
XN(~) = ~
~
~o(~)--~-io(~) To
,
(8)
where functions eo (t), es (t) satisfy the equations 02 ~
2 (t) e~ = 0, 02 ~o
at--~ + ~s
- ~ - + Oo~ (~),o = 0
(9)
with the additional conditions 0¢. ¢; _ Oe~
0"-¥
•
&o
~ r e~=21,
.
~eo-eo~-r
Oe~
=2i .
(10)
The dimensionless times *o = a0 (0) t, ~ = a, (o) t are used, when one differentiates with respect to time in formulae (8) and (10), in the next formulae the dot means the same differentiation. The numbers lo, ls are amplitudes of the oscillations in ground oscillation states. They are equal to m ~20 (0)
, Is =
m I2s (0)
The commutation relations of the boson creation and annihilation operators take place for the operators (8) [Xs, A+ +] ^+
Ar]=6~r,[K,fi
=~r,[XN,AN]=I,
[Xs, K I n [X~, Br ]= [Xs,XN]= [X~,AN] ^+
= [ K , X ~ ] = [K, AN]=0. It can be shown that the operators .&~ (t), % (t), AN (t) and their hermitian conjugate A S (~), B : (~), A N (~) satisfy the equations dX5
OXs
i [~,
d~-- o-~-+~
X~] = 0
dB~ - 0
'--a-v-
dAN - O,
' at
where H is the Hamiltonian (1), and, consequently, they are integrals of motion. II The ground state of the parametric chain can be obtained from the equations
[3] X, I~o)=0, K / ~ o / = 0, Xr¢ I ¢ o ) = 0 464
and the condition
(~01~o) = 1. The ground state in the coordinate representation is equal to ~'o (ql, ..-, qN, t) --
v~
2%(')zg
l-I,=~ ~,
+x--" i~, (t)
.,=~
~
s=1 2es (g)
m=l
27rsm qm COS f
+(~=~ (N)½q'~sin (-~))2]
} • (11)
The coordinate distribution function has the form Wo = ~PO~'0 = ~
7r-P~p
exp
-
I¢ol I-l,=~ I~.l =
,=~
x
2qm qm,
'=~
l'rt I 71rL
-2tg - - -I~ol - ~ + cos
N
N
~
.
Using the characteristics of Gaussian distribution [4], the determinant of the matrix of coordinate dispersions and elements of the inverse matrix of coordinate dispersions can be obtained. They are P
det
O'q - =
}~01 FI,=I I~,1~ 2N/2 (12)
(~,-~)q,,,q,, = Nlo21~oj ~ + ~ Nl2 iE,[~------~ cos s=l T h e ground state in the m o m e n t u m
representation e
m
N
is
4
~'o (Pl,--., PN, t) = ~rN/4~1/2
P
Hs=I is 2
x exp
2~o
h2
=
-~
"
p,~ cos
(2~ms)) : (la)
465
T h e m o m e n t u m distribution function for the chain is
1
P ~ I~ol-~ 1~ I~,1-= exp [ --
Wo = ~
s=l
N ~
Pm Pro'
~a, r n t =1
and one can obtain, that the determinant of the matrix of m o m e n t a dispersions and elements of the inverse m a t r i x of momenta dispersions are P
net ~p = It0l ]-[ It, I= 2 -N/= ,
41~a=Nit, l=]
(o --~ )p,,.p..,, = ~tt=Nltol + ~ s=l
cos
~2~'s(,~ _ m')
(15)
One should remember that the dimensionless times are used when one differentiates with respect to time in formnlae (S), 0 0 ) , (11), (1~) - (15). One can see, that by varying in a definite manner the functions es (t), ~0 (~) (9)~ the q u a n t u m dispersions of coordinates and m o m e n t a of the parametric chain can be controlled. The dispersions of coordinates can be decreased at the cost of increasing the dispersion of momenta, and vice versa. So if we can change the frequency of interaction I2 (g) or the frequency of all oscillators in our chain $20 (g) then we can have squeezings and correlations in the oscillator system of the chain. Acting by varying the frequency on oscillators we can generate correlated states of the chain. The correlated states off" the parametric chain can be found from the ground state (11) with the help of the displacement operator [4,5,2] P s-,----.1
P
where exs,/3s, ~ , are complex numbers. So, the entire family of correlated states of parametric chain is of the form
P
+x-" s----1
[
[
___ ~ ~= %* + _ '¢~
I~1=
~,(q~,...,qN,t)=~oexp
2
I,~,1=
I~,1=
2
2
2
2 ~o
.
~, ~;
2 es
2 ~s
o,, q
Eo Io
=
2
---E-
466
q~
The correlated coherent states obey the eigenvalue equations
Using the property of correlated states to be a generating function for discrete Fock's states [4]
k~a (ql, . . . , qN, t) : e x p
--2 ~
(1~1=+ i~,1 =) _ I =
s=l
× fi ~=o
~ I (Oq)'~"(/3s)n='(~)n°@no,nl~ s
(nl~!n2~!no!)
(ql ......
~"~"
......
'~
qN t) '
"'"
'
'
'
one can obtain, that the Fock's states are T, o
~P,~o,n,1..... n,p,~2 ...... n : r ( q l , ' ' ' , q N , t ) x
II
×H,~°
( no !
~=1
qm
ni~!n2s ! ) 1/2
x H~,.
xHn2.(fi (2) I/~ q~ m_-i
=@0(ql,..-,qN,t)
q~
sin ( 2 7 r s m ) )
I~,lZ---~
-Y-
] '
where H,,, (z) - Hermitian polynomials. Thus, we have constructed explicitly the wave functions of Fock and correlated states for nonstationary quantum string. It should be noted that the dynamical symmetry of this string is described by inhomogeneous sympleetie group ISp ( 2N, R) in accordance with the statement for arbitrary quadratic quantum system [3]. It would be interesting to calculate the Wigner function and density matrix for the string in thermodynamic equilibrium state and the corresponding density matrix taking into account the influence of the string nonstationary. I would like to thank V. V. Dodonov and V. I. Man'ko for the useful discussions. References 1. E. M. Henley, W. Thirring: Elementary Quantum Field Theory. N. Y.: McGrawHill Book Company, inc. 1962 2. V. V. Dodonov, V. I. Man'ko, O. V. Man'ko: Trudy FIAN1 1989, v.200 3. I. A. Malkin, V. I. Man'ko: Dynamicheskye S~jmmetrii i Kogerentnye Sostoyanya
Kvantovykh System (Dynamical Symmetries and Coherent States of Quantum Systems) Moscow, Nauka, 1979
467
4. V. V. Dodonov, V. I. Man'ko: Trudy FIAN, 1987, v.183, p 140-145. Translated into English by Nova Science, 1989, N. Y.: lnvarlants and Evolution of Nonstationary Quantum Systems, ed. by M. A. Markov. 5. V. V. Dodonov~ V. I. Man'ko, O. ~/. Man'ko: Trudy FIAN, 1989, v.191, p 185-224. Translated into English by Nova Science, 1989, N. Y.: Theory of Nonstationary Quantum Oscillator, cd. by M. A. Markov
This articlewas processed using the D T E X macro package with I C M style
468
Interaction of Weak Coherent System of Two - Level Atoms Cavity
Light with a in a Lossless
M. Kozierowski 1, A. A. Mamedov 2 and S. M. Chumakov 3 1 Institute of Physics, A. Mickiewicz University, 60-780, Poznan, Poland Institute of Physics, Academy of Science of the Azerbaijan SSR Baku, 370 143, Prospect Narimanova 33, USSR Central Bureau of Unique Device Designing, Moscow, 117 432, Butlerova 15, USSI:t; P. N. Lebedev Physical Institute, Moscow, ]17 924, Leninsky Prospect 53, USSR
1 Introduction The 3aynes-Cummings model (JCM) of light-matter interaction despite its simplicity demonstrates a number of interesting phenomena such as collapses and revivals [1], sub-Poissonian photon statistics [2] and squeezing [3]. Early studies showed the appearance of the so-called Cummings collapse [4] at coherent quantum pumping. Eberly et al. [1] have later found a revival of the collapsed oscillations, in fact an infinite sequence of collapses and revivals with Gaussian decrease of the revival maxima. The origin of collapses and revivals in the 3CM is connected with the photon number distribution which produces spread in Rabi frequencies. The Rabi oscillations, initially all in phase, periodically dephase and rephase which leads to collapses and revivals, respectively. Barnett and Knight [5] studied numerically collective collapses and revivals for a group of two-level atoms. The atoms were assumed as initially unexcited or excited (a maser case). In general, there are two sources of spread in Rabi frequencies: the photon number distribution (as previously) and the collective atomic evolution. The origin of the collective collapses and revivals is related with a non-equidistant spectrum of the eigenfrequencies of the system. Recently, a new solution to the problem of interaction of a system of N twolevel atoms with a single quantized field mode has been proposed [6]. Strictly speaking, cooperative spontaneous emission of a small number s of initially excited atoms in the presence of a large number of N - s unexcited atoms (8 << N) has been considered in terms of the SU (2)-group representation. Our method consisted in construction of the perturbation theory with a small parameter e, = ( N - ~ + ½)-1. The results obtained in this way are valid for an arbitrary time ~. In the first-order approximation in e, it was found that the atomic inversion evaluates as follows [6]: s
e,
~/
S
1
E(t) = ~ cos(212t) -t- ~-~ s(s - 1) (1 - cos(4f2t)) ,~2 = g _ _ g - ~ ÷ ~ ,
(1)
where g is the atom-field coupling. The time evolution of the system is truly periodic since the spectrum of the eigenfrequencies is equidistant in the linear 469
approximation in e,. The second term in (1) appears for s # 1. The collectivity of the system adds in this approximation only the harmonic Rabi frequency 412 in comparison with the 3CM. The above solution is suitable for the description of the dynamics of the system even for the moderate values of the ratio ~; according to the computer calculations the agreement with the real behaviour is then particularly good for relatively short times. For the sufficiently small values of ~ the dynamics of the system is almost exactly described for all times by the first term of the above fornmla. The time evolution of E(f), calculated within an accuracy of e~, may be aperiodic if J is not small enough. This is because the eigenfrequencies are non-commensurate in this approximation. Then, depending on the magnitude of the ratio -~ beatings between the terms with different frequencies, resulting in modulation of E(i), appear sooner ox later. So, the second-order approximation of our theory is responsible for the collective spread in lZabi frequencies. In the present paper we discuss a system of N two-level initially unexcited atoms interacting in a high-Q cavity with a weak, initially coherent, single-mode field. We perform our calculations in terms of the SU (2)-group representations.
2 Results The Hamiltonian for the model in the rotating wave approximation reads (h = 1): /v a
-
H0 + v ,
H0 =
!a÷a ÷
N
S?', j=t
V -- g
+
(2)
j=t
a(a +) is the photon annihilation (creation) operator. The j - t h atom is described
by the pseudospin operators S ~ ) (k = 3, +, - ) . Since we consider a small-sample approximation the coupling coefficient g is the same for all atoms. Moreover, it is implicit that the transition dipoles are aligned with the mode polarization. In what follows, we assume exact resonance (the field frequency Wl is then equM to the transition frequency w) and choose the scale such that w! = w = 1. Let us recall that the excitation number operator N : / V = a + a + ~-~7=t $3(j)+ N • -V is an integral of motion. Hence, if the initial state of the system belongs to the subspace with the fixed eigenvalue /V, the time evolution of the system is restricted to this subspace. It is convenient to introduce the following basis vectors:
I~,ra)(°)=l.--ra>a~lm)l,
~ln, m)(°)=~l~,m>(°),
0~m~n.
(3)
Here, Ira)] denotes the Fock stage of the field, while In - m)= is the state of the atomic subsystem, symmetrical with respect to the permutations of the atoms. The dimension of the subspace corresponding to the eigenvalue n of the operator fi? is n + 1. The initial condition is ra = n.
470
In general, the time evolution of the average photon number is calculated through formula
~(~.) = Z .v(~)co)<~, ,.,,le~Ht a+a e-m~ I ~, ~>Co) ~----0
- ~ n=O
PCn) ~
A(~)AC' i ' ' ( A~q ~')e' ' ' - A ' ' ' ' ) ~ ,
~
p,q----O
mA('~)A ( ~ ) , ~,~q p •
C4)
m--O
P(n)
= e x p ( - f i 0 ) ~ is the Poissonian photon number distribution and no is the initial average number of coherent photons. A r('~) a p denotes the components of the eigenvector of the Hamiltonian (2), while Ap,n is the eigenvalue corresponding to this eigenvector: A ~ ) = (0)( n, m I n , p ) , Hi n,p) = Ap,,~[n,p). The approximate forms of the quantities A(~) and Ap,,~ have been found by us in [6]. Here, we construct the perturbation theory with a small parameter e,~: en = ( N - ~ +
,
(5)
i.e. the initial photon number is assumed to be much less than the total number o f the atoms. In particular, in the zeroth-order approximation in en the spectrum of the eigenfrequencies is equidistant within each subspace with the fixed n and has the form: n 1 A~,~ = ( N - ~ + ~)½~(0)~,~,
~(0~ = n - 2p,
0 _< p _< n .
(6)
It is worth noting that due to our choice of the form of the parameter e,~ the firstorder corrections to the eigenfrequencies vanish, i.e. A]01n (1) = 0. In consequence, the spectrum of the eigenfrequencies remains equidistant in this approximation as well. Using the eigenvectors found by us in [6] and the properties of the matrix elements of the SU (2)-group representations, in the zeroth-order approximation from (4) we get that the mean photon number evaluates as follows:
"n,=0
With respect to the assumed condition fi0 ~ N and to the properties of the Poissonian distribution we can abbreviate summation in (7) on n less than N. Hence, the term under the square root is always positive. The spread in Rabi frequencies is solely related with the graininess of the quantized field mode. As in the case of the 3CM we deal here with one series of revivals and in consequence the envelope of the quantum collapse of the mean photon number remains Gaussian in form in this order of approximation. The above approximation is valid for the sufficiently small values of -~. The cooperavity of the system changes, in this approximation, the magnitude of the Rabi frequencies only.
471
The quasisteady-state values of the mean photon number (7) reached either at very long times or between collapse and revival is f~(~) = ~ . In turn, accurate to en, we find:
+ 8 (/~-- ~)
1-cos
4g~
.
(8 /
In this approximation a new collective term oscillating at double Rabi frequency 4/2 = 4 g t v / N - -~ appears. However, as previously, its spread depends on the photon number distribution solely. We now deal with two series of revivals and both of term are due to the photon statistical mechanism [5,7]. Each series is Gaussian in form but they have different width. The width of the first series (2~) is obviously twice the width of the second series (4~). Moreover, both series contribute to the mean photon number with different weights. The amplitudes of the second series are seriously diminished by the factor ~ in comparison with those for the first series. The total collapse, which is a linear superposition of these two series, is no longer Gaussian in form in this approximation in e. Both series of revivals are observable in Fig.2. The second series of revivals (4~) leads to weak enhancement of the oscillation amplitudes between strong revivals of the first series (2£2). In order to include the collective mechanism of revivals we have to make calculations within an accuracy of e:. For this purpos e it is sufficient to take into account the terms calculated in the zeroth-order approximation for the eigenvectors and in the second-order approximation for the eigenfrequencies. Then, obviously, only revivals of the significant first photon statistical series will be modulated by the collective mechanism. Thus instead of (7) we have ,~(~) =
'n=O
+ s
[
I + ~ ] P(,~) 2 ~ p ! ( n _
(.~- ~)
p) ! cos g~ ( a . + ~ , . + ~ - A.,~+~) +
~ - cos 9 t
,
(9)
where the eigenfrequency At,= within an accuracy of e2 reads:
I. ½ {A(o) . 2 - ( ~ . ) '~ Av;" = (N - 2 + 2) k P ' " +~.,,~..]
A(~)=,,,.
(~~:')[5.(~-
.).
(. - 1)(.2)]2
(Io) ,
(11)
)~(0,) is given by (6). The time dependent part of the pure zeroth-order approximation (7) is certainly implicit in the formula (9). Namely, it is obtainable from the first term 2 = 0 in (10). Then, in fact, the difference in square brackets of (9), if we put e,,
472
Ap+l,,~+l - Av,,,+l becomes independent of p and after some simple algebra we find that this term goes over into cos (2 g ~ ~ as it should be. In general, it is seen from (10) and (11) that the spectrum of the eigenfrequencies is non-equidistant in the second-order approximation in e. Since the eigenfrequencies are now non-commensurate beatings between the terms with different atomic frequencies, resulting in additional modulation of fi(t), will occur. This is simply the collective mechanism of the spread in Rabi frequencies. The above calculated correction contributes to the collective mechanism with the highest weight and, as mentioned, is responsible for the saddles in the revival series 2~. In Fig.2 the dynamics of fi(t) for N = 15 and no -- 4 is presented. The agreement between the exact numerical solution and our analytical one is excellent. Both envelopes manifest saddle-like forms of the revival series 2~. The
"% . °
• °
•
/ Fig. 1. The envelope of the mean photon number n(t) : N = 15,n0 : 4. exact (computer simulation) oooo
from
(9)
128 periods of oscillations are presented.
main results of this paper are contained in (7)-(9). Our comparative computer calculations allow us to conclude that (7) describes correctly the time evolution of fi($) for the extremely small values of the ratio ~ . . I n turn, (9) is sufficient to describe the evolution of the system even for two moderate values of "N-" ~o
Finally we want to point out that the system under consideration may be viewed as possessing the approximate dynamical symmetry (in a sence of the works [8,9]) with SU(2) as the appropriate dynamical symmetry group. In the case -~ ---* 0 this approximate dynamical symmetry becomes an exact one and the Hamiltonian V from (2) becomes the generator of SU(2) group representation. Our example shows that the presence of approximate dynamical symmetry gives the possibilities for qualitative and quantitative description of the system dynamics. We would like to thank Prof. V. I. Man'ko for helpful discussions.
473
References I. 3. H. Eberly, N. B. Narozhny and 3.3. Sanchcz-Mondragon : Phys. Rev. Lett. 44
323 (198o); N. B. Narozhny, 3. J. Sanchez-Mon&agon and J. H. Ebefly : Phys. Rev. A 23
236 (1981). 2. S. Singh : Phys. Rev. A 25 3206 (1982).
3. 4. 5. 6.
P. Meystre and M. S. Zubairy : Phys. Lctt. 89 A 390 (1982). F. W . Cummings : Phys. Rcv. 140 AI051 (1965). S. M. Barnett and P. L. Knight : Optica Acta, 31 435, 1203 (1984). M. Kozierowski, A. A. M a m e d o v and S. M. Chumakov: Phys. Rev. A 43 (1990; in press). 7. Z. Deng : Optics C o m m . 54 222 (1985). 8. V. V. Dodonov and V. I. Man'ko :Lchedcv Physics Institute Proceedings 183 263 (1989); (English translation : Nova Science, Con~nack). 9. V. V. Doclonov~ V. I. Man~ko and S. M. Chumakov : Lehedev Physics Institute Proccedings 167 209 (1987); (English ~anslation : Nova Scicncc, Commack).
This article was processed using the I ~ j X macro package with ICM style
474
TIME DEPENDENT QUANTUM TUNNELLING
G J Papadopoulos Department of Physics, Laboratory of Mechanics, University of Athens Panepistimiopolis GR 157 71 , Zografos Athens-Greece
Abstract The propagator K(xt I x'o) for a particle in a potential field is shown to be derivable from a single classical path evolving under the field and which at time 0 starts from x" and reaches x at time t. The wavefunction of a particle represented initially by a wavepacket lying mainly on one side of a barrier is then propagated and supplies information about the particle's probability and current densities on the other side. This approach
to
tunnelling is seen to be performed via energetically crossover classical flights. Results relating to the parabolic repeller and an eta potential are presencad. 1.
Introduction
In a recent paper [1] we have drawn certain comparisons between the customary WKB treatment of the tunnelling phenomenon and the time dependent approach which we adopt here. In this lecture we briefly review the essential points leading to the construction of the quantal propagator from only a single particular classical path. We further enrich the applications, presented here, by the case of an eta potential. In addition we show, through an example, how the principle of superposition can be held responsible for the tunnelling effect. The discussion for the tunnelling problem will be based on the evolution in time of an initial wavefunction taken to be a wavepacket in the form O(Xo,Po)(x) = (21~o2)1/4exp [ -
4o .2
i (x-xo)2+ -~--PoX]
(1.1)
which by appropriate choice of Xo lies essentially on the LHS of a barrier which is represented by a potential U(x). At the expectation value level, (1.1) represents a particle at Xo with momentum Po • in order to restdct ourselves to tunnelling problems the expected energy associated with (1.1) in. conjunction with U(x) has to be smaller than the barrier's height. This is attained by an appropriate
choice of the parameters xo,Po,a.
The evolving wavefunction on the RHS of the barrier provides all essential information about the tunnelling behaviour of our particle. In this frame we accommodate the notion of transmission coefficient by considering the probability of finding the particle initially on the LH$ of the barrier, let it be A, and the probability that at time t the particle has
475
migrated onto the other side of the barrier, let it be B. The transmission coefficient, T , is then
given as the ratio B/A. It is in general time dependent and may reach a fixed
value after a long time. With this understanting for the transmission coefficient we procceed to see the tunnelling effect in the light of the superposition principle of quantum mechanics. We consider a situation with a single barrier bounded by perfect reflectrors at minus and plus infinities.
The requirement of perfect reflectors enables use
of periodic eigenfuncUons. It is clear that if the state of our system at a given moment is an eigenfunction the probability of finding the particle in a specified region does not change with time and so no tunnelling takes place. Consider, now, the case where the particle's initial state ,~(x,o), is a superposition of , say, two eigenfunctions
(])1,(])2
LP(x,o)= Cl~)l(X) + C2(D2(x)
i.e. (1.2)
As time proceeds the evolving wavefuncUon takes the form LP(x,t)= Cl~l(x)exp(-i001t)+ C2~2(x)exp(-i~2t )
(1.3)
where o~1 and ~2equal correspondingly E1/h and E2Afi with E1,E 2 being the associated energy eigenvalues. The transmission coefficient at time t pertaining to our situation is given by T(t) =
iLP(x,t)l2 _ i~(x,o)12]dx / Xm
f'°
i~(x,o)12dx
(1.4)
-CO
where Xm is the point at which the potential barrier has its maximum. The numerator in (1.4) represents the amount of probability B, that has migrated in time t onto the RHS of the barrier. This quantity may be negative, as well, and in the case of the state considered becomes Go B= 2{cos[(¢Ol-O}2)t]-1 } R e C 1 C ; I
oo
"1¢2dx + 2sin[(°°l-°)2)t] m
,mC,C ~x
dx
(1.5)
m
It is the variation with time of the migration probability B that accounts for the tunnelling effect. Note that in our considerations we have not made any assumptions as to the magnitude of the energies El,E2. Both of these energies and, therefore, the expected energy associated with the initial state ~(x,o) can be smaller that the barrier's height, and yet produce a nonzero value for T. 2.
The
propagator
The frame of the time dependent tunnelling is based on following the evolution of a particle's wavefuncUon that derives from a given initial state. An effective way for
476
obtaining the evolving wavefunction is by use of the associated propagator, which serves as a linear transformation. In what follows we shall sketch how we can obtain the full quantum propagator utilizing a single particular classical path. Our considerations will refer to one dimension, for simplicity, while generalization to many dimensions is straightforward. We begin with Van Vleck's expression for the semiclassical propagator [2] for a particle whose potential energy is U(x). It has the form Ko(xt I x'o)= D t I x'o).i,2 t ( x 2rdl~ J exp [
$o(xt I x'o)]
(2.1)
where So is the classical action, associated with the potential energy U(x), between the space-time points (x',o) and (x,t) and
as
D(xt I x'o) = - a-~aSo(xt I x'o)
(2.2)
According to Dirao [3], D obeys a continuity equation. Although (2.1) satisfies Schro'dinger's equation approximately, it does obey, according to Pauli [4], the right initial condition required of the exact propagator, namely K(xtlx'o)
)
i5(x-x')
as
t----~
0
(2.3)
If we now write the full quantal propagator in the form (2.4)
K ( x t l x'o) = Ko(xt I x'o)exp [~-Q(xt I x'o)] and employ Schrt~dinger's
equation associated with U(x) we obtain the following
equation for the quantity Q which, at least for reasons of communication, we shall call purely quantum action, aQ+laSo
at
ax
o~Q
1 .aQ.2
i~ a=Q
1 ODaQ_
ax =2~(~-) +2-~[a---~ D ~ - J *
~2
1
(3D,2
1 o~2D
~"~[(~ "~~ " ~6 ax2] C2.5)
Since K¢ satisfies the initial condition
required of the propagator K (2.5) must be
solved for Q under the initial condition Q (xt I x ' o )
~
0
as t ~
0
(2.5) can be solved by iteration. Utilizing an expression for Q in the form co Q =,~, ~ln Qn n-2
477
(2.6)
(2.7)
we are led to a hierarchy of equations all of which take the form (3Qn
1
aS o (3Qn
(2.8)
a---E- + m a - ~ T x - = F.(xt I x" o)
with F 2 = ~ [ (1
2_~ "a,D ~') 2
10~2x2D " 2"-D a ]
known as well as the rest of the Fn'S
can be made known. Details of the solution can be found in [1] and
attention is drawn
to a correction made to the term F2 as above. Here we shall point out certain results. The solution of (2.8) with the appropriate boundary condition (2.6) can be obtained through the propagator of the equation &Q + 1 (3 S° (3 Q= 0
~t If
X c (1.) = X (xt I x ' o ; 1")
m
(3x
(2.9)
o~x
is the classical path which our system follows form time
0 to t starting form x" and reaching x then, as has been shown in [1],the required propagator G(~T; xt I x'o) is given as G = ~S(~;-Xo(T))
(2.10)
which leads to the required solution of (2.8) as t
Qn (xt I x ' o ) = f d~ Fn(Xc(T ) T I X'O)
(2.11)
O
From the above we infer that the quantal propagator can be obtained by use of a single classical path, the path joining the end space-time points of the propagator. 3
Applications
We wish to deal with two cases; namely the parabolic repeller and a case of an eta potential. The first instance, although a simple one, is exaclty soluble and furthermore predicts, contrary to expectations from hitherto experience, a form of negative conductivity. The latter
being in the sense that whilst the particle in its classical motion is receding
from the potential barrier a tunnelling
current may develop on the other side in the
opposite direction. The case of an eta potential has interest in that one can study to a certain extent the ecsape through the barder of a metastable potential for a particle initially trapped in a potential valley.
The parabolic repeller The parabolic repeller is represented by the
potential energy
1 2 U(x)= - ~-m~x
478
(3.1)
We cite below the expressions for the probability and current densities associated with (3.1), while the reader is refered to [1] for the various steps leading to these expressions. We have
p(x,t) = ~
j(x,t) =
1 o
exp I['(t)l
(x-X(t)) 2 ['2o 2 II-(t)l 2]
(3.2)
1 {P(t) * [l+(-~-)2}sinhDt '~' Pc(xt I XoO)}p(x,t ) mll-(t)l 2
where X(t) and P(t) are the particle's classical position and momemtum
(3.3)
respectively,
under the intitial conditions X(o)--Xo, P(o)= Po, and are given by X(t)=XoCOShOt + P°sinh£Jt m P(t) = Po cosh="3t + mOxo sinhQt
(3.4) (3.5)
and Pa(xt I XoO) = ~
mO
(x coshQt - Xo)
(3.6)
is Hamilton's momentum under the end space-time conditions (xo,o) and (x,t). ~ is a charactedsUc length associated with the scattering processes induced by the potential (3,1) and is given by ~.=(~/2mQ)l/2
(3.7)
and finally r'(t) is given by I-(t) = cosh~t + i(~-)2sinhf2t
(3.8)
Figure 1 depicts the appearance of negative conductivity, a situation obtained with Po negative leading to a classical motion for the wavepacket away from the barrier whilst on the other side the current initially follows the wavepacket's reverses direction.
motion but after a while
In figure 2 we have a situation in which particles enter one side of the barrier at intervals in succession and generate a current on the other side; ballistic current
479
0 -0.2-
2
t,
6
9t
Figure I. Evolution of probability and current densities for a particle entering the field of the parabolic repeller at xo = -2A. The particle's expected energy equals the corresponding classical energy, in this case -Tmf~2A2/8. Curve (a) is probability density in units of 103 X-t and (b) is current density in units of 10-~G. Although the particle classically moves away from the barrier, after a while a tunnelling current in the opposite direction gets established. The probability density goes down initially and then rises.
0.15
0.10,
~
(b)
0.05,
2
L,
gt
6
8
10
Figure 2. Current density for ballistic tunnelling against a parabolic repeller. Entry of panicles at x o=-2~, at regular intervals with zero speed. Observation at x = 2A. The expected energy for each particle equals its classical value which is -2rnll-'A 2. Curve (a) is entry rate of 1 particie/fl -t and curve (b) is 2 particles/ft -I. The current density (units G) reaches a saturation value proportional to the entry rate.
Eta potential The eta potential we consider here is made out of an oscillator well joint to a parabolic repeller, both appropriately truncated, in the way expressed by the associated potential energy
480
m~2 2 ~--Ll X
X ~< b/2
Ue(X)= {
(3.9) -~--k~m2b2 . . ; Q ( x . b ) 2
x >I b/2
The point at which the two potentials are linked is b/2 and b is the point at which the top of the parabolic repeller portion of the combined potential is located. For the purpose of studying the tunnelling of a particle, initially trapped in the oscillator valley, through the hump of the repelling portion we require the classical path starting from x" and reaching x in time t with x located on the right of b. If x" is greater than b/2 the situation is straightforward and here we shall restrict ourselves to the case when x'< b/2. Denoting by tm the time at which the particle passes through the matching point b/2 on its way from x °, at time 0, to x ,at time t, we have the required path, obtained by Newton's equations, in the form Xl('r)
0 ~< T ~< tm
X~(T)={
(3.10) X2(T )
t m ..< T ~< t
where b sinQ'r Xl('r)=x" (cos£~'r -cotQtm sinQT)+ 2 sinQt m b X2(T)=b-~-
[cosh£~('r-tm)- coth£2(t-tm) sinhQ(T-tm)]+(x-b)
(3.11)
sinh•(T-trn) sinhC2(t-trn)
It is clear from (3.11) and (3.12) that at "r=trn Xl(tm)=X2(tm)=b/2.
(3.12
The matching time trn
is fixed by the continuity requirement that the left and dght velocities be equal at the point b/2, i.e. d x 1(T)]T-tm =[ddX2(T)]T=tm [a~
(3.13)
Equation (3.13) supplies by computation tm=tm (xtl x'o). The corresponding classical action for the eta potential is obtained by use of the action's additivity property as b b Sce(Xt I x'o) = S2(xt I -~-tm) + Sl(~--t m I x ' o ) To obtain S1 and S 2
(3.14)
is a simple matter and for lack of space their expressions are
omitted. For obtaining the semiclassical propagator we need, in addition to the classical action = its dedvative a Sce / a x 0x" .This involves the dedvatives of tm with respect to x and x',
481
which are explicitly obtained as functions of tm. The semiclassical propagator can now be obtained utilizing (2.1) by inserting the appropriate classical action. We have employed the above approximate propagator for obtaining the evolution of an initially still wavepacket centred at the bottom of the occillator well valley. The particle's energy was chosen to fulfill the requirement for tunnelling and a numerical evaluation of the evolution of the probability density has been carded out, and shown in figure 3. Reliable results for the probability on the t
RHS of the barrier have been found for times
strange things are happening with the classical path; in fact
depending on the initial and final positions, for t very near
,
n/2Q there may not be a
classical flight. Beyond the time t=nf2Q the situation regularises again.
3.
2-
f~f Figure 3. Probability density times 103at x-3b as a function of time. Initial state: still wavepacketcentred at well's bottom with energy 2/3 barrier's height. Acknowledgements The author should like to thank the organizers, Academician Professor M A Markov, Professor V I Man'ko and Professor V V Dodonov for their invitation to a most enjoyable and
profitable
Colloquium.
He,
further,
greatfully
acknowledges
assistance
tirelessly
rendered by the Man'ko family, Mrs M A Man'ko and daughter O V Man'ko. References [1] PapadopoulosQ J 1990 Phys. A: Math. Gen. 23 935-47 [2] Van Vleck J H 1928 Proe. Natl Acad. Sci: USA 14 178 [3] Dirac P A M 1958 The Principlesof Quantum Mechanics (Oxford: Oxford UniversityPress) PP 121-5 [4] Pauli W 1952 Ausgew~lteKapitel der Feldquautisierung(Lecture notes) (ZUrich: ETH) pp 139-52
482
NON-SPUR/OUS H A R M O N I C OSCILLATOI% STATES W I T H A R B I T R A R Y PERMUTATIONAL SYMMETRY
Akiva Novoselsky Department of Nuclear Physics The Weizmann Institute of Science, Rehovot 76100, Israel and Jacob Katriel Department of Chemistry Technion- IsraelInstitute of Technology, Haifa 32000, Israel
Abstract An algorithm for the construction of non-spurious harmonic oscillator (h.o.) wave functions with arbitrary permutational symmetry is presented. The h.o. wave functions, expressed in Jacobi coordinates, are calculated recursively using a new type of h.o. coefficients of fractional parentage. These coefficients are the eigenvectors of the two-cycle class operator of the permutation group in the appropriate basis, The matrix elements of the class operators are evaluated by using a specific version of the h.o. brackets. A procedure is developed to transform the resultant h.o. states from Jacobi into single particle coordinates. The procedures proposed are expected to enhance the effectiveness of computations involving h.o. basis sets. I. C o n s t r u c t i o n of non-spurious h a r m o n o i c oscillator states with arbitrary p e r m u t a t i o n a l s y m m e t r y Harmonic oscillator wave functions have been widely used in computational molecular~ atomic and nuclear physics, and recently also in non-relativistic quark calculations[l]. In all these applications the eigenvectors of a translationally invariant hamiltonian are evaluated in terms of h.o. eigenstates. The h.o. states used in these calculations should be constructed in such a way t h a t the trivial center-of-mass (c.m.) motion is explicitly separated: Spurious states, in which the c.m. is excited, must be eliminated. In order to construct non-spurious states for n identical isotropic three-dimensional h.o.s we must use a set of coordinates where the c.m. is separated from the n - 1 internal coordinates. Among the various sets of coordinates satisfying this requirement the normalized Jacobi coordinates
were found to be preferable because each internal coordinate/~ i -- 2 , . . . , n depends on the first i single particle coordinates only. This property enables the formulation of a recursive procedure for constructing the set of h.o. non-spurious states[2]. The h.o. wave functions expressed in Jacobi coordinates are naturally separated into an internal and a c.m. wave function. The c.m. wave function is totally symmetric with respect
483
to permutations of the particle coordinates. On the other hand, the internal coordinates do not have simple symmetry properties with respect to permutation of particle indices. Our aim is to construct internal wave functions, consisting of n - 1 h.o.s, which belong to an irrep of the permutation group, S~. The permutational symmetry of an internal wave function of n-particles can be specified by a sequence of Young frames r 2 r 3 . . , rn, where ri is the/-particle Young frame. This sequence is equivalent to the Yamanouchi symbol Yn[3]. Additional good quantum numbers are the resultant internal angular momentum An and internal energy ( hw (en + ~(n - 1)) where en is an internal energy parameter ). However, the angular momenta and energies of less than n particles are not good quantum numbers. One can construct a complete set of states labeled by IYnAnenan > where an is an additional label that takes care of the remaining degeneracies. For simplicity we denote the combination o f quantum numbers Anenan by On. ~(1) stands for the individual h.o. radial and angular quantum numbers Ni and Li corresponding to the i'th Jacobi coordinate. The two particle internal wavefunction 9an 15'ewritten as IF2 A2 e2 ; ~2 > =
Ir2 02 ; y~ > = ~N2L2(Y2) = 17¢2);Y2 >
(2)
where e2 = 2N2 + L2, A2 is the internal angular momentum and L2 = A2. F~ is determined by L 2 : r 2 = [2] for even L2 and r2 = [15] for odd L2. The value of the z component of the angular momentum is suppressed. Let us assume that the (n - 1)-particle wave functions, symmetry adapted to Sn-1, have already been constructed. The general expression for the n-particle internal wave functions symmetry adapted to Sn, can than be written in the form IYnOn; Y2Y3. . .Yn
>= ~n_l rj(n) (en = en-1 + 2Nn + Ln)
[Y.-lOn-lV(n)Anl)rnon] IYn-10n-xn("lA.;ffzy3...y.
>
(3)
where Yn = Y,~-xr,~ and the coefficients [ I) ] are the h.o. coefficients of fractional parentage (hocfps). The hocfps defined in Eq. (3) satisfy orthogonality and completeness relations[2] similar to those satisfied by the single shell cfps for arbitrary permutational symmetry, defined in ref. [4]. On the other hand, here, the n ' t h particle state I~l(n);ffn > = INnLn;ffn > is not unique. We have to sum over all the different single particle states consistent with the angular momentum coupling/~n.--1 + ~n = /~n and the energy relation en = en-1 + 2Nn + Ln, since the elements of Sn couple all those states. This is the price paid for using the :Iacobi rather than single particle coordinates. The hocfps are evaiuated recursively by diagonaiizing the transposition class operator (the sum of all the different transpositions) n
c 2 [s.] = ~ ( i , ¢)
0)
i
within sets of states having common en, An and Yn-1. The eigenvalues of this operator uniquely determine the various irreps of Sn obtained from a given irrep of Sn-a by adding one box[4]. The eigenvectors are the desired hocfps. The evaluation of the matrix elements of the transposition class operator is presented in ref. [2]. It involves a passage to a new set of coordinates ~n-1 and ;~n, which are either symmetric
484
or antisymmetric in the coordinates r'~-1 and ~'~. This passage is achieved under a rotation by angle
satisfying
Pn
=
~"
--
and
~ n - l s i n f l + fi,,cosfl
--
~"
n - 2
@*n--I "~- ~'n)
(5)
2
n -- 2 i=1
The phase problem associated with degenerate irreps is discussed in ref. [2] II. Transformation of the harmonic Particle coordinates
oscillator states from Jacobi
into single
The internal h.o. states with arbitrary symmetry, derived in the previous section, are expressed in terms of the normalized Jacobi coordinates (Eq. (1)). However, in many calculations in atomic and nuclear physics it is desirable to have expressions for the wave functions in terms of the single particle coordinates. This is particularly important when the h.o. states are used as a basis set in a calculation involving non-harmonic potentials, which are not easily expressible in Jacobi coordinates. The total n-particle h.o. wave function is obtained by coupling the c.m. wave function to the internal wave function, obtaining
W.~,~c(")~z.; ~2Y3...Y./") >
(8)
where £ , is the total angular momentum and c(n) stands for the u-particle c.m. quantum numbers N(n) and L(n). The permutational symmetry of this coupled state is determined by that of the internal state, as was shown in the previous section. For n-particle non-spurious states the c.m. wave function is always in the ground state and therefore L(n) = 0 and £n = An. The coordinates /Yn and ~(~) can be rewritten in terms of the first (n - 1)-particle c.m. coordinate, p (,-1) = ~ (r'l + . . . + r'n-1), and the n ' t h particle coordinate r'n. Inserting this relation we obtain[5] y(.-1)
=
~'. =
fi(")cos~ - y . s i n f l y(n)sinl9 + ~,~cos~
(7)
where co,,8 = V / - ~ . The h.o. wavefunctions expressed in terms of the coordinates ~, and ~-(n) can he transformed into h.o. wave functions expressed in terms of the coordintes f ( , - 1 ) and r'n by using the h.o. brackets for different masses[6]. After separating the wave function of the last coordinate ( ~',, ) by using a change of coupling transformation[7], we obtain
]Y"')'~e('O£";Y2Y3"""AY('~) >=
~
C,Y. ¢.e(")~Cn ~._,~--,)~._,n(-)
,I~._ad--1LC._a h(-) [Yn-x ~,~-x c('~-l) £ . - l h ( " ) £n; ~21Y3...~n_ly(n-1)~'n >
485
(8)
where the coefficients are C •Y"~"~"~" = ~/(2A, + 1 ) ( 2 ~ + 1) ,~-z d n - 1 ) £ n - 1 h('q
E
Z(2 + 1)
~(,~)
L(~)
A
L~ A~
]~n
L (~-1) £n-1
<
(9)
and where the h(n) stands for quantum numbers Nn, Ln of the h.o. wave function in the single particle coordinate r'n. The summation over the quantum numbers Nn and Ln (denoted by z/(n)) is restricted by the condition e~ = ~ - t + 2N. + L~, where ~. and en-1 are specified by ~ and ~ - 1 . Note that the Yamaaouchi symbol Yn determines the Yamanouchi symbol Yn-t. In order to transform completely to the single particle coordinates we have to apply Eq. (8) recursively (n - I) times. In conclusion we point out that the straightforward construction of an n-particle basis in terms of h.o. wave functions generates a large number of spurious states~involving c.m. motion. A c o m m o n device employed to eliminate these states is the addition of an appropriate operator with a relativelylarge coefficient to the hamiltonian, in order to push them up in energy. A n obvious drawback is that a huge basis set is employed, a substantial part of which is totally ineffective. The explicit elimination of the spurious states presently proposed resultsin a very significant reduction in the size of the basis employed. Moreover, the states constructed in our method
have a definite permutational symmetry. This property is essential for calculations involving multiple angular momentum quantum numbers, such as non-relativistic quark calculations.
References [1] M. Oka and K. Yazaki: Baryon Baryon Interaction frora Quark Model Viewpoint in Quarks and Nuclei, W. Weise, Ed. (World Scientific, Singapore, 1985). [2] A. Novoselsky and J. Katrieh Ann. Phys. (N. Y.) 196 135 (1989). [3] M. Hamermesh: Group Theory (Addison-Wesley, MA, 1959). [4] A. Novoselsky, J. Katriel and R. Gilmore: J. Math. Phys. 29 1368 (1988). [5] A. Novoselsky and J. Katrieh J. Math. Phys. 31 1164 (1990). [6] A. Gal: Ann. Phys. (N. Y.) 49 341 (1968). [7] A. De-Shalit and I. Talmi: Nuclear Shell Theory (Academic Press, New York, 1963).
486
S Y M M E T R Y A N A L Y S I S OF T H E Q U A L I T A T I V E INTRAMOLF.CULAR PHENOMF.NA B.I.Zhilinskii Chemistry Department~ Moscow State University, Moscow 119899 USSR
Rapid modernization of spectroscopic methods produced a considerable increase of the accuracy and the number of known molecular energy levels. The main difficulty in the study of highly excited states of finite particle system (molecules, atomic nuclei) is the necessity of the description of a large number of admissible states. Such an analysis of quantum systems may be based on the same ideas as that used by Poincard, Lyapunov, Hopf, Andronov for the qualitative study of classical mechanics problems. In a series of recent works realized by Moscow physicksts, a systematic study of qualitative effects in finite particle systems under the variation of integral of motion was performed [1-9]. The subject of the theoretical analysis is a family of effective operators depending on one parameter which has the physical meaning of an integral of motion. The purpose is to describe qualitative changes under the parameter variation. The general procedure of the theoretical analysis includes the following stages. i) Construction of the effective Hamiltonian or its phenomenological expansion depending on some parameters which may be considered as strict or approximate integrals of motion. ii) Introduction of the classical limit manifold and determination on it of the symbol corresponding to quantum Hamiltonian. iii) The determination of the action of the symmetry group on classical limit manifold and indication of all possible local symmetry groups, i.e. the stratification of the group action on the classical limit space. iv) Analysis of the singularity points for different local symmetry groups for any isolated energy surface (degenerate stationary points or bifurcation points). v) Analysis of the singularity appropriate for a system of energy surfaces (the degeneracy or diabolic points). Several examples of the effective molecular Hamiltonians and corresponding classical phase spaces and classical symbols are as follows. 1) Effective rotational Hamiltonian for a nondegenerate vibrational state, H = /-/(J=, J~, J , ) , where Ja are the components of the quantum angular momentum operator. Classical symbol H J (8, ~o) is a function defined on a sphere and depending on the angular momentum value J which is the integral of the motion. Classical phase space is a bidimensional sphere S 2 . From the physical point of view the point on a sphere specified by the coordinates 0~~o defines the orientation of the vector J in the body-fixed frame. 2) The effective rotational Hamiltonian for degenerate or for a system of quasidegenerate vibrational states m a y be represented in a form of a matrix operator / / = / / , b ( J , , J~, J~), a, 5 = 1 , . . . , k, where k is equal to a number of vibrational states considered. The classical limit manifold is again the bidimensional sphere and the symbol
487
of Hamiltonian in the classical limit is a matrix H = H~b (0, ~o) the elements of which are functions defined on S 2 and depending on one parameter J . 3) The effective operators describing pure vibrational states (their relative positions within the polyads) formed by N degenerate or quasidegenerate modes. If one suppose the total number of quanta to be an integral of'motion the effective operator may be written as
where ~ and re~ satisfy the fo||owing condition
For N-dimensional vibrational problem the effective operator may be rewritten in termS of generators of the SU(1V) dynamic group. Vibrational states forming a given polyad belong to the degenerate completely symmetrical irreducible representation of S U ( N ) . The corresponding classical limit manifold is the complex projective space C P ~ - * 4) Effective rotational Hamiltonian for vibrational polyads leads to S = x C P ~ - 1 classical phase space. 5) Effective Hamiltonian describing the fine structure of molecular l%ydberg states results in S ~ x S ~ classical phase space. If rotational and fine structure are simultaneously studied the corresponding classical limit manifold is S 2 x S 2 x S 2. The important step for the qualitative analysis is the evaluation of the group action on the space of dynamic variables (the stratification of the phase space). One should remember that the stratification of the phase space for different molecular symmetry groups and different types of dynamic operators may be identical because it depends on the group images in a given representation rather than on the initial symmetry group and the representation realized on the dynamic variables. As soon as the stratification of the phase space is given one can study the local singularities for each local symmetry group in classical problems and study the corresponding effects in quantum problems. The correspondence between classical equivariant bifurcations and critical phenomena in the energy level patterns for quantum problems (quantum bifurcations) was studied in recent papers [1, 2, 8]. The relation between the formation of diabolic points (conical intersection points of different energy surfaces) and phenomenon of the redistribution of energy levels between different branches in the energy spectrum was demonstrated in [3, 7, 8]. New interesting class of qualitative phenomena is the organization of elementary bifurcations caused by symmetry. Such phenomena exists for problems whose classical phase space possesses a system of one-dimensional strata. Let us consider S 2 as a classical phase space and the stratification produced by the natural action of different three-dimensional point symmetry groups on it. Let us suppose also that after the first bifurcation the stationary points move monotonously along the one-dimensional strata. Under these suppositions all types of organization of bifurcations for different point groups on S 2 phase space which do not change the number of stationary points may be given. We list here only those which do not include C1 unsymmetrical bifurcation. The following notation for bifurcations is used [1].
488
C~x~ , i = 1 , . . . ,oo, X = L,N, a = + , - , is local (L) or nonlocal (N) bifurcation with C~ broken local symmetry which produces (+) or annihilates ( - ) stationary points. The detail description of different bifurcations is given in [1]. The list of the organization of bifurcations for different point groups is as follows. C,,, $2,~, D,,, T, O, I, C,,h, D,,d, Th groups give no organization under the suppositions mentioned above. C,~ . The organization C L+ --* C2- exists for n > 4. The result is the crossover. D,h. Two sequences C x + --+ C x - , X = L, N give finite rotation. Other sequences C~ + -.., Off ~ C ~ - , n = 3, 4, a n d e S - ~ C~ + ~ C ~ - , n > 4 r e s u l t in crossover plus finite rotation. Td. C~ + ~ C~ ~ C~ ""+ C ~ - gives c r o s s o v e r . Oh. All sequences result in crossover. C~ + -* C~ --~ C~ --* C ~ - , C~ + -+ C~ --4 C~ -~ C ~ - , C~ + --* C~ --* [C~-, Cff-]. Two bifurcations in brackets take place on the same zero-dimensional stratum. Ih. The only possible sequence C~ + -+ Cff -* [C~-, C y - ] results in crossover. There is a number of experimentally studied spherical top molecules (CH4, Sill4, CD4, SnH4, CF4) which demonstrate clearly the organization of bifurcations, appropriate for the effective rotational Hamiltonian with Oh symmetry group [6,9]. Other types of organization have not yet been studied experimentally. 1. I.M.Pavlichenkov, B.I.Zhilinskii: Annals Phys.(N.Y) 184 1 (1988) 2. D.A.Sadovskii, B.I.Zhilinskii: Mol.Phys. 65 109 (1988) 3. V.B.Pavlov-Verevkin, D.A.Sadovskii, B.I.Zhilinskii: Europhys. Left. 6 573 (1988) 4. B.I.Zhilinskii: Chem.Phys. 137 1 (1989) 5. I.M.Pavlichenkov: ZhETF 96 404 (1989) 6. G.Pierre, D.A.Sadovskii, B.I.Zhilinskii: Europhys.Lett. 10 409 (1989) V.M.Krivtsun, D.A.Sadovskii, B.I.Zhilinskii: J.Mol.Spectrosc. 139 126 (1990) 8. D.A.Sadovskii, B.I.Zhilinskii, J.P.Champion, G.Pierre: J.Chem. Phys. 92 1523 (1990) 9. O.I.Davarashvili, B.I.Zhilinskii, V.M.Krivtsun, D.A.Sadovskii, E.P.Snegirev: Pis'ma ZhETF 51 17 (1990) .
489
ALGEBRAS
OF
THE
SU(n)
VECTOR
INVARIANTS
AND
SOME
OF
THEIR
APPLICATIONS
Valery P. Karassiov P,N. Lebedev Physical
Institute
of the USSR Academy of Sciences,
Leninsky prospect 53, Moscow 117924, USSR.
ABSTRACT:
We
generated
by
examine the
new
SU[n)
algebraic vector
structures
Invarlants
oscillator creation and annihilation operators. bosonlc
oscillator
systems
with
internal
introduce
both
infinite-dimensional
nonstandard
polynomial
deformations
dimensional
oscillator
Lie
to
the
SU[n)
invariant
algebras
symmetry
A spectral
the rock spaces of initial oscillators
of
algebras
and mutations
algebras.
are
For analyzing
SU(n) Lie
which
consisted
we and
of finiteanalysis
of
is given with respect
under
consideration.
Some
physical applications in composite models of many-body systems are pointed out.
l. Introductlon. The symmetry approach based on the use of mathematical formalism of represantation
theory of Lie groups
and Lie algebras
is
widely and successfully used in quantum theory of many-body systems (see, e.g.,
[i-6] and references therein). Specifically,
ana-
lysis of many-body problems within the second quantlzation method introduces
in a natural way a symmetry formalism associated with
oscillators of bosonic and fermionic types: of oscillator
Lie algebras
represantatlon
and superalgebras
in the Fock
theory spaces
[1,4-S]. Such an approach is especially
fruitful
in examlnin E compo-
site models with an internal symmetry since it allows to display some hidden consideration
symmetries
and other peculiarities
of
systems
under
[8-II]. Besides, within this analysis we obtain some
new alEebraie structures which differ from usual Lie algebras and and
groups
mutations
and
repreesent
their
specific
deformations
and
(cf. [12-15]).
Indeed, [ of bosonic
let us consider many-body quantum oscillator systems or fermlonlc
types)
which
are associated --~
with
the
~+
creation and annihilation operators x~i and xi=(x i) , respectively (~=i,2 ..... n; i=I,2 ..... m<m, the superscript "+" denotes the Her-
493
mitlan
conjugation).
Here
the
components of one-partlcle with the vector
"~"
superscript
labels
"internal"
states that transform in a accordance
(fundamental)
irreducible
representation
(irrep)
DI(G) of a classical group G:
x", 9
x?
÷,
, D'(G).
where from here on the summation superscripts.
(i I)
is implied over repeated Greek
The operator x ~ x~ satisfy the standard commutation I' J
relations (CR)
(i.2)
ix i,xj] (~)=xtx]+Ax]xt=0=[xl,xj]~CX),
[x~.x~]cr()O=~
j. o-(;~)=sgr~,
where A=-I and I for bosonlc and fermlonic systems,
respectively.
The Hilbert space for these systems are the Fock spaces L F spanned by the basic vectors
I{n~}>=N({n~})~. G (x11)
where
I0>
nll
g2 n22
(x 2 )
is the vacuum vector:
constant;
g
n m
. . . ( x m) m I0>, x=lO>=O |
the range of the exponents
(1.3)
~,i, N is a normalization
{n7}
depends on the type of
the oscillator statistics. All physical operators includin E Hamiltonian H are polynomials in variables x ~, x~, e.g. i ] H=.Z.~..x.x.+~(c.x.+c. ,,j
,j
,
j
,
i
,
,
x.)+higher powers, ,
(1.4)
where the asterisk ~ denotes the complex conjugation. Now we suppose that Hamiltonlan H is Invarlant with respect to the action (i.I) of the "internal" symmetry group G. Then, according to the vector invariant theory [16], H depends polynomially only on some elementary G-invarlants I ({x?,~.}) constructed in r
i
j
terms of G-vectors xi=(x ~) and xi=(xT). Further, of H provokes
a possibility
this G-lnvarlance
of picking out the G-invarlant
sub-
spaces in L F that one may interpret as a existence of kinematically coupled
("confined")
G-invarlant dynamics.
in internal variable subsystems with the
In order to examine such composite subsys-
tems within the general
symmetry approach
[3,4] we need
in con-
m
structing
C -algebras
[17]
of
the G-invarlant
observables
and the G-Invariant dynamic symmetry algebras k(h)(G) m
494
k (G) m in terms of
{Ir({X?,X~})}
as
well
as
studying
representations
of
these
al-
gebras in the spaces L F. Efficient tools for solving these problems are the vector invariant theory [16] and the conception of complementary groups and algebras
[10,18].
Specifically,
the complementarity
theory allows
us to decompose the space L F into direct sum (with a simple spectrum)
LF= ~ LF,
(1.5)
where the subspaces L_ are irreducible wlth respect to an action of the
algebra
gek~)(G)-
("g" being
the Lie algebra of G) and
furthermore the label "~" determines simultaneously both an irrep of k(A)[G). D~[g) of g and an dual irrep D~(kLA)(G))m m physical point of view the decomposition superselection rules
From
the
(i.5) gives rise to some
[19] since the single spaces L F with diffe-
rent "~" do not "mix" under the time-evolution governed by a Hamiltonian H E k(A)(G). Thus the "internal" symmetry algebra g "inm duces" the "hidden" dynamic symmetry algebra k(A)(G). m This program is simply and fruitfully realized in many-body physics for the groups C=O(n) and Sp(n)
since in these cases the
basic invariants Ir({...}) are bilinear combinations of the operators x ~ and x~
i
and therefore algebras k(A)(G) are well-known fi-
J'
m
nlte-dlmenslonal
Lie-algebras
(see,
rences
However
the
therein).
for
situation is more complicated. k(-1)(SU(n)) m dimensional
and k(-1)(SO(n)) m Lie algebras
deformations
of the universal
oscillator algebras
e.g.,
groups
[18,20-22] G=SU(n)
Specifically, belong
refe-
SO(n)
the
for nm3 the algebras
to new classes
[8,9,23]
and
and
of
associated
enveloping algebras
infinite-
with
some
of generalized
[Ii]. A theory of these structures has as yet
been developed not quite enough. The main aim of the present paper is to examine the situation in more detail [pl,...,pn ]
for the case G=SU(n),
is
D(pl ..... pn_1)and
the
highest
DI(G)=D(IOn_2) , A=-I,
weight
of
the
SU(n)
where irrep
dot as a superscript over "a" in "~ " means the r
repetition of "a" r times. Sec. 2
we
investigate
k(-1)(SU(n))=k(-)(n) m
The paper is organized as follows. some
and associated
properties structures.
of
In
algebras
In See. 3 we
study
m
their representations
in the spaces L F. See. 4 is devoted to cer-
tain physical applications of the algebras under consideration. See. 5 some problems and generalizations are discussed.
495
In
2. Bosonic algebras of the SU(n} vector invarlants. So,
specialize
our
further
analysis
for
bosonic
systems
(A=-I). As is well known [8,16] the set of the basic vector invariants
Ir({Xl,Xj})
for the group SU(n)
consists of the following
constructions:
Etj~(XlXj)=X?X~=(Ej~)
m[Xl . . . X 1 ] = E 1
Xl . . . I 1
i,j=l
+,
n
1
n Xl
n
.....
m,
(2.1a}
1. . . X ! n ~ i 1
(2.1b)
n
Xi1""tnE[xil'"xt]=(Xln 1""In )+ for A=-I,
where
c''"
(2.1)
are
is the
[nvariant
generators
SU(n)-invarlant
of
conceptual
the
ant£symmetrlc C -algebra
observables
formal power series in the variables dering.
(2.1c)
tensor•
The
ent£ties
km(SU(n))mkm(n) [17]
whose
of
the
elements
are
(2.1) with their certain or-
The ordering is dictated by existence some relations bet-
ween the quantities Specifically,
(2.1). from the second Hilbert theorem of the vector
invariant theory [16] for A=-I we have the identities ("syzygles")
X
X -X X +... + ll...in Jl...Jn J112,.'In l l J 2 . ' ' J n +(-l)nx X =0, ill[• •i n-1 tnJ2"" 'in
(2.2a)
•
X| I.. .InErj-Xrl2" '" InEilJ+•''+C-l}nXr|l'"In-IEinJ=O'
(2.2b)
XII., .inxjl... jn=Pn{ {Eli})
(2.2c)
and those obtained by the Hermit[an conjugation of eqs (2.2}. Here P ({EH}} whose
are polynomials
explicit
of
the n-th order
form can be found
from
in variables
the well-known
(2.1a)
algebraic
Identity [16]
E
1
n c
~,...~=
detU[Jll.
(2.3)
~j
For example, for the case n=2 we have
P2({Eij})=E
E -E E -6 E +~ E . ljJ 1 12J 2 JlJ 2 12J | 12J 1 l l J 2 i2J 2 i i J i
496
(2.4}
The identles
(2.2) allow us to identify the algebras k (n) as PIm [24] on the Grassmann manifolds with the Plucker coordi-
algebras nates X
[25,26]. i ...i I n Further, from the CRs (~.2) with k=-I we easily find CRs for
the quantities
(2. i):
[Elj, Ers ] m [El j, Ers]_= ~ jrEis-~IsErj'
(2.5a)
. J ]=0=[Xt I ,X ], [Xll ''in'XJI"" n 1 ' ' " n Jl'''Jn
(2.5b)
[Erj'Xl i'"In]=~J|IXri2""In+~Ji2Xilri3 "''In÷ ....
(2.5c]
[Erj'Xt ...i ]=-(~rt ~
(2.5d)
1
n
+~
1 Jl2'''ln
X
+'"" )'
rl2 llJl3"''ln (2.5e)
[X|I' ..In'XJl... Jn ]=P~({EIj})' pt n({Eij}) are polynomials
where E
of the {n-l)-th order in variables
which are obtained by using the explicit
lJ mlals
Pn[{Eij})
in eq. (2.1c).
Specifically,
form of the polynofor
the case
n=2 we
have i [Xlj,Xkl] = P2({EtJ}) = - 2(~]k~il-~Ik~Jl) +
~
E
-
lJ k!
~
- ~ tlE k]+~ IkE I] +
E
(2.5e')
Jk I t '
that allow us to close the CRs (2.5) and to introduce the S0e(2m) Lie algebra structure on the set Im(2)m{Xlj,Xkl,Eij} It is not the case, however, P;({EIj}),
with
[8,11].
for nZ3 because repeated CRs of
elements
of
the
Im(n)E{Etj'Xl ...I 'Xi ...i li=1 .... m} contain elements 1 n 1 n k (n) algebras wlth higher powers of EIj,X i i 'Xj thus result in infinite-dimensional
Lie algebras
k- On)
set of
the and
[II]. But
if we restrict ourselves by considering only the initial CRs [2.5) we obtain
a new class of Lie-algebralc
structures
are some deformations
of usual Lle algebras
the CRs
are similar
bosonic (2.5e}
(2.5a)-[2.5d) oscillator represents
Lie
algebras
a polynomial
for usual bosonlc oscillator
to those for elements u(m)eh(m)
deformation
operators.
n-m=3 we have
497
I (-){n) which m (cf, [12-14]). Indeed,
[2-4]
while
of usual the
CR
of the canonical
CR
For example,
in the case
[X123, X123 ] = 3 =3!+3 Z E 1=1
Therefore,
+E
il
E
11 22
-E E
21 12
+E E
22 33
-E E
32 23
+E E
33 11
-E E
(2•6)
13 31
taking also into account the eqs•(2.2), we may call al-
gebras I(-)(n) as Grassmann deformations of bosonlc oscillator alm
gebras. We also note that with each algebra I(-)(n) one may associate m
a Lie algebra k(')(n) if instead of the usual Lie bracket
[.,.] we
m
use
a
new
Lie
bracket
[',.]~mPrlm[',.
] where
stands for the projection onto Spanlm(n)U{cl} identity operator)•
the
symbol
Prlm
(with "I" being the
In a sense the algebras k(')(n) may be conslm
dered as peculiar
("linearized")
mutations
[15] of the algebras
k(-](n)whlch can be used for a descriptions of a generalized dynam
mics of the SU(n)-clusters
(cf.[ll,15]).
Thus, the set (2.1) generates raic
structures
mutually
k~-)(n), I(-)(n) and m
related
k(')(n)
of
Lie algeb-
three
types which are connected with Grassmann oscillators•
Any of these
algebras has two mutually conjugate finlte-dlmensional gebras
b(m'n)=span{Etj,X i +
}
the
and
•..| 1
Grassmann
oscillator
different
m
Lie subal-
b(m'n)=Span{Ei.,Xi -
.
1
n
1
algebras.
In
addition,
the
..i
}
of
n
algebras
k(-)(n) have a characteristic property of nllpotency m
a-'n+tQAB=O,
AeX_=Span{X? •i.t }'n BeX+=Span{Xl n
i"'i }'n
(2.7)
n-i
adAB=[A,B], adAB=adA (adAB}, which is useful
for summing up the Baker-Campbell-Hausdorf
series
[ll]. 3.Representatlons
of
the
algebras
k~-)(n},
l~-)(n} and
k(')(n}m in the Fock spaces L F. For the physical applications we need In constructing representations
of above
riants, particularly,
defined
algebras
of
the SU(n)-vector
inva-
in the spaces L F. Below we outline a general
scheme of the spectral analysis of the spaces L F with respect to actions of the algebras su(n}®k(')(n) that determine also approp=) rlate Irreps of the algebras I ~- (n) and k{')(n}• For this aim we m m use the concept of complementary algebras and groups [18,22]• We
start
from
the
simplest
case
n=2
when
k(-) (2)=k(*) (2). As is known the algebra k(-)(2)=som(2m) m
m
m
498
we
have
acts com-
plementarily
to the
algebra su(2)~sp(2)
27] and the decomposition
on the space L F [11,22,
(1.5) takes the form
LF=j~ ° L(J), where
J specifies
the label
appropriate
(3.1) both
the SU(2)
so (2m) irrep DJ(sot[2m))
The subspaces
L(J) are spanned
irrep D(2J)
and
the
[II]. by the basic vectors
]J;M;v>
where the lables "M" and "u" distinguish basic vectors within irreps D(2JO
and DJ[so*(2m))
respectively.
(3.1) represents a vector bundle near combinations a simple
of the Fork states
algorithm
Thus,
the decomposition
[28]. The vectors (1.3).
has been developed
IJ;M;v> are li-
In the papers
for explicit
constructing
these vectors by using the techniques of the generating and generalized
coherent
for the case m=2 which,
states. however,
We
consider
elucidiates
[10,29]
such
invariants
constructions
the situation
in the
general case.. For m=n=2
the algebra k(-)[Z)=soI(4) decomposes into the dl2 rect sum so (4)=SUlnv(2)®su(l,l) (with generators x12, x12 , (I/2)[EII+E22)+I
and
su(l,1) and su(2),
El2, E21, [1/2)(Eli-E22)
respectively)
the
subalgebras
and the basis vectors
for
IJ;M;u> ha-
ve the form • -,J-M, -,J÷M J÷t r IJ;M;v>~IJ;M;{T,t}>=N(J,M,T,t)telu) te2u) [xtu] tx2u]J-tx X[XI2)T-JIo>,
where N is a normalization diate boson operators, fically,
for
(elu)meTu~
factor,
(3.2)
"u" and "u" are some interme-
e1=(~ 7) are the reference vectors.
the G=SU[2)-scalar
subspace
L(O)
Speci-
the vectors
(3.2)
take the form [Ii]
IT>~IO;O;{T,O}>=[T!(T+I)!]-t/2[x which
is generated
invariant cluster the vacuum vector (1.3).
Comparing
by measns
12
)TIo>,
of action
(Grassmann oscillator)
(3.3) of powers
of the SU(2)-
creation operators x
I0> by analogy with usual one-mode the appearence
of the vectors
observe that they have a similiar structure. is in that the vectors
(3.2) are generated,
on 12 Fork states
(3.2) and (3.3) we
The only discrepancy unlike
(3.3) by action
of the operators
X s on the (2J+l)2-dimensional "vacuum" subspace 12 Lu(J)=Span{IJ;M;{Jt}>:J=const} with characteristic property X121~>=0,
(3.4)
iv> E Lu(J).
499
In turn the space LIJ) is generated by means of action the lowe. . ~ 2-1 - - 2 1 operators.~ x x.=~ and E of two subalgebras su(2) c
rlng
1=1
1
21
1
su(2)eso (4) on the highest vector IJ;J;{JJ}>. Now we consider an action of the above algebra so"(4) on the vectors (3.2) using the CRs (2.5). We note that because of the definition of k(-)[2)=som(4) its action does not change the values 2 of numbers J,M "controlled" by the "internal" algebra SUlnt(2). Hence each space L(J), J~O, decomposes into the direct sum
L(J)=eL(J,M)=eSpan{IJM;{Tt}>:J,M=const} M
(3.5)
M
of the disjoint spaces L(J,M) which are equivalent with respect to the action of the algebra k{-)(2)=so~(4). Further, the action of ~2 the subalgebra su(2)invC so (4) does not change the quantum number T while so~(4)
the operators
X
]2
and X
12
of
the
subalgebra
raise and lower its value by one respectively.
space L(J,M)
is a conjuction of the disjoint
subspaces L(J;M;T)=Span(IJ;M;{Tt}>:J,M,T~J
su(l,l)
c
Thus each
sUlnv(2)-equivalent
=const} which are "in-
tertwined" by the operators XI2,X12. Such the action of the algem bra k(')(2)=so (4) on the space L(J,M) resembles that of usual 2 oscillator algebra on the Fock space (cf.[2,4]) and allows us to m obtain the space-carrier of the so (4) irrep DJ(soQ(4)) starting from any vector of the "vacuum space"
Lv(J).
Similiarly,
show that all spaces L(J,M) are the carrier-spaces
one can
of equivalent
irreps of the algebra I(-)(2). 2 The above analysis provides a sample for realizing spectral analysis of the spaces L F in the case of arbitrary
"m" and "n"
[II]. Therefore we outline its logical scheme and point out some peculiarities in the general case. For arbitrary "m" and "n" the algorithm consist of determinin E
"vacuum
spaces"
su(m)-equivalent means
(su(m)
action
of
operators
of
lowering
susnt(n)=Span{EiJsr=1~ x|xJ'r r SUlnv(m)=Span{Eij, mon highest vectors
X
next
constructing
their
c k~-)(n), I(-)(n)m' k(')(n))m by
on the vectors iv> I n LuCPl ..... Pn_1). In turn the spaces Lu(...) are generate~ by means of
action
L (pl...pn_ I) and
replicas
! ...i
operators i~j,
i~j, ~II=EII-EI+l,I+1}
of
the
algebras
£11=E11-Ei÷1'l**}
and
c kC-){n)m
on the com-
~pl...pn_1;max>ml>satisfyinE
the following
equations
500
a) Xi ...1
I>=O'
1
(3.6a}
n
b) ~ il>=pil>=E111>,
c) EtJt>=O=Eijl>,
(3.6b)
i=l, . . ,n-1 . .
(3.6c)
i<j.
As a result we obtain at final step of the algorithm the following specialization of eq.[l.5)
[II]:
LF=<~I>L(<~ >)=,~e,,~,,, L(<~ >;~';{~';~}), where
(3.7)
L(;~';{p'';~})=Span{f;~';{~'';T}>}
are
carrier-
spaces of the SUlnt(n) irreps D() and of associated D(]) and
9''
SUint(n)
irreps are and
the
Gel'land
k(-)(n)'Im - ) ( n } m
-Tsetlln
respectively;
SUinv(m),
distinguishing vectors
of the algebras
vectors
within
[Spt>;~';{~'';~}>
(dual to
and k(')(n);m ~'
patterns
for
the
algebras
~ is an extra label for
irreps of k(-)(n) etc m
[IO]. Basic
resemble in their appearance the struc-
ture of the vectors some polynomials
(3.2) but instead of monomlals X s we obtain 12 in variables X i ...t " An algorithm for obtaining 1
an explicit
(quasimonomial
n
in vector
invarlants consisted of x i and some intermediate boson vectors u i, uj) form has been developed in our papers [10,29].
4. Some physical applications. A natural area of applications of the above results is in developing
composite
models
with
internal
SU(n)-symmetries
within
both quantum mechanics and quantum field theory [2,11,30-32]. Such models are governed by SU(n)-invarlant Hamiltonian H Inv formulated in terms of elements of the algebras k (n}: m
Htnv=CI+ ~ (~iEi t + i Z c tJ E iJ + ,J
~
di
1
,..i
n
Xi
1
...i
+ n
o
+
Z
d
I
1
...i
X n
Specifically,
%
1
...1
+higher powers.
(4. I)
n
some effective Hamiltonlans
in quantum polari-
zation optics have this form [11,32]. The quantities
X i ...i I
operators
of creation
Invarlant
clusters.
and Xt ...I n
I
and annihilation,
But,
unlike
usual
501
may
be
interpreted
as
n
respectively,
of SU(n)-
quantum particles
(bosons
and fermions) these clusters have unusual statistics as it follows fromthe CRs (2.5). In particular,
in the case n=2 we obtain from
(2.5) trlllnear CRs
[Xr,[Xlj,Xkl]]=(~jl~rk-~jk~rl)Xls+(~jl~sk-~jk6sl)Xr1+ . . . .
(4.2]
which generalize the Green's trlllnear CR for paraflelds and paraparticles
[2]. The CRs (2.5] imply
also the general form of the
number operator Ncl of such clusters [11]
N
=(I/n)ZE
cl
I
II
-C({E~})=(I/n)ZE I
il
-C({E
}),
lj
where C(...] are some SU(n)-invariant
(4.3]
nonlinear functions of the
SU(n] generators E °~ which are multiple to the identity operator I on each subspace L()
from
(3.7]. Specifically,
for m=n=2 we
have
C({EC~8})=-I/Z+(I/2)(l+2(EI2E21+E21EIe)+(EII-E22)2) I/2. Thus,
(4.4)
taking also into account (2.2], we see that internal SU(n)-
symmetry yields us a scheme of a generalized paraquantlzatlon with constraints nontrlvlal develope
[cf.[30,33]) dimensions
models with
on
the
spaces
LF=eL().
of the "vacuum subspaces" spontaneously
Because
Lv[)
broken and hidden
of
we can
symmetries
(cf. [31]] within above formalism. Another interesting llne of InvestlEatlons here is in examlnlng posslb111tles of constructing canonical bases of observables Ya' Yb ([Yb'Y,]=~ab]
in terms of elements of algebras km(n).
way seems to be perspective [34], we obtained
since,
followln E the general
In [11] explicit expressions
Thls
scheme
for Y, Y in the
case m=n:
Y=
Z
Cj(X12...n)J+l(x12...n]J,
Y=(Y]+,
(4.5]
JZO
where the coefficients C
r are determined from a set of reccurence
relations depending on signatures of subspacles L(). Such developments can be useful in analyzlnE composite models of many-body quantum systems of arbitrary physical tons, phonons etc.).
nature
(pho-
Some examples of solving certain problems in
polarlsatlon quantum optics have been considered wlthln this approach in [11].
502
5.Conclusion.
In conclusion we point out
some problems and generalizations
of the above developments. The results obtained provide a mathematical
tool for analy-
zing composite models with internal SU(n)-symmetry only at alEebtalc level. However, miltonians
for examlnln E time evolution governed by ha-
(4.1) we need
of the theory,
in developing group-theoretical
in particular,
generalized
coherent
aspects
states of al-
Eebras k(-)(n) etc. m
It is also of interest to extend our analysis by common considering both internal and the space-tlme Polncare symmetries. "Grassmann nature"
The
of the SU(n)-clusters
X! ...i Elves hope that I n we can obtain alone this llne certain results which are useful for some developments in strlnE theory (cf.[25,26]) and for analyzing nonlinear phenomena and coherent structures in stronEly interacting many-body systems [35]. Finally we note that formal aspects of the above analysis may be extended completely for the case G=SO(n). tlonls
obtained
by
involving
in
Another Eenerallsa-
consideration
other
than
DI(G)
irreps of "internal" groups G.
References.
1.H. Weyl. The theory of Eroups and quantum mechanlcs.(Dover Publ.,New York, 1950). 2. Y. Ohnukl and S. Kamefuchl. Quantum field theory and parastatlstlcs.(Unlv. Press,Tokyo, 1982). 3. I.A. Malkin, V.I.Man'ko. Dynamics symmetries and coherent states of quantum systems.(Nauka, Moscow, 1979). 4. A.M. Perelomov. Generalized coherent states and their applications.(Nauka, Moscow, 1987). 5.P.Jordan: Zeits.f. Phys. 94 531 (1935). 6. B.R. Judd. Second quantization and spectroscopy.(Maryland Univ. Press,Baltimore, 1967). 7. I.Bars and M. Gunaydin: Commun. Math. Phys. 91 31 (1983). 8. V.P. Karasslov. In: TopoloEical phases in quantum theory.(World Sci., Singapore, 1989), p.400. 9.V.P. Karasslov: Sov. Phys.-P.N. Lebedev Phys. lnst. Reports No.9 3 (1988). lO.V.P. Karasslov. In:Group-theoretlcal methods in fundamental and applied physlcs.(Nauka,Moscow, 1988),p.54. 11.V.P. Karasslov. P.N. Lebedev Phys. Inst.preprlnt No137 (FIAN,Moscow, 1990); J.Sov. Laser Res. 12 No.2 (1991). 12. E.K. Sklyanln: Funkt. Anal. Prll. 16 27 (1982); 16 263 (1982). 13. L.C. Biedenharn: J.Phys. A22 L873 (1989). 14. A.J.Macfarlane: J.Phys. A22 4581 (1989). 15. H.C. MyunE and A. Sagle: Hadronlc J. 10 35 (1987). 16.H. Weyl. The classical groups.(Unlv. Press, Prlnceton, 1939). 17. G.G. Emch. Algebraic methods in statistical mechanics and quantum field theory.(Wiley-Intersclence, New York, 1972). 18. M. Moshlnsky and C. Quesne: J. Math. Phys. 12 1772 (1971).
503
19.A.O. Barut, R. Racka. Theory of group represantatlons and appllcatlons.(PWN-Pollsh Sci. Publ.,Warszawa, 1977). 20. P. Kramer: Ann. Phys.(N.Y.) 141 254 (1982). 21.E. Chaco~, P. Hess, M. Moshlnsky. J.Math. Phys. 30 970 (1989). 22. S.I.All~auskas: Sov. J. Part. Nucl. 14 563 (1983). 23. G. Couvreur. J.Deenen and C.Quesne: J.Math. Phys. 24 779 (1983). 24. Yu.A. Bakhturln. Identities in Lle algebras.(Nauka, Moscow, 1985). 25. A. Pressley and G. Segal. Loop groups. (Clarendon, Oxford, 1986). 26. Y. Matsuo. Preprlnt UT-523(Unlv. Press,Tokyo, 1986). 27. H. Le Blanc, D.J.Rowe: J.Math. Phys. 28 1231 (1987). 28. A.S. Mishchenko. Vector bundles and their applications. (Nauka, Moscow, 1984). 29. V.P. Karasslov: J.Phys. A20 5061 (1987). 30. C. Itzykson, J.-B. Zuber. Quantum field theory.(McGraw Hill, New York, 1980). 31.L. Mlchel:Rev. Mod. Phys. 52 617 (1980). 32. V.P. Karasev(Karassiov) and V.I.Puzyrevskli: J.Sov. Laser Res. I0 229 (1989). 33. P.A.M. Dirac. Lectures on quantum field theory.(Yeshlva Unlv. Press, New York, 1967). 34. R.A. Brandt, O.W. Greenberg: J. Math. Phys. 10 1168 (1969). 35. J.A. Tuszyn'skl and J.M. Dixon: J.Phys. A22 4877,4895 [1989).
504
P R O D U C T FORMULAS FOR. Q-REPRESENTATIVES
P.Kasperkovitz Institut £dr Theoretische Physik, Technische Universit£t Wiedner Hauptstr. 8-10, A-1040 Wien, Austria
Introduction. Let Q be a Lie group whose elements - up to a set of measure zero - are uniquely labelled by global coordinates 7/= (r/l, T/2,... ). Furthermore, let g(T/) ~ gr(~/) be a representation of ~ by unitary operators in a Hilbert space whose elements are identified with the states of the quantum system under consideration. If one of the states is selected by a convention then the states I ' > = O(~)l fixed >
(1)
span a linear space that is, in general, a proper subspace of the state space. If this space coincides with the original Hilbert space, which is especially true if the representation is irreducible, the states (1) are called group-related coherent states [1-3]. For historical reasons the expectation value of an operator A with respect to these coherent states, A(~) = < , l d l , > ,
(2)
is then called the Q-representative (or Q-symbol) of A. Equation (2) assigns to each operator a function of the group parameters. Under certain conditions this mapping is invertible. In the following it is assumed that ~ is compact and that the fixed state from which the coherent states are generated is an eigenvector belonging to the highest weight A of an irreducible representation [3]. In this case the Hilbert space is finite-dimensional and the mapping A ~ A is one-to-one [4,5]. As the functions A, B , . . . contain the same information as the operators ,4, J~,... it is possible to perform all quantum mechanical calculations with functions instead of operators provided that the counterparts of the following operations are known: linear combinations, products, transition to the adjoint operator, and trace operation. Except for the product operation,
i#=O,
, AoB=C,
(3)
the corresponding operations for Q-representatives have been known for a long time. In the present contribution formulas are given for the RHS of (3).
The general product formula. Under the assumptions stated above, i.e. ~ compact and I fixed > = ] A >, the binary relation that corresponds to the product of two operators reads as follows.
(4)
A o B = P
505
In this equation the symbols p represent Gel'fand-Tsetlein patterns that label the basis vectors of the irreducible representation characterized by A. To each label p belongs a sequence of shift operators that transforms the vector [ A > -- [ 0 > into the basis vector [ p > (up to a complex factor). The sum over all p's, including the trivial pattern p = 0, reflects the matrix product of the two operators. However, the number of nonvanishing terms, starting with the product of the values of the two functions at position rl, depends on the functional form of A and B. For given A and p the real number (Alp) can be calculated from the commutation relations of the Lie algebra. The differential operator 0p and its complex conjugate 8~ are uniquely determined by the pattern p and the group parameters ~?. The highest order of derivatives occurring in 8p is equal to length of the pattern, i.e. to the number of shift operators contained in p. A detailed derivation of (4) is given in [6]. It rests on the fact that every Qrepresentaive may be expressed as a linear combination of products of matrix elements of the irreducible representation and its complex conjugate. These matrix elements are solutions of partial differential equations and related by recurrence relations; this follows from the special functions approach to group representations where the elements of the Lie algebras related to the left and right regular representation of ~ are represented by differentiM operators.
Cohcrcn~ Spin Sta~cs. Coherent states related to SU(2) have been considered in [7-9]. For a spin of magnitude 8 the Q-representatives comprise all linear combinations of the spherical harmonics YLM with 0 _< L <: 28. The argument of these functions consists of two of the three Euler angles used to label the elements of SU(2); variation of the third angle adds only a phase factor to the coherent states that drops out in the expectation values (2). If the general product formula (4) is specialized to this case one finds A o S =
(81g) -1
K where
K = O,... , 2 S ; (28 - g)! _ O(S_K) -- K!(28)! OK : [ ( g - 1) cot~-~-i CSC~8~b-[-80] 8K-1 ; 8o=1.
(Si/~)_ 1
(6)
Canonical Cohcren* S*a~e,. These states, extensively studied by SchrSdinger [10], Glauber [11], and many others [1], are the most familiar coherent states. Because of their intimate relation to classical mechanics it is convenient to label them by the 'phase space' variables p and q. Group-theoretically these variables and the coherent states are related to the Weyl-Heisenberg group. It is well known that this non-compact group and the related coherent states may be obtained from SU(2) and coherent spin states by means of a group contraction [12-13]. Not surprisingly this procedure allows one to obtain also the corresponding product formula.
506
In this example the Q-representatives are of the form
A(p, q) = P(p, q) exp { p2+q2}2h P(p, q) = polynomial in p, q
(7)
or may be defined through sequences of such functions. Group contraction transforms (5) and (6) into A o S = ~ (h]k) -x (O~A)(OkB) (8) k
and k = 0,1,2,...oo ; ( n l k ) - * = kU
= °(n~) ;
ok = (p2 + q2)-~ [ ( k - 1)+ (p + iq)(op -ion)] ok-1 ; (9)
00=1.
Conclusion. To obtain a closed and consistent mathematical description of quantum systems in terms of group-related 'phase space' functions one needs counterparts for all the operations usually performed with operators. For functions related to coherent states, so-called Q-representatives, the relation corresponding to the product of two operators is given for compact Lie groups, especially for SU(2), and for the non-compact Weyl-Heisenberg group. Equations (5-8) show that these formulas, as well as the resulting commutator relation, are especially suited to discuss quantum corrections to classical results. In addition they should simplify the calculation of expectation values for coherent states as they relate the Q-representatives of more complicated operators to those of simpler ones. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
J.R.Klauder, B.S.Skagerstaxa : Coherent States, World Scientific, Singapore (1985) A.M.Perelomov : Commun. Math. Phys. 26 222 (1972) R.Gilmore : Rev. Mex. de Fisica 23 143 (1974) J.Klauder : J. Math. Phys. 5 177 (1964) B.Simon : Commun. Math. Phys. 71 247 (1980) P.Kasperkovitz : J. Phys. A.: Gen. Phys. to appear (1990) J.R.Klauder : J. Math. Phys. 4 1058 (1963) J.M.Radcliffe : J. Phys. A.: Gen. Phys. 4 313 (1971) F.T.Arecchi, E.Courtens, R.Gilmore, H.Thomas : Phys. Rev. A6 2211 (1972) E.Schrhdinger : Naturwiss. 14 664 (1926) R.J.Glauber in Quantum Optics and Electronics, ed. C.DeWitt, A.Blandin and C.Cohen-Tannudji, Gordon and Breach, New York (1964) R.Gilmore : Lie Groups, Lie Algebras, and Some of Their Applications, John Wiley, New York (1974) R.Gilmore : Ann. Phys. 74 391 (1972)
507
T E N S O R P R O D U C T S F O R A F F I N E KAC-MOODY ALGEBRAS R.C. King and T.A. Welsh Mathematics Department, University of Southampton, Southampton, SO9 5NH England Following Kac [1], let G - G(A) be the affine Kac-Moody algebra of rank ~ associated with a symmetrisable generalised Caftan matrix A having matrix elements A,~ = (a,,a~) = 2((a,,a~))/((cg,a~) ) for i,j E I where I = { 0 , 1 , . . . , ~ } . Each simple root ai, with i E I lies in 9/', the dual of the C a f t a n subalgebra 9t of G. The corresponding co-roots are defined by a v = 2a~/(a,, a,) for i E I. It is convenient to introduce vectors 6 and wi, with i E I, which together span 7"(*. In terms of the integer marks cl and co-marks cv for i E I we have & = L The ma ks are chosen [1] so that - - 0 for j e I. In addition (w,,a~) = 6,j for i,j e I, (w,,w0> = 0 for i e I, and (6,wo> = c~. Every weight ), e ~ " m a y then be expressed in the form )~ = ~ l~=0A~w~ - k6 = (A0, A I , . . . A I ; k ) where the Dynkin components of A are given by As = (A, a~ ) for i E I. The level and depth of the weight A are defined by L(A) = (A, 6) = E , =*0 c , v a n d = = k. A weight A E "H" is said to belong to the set of integral weights, P, if (A,a v ) E Z for i E I. Such an integral weight A is dominant and in P+ if (A, a~' ) > 0 for i E I, and strongly dominant and in P + + if (A, a~' ) > 0 for i e I. The Weyl group, W, of ~ is generated by the Weyl reflections whose action on 7-/* are defined by ri(A) = A - ()~,a~)a~ for i E I. The orbit of A under the Weyl group action is the set W~ = {wA [ w E W}. For each A in P there exists a unique dominant weight A+ in P+ such t h a t A+ = wx A for some wx E W. If A+ is in P + + then all the elements wA with w E W are distinct, and in particular wx is unique. The null depth of A is then defined to be d()~) = D(A) - D(A+). By exploiting the Weyl group invariance, which implies (),+, ,k+) = (w~ ~, w~ ,k) = (~,)~), it is possible to show that d(A) = (1/2L(A)) E t,,j=x (a,~ ()~,)~j -~+)~+)), where a = S_I w i t h So' = c,(c'[)-lA,i for i,j E I + = { 1 , 2 , . . . ,t}. Each irreducible highest weight integrable module V ~ of ~ is labelled by a dominant integral weight A. Such a module has a weight space decomposition V ~ = @,eu" V~ and the character of this module is formally given by ch V ~ = ~ ,e P m ,A e , where the weight multiplicity m , is the dimension of V,x. Kac [1] has established the character formula:
chV
Z: w~W
O)
w~W t.
where p ~ ~* is defined by p = ~ ~=0 w~ so that (p, a~ ) = 1 for i E I. The tensor product V" ® V ~ of two irreducible integrable highest weight modules of G is fully reducible into a direct sum of such modules. If the multiplicity of occurrence of modules V ~ in this tensor product is denoted by g~, then ch V" ch V ~ =
508
~¢ P+ g~v ch V ~ . Two problems which immediately present themselves are the explicit evaluation of the weight multiplicities rn,~ and the explicit evaluation of the tensor product multiplicities g~v x • In what follows it is demonstrated not only that these problems may be solved algorithmically but also that they are intimately connected. Illustrations ave confined for simplicity to the case ~ = A~1). The formal definition of chV" and the character formula (1) for c h V ~ and c h V ~ imply
E: re:e" E ~EP
yEW
E]
(2)
,~EP+
wEW
For any A E P+ it follows that A + p E P++. Moreover w(A + p) E P + + if and only if w is the identity element of W. Setting ~ = vg, using the fact that m~¢ -- rn~ and picking out those terms on both sides of (2) involving e~ with r1 E P + + leads to the identity
Z:
= Z:
fEP
(3)
)~GP+
(,. -F ~,+ p)+ E P + +
This identity provides a geometric procedure for determining the tensor product multiplicities g ~ from the weight multiplicities rn~ of just one of the constituent irreducible modules in the product. Its use is a straightforward generalisation of a very well known technique [2,3,4] developed in the context of finite dimensional Lie algebras. The identity (3) actually provides an explicit formula [3] for tensor product multiplicities:
g.t =
Z:
= Z:
(4)
wEW
wE~' ( r + . + p ) + =~+A,
where the sum over a has been replaced in the second expression by a sum over w E W since the set of elements a + u + p such that (a + v + p)+ = A + p is precisely W(,~ + p). Use has also been made the fact that m~ = r n ~ . This formula not only allows the explicit calculation of tensor product multiplicities from a knowledge of weight multiplicities but also the converse. Indeed setting u = 0, so that V ~ = V 0 is the trivial one-dimensional module and g ~ = g~0 = 6~, and taking ~ ~ / ~ in (4) gives Racah's familiar recurrence relation [3] for weight multiplicities: ~.wew e(w)m~+p-wp = 0. However, (4) can be used as it stands as a tool for determining weight multiplicities from tensor product multiplicities. By way of illustration, in the case of g = A~1) we have A = (_2 - 2 ) , so that S = (2), G = (½) and d(/~) = ~ 7 ~ ( # i 2 - #i+2). For the simplest non-trivial module, V(i,°;°) with highest weight # = w0, the weights are all of the form a = (1 - 2re)w0 + 2row1 - p6 with m E Z and p E Z+. The top ten rows of the infinite weight diagram are shown below. Ignoring for the moment the foot of the table, the Weyl reflection planes are the vertical lines through 80 = (0,1) and 8i = (1, 0). All weights (r = (1 - 2m, 2m; p) on any vertical string in a column labelled by m are Weyl equivalent to those on the one string in the dominant sector labelled by rn = 0. In fact or+ = (1, 0; k) with k = p - m 2, since 2 = m 2 . The weight multiplicities themselves are given by m~ = ap_,~ :I(al2 --Pl) where a, -- p(n), the number of partitions of n, as can be shown through the use of Racah's recurrence relation [5] or otherwise [1].
~09
m
--3
o°o
(7,-6) 9
, , ,
p=0 p=l p=2 p=3 p=4 p=5 p=6 p=7 p=8 p=9
d(C) (Co,C1)
1
--2
(5,-4) 4
1 1 2 3 5 7
--I
0
(3,-2) 1
1
(1,0) ( - 1 , 2 ) 0 1 61 60
I 1 2
1 1 2 3
I ! 2
3 5 7 11 15 22
5 7 11 15 22 30
3 5 7 11 15 22
2
-
0
(8,-3)
-. •
...
1
T ¢0
0
(6, - 1 )
(-5,6) 9
1 1 2 3 5 7
T ¢1 (10, - 5 )
3
(-3,4) 4
0
0
(4,1)
(2,3)
-
2
(0,5) ( - 2 , 7 )
...
The above weight diagram may then be used to calculate, for example, the tensor product multiplicities for V(1,0;0)® V(2,°;°). The procedure based directly on (3) involves shifting the weight diagram of V (1,°;°) through u+p = (3, 1; 0) so that a = ( 1 - 2 m , 2m; p) goes to ¢ = (4 - 2m, 1 + 2re;p) with L(¢) = 5. This can be effected by the relabelling given at the foot of the above diagram. The reflection planes are now at the positions ¢0 = (0, 5) and ¢1 = (5, 0). All weights ~ on any vertical string either lie on a Weyl reflection plane or are such that ~+ = (4, 1; k) or (2,3; k) for some k, with d(~) = ]rn(rn + 1) or ](rn 2 + m -- 2), respectively. Carrying out the Weyl reflections for each vertical string, taking signatures into account and subtracting p = (1, 1; O) gives the .(~o,~,;k) . following tensor product multiplicities u(1,0;0)(2,0;0)"
= (3, 0) k=O k=l k=2 k=3 k=4 k=5 k=6 k= 7
1 1-1=0 2-1=1 3-2=1 5-3=2 7-5=2 11-7=4 1 5 - 11 = 4
= (1, 2) 1 1 2 3-1=2 5-1=4 7-2=5 11 - 3 -
1= 7
Since tensor products are commutative, the same multiplicities must arise if the problem is approached in the same way but starting from the weight diagram of the module V(2.°;°). This takes the following form in which reflections in the planes signified by ¢0 = (0, 2) and ¢1 = (2, 0) have been used to parametrise the weight multiplicities
510
in t e r m s of those in the dominaalt sector. T h e weights are all of the f o r m T = (2 -2 m , 2 m ; p ) , w i t h L ( r ) = 2 a n d d(7) = ½m 2 or ½(rn 2 - 1), a c c o r d i n g as m is even or odd. m = -4 (~o,~t) "." d(r) 8
p=0 p = p = p = p = p = p = p = p =
1 2 3 4 5 6 7 8
ao
-3 (8,-6) 4
-2 (6,-4) 2
bl b2 b3 b4
ao al a2 a3 a4 a5 a6
-1 (4,-2) 0
O (2,0) 0
1 (0,2) 0
¢1 l
¢o 1
ao at a2 a3 a4 a5 a6 a7 a8
bl b2 b3 b4 b5 b6 b7 b8
ao at a2 a3 a4 a5 as
0 (2,3)
¢0 (0,5)
bl b2 b2 b4 b5 b6 b7 b8
T 2 ...
0 (10 , - 5 ) ( 8 , - 3 )
o (6,-1)
3 (-4,6) 4
4 (-6,8) 8
bl b2 b3 b4
ao
2 (-2,7)
4 (-4,9)
T
¢1 d(¢) (¢o,¢t)
2 (-2,4) 2
0 (4,1)
P r o c e e d i n g as in the p r e v i o u s e x a m p l e on the basis of (3) with # a n d v i n t e r c h a n g e d , n o w involves a d d i n g # + p = (2, 1; 0) to t h e weights ~- to give ~. Of course the reflection planes signified b y ¢0 a n d ¢1 a n d d(~) are exactly as before. C a r r y i n g out the reflections, t a k i n g s i g n a t u r e s into a c c o u n t a n d s u b t r a c t i n g p leads to expressions for the tensor p r o d u c t multiplicities which m a y be solved recursively for the weight multiplicities ak a n d bk of the d o m i n a n t weights (2, 0; k) mad (0, 2; k) of V (2,°;°) as s h o w n below.
k = k=l k = k = k = k =
0 2 3 4 5
(~0, ~1) = (3, o)
(~o, ~t) = (1, 2)
1 = ao 0=at-bt 1 = a2 1 = a3 2 = a4 2 = as-
1=bl 1 --- b2 2 = b3 2 = b4 4 = bs-
b2 b3 b4 - a0 b s - az
b0 bl b2 b3-
a0 at a2 a3
a0 = al=l a2 = a3 = a4 = a5 =
1 3 5 10 16
bl=l b2 = b3 = b4 = b5 =
2 4 7 13
REFERENCES [1] V . G . K a c , Infinite dimensional Zie algebras (Boston, Mass.: B i r k h a u s e r , 1983) [2] A.U. K l i m y k , Amer. Math. Soc. Transl. Series 2 Vol 76 ( P r o v i d e n c e , RI.: A m e r . M a t h . Sot., 1968) [3] G. Ra~zah, in Group theoretical concepts and methods in elementary particle physics, Ed. F. G u r s e y (New York, NY.: G o r d o n a n d Breach, 1964) p p l - 3 6 [4] D. Speiser, in Group theoretical concepts and methods in elementary particle physics, Ed. F. G u r s e y (New York, NY.: G o r d o n a n d Breach, 1964) pp237-246 [5] A.J. Feingold a n d J. Lepowsky, Ado. Math. 29, 271-309 (1978)
511
ATYPICAL MODULES OF THE LIE SUPERALGEBRA gl(m/a) J. Van der Jeugt ~ (University of Ghent, Belgium), J.W.B. Hughes (Queen Mary and Westfield College, U.K.), R.C. King (University of Southampton, U.K.) and J. Thierry-Mieg (University of Montpellier, France) ~ Let G --- G o S G ~ be the genera/l/near Lie superalgebra gl(rn/n) [2] consisting of complex matrices (vA BD) of size (rn -b n) 2. The even subspace G O of G consists of the matrices (oA0D) and the odd subspace G i consists of the matrices (o s° ) . The bracket between homogeneous elements is defined by [a,b] = a b - (-1)=Zba for a e G=,b e G~ (a,~ e (0, i} ---- ~2). Thus the even subalgebra is isomorphic to gl(rn) ~ gl(n). G admits a consistent ~-grading G = G-1 ~BGo (9 G+I where Go = Go, G+x is the space of matrices of the form (0°0 s ) and G-1 is the space of matrices of the form ( o 0 ) . The special linear Lie superalgebra sl(m/n) is the subalgebra of gl(m/n) consisting of matrices with vanishing supertrace. In what follows we put G = gl(m/n), but all of the results can be reformulated for ~l(m/n) as well. The Cartan subalgebra H of G consists of the subspace of diagonal matrices. The roo$ or weight space H* is the dual space of H and is spanned by the forms ei (i 1 , . . . , m ) and 6j (3" -- 1 , . . . , n ) . The inner product on the weight space H* is given by [5] (ei[(i) = 61j, (ei[6i) : 0, (6~[6y) = -6ii, where 61i is the usual Kronecker symbol. In this e6-basis the even roots of G are of the form e l - ei or ~ i - 6i, and the odd roots are of the form q-(el - 6j). Let A denote the set of all roots, Ao the set of even roots and Ax the set of odd roots. As a system of simple roots one takes the so-called distinguished set [3] ez-e~, e 2 - e s , . . . , e=-61, 61-62, ..., Q - 1 - Q . Then the set A + of posi~ive roots consists of the elements e, - ej (i < j), 61 - ~ (i < j) and e, - ~j. Now the notations A+ and A + are obvious; in particular : A + = {~,.~. = c,. - 6~.,
i = I , . . . , m,
j = i,...,,~).
(i)
All simple modules (i.e. irreducible representations) of the classical simple Lie su-
peralgebr were classi ed by Kac [S]. Kac's result speci ed to sICm/ ) implies that every finite-dimensional simple G-module V is a highest weight module V(A) specified by an integral dominant weight A. A weight A E H* is said to be integral dominant if and only if its so-called Kac-Dynkin labels A = [al, a 2 , . . . , a.~-l; a.~; am+l,..., a.~+.-i] are such that al E ]N for i ~ m whereas am can be any complex number. For our purpose it is sufficient to consider only those A for which a,~ E 2~. If A is expressed in terms of the eS-basis as A = ~ ~iei + ~ uj6j., then the Kac-Dynkin labels of A are given by a~ = /~i-/~i+1 (i < m), am = /~,~+uz, a~+y = u y - u y + 1 (3" < n). Note that the coordinates in the e6-basls represent a unique weight of gl(m/n) whereas the Kac-Dynkin labels represent a unique weight of sl(rn/n) rather than gl(m/n). Often it will be useful to represent a weight A by a composite Young diagram, consisting of 1Research Associate of the NFWO (National Funds for Scientific Research of Belgium} 2Talk presented by J. Van dcr Jeugt
512
the diagrams of {g} and {~} in appropriate positions [5]. For example, for gtC4/6 ) and A = (7, 6, 6, 3[i, i, 3, 3, 5, 5) in the e~-bazis (where k stands for - k ) , the composite Young diagram is shown in (5). In this case, for example, the Kac-Dynkin labels of A are [1,0,3;2;0,2,0,2,0]. The basic problem we are concerned with is the determination of the weights and weight multiplicities of V(A). Such information is contained in the so-called character of V(A), which is by definition equal to chV(A) = E~(dimV~) e", where dimV, is the multiplicity of the weight ~ appearing in the weight space decomposition V(A) = ~ , V,. Recall that for a (reductive) Lie algebra Go (in the present case we can think of Go as the even part gl(m) (9 gl(n) of gl(m/n)) the character formula of a G0-module V0(A) with highest weight A is given by Weyl's character formula :
chVoCA) = Lo E
Lo = H
,.uEW
C2)
aEA+
where W is the Weyl group of Go, ~(w) is the signature of w E W and Po = ~ ~.~e,,+ o~. A very important finite-dimensional highest weight module V(A), the so-called Kacmodule, was introduced in [3]. For given integral dominant weight A, the G0-module V0(A) is uniquely determined (up to isomorphism), and can be extended to a Go (9 G+Imodule by putting G+xV0(£) = 0. Then one defines the induced module V(A) = IndaaoOa+, Vo(A) ~. U(G-z) ® Vo(A).
(3)
It follows from the structure of U(G_t) that
car(A) = x CA) = Lo
II (1 + ~ew
(4)
ae~+
When is V(A) a simple module? The answer to this question was given by Kac [3] : only if (A + Plfl) #- 0 for all fl in A +. Herein p = P0 - P,, where P0 has been defined previously and p, = ½E~e~tfl" In the case that all (A + pl/~) # 0, A and V(A) = V(A) are said to be typical, otherwise h and V(A) # V(A) are said to be atypical If A is atypical, V(A) contains a unique maximal submodule M(A) and V(A) -~ V(A)/M(A). Our main aim is to determine characters for such atypical modules. One of the useful tools in studying atypical modules is the so-called atypicality matrLx A(A) consisting of the rnn integers A(A)~i = (A + P]fl~i) [4J, where fl;i has been defined in (1). In terms of its components in the e6-basis, A(A)~i is given by #i + vi + rn - i - j + 1. The m x n atypicality matrix fits nicely into the compositie Young diagram, as is illustrated here for our example, A = (7,6,6,3]i,i,:],:~,5,5) for g/(416) :
V(M is simple if and
III
]
513
This A is atypical of type fix,6, and is singly atypicM. Various character formulae for atypical V(A) have been proposed, most of which are of the following form (see [4,5] and references therein):
= 4'
,C,,,)',, (, "+'° I I
flEA(A)
wEW
+,-")).
(6)
where A(A) is some subset of A +. In particular, Bernstein and Leites [1] proposed A(A) = { f l e A + ](A + p]fl) ~ 0), in which case Xa(A)(A) in (6) is replaced by XL(A). However, counterexamples were found to their formula. Similarly, counterexamples were found to other formulae of the type (6) [5], and in particular we were able to prove that for G = gl(3/4) and A = [1, 1;0; 0, 1,0] no set A(A) exists yielding the correct character of V(A). Hence no formula of the type (6) can give correctly the characters of all simple modules V(A) of glCm/n). There is, however, the important class of singly atypical modules where the problem of finding character formulae has been solved [4]. When there is only one ~ in A + with ( A + p l " / ) = 0 (and (A+plfl> # 0 for all fl ~ ~), A is singly atypical. In this case, we proved that the maximal submodule M(A) is itself a simple G-module, and that M(A) ~ V ( ¢ ) , where ¢ = w . (A - k~) = w ( A - k~ + p) - p and A - k~ is the first element of the sequence A - % A - 2 % ... that can be mapped into an integral dominant weight ~ by means of a w. action. In terms of the composite Young diagram, with the zero in the atypicality matrix at position (i,j), we move to the end of row i in the ~ p a r t of the diagram and to the end of column j in the v-part of the diagram, and perform a strip removal of length k in both parts of the diagram, removing one box at a time until the composite diagram is standard [5]. In our example (5) this leads to the following strip removal :
I I xx 9 8 5 4
N[ol
7 6 3 2[i]~ 6 5 [2l[11[2] 2 1~ 3 6 ?
IIxx
xxl
Note that only after 6 box removals, the remaining Young diagrams are standard. Thus ~I, = w . (A - 6fll,6) for some w E W, and it follows that @ = A - (fix,6 + fll,s + fl2,5 + fls,~ + fl3,4 + &,s) = (5, 5, 3, 31i , i, 2, 2, 2, 7i). Both strip removals (indicated by X's) are necessarily of the same shape, and the positions of the brackets [ ] in the atypicality matrix (which constitute the same shape again) determine the flij one has to subtract from A in order to obtain @. Also, ~ is atypical of type fiB,a, which corresponds to the "tail" of the removal strip. Making use of these properties, of combinatorial properties of the atypicality matrix, and of recursion, one is able to prove [4] that for the singly atypical case chV(A) = XL(A). Then, making a formal expansion, one can rewrite the
514
character as an infinite alternating series of Kac-characters XK(~.) : oo
chV(A) =
x
x (A-
(A) =
(8)
t=0
Let us now return to the more general case of mul~iply atypical modules. For reasons of presentation we shall illustrate here the case of doubly atypical modules. Thus CA Jr" P[fll) = 0, C-h- -~- P[fl2) = 0, and CA + plfl) ~ o for every f l ¢ / ~ l , & . Similarly as in (8) one can formally expand the Bernstein-Leites formula as an infinite alternating series of Kac-characters :
(9) tl,t2=O
CA
where CA = {,~ = A-tiff1 - t2/?~} is the "cone" with vertex A and ( - 1 ) IA-~l = ( - 1 ) t'+$2. Let ~1 = e l - 6 j andfl2 = e k - ~ w i t h i > k a n d j < I. Then there is a u n i q u e w l ~ in W which permutes the components i and k, m + j and m + l, and leaves all the other components of a weight in the e~-basis invariant. Let H12 = {~? e H*lw12 • (71) = r/}. Clearly, such a hyperplane splits the weight space H ' into two half-spaces. The truncated cone C + is defined to be the set of weights of CA that are in the same half-space as A. Then we conjecture : chV(h) = XL(A) = ~c~(--1)[A-~IxK(A) if A is not critical, and chV(A) = ~c+(-1)IA-~l XK(A) if A is critical, where A is c r i t i c a / i f and only if the entry A(A)kj in the atypicality matrix is equal to the "hook length" connecting the two zeros (at positions (i,j) and (k,l)) in the atypicality matrix, i.e. equal to i - k + l - j - 1 [5]. The ways in which this conjecture has been tested, and how it works for atypical modules with degree of atypicality > 2 is described in [5]. Let us emphasize that the given formulae are expansions of chV(A) in terms of the formal characters XK(A), which are characters of Kac-modules when A is dominant integral. One may also consider the inverse problem : given the Kac-module V(A), how can chV(A) be expressed as a Cnecessarily finite) sum of characters of simple modules chV(a)? In other words, what are the non-zero multiplicities n= in the expression chV(A) = ~= nachV(a)? This is known as the problem of the determination of the composition series of V(A). Recently, we have made a lot of progress in solving this question. Our results concerning the determination of the composition factors of the Kac-module V(A) were presented at this Colloquium by R.C. King, who reports on it elsewhere in this Volume. REFERENCES [1] I.N. Bernstein and D.A. Leites, C.R. Acad. Bulg. Sci. 33, 1049-51 (1980) [2] V.G. Kac, Adv. Ma~h. 26, 8-96 (1977) [3] V.G. Kac, Lecture Notes in Mathematics 676, 579-626 (1977) [4] J. Van der Jeugt, J.W.B. Hughes, R.C. King and J. Thierry-Mieg, "A character formula for singly atypical modules of the Lie superalgebra $l(m/n),", Commun. Algebra, in press (1990) [5] - - , "Character formulae for irreducible modules of the Lie superalgebra sl(m/n)," J. Ma~h. Phys., in press (1990)
515
Vector coherent state theory of the non-compact orthosymplectic superalgebras C. Q u e s n e *
Physique Nucl~aire Th~orique et Physique math~matique Universitd Libre de Bruxelles C a m p u s Plaine, C.P. 229 - B1050 Bruxelles - Belgium
Abstract The vector coherent state and K-matrix combined theory is applied to construct matrix realizations of the positive discrete series irreps of the orthosymplectic superalgebras osp(P/2N,R) (P = 2M or 2 M + l ) in osp(P/2S,R) D so(P) @ sp(2N,R) D so(P) ~ u(N) bases. As an example, the case of osp(4/2,R) is treated in detail.
1
Introduction Vector cohe.rent states (VCS), also called partially coherent states, were independently intro-
duced by Rowe [1], and by Deenen and quesne [2] as a natural extension of generalized coherent states [3,41. At the same time, it was noted that coherent states provide a very powerful method for constructing matrix realizations of Lie algebra ladder irreps in bases symmetry-adapted to some maximal rank subalgebra [5,6]. Such a construction is carried out by the so-called K-matrix technique [7,8]. Since then, the VCS and K-matrix combined theory has been applied to a lot of algebrasubalgebra chains (Refs. [7,8] and references quoted therein}. Recent extensions have allowed the method to be used for non-semisimple Lie algebras [9] and for Lie superalgebras [10]. In the present communication, we report on a new application to the positive discrete series irreps of the non-compact orthosymplectic superalgebras osp(P/2N,R), where P - 2M or 2 M + l . In *Directeur de recherches FNRS
516
Refs. [11] and [12], a general method is provided for determining the conditions for the existence of star irreps (and of grade star irreps in the osp(2/2N,R) case), the branching rule for their decomposition into a direct sum of so(P) ~ sp(2N,R) irreps, and the matrix elements of the odd generators in osp(P/2N,R) D so(P) @ sp(2N,R) D so(P) ~ u(N) bases. The cases explicitely worked out include the most general irreps of osp(1/2N,R), osp(2/2,R), osp(3/2,R), osp(4/2,R), osp(2/4,R), and the most degenerate irreps of osp(2/2N,R). We shall review here the osp(4/2,R) example.
2
The positive discrete series irreps of o s p ( 4 / 2 , R ) The osp(4/2,R) superalgebra is spanned by the so(4) generators A~2, A 12, C b, a, b = 1, 2,
the sp(2,R) generators Dr, D, E, and the odd generators G =, //=~ I~, Jo, a - I, 2. We choose to enumerate the weight generators in the order E, C1, C~. Then the lowering generators are A 12, C~, D, G =, and Ja, and the raising ones A~, C~, D t, H ° and I=. The adjoint operation in so(4) (B sp(2,R) can be extended to an adjoint operation in osp(4/2,R) in two ways differing in a sign choice : (Ga)t = ±I~, (Ja) t -- ± H a. On the contrary, it cannot be extended to a grade adjoint operation. Hence, osp(4/2,R) may have star, but no grade star irreps
[13]. The positive discrete series irreps of osp(4/2,R) can be induced from a lowest-weight so(4) sp(2,R) irrep [~1::~] (B (l~) or, equivalently, from a lowest-weight so(4) (~ u(1) irrep [~IE=] (~ {~'1}. They will be denoted by [~xE~n/. Here, ~1,~2, and fl are some integers subject to the conditions ~1 > IE~I~and n > 1. To construct a basis of the [~lE~n) carrier space, symmetry-adapted to the chain osp(4/2,R) D so(4) ~ sp(2,R) D so(4) E~ u(1), one may start from a basis (I[~l~2]{n}c~)} of the lowest-weight so(4) (B u(1) irrep. Since the raising generators H a and Ia (Dr) are the components of an so(4) (B u(1) irreducible tensor ~ (Dr), transforming under the irrep [10] ~ {1} ([00] E~ {2}), one can construct polynomials in ~ (Dr), transforming under an so(4) @ u(1) irrep [A1A2] (B {p} ([00] {v}). Here, v runs over all even integers~/~ over the set ~0,1 .... ,4}, and [~,1A2] over those so(4) irreps contained in the u(4) irrep {1"6}. By acting with these two sets of polynomials on the states I[~r~.=]{N}cx) and by performing so(4) couplings, one can form the set of states
=
[PI°°J~(Dt) × [Q[~,~,J<.~(~) × [[.~,~,]{a})]I~'~'l~=)]~ '~:]~ , (x)
characterized by a given so(4) E~ u(1) irrep [ ~ 5 ] E~(h}, and where # = co - fl, and ~, = h - ~.
517
In general, however, the states (1) corresponding to (z/} -- {0}, thence to {h} = {w}, do not belong to a definite sp(2,R) irrep. To obtain the lowest-weight state 1[~l)~2][~,~2](w){w}X) of an sp(2,R) irrep {w), one has then to combine 1[~lA2][~,~2]{w}{w}xI with some states (i) for which {h} ¢ {w}. Once this has been done, it still remains to calculate and diagonalize the overlap matrix
([,~.&~][~2](w){w}Xl[,~l,~=][~t~2](w){w}X)
= (KK't([¢~2]{w})){~:x&],[~,~,],
(2)
since neither the states (1), nor the states l[,~lA,][~l~2](w){w}x) form orthonormal sets. This painful calculation was actually performed by Schmitt et al. [14], who determined in this way the branching rule osp(4/2,R) J. so(4) (9 sp(2,R). The construction of the corresponding osp(4/2,R) matrix realization would necessitate the computation of some additional complicated scalar products and was not carried out in Ref. [14]. In the next section, we shall see how the VCS and K-matrix combined theory allows the same problems to be solved in a much more elegant and efficient way.
3
V C S a n d K - m a t r i x c o m b i n e d t h e o r y of osp(4/2,R) The VCS corresponding to the osp(4/2,R) irrep [~1~2t2) are defined by 1
Iz, a , r ; a ) = expCZt)l[-Ul~%]{n}a),
Z -- ~zD + a~G a + raJ~.
(3)
Here, there is a summation over repeated covariant and contravariant indices, z is a complex variable, and an, r a, a -- 1, 2 are Grassmann variables. The set of variables {z, an, ~a} parametrize the complex extension of the super coset space OSp(4/2,R)/[SO(4) ® U(1)]. The VCS (3) differ from standard generalized coherent states [3,4] by the replacememt of a single reference state by a set of such states, spanning the lowest-weight so(4) (9 u(1) irrep carrier space, which will henceforth be referred to as the intrinsic subspace. The VCS representation of an arbitrary state I~/, belonging to the irrep [-~1:~21~carrier space, is given by a function ¢J(z,a,r) taking vector values in the intrinsic subspace. Its components kD~(z, a, r) are holomorphic functions in the variable z, and polynomials in the Grassmann variables aa,~ a. The carrier space of the osp(4/2,R) VCS representation is defined as the graded Hilbert space of all such vector-valued functions which are square integrable with respect to the VCS scalar product (¢Y'lqJ)vvs. K-matrix theory replaces the difficult calculation of the integral form of ( ~ l ~ ) v o s by an implicit determination through the construction of an orthonormal basis with respect to this scalar product.
518
The VCS representation F(X) of an osp(4/2,R) generator X is a differential operator on • (z,o,r) depending in addition on the intrinsic representation ~r~2 , ~.12 , Cba , a , b = 1, 2, and IE of so(4) ~ u(1). Its explicit form can be easily found by using Baker-Campbell-Hausdorff formula. To apply the K-matrix technique to the osp(4/2,R) irrep [==IE~F~),we start by considering a vector Bargmann-Berezin (VBB) space. The latter is defined as the space of vector-valued functions
• (z,a,r)
which are square integrable with respect to a Bargmann-Berezin (BB) scalar product
(~'l~) [15, 16]. With respect to such a scalar product, the differential operators
20/az, O/Ooo,
and 0 / 0 r = are adjoint to the corresponding variables z,a=, and r =. Starting from the intrinsic subspace, the F representation of the raising generators generates an irreducible invariant subspace of the VBB space, which is by definition the VCS space. Although the domain of the operators r ( x ) is restricted to the latter, one can extend it in a natural way to the whole VBB space. As a result, one obtains the so-called extended r representation [10], which may be reducible, and even not fully reducible, although the VCS representation is irreducible. Since the variables z, ao, and r = transform under so(4) q~ u(1) in the same way as the generators
D*, Zo, and H =, the set of states { f [ ~ , ~ l [ 6 6 1 { ~ H h ) × ) } , obtained by substituting z and ~ = (aa, r =) for D t and ~ =
(I,,, H =) in
(1), form a VBB basis reducing the subalgebra so(4) @ u(1).
Contrary to the set (1), the VBB basis is orthonormal (with respect to the BB scalar product). Let us now introduce a transformation K mapping the VBB basis (1[~,~=][6¢,1{~}{h}×)} onto a VCS one { K l [ ~ , l [ 6 6 ] { ~ } { h ) × ) } , orthonormal with respect to the unknown VCS scalar product. Instead of using this VCS basis and the VCS representation F, which would have to be a star representation with respect to the VCS scalar product, it is much more convenient to keep on working with the VBB basis and transform the VCS representation F into an equivalent one % defined by ~,(X) =
K-~F(X)K,
and satisfying star conditions with respect to the known BB
scalar product, i.e. "7(Z t) =. ~I (X). We may restrict ourselves to the submatrices K (][~,~2]{w}) of the full K matrix, defined by
(K([66]{,.,}))[.~l,t~,~=~
= ([A" A'~I[6~=]{.,~{,~)xlKIIA,A=]I6~=I(~}{,~}X)
.
(4)
By imposing star conditions to "Y(X), it can be shown that the matrix KKt([~1~2]{w}) -K([~2]{co})Kt([~1~=]{w}) satisfiesa recursion relation, whose explicit form can be easily obtained from the r representation by using tensor calculus with respect to so(4) ~ u(1). In addition, it can be proved that KJ£t([~2](w}) is nothing else but the overlap matrix defined in (2). Hence, K-matrix theory provides a simple and systematic method for evaluating the scalar products (2) without having to construct the sp(2,R) lowest-weight states
519
I[A1)t.~][~z~=](w){w}X).
There are at most 15 different irreps IriS2] $ {w} in the VBB basis. The conditions for their existence, to be referred to as the VBB conditions, can be easily determined from the coupling rules of so(4) irreps. All submatrices K([~l~2]{w}) are one-dimensional, except for K([E1E2]{fl + 2}) which is two-dimensional whenever E1 ~ 1-21. There are at most 16 matrix elements (2) to be determined from KKt([~l-~2]{l~}) = 1, corresponding to the intrinsic subspaee. The recursion relation provides 40 equations to be satisfied by these 16 unknowns, hence allowing the calculations to be cross-checked. By definition, the matrices K K t ( [ ~ 2 ] { w } )
are positive semi-definite. The solutions of the
system of 40 equations have such a property if and only if : (i) the plus sign is chosen in the adjoint relations for the odd generators, i.e. (Ga) t = Ia, (J=)t = H a, and (ii) the irrep labels satisfy the condition n _> El. If 12 > S~, then all the matrices KKt([~l~2]{w}) are positive definite, and all the VBB basis states are m a p p e d onto VCS ones. On the contrary, if 12 -- El, then not all the matrices
KKt([~I~2]{w}) are positive definite, showing that the VCS space is a proper subspace of the VBB one. The linear combinations of VBB basis states, corresponding to vanishing eigenvalues of KKt([~z~2]{w}), have to be eliminated. The conditions for the existence of the remaining linear combinations are referred to as the VCS conditions. The branching rule osp(4/2,R) J. so(4) $ sp(2,R), obtained by combining the VBB and VCS conditions, is given in Ref. [12]. The so(4) ~ u(1) reduced matrix elements of the odd generators between two lowest-weight so(4) (B u(1) irrep basis states can be easily determined from those of ~ in the VBB basis, and from the matrix elements of K'([~1~2]{w}) corresponding to non-vanishing eigenvalues. Finally, by applying the Wigner-Eckart theorem with respect to sp(2,R) D u(1) [17], the so(4) • sp(2,R) (triple) reduced matrix elements of the odd generators can be calculated and are tabulated in Ref. [12].
4
Conclusion The VCS and K-matrix combined theory provides a simple and systematic procedure for
determining matrix realizations of osp(P/2N,R) by exploiting the full power of tensor calculus with respect to so(P) ~ u(N). Its only practical limitation lies in the necessity for an explicit knowledge of some so(P) and u(N) Racah coefficients. Note however that in addition to the cases treated in Refs. [11] and [12], many other examples might be worked out. Among them, let us mention the most general irreps of osp(3/4,1~) and osp(4/4,R), for which only u(2) Racah coefficients are needed.
520
References 1. 2. 3. 4. 5. 6. 7. 8. 9.
I0. 11. 12.
13. 14. 15. 16. 17.
D.J. Rowe : J. Math. Phys. 25 2662 (1984) J. Deenen, C. Quesne : J. Math. Phys. 25 2354 (1984) A.M. Perelomov : Commun. Math. Phys. 26 222 (1972) R. Gilmore : Ann. Phys. (N.Y.) 74 391 (1972) D.J. Rowe, G. Rosensteel, R. Cart : J. Phys. A 17 L399 (1984) J. Deenen, C. Quesne : J. Phys. A 17 L405 (1984) K.T. Hecht : The vector coherent state method and its application to problems of higher symmetries, Lecture Notes in Physics 290, Berlin : Springer 1987 D.J. Rowe, R. Le Blanc, K.T. Hecht : J. Math. Phys. 29 287 (1988) C. Quesne : J. Phys. A 23 847 (1990) R. Le Blanc, D.J. Rowe : J. Math. Phys. 30 1415 (1989) ; 31 14 (1990) C. Quesne : J. Phys. A 23 L43 (1990) C. Quesne : Vector coherent state theory of the non-compact orthosymplectic superalgebras: I. General theory, submitted for publication ; Vector coherent state theory of the non-compact orthosymplectic superalgebras: II. Some selected examples, submitted for publication M. Scheunert, W. Nahm, V. Rittenberg : J. Math. Phys. 18 146 (1977) H.A. Schmitt, P. Halse, B.R. Barrett, A.B. Balantekin : J. Math. Phys. 30 2714 (1989) V. Bargmann : Commun. Pure Appl. Math. 14 187 (1961) F.A. Berezin : The method of second quantization, New York : Academic 1966 H. U i : Ann. Phys. (N.Y.) 49 69 (1968)
521
T H E COMPOSITION FACTORS OF KAC MODULES OF s l ( M / N ) R.C. King (University of Southampton, U.K.), J.W.B. Hughes (Queen Mary and Westfield College, U.K.), J. Van der Jeugt * (University of Ghent, Belgium) Throughout this article we adopt the notation and conventions of the article [1] entitled Atypical modules of the Lie superalgebra g l ( m / n ) based on the Colloquium talk by Dr J. Van der Jeugt and published elsewhere in these Proceedings. The complex Lie superalgebra s l ( m / n ) is the subalgebra of g l ( m / n ) consisting of matrices x = ( ~ os ) , with A, B, C and D matrices of size rn x rn, m x n, n × m and n x n, respectively, with s t r x = t r A - t r D = 0. G = s l ( m / n ) admits a grading G = G-1 $ Go $ G1 with Go = sl(m) @ C $ sl(n), the even subalgebra of sl(rn/n). The universal enveloping algebras of G, Go and G_z are denoted by U(G), U(Go) and U(G_I), respectively. The weight space H ° is the dual of the Caftan subalgebra H of sl(m/n). It is spanned by the forms e, (i = 1 , . . . , m ) and6~ (j = 1 , . . . ,n), with ~ , ' _ z e, -~v'"i=l usx _- 0, and is equipped With an inner product such that (e, [ e j / = 6,~, (e, [6~) = 0, (~5,[6j) = -6,~, where 6t~ is the usuM Kronecker symbol. The set of simple roots is taken to be the distinguished set {e~ - ei+l,i = 1 , 2 , . . . , m - 1; e,, - 61; 5~ - 5~+l,j = 1,2 . . . . , n - 1}, so that the sets of positive even and odd roots are given by A0+ = {el -- ej, 1 < i < j m; 6, -- hi, 1 < i < j < n} and A+I = {~,~ = e, - 6~, 1 <_ i <. m, 1 <_j < n}, respectively. It is convenient to define p = p0 - pz with p0 = ½ ~ e a ~ or, and Pl ½ E z e A t ft. In the e6-basis a weight A E H" takes the form A = ~ i~l#iei + ~ "j = l vj 6j. The corresponding Kac-Dynkin labels are defined by ai =/~i - #,+1 for i = 1, 2~... , m - 1; a., = #m + Vl and am+j = vi - / J i + l for j = 1, 2 , . . . , n - 1. With these two conventions we write A = (#l~u2. • • #., Ivlv2... v.) = [ala2... am_ 1;am ; a m + l . . . a.~+._ 1]. The e6notation has a built-in redundancy thanks to the identity ~ ~=l " e~ - E j=l " 6~ = O. This m a y be exploited to ensure that #,, > 0 and Vl _< 0. In what follows it is convenient to denote all negative integers - k by k. A weight A E H* is said to be integral dominant if and only if a~ 6 N for i ~ m and a= E C. Corresponding to each integral dominant weight A there exists an irreducible finite-dimensional highest weight module V0(A) = U(Go)vA of sl(m)@C$sl(n). Extending this to a G 0 $ G + I module by setting G+IV0(A) = 0 and inducing to G then gives the Kac-module [2] "V'(A) of at(re~n). This is isomorphic to U(G_ l)® V0(A) and is generated through the action on V0(A) of the exterior algebra over e ( - f i ) with ~ E A~. Thus V(A) and V0(A) share the same highest weight vector vA, and dimV(A) = 2 m" dimVo(A). In general the Kac-module V(A) of s l ( m / n ) is indecomposable but reducible, with composition factors isomorphic to various irreducible modules of 81(re~n). Our aim is to present a prescription for determining all these composition factors. First we have two theorems: * Research Associate of the N F W O Belgium t Talk presented by R.C. King
522
T h e o r e m 1. (Kac [2]) Let M(A) be the unique maximal submodule of V(A), then V(h) = V ( A ) / M ( A ) is irreducible. T h e o r e m 2. (Gould [3]) Let ve IIaeA+c(--fl)VA, then X(A) U(G)vn is irreducible and X(A) = V(F) for some F. As pointed out elsewhere [4] a key construct in discussing the structure of V(A) is the atypicality matrix A(A). Its matrix elements are given by A(A)~ = (A + p[fl~ ) with flq = el - 6~ 6 AI+ for i = 1 , 2 , . . . , m and j = 1 , 2 , . . . ,n. The integral dominant weight A is said to bc typical if A(A) contains no zeros, and atypical of degree r if A(A) contains r zeros. With this definition we have two more theorems: T h e o r e m 3. (Kac [2]) If A is typical then V(A) = Y ( h ) is irreducible, and thus consists of a single composition factor. T h e o r e m 4. (Van der Jeugt et al. [5]) If A is singly atypical of type fl then V(A) is the semi-direct sum of V(A) = V ( A ) / M ( A ) and V(q') = X(A) = M(A). The algorithm for determining ¢ is as follows: Construct the sequence S~ = (fllfl2... flk ) of positive odd roots with fll = fl, such that (A + Plfli) = 0 with A - fli non-dominant; (A + p - flllfl2) = 0 with A - fll - f12 non-dominant; and so on, until the sequence terminates with (A + p - fll - f12 . . . . . ilk-11fl~) = 0 with A - fll - f12 . . . . . flk dominant. Then • = A - S(fl), where S(fl) = fll + f12 + " " + ilk. This procedure can be implemented diagramatically [1,4,5], see for example the singly atypical case A = (76631113355) = [103; 2; 02020] of 8/(4/6) illustrated in [1] for which ¢ = (55331112227~) = [020; 2; 01002]. S(fl) defines, and is defined by, the removal of a continuous boundary strip of boxes from each portion of the composite Young diagram F~";" with the row lengths of F " and the column lengths of F ~' determined by the parts of the partitions # = ( # 1 # 2 . . . . #m ) and ~ = ( - v . . . . - v2 - ~l). The problem is to find the generalisation of these results appropriate to the multiply atypical case. Consider the case for which A is doubly atypical of type fll = e, - 6~ and ~2 = ek - 6t, with k < i and j < l, so that (A + p[fl) = 0 for fl 6 A1+ if and only if fl = fll or f12. Let A(A)k~ = x = -A(A)~, and h = i - k + l - j - 1, the hook length between the two zeros of the atypicality matrix, then A is said to be normal if x > h + 2, quasi-critical if x = h + 1 and critical if x = h. On the basis of extensive investigations we conjecture the following: C o n j e c t u r e 5. (i) If A is doubly atypical and normal then V(A) contains four composition factors isomorphic to irreducible modules with highest weights A, A - S ( f l l ) , A S(fl2) and A S(fll) S(fl2). (ii) If A is doubly atypical and quasi-critiCal then V(A) contains five composition factors isomorphic to irreducible modules with highest weights A, A - S(fll), A - S(fl2), A - S(fll) - S(fl2) and A - S(fllLfl2), where S(fllLfl2) is defined by the removal of continuous boundary strips starting from the position specified by f12 a~ld continuing until they link with and include the strips associated with S(fll). (iii) If A is doubly atypical and critical then V(A) contains three composition factors isomorphic to irreducible modules with highest weights A, A - S(fll) and A - S(fllWfl2), where S(fllWfl2) is obtained by first removing the strips defined by S(fll) and then removing further strips starting this time from the positions specified by f12 which wrap around the first strips and continue until the resulting diagram is once more regular. These three possibilities are illustrated in the following examples: =
-
-
=
-
523
(i) The a/(2/3) case h = (521~71) = [3; 0; 11] is doubly atypical of type #1 = ~21 and f12 = ill4. The atypicality matrix and the starting points of the strips to be removed ave indicated in the following diagram:
a 2 ol I I I 121 o 2 ~1 Ill A is normal since x = 4 and h = 2. In this case S(fll) =/921 and S(fl2) = /3~.4. The four composition factors have highest weights: (521~71) = [3; 0; 11], (5 l li3Y~) = [4; 0; 21], (421999) = [2; 0; 10] and (411199) = [3; 0; 20] corresponding to the strip removals:
~~oOoOl II Ii I I I o~oo
I I I I I o
1 o ol Ill
o
21 I I I 121
o o oll
I
I I I 121
1 o ol Ill
(ii) The s/(2/2) case A = (32123) = [1; 0; 1] is doubly atypicaa of type # = #21 and # = #12:
%
In this case A is quasi-critical since x = 2 and h = 1. Now S(fll) = h i , S(fl2) = fl12 and S(fllL/~2) = ~11 + B12 + ~22. The five composition factors have highest weights (321~.~) = [1;0; 1], (31113) = [2;0; 2], (221~) = [0;0;0], (211i~) = [1; 0; 1] and (111ii) = [0; 0; 0] corresponding to the strip removals:
i oL__~
o 2L.2~
(iii) Finally, the si(314) case A = (4321~9) = [II;0; 010] is doubly atypical of type fll = fl31 and/~2 = fl14: IIII 4 3 1
II
121
013
A is critical since x = 4 and h = 4. This time S(fll) = fl31 and the usual method of determining S(~2) leads to a strip which reaches and wraps around that associated with ill. There ave now only three composition factors with highest weights (4321~33) =
524
[11; 0; 010], ( 4 3 1 l i ~ ) the strip removals:
= [12; 0; 110] and ( 4 3 2 l ~ 3 ) = [11; 0; 010] obtained by means of
IoIoIoI 0 [ III °o °o o° ~ t - - I
lit m
12121 12 o o 2 I 12l~.t
Iol I I ooIo[ I I I
2 22 ~ 120
° ° ° °~Zd-I lOO
To deal with cases for which the degree of atypicality is greater than two an algorithm has been developed, based on the notion of a strip removal scheme in which the question of linking and wrapping is determined from a criticality matrix. The whole process is codified, and an algorithm has been constructed and implemented on a computer. M a n y checks have been carried out covering cases of atypicality degree as large as five, for which the number of composition factors rises as high as 132. The converse problem of determing all those Kac-modules V(A) which contain a specific irreducible module V(~) as a composition factor turns out to have a simpler solution. The algorithm for its solution starts from the atypicality matrix A(~). The first step is to determine those fl's which belong to a set /ks (~) __ ~ + . This m a y be done in several ways [4] but for our purposes here a diagramatic way is preferable. It is illustrated in the following diagram in which the entries * specify the fl's belonging to A s ( ~ ) for si(3/4) with ~ = (43212233) = [11; 010].
[ I ~T~ I IIII 431ol
.I.10 0 II
I I
2 i i ~.~J_J
.Iooo
oo.. l
The entries • are those covered by boxes of F ~ and F ~' positioned so that the ith row of F~ and the j t h column of F ~' terminate just to the left and just below the position of the leftmost 0 in A(~..). If overlapping had occurred it would have been necessary to truncate the diagram and reposition a new portion around the position of the next zero in the atypicality matrix. In the case of no overlap~ as above, the top boundary of F " and the right hand boundary of F z' are extended until they meet. The algorithm is then as follows. Each connected set of zeros in the matrix of • 's and O's constructed as above is renumbered consecutively step by step in a shifting process whereby at every stage F " and F ~' either slide one step south and one step west~ respectively, or one step east and one step north, respectively. In these two cases the zeros covered in this way are all to be renumbered either 1 or 1~, appropriately. The process is then repeated until all zeros are covered. At each stage there is a choice of a south-west (SW) or north-east (NE) slide leading to new unprimed and primed entries. The process terminates after precisely r steps, leading to a total of 2 r distinctly labelled matrices CF(~C). The significance of this labelling lies in the following: C o n j e c t u r e 6. Let ~ be multiply atypical of degree r. Each of the 2 r matrices CF(~..) defines A such that V(~) is a composition factor of V(A). A is found by adding to ~ those fl's associated with the positions of the unprimed numbers.
525
The procedure is exemplified as follows in the doubly atypical M(3/4)case P, -~ (43212233) = [11; 0; 010]. NE NE
/
*
*
I'
0
*
i'
1
1
1'
1'
*
*
/
\ SW
• • 0
• 0 0
0 0 *
0 0 * NE
\ SW
• • 1
• 1 1
1 0 •
1~ 1 •
/ ",~ SW
*
*
1'
2'
*
1'
1'
1'
I'
I'
*
*
*
*
1'
2
*
1'
1'
1'
1'
1'
*
*
•
•
1
1
•
1
2'
1
1
1
•
•
•
•
1
1
• 1
1 1
2 •
1 •
It can be inferred from the final four diagrams that V(P.), with 1i2 = (43212233) = [11; 0; 010], is a composition factor of four Kac-modules V(A) having highest weights:
(4321~:~) (43212233) (4321~.~) (4321'2'233)
+ (00010000) -t- (10010001) q- (2221~i~) -b (2321T222)
= = = =
(4321~) (5321~23a.) (654]~afi.5) (664137t55)
= = = =
[11; 0; 010]; [21; 0; 011]; [11; 1; 101]; [02; 1; 110].
This procedure is very easy to program and all our checks to date indicate that the results are entirely consistent with the determination of composition factors of Kacmodules by means of the algorithm based on criticality and strip removals. Indeed it was the nice combinatoriM features of this algorithm in its identification of/~'s to be subtracted from A to give E that led to the discovery of the very simple converse procedure just described for obtaining all 2r possible A from a knowledge of the r-fold atypicM P.. References [1] J. Van der Jeugt, J.W.B. Hughes and R.C. King Atypical modules of tl~e Lie superMgebra g l ( m / n ) (to be published elsewhere in this volume). [2] V.G. Kac, Lecture Notes in Mathematics 676, 579-626 (1977) [3] M.D. Gould, J. Phys. A22, 1209-1221 (1989) [4] J. Van der Jeugt, J.W.B. Hughes, R.C. King and J. Thierry-Mieg, "Chaxacter formulae for irreducible modules of the Lie superalgebra s l ( r n / n ) " , J. Math. Phys., in press (1990) [5] J. Van der Jeugt~ J.W.B. Hughes, R.C. King and J. Thierry-Mieg, "A character formula for singly atypical modules of the Lie superalgebra 81(re~n)", Commun. Algebra, in press (1990)
526
CLIFFORD ALGEBRAS, SPINORS AND F I N I T E GEOMETRIES R o n a l d Shaw of Mathematics, University H u l l , HU6 7RX, E n g l a n d .
School
of Hull,
ABSTRACT The p l e a s a n t order
to
incidence
handle
Clifford
properties
nicely
algebras
of the
certain
Cl(0,d),
d =
finite
projective
geometry
PO(m,2)
oommutattvity/antl-oommutattvity
[PG(m,2)I=
2 m+l
- I
,
are
aspects
invoked
of
the
in
real
m = 2,3 . . . .
l~Introduotlon As in
[i],[2]
(m ~ 2),
w e deal
in w h i c h
with
an i r r e d u c i b l e
the o p e r a t o r s
(rp) 2 = -I
FI,
rprq
,
representation
F2 . . . . .
= -rqrp,
Fd
p # q
of C l ( 0 , d ) ,
d = 2 m+1
- I,
satisfy
,
(1.1)
and
T~T2 , , . Let
S
=
associated
r(a)
Fd
{1,2 .....
.
0f
c~)
addition,
observe
of order that
denotes
2.
the
that
is
aeP(S)(=
representation.
F({p})
= rp
r(s)
,
power For
.
set
example
It f o l l o w s
from
of
S)
if
we
can
define
(1.1),(1.2)
then
that
(I.3)
difference
a vector
an
= {2,3,7,8}
a
= +I
of
space,
the
subsets
a,~
of dimension
aA~ = ~
(= t h e
zero
, we s e e
that
(P(S),A,n)
= (o/3~)A(cdl~)
(projective) implies
space
V
of
Let
Cr
S
as t h e s e t
dimension A = I
m
of
over
, we may v i e w
p+q+r = 0 . Sr
denote
denote, that
Now
for
points
F2. S
d i m e n s i o n m+l o v e r Fz,
and o n l y i t
above. let
each
vector
d,
of
over
of P(S)). is
S.
the
Using
field
What
a F2-algebra
is
A
as
F= = { 0 , 1 ) more,
having
noting S
as
1
aRS = a).
A ~ 0
if
our
symmetric
P(S)
L e t us now i n t e r p r e t of
(1,2)
for
,r(~a~)
=
In particular
c~(/]A~)
(since
in
course
r(~)r(~) where
I.
+
Then
F(a)
element
= Fzr~r~ro
=
d}.
a finite
as c o n s i s t i n g
three
distinct
JV[ = 2=*~
r = 0,1 . . . . .
vector
of
subspace o f
projective
Because o f t h e p e c u l i a r of
, and so
m,
the
points
which is
of
S
,
in which
of
a vector
, as announced
the r-flats
spanned by t h e
PG(m,2)
F2,
being colltnear
= 2m. 1 - Z = d
the set of all
P(S)
of
nonzero Vectors p,q,r
iS[
geometry
nature
o f PG(m,2),
and
complements o f t h e
r-flats:
(1.4) Observe t h a t
subsets
of
Co
coincides
with
in
¢c
B(S)
of
P(S)
consisting
of
all
the
even
S: co
Since
t h e subspace
= -AS
[1],[2]
multtplicattve
, we have
we v i e w e d notation,
Co
=
P(S)
E(s)
slightly
with
Co
=
(~ep(s)
= C== ~ S •
:
.
differently, isomorphic
I~lezZ)
•
In particular
to
527
as the
the
(1.s)
d i m Co = d -
quotient
elementary
1.
P(S)/~S~
abellan
group
(Caution: ,
and
used
(Z2)d'~.)
2.
Some abellan results
Denote
by
F(S)
the
functions
S~F2).
consisting
of all
flf=
... f~
F2-vector For
the
r>0
space
fl
of
= Fr(S) r.
V ffi L(V,F2).
form,
For
yields
the
F~
forms
the
on
vector
S
(i.e.
all
subspace
of
(by restriction
feF(S) we define = I}
to S)
from an
~(f)eP(S) by
.
(2.1)
an isomorphism
(F(S), + , • , 0 , I) ~ (P(S), a , n , ~ , S) of
F2-algebras.
forms, and
0, I denote the forms taking the constant values ~
for
c o n v e n i e n c e , we d e f i n e
we h a v e t h e
,
r
ffi 0,1 . . . . .
m,
(2.3)
f2 = f, fag4h2 = f g h ,
and
etc.,
nesting
~ F m ~ Fm-~ ~
F(S)
Upon
Fo t o be ( 0 ) ) .
on account of the peculiar nature of F2, we have
consequently
0, I respectively.)
yields the isomorphlsms of F2-vector spaces
Fr ~ C m - r
Now,
(2.2)
(In the algebra F(S) the multiplication is polntwlse multlplication of
restriction the algebra isomorphism
(where,
the F(S)
Is spanned by forms of the kind
arising
= {pGS : f ( p )
~ : f ~ ~(f)
all denote
Thus
is a llnear
~(f) The m a p p i n g
Fr
forms of degree
where each
element of the dual space
LZK~A A
consisting
, let
."
(2.4)
~ Fo = (0) ,
and also the equalltles Fr = F(S)
, for
(2.~)
r > m .
From (2.3), (2.4) we obtain immedlately the next 1emma. C¢ ~ Cr+i
the i n c l u s i o n s
Alternatively
follow from theorem 2.3 of [1], and the fact that, for
r = 0,I . . . .
,m-l, the
£ncluslon is proper follows, for example, from (2.7) below. LBZZA B THBOR~
E(S) = Co ~ CI D ... D Cm = (~) . C
For
r = 1,2 . . . .
,m
(2.6)
there exists a unique linear isomorphism ^
~r
such that, for arbitrary ,..
~r(fI^ By i n v e r t i n g Potncar~
the
isomorphism
isomorphism
THBOREM D
For
fl . . . . .
Afr) ffi f l ( f l ) n
freV ...
m
and
mod Cm-r÷ I
using
we o b t a i n
there
(2.7)
, nfl(fr)
~m#*-r
^m+~-r ~ ~ ^r V
r = 1,2 .....
Cm-~/Cm-r+~
: ^rV ~
the also
exists
(2.8)
.
properties the
a unique
next
of
the
(unique,
over
F2)
theorem.
linear
surJection
@r : Cr-~ * ^rV such t h a t @¢(Join(vl . . . . . holds
whenever
vt,
...
, Vr
are
(Thus the usual PlUcker map,
v~)~) = el^ ...^v~
lndepe-dc.L
points
o f S,
from Sr-~ on to those rays of
^rV
decomposable r-vectors, "extends" to a linear map from the whole of of
^rV .)
Moreover
which
are
spanned
Cr-1
on to the whole
ker ~r = Cr , and we have the linear isomorphism Cr-i/Cr
Z ArV , r = 1 . . . . .
528
m .
(2.9)
by
Convenient bases for the subspaces Cr can be displayed Simplex of reference for PG(m,2). let
C, = {~c
: ~e~,}.
Let
~,
In terms of the faces of a chosen
denote the set of s-faces of the simplex and ^V, o r
By appealing to standard bases in the exterior algebra
otherwise, one obtains the next lemma. LEMM* E
C=., UC=-2 U . . .
Finally, recalling that
U~r is a basis for Cr • Consequently m+l. m+l m+l dim Cr = ( i ) * ( 2 ) + ''' + (m-r) '
^rV
(2.10)
is known to be irreducible under the natural action of GL(V)
GL(m+I; F=), the leomorphlsm (2.9) yields the following result. 2REOK~M F
Under the natural action of GL(V),
the subspace chain
(2.6) is a composition
series. REMARK
If
m = 3
figures of
Ci
then,
by ( 2 . 1 0 ) ,
dim C1 = 10 .
As d e s c r i b e d
fall into seven GL(4;F2)-orbits.
in
[1],
the
21o = 1024
At the tlme of wrltlng the paper
[1]
the author was not aware of the isomorphlems (2.3), and so did not see the tle-up with the classlflcatlon
of
quadrles
In
PG(3,2),
as
given
In
Tables
Similarly, in the case m=4, the classification in [2] of the into eight GL(S;F2)-orhlts tles in, via the isomorphism of quadrlce in PG(4,2).
For example,
each
153
and
15.9
of
[3].
figures of
Ct
F2 ~ Ca , with the classiflcatlon
figure,
degenerate quadrlo whose equatlon can be taken to be
15.4
2 ts = 32,768
see
[2],
in
C2
Is a non-
xlx2 + x3x4 = (xs) 2 , and one finds
that there are 13,888 such quadrlcs in PG(4,2), in agreement with equation (4.10) in [2].
~.
Some Clifford algebra consequences
Loosely speaking, we now deal wlth m-dimensional projective geometry In which the "points" rp
anticommute.
The chief llnk-up of the incidence properties of PG(m,2) wlth commuta-
tlvlty/anti-oommutatlvlty Part
of thls
lemma
properties of Cl(O,d)
follows
from
(1.I),
is by way of the next lemma.
(1.2)
upon
using
the
fact
that
The first
a
projective
subepace has an odd number of points. L_LEMM* G
If
~eSr
, ~S,
, with
r Z 0 , s Z 0 , then r(~)r(a),
if ~ meets
r(~)r(~) =
(3.1) if
[r(D)r(~), Also, for
r Z I , we have
a i s skew t o
r(~) 2 = +I .
For
r = 0,1 . . . . .
m
we shall be interested in the finite groups
For
r E 1 ,
is a proper subgroup of the finite group Go generated by the rp.
Gr = ~ ± r(~)
group
is
Gr
of order
2d,
and
is
of an even number of elements generating
Cl(O,d),
isomorphic
the
the
"even
drawn from a usual
Dlrac
orthonormal
(3.2)
group" set
coflststlng
(e, .....
of
This
products
ea} of vectors
Clearly Gr/{
(Incidentally,
to
: ~eSr > .
fact
that
the
± I}
commutator
~ Cr .
subgroup,
(3.3)
Frattlnl
are all equal to (± I) means that 0o is an extra-speclal
529
subgroup
2-group;
and
see for
centre
o f Go
example [4],
where
further
references
can be found.)
Consequently,
f r o m lemma B, we h a v e
the
subgroup
chain Go ~ G~ ~
LEnA
H
This
For
follows
r = 0,Z ..... from
lemma G,
can be strengthened a
fair]y
easy
appeared T~Eo~
only I
as
consequence
each
For r = 0,1 ..... If
In the
case
x K2,
where
of
Kz I s
{± I )
accordingly, and five
by
S-faces
label
the
then
m = 4,
and
2 zs
i.e.
a,~
denote
E,
every
lemma
r(a) 2 = (-I) q(aJ," where
il)
r(a)r(~)
of em-r within
(m-r)-flat. out
H.
(In
+ 2Z e Z / 2 Z
= F2
ill)
is
However,
in
section
[1]
Go.
our
VI
lemma H
of
present
[1],
is
theorem
F
set
of
± 1)
the
of
subsets
of
subgroup
o f Go.
abelian
normal
flfteen
independent
of
: ae~
U ~=}
reference
for
fifteen
S.
] + I)
e(a,~)
normal
choice
independent
= ~]al(]a
, where
abelJan
Go.
a maximal
(r(a)
simplex
o f Gm-r w i t h i n
associated
Go
the
ten
The 2 zs s e t s
commuting
states
of
generators
with
PG(4,2).
mutually
spinor
subgroup
of
of
involutions
our
irreducible
Then * 2Z,
= (-l)v~,~
~, w i t h
b(a,p)
= rafSpl + [ a ~
•
an alternating
LBmXA L
Let
bo d e n o t e
Then
is
a non-degenerate
subspaee to C ~ - r
centralizer
a maximal
linearly
q(a)
= e(a,~)r(~)r(a)
ful~
Cl(0,31),
the
arbitrary
I)
the
blllnear restriction
scalar
form on P(S), of
b
to
product
Co x
Co
o n Co a n d ,
(So
within
bo(a,~)
Co,
Cr
=
is
[a~[
the
+ 2Z.)
orthogonal
: C~ = (Cm-~) ± ,
equality
(3.4)
.
as pointed
A possible
chosen
= 32,768
L~XMA K
The
I}
centralizer
meets
which,
G~ i s
(± 1 . . . . .
of C1(0,31).
b(.,.)
F
K2 ~ C2. lemma
representation Let
the
r-flat
Gr i S t h e
even,
of the
etgenvalues
will
inside
theorem
theorem
m,
m = 2~ i s
G2 ~
simultaneous
next
of
I~LUST~ATZOS
bo
since
in the
is
r(a)
Gr l i e s
~ Gm = ( ±
as a conjecture.)
CO~OLLX~Y J
2-faces
m,
...
(3.5)
follows
by
dimensions
r = 0,I .....
m .
(lemma
after
E),
(3.5) noting
that
we
have
the
inclusion Cr ~ ( C = - r ) 1 (because each r-flat meets every (m-r)-flat). REMARK
Since
bo(a,~)
= 0 if and only if £(a) commutes with r(p),
observe that
provides us with a second proof of the full centralizer property of theorem I.
References I, 2. 3. 4.
R. S h a w : J. Math. Phys. 30, 1971 (1989). R. S h a w , T. M. J a r v l a : J. Math. Phys, 31, 1315 (1990). J.W.P. Htrschfeld: Finite Projective Spaces of Three Oxford, 1985. H. W, B r a d e n : J. Math. Phys. 26, 613 (1985).
530
Dimensions.
Clarendon,
(3.5)
Reducibility
of Euclidean
Motion
Groups
Vojt~ch Kopsk~ Institute of Physics, Czechoslovak Academy of Sciences, Na Slovance 2, POB 24, 180 40 Praha 8, Czechoslovakia Introduction There exist several views of the concept of reducibility and of decomposability in algebra. Reducibility of matrix groups and of groups of linear operators is now about a century old and forms a background for reducibility theory of representations which found its applications in determination of selection rules and classification of spectra. The reducibility in physics is usually the reducibility of linear spaces under the action of some symmetry groups and over the field R of real numbers; the so-called physically irreducible representations are exactly representations irreducible over R. The results of more general reducibility theory which includes reducibility of Z-modules are exposed in the book of Curtis and tteiner [1]; let us only remind that reducibility over fields is equivalent to decomposability which is not true over the rings. The translation subgroups of space groups are exactly ZG-modules and arithmetic classes (G, T) may be either reducible or irreducible; a reducible class may still be either indecomposable at all or its reducibility and decomposability patterns may not coincide. Reducibility of arithmetic classes implies a certain reducibility of space groups. Though reducibility patterns over various fields are given already in the book on four-dimensional space groups [2], this implication has been considered much later [3,4,5,6]. We want to show now that the concept of reducibility and the results of its theory for space groups, especially the factorization and intersection theorem can be applied as well to more general cases of Euclidean and even affine groups.
Extension of the concept o f the reducibility o f space groups We denote by E(rt) an Euclidean space and by V(n) its difference space, the orthogonal vector space,, where 7t is the dimension. Then every Euclidean motion can be expressed by Seitz symbol {glt}p with respect to a certain origin P and every Euclidean motion group Q by a symbol {G,T,P, uG(g)}, where G is a subgroup of the orthogonal group O(n) acting on V(n), T is a G-invariant translation subgroup of V(n) and uo(g) the system of nonprimitivc translations. By the latter we mean a function ua : G --~ V(n) which satisfies Frobenius congruences:
wG(g, h) = uG(g) + gum(h) - u~(gh) = 0 rood T. We can extend the concept of arithmetic class (G, T) to any kind of Euclidean groups and it is also suitable to use the word space group for any group, the translation subgroup T of which spans the whole V(~) over R, while other groups are called subperiodic. It is easy to realize that the same scheme is valid also on the level of affine groups, where g in the Seitz symbol is from the general linear group ~L(n) acting on the linear space L(Tt) which turns into orthogonal space V(~t) with introduction of an orthogonal scalar product. Every aifine group 9, the point group G of which is orthogonalizable by a 531
suitable choice of scalar product is affincly equivalent to some Euclidean group and hence all further considerations apply to it as well. W e have defined reducibility of space groups [3,4,5]as a consequence of Q-reducibility of the action of G on T. It would be more appropriate to distinguish this reducibility in a wider context as a crystallographic(or arithmetic) reducibility. To extend the concept of reducibility to arbitrary Euclidean groups, we have to realize, that the translation subgroup T itself as well as its reducibility under the action of G m a y have various features. In particular, T m a y b c a direct sum of G-invariant modules or even spaces, each of which spans V(n). Figuratively expressed, the algebraic structure of T does not reject the geometric meaning of T as a subset of V(n). We should therefore distinguish between algebraic reducibility which is simply the reducibility of T as ZG-module and geometric reducibility, which is a consequence of the reducibility of the action of G on the space V(n). The geometric reducibility implies at least partially the algebraic one and both reducibilities imply a certain reducibility of Euclidean groups of the class (G, T). W e shall say that the Euclidean group G is geometrically reducible,if the action of its point group G on V(n) is reducible. Subperiodic groups with nontrivial T are naturally always reducible. The reducibility of crystallographic space groups is defined as a consequence of geometric reducibility with an additional requirement of arithmetic reducibility. The latter is automatically fullfilled for orthogonal reductions, while inclined ones may create some problems, which have been considered in more detail in the study of reducible space groups in arbitrary dimensions [5]. To avoid complications, we assume further that we are dealing with orthogonal reducibility.
Consequences of geometric reducibility Geometric reducibility of Euclidean groups is a reducibility of the action of a group on a point space and the main construction connected with it is the subdirectproduct or subdirect sum. This construction is of frequent use but it occurs rarely in textbooks. We can trace it to Goursat [7] and recognize it in m a n y recent constructions. In the book by Huppert [8] it appears under the name das direkte Produkt yon Gruppen mit vereinigter Faktorgruppe. The subdirect product is also used in consideration of transitivity [9], which is a kind of reducibility of the group action on sets. The importance of this construction is realized by Opechowski [10] who uses it throughout his book. We gave an overview of its use and an analysis for cases of more then two components, when we prefer the term multiple subdirect product (sum) [11]. The work with subdirect products is based on a theorem which says that a subgroup of a direct product of groups Oi is either a direct or subdirect product of subgroups G~ of groups Oi. These groups are obtained by homomorphisms ai which map G onto its components in Oi and the greatest direct product of subgroups of O~ contained in G is the product of intersections Gi = G R Oi. T h e m a i n t h e o r e m on r e d u c i b l e E u c l i d e a n g r o u p s . The geometric reducibility of (G, T) and hence of all Euclidean groups ~ of this class means that V(n) splits into a direct sum of G-invariant subspaces V~(k¢) of which T is a subgroup. Further, G is a subgroup of a direct product of orthogonal groups Ok which act on spaces V~(k~) and G is a subgroup of a direct product of corresponding Euclidean groups Ei acting on Euclidean spaces E(/¢i). It follows immediately, that T is a subdirect sum of certain groups T~ C_ V~(/¢~) , G is a subdirect product of groups G~ C_ 0¢ and g is a subdirect product of groups ~ ' _C E¢.
532
Compare this result with the splitting of reducible representations into a direct s u m of irreducible components and let us observe that to express it in terms of operator or matrix groups we have to use again the subdirect products [11]. F a c t o r i z a t i o n t h e o r e m a n d p r o j e c t i o n h o m o m o r p h i s m s . If (G,T) is geometrically reducible, then there appear neccessarily G-invariant subgroups T~ = T N V~(k~). Each such subgroup is normal in every group g of the class (G, T). The factorization theorem a~serts that the factor group G/T~ is isomorphic to a certain subperiodic group. The group G is mapped onto so-called contractedsubperiodic groups by projection homomorphisms which are unique for orthogonal reductions. These projections, described in detail in [5] can be applied to any geometrically reducible Euclidean group. I n t e r s e c t i o n t h e o r e m . This theorem is valid in each case when T splits into a direct sum of G-invariant subgroups 2~ = T N V~(k~) and its meaning is very transparent. The system of nonprimitive translations u ~ can be uniquely expressed as a sum of its components ua~ in individual subspaces V~(k~). Since these subspaces are G-invariant, each of the components uai satisfies l~obenius congruences h) = u a i ( g ) + g u a , ( h ) - u
(gh) = 0 rood
We can introduce now either subperiodic groups L~ = {G, T~, P, ugh} or, since congruences rood T~ imply the same congruences rood T, we can as well introduce groups ~ = {G, T, P, ua~} and classify groups of arithmetic class (G, T) into subperiodic classes £i of which the groups ~ are the symmorphic representatives [4,5]. The essence of the intersection theorem lies in the statement that each space group of the class (G, T) lies on the intersection of subperiodic classes £i. An example. The following table shows how intersection and factorization theorem work together in practice. The upper row of this table lists rod groups, the first column 4romp [ p4mm p4mm I P4mm p4bm I P4bm
p42cm P42cm P42nm
p4cc P4cc P4nc
p42mc P42mc P42bc
lists the layer groups and on intersections of rows and columns stand the eight space groups of the arithmetic class 4mmP. In these symbols P denotes the translation subgroup r ( a , b, e), p stands for T(a, b) and p for T(¢). Each layer group of the first column is a common factor group by T(c) for all the space groups of the row and each rod group of the first row is a common factor group by T(a, b) for all space groups of the column. Views
for the future
development
We can see already now a few of valuable ramifications and consequences of reducibility theory of Euclidean groups. Its first natural use concerns the crystallography in spaces of arbitrary dimensions. It is easy to see that irreducible space groups and the laws of their composition into reducible ones are of primary interest. We believe, however, that algebraic reducibility is more adequate from the viepoint of applications to incommensurate structures and/or quasicrystals while geometric reducibility is of rather academic interest. The mentioned structures are after all structures of three-dimensional space. 533
Factorization and intersection theorems are very valuable in three-dimensional crystallography. Both two types of subperiodic groups in three dimensions, the layer and the rod groups clearly appear as subgroups of space groups. Now we know that they appear also as factor groups of reducible space groups. In this role it is suitable to consider them as contracted layer groups acting on E(a, b) x V(c), and contracted rod groups acting on V(a, b) x E(c). The relationshipbetween these contracted subperiodic groups and subperiodie groups acting on the ordinary Euclidean space E(a, b, c) is in all respects analogous to the rdationship between point groups G acting on V(n) and site-point groups acting on
Factorization theorem enables us to classify space groups into layer and rod classes [6] and introduce their standards not in an ad hoc manner but in correlation with standards of space groups. This is a firststep in a solution of an important problem of bicrystallogr a p h y scanning of layer and rod groups through the space for defined space symmetries and plane or line directions, definition and classification of Wyckoff types of orbits for planes and lines in a crystal. Such problems can be easily solved for significant directions of reduciMe space groups and, as we have shown [12],this solution can be extended to arbitrary groups and directions with use of so-called scanning theorem and scannin# -
#roup. LAST BUT NOT LEAST. Since layer and rod groups appear as factor groups of space groups, their representations are connected with certain representations of space groups via the well known process of engender-in#. Every expert in representation theory will probably realize at once the importance of this fact for systemization of our knowledge of representations of space and subperiodic groups. But this is quite a new story.
References
[1] C.W. Curtis, I. Rdner R e p r e s e n t a t i o n T h e o r y of Finite G r o u p s a n d Associative Algebras, Wiley Interscience New York (1966). [2] H. Brown, It. Billow, J. Neubiiser, H. Wondratschek, H. Zassenhaus C r y s t a l l o g r a p h i c G r o u p s of F o u r D i m e n s i o n a l Space, Wiley Interscience New York (1978). [3] V. KopskSr, J. Phys. A. (Math. Gen.) 19 L181 (1986). [4] V. Kopsk~, Lecture Notes in Physics 313 352 (1988). [5] V. Kopsk~, Acta Cryst. A 45 805 (1989). [6] V. Kopsk~, Acta Cryst. A 45 815 (1989). [7] F~. Goursat, Ann. Sci. ]~cole Norm. Super. Paris (3) 6 9 (1889). [8] B. Huppert Endliche G r u p p e n I., Springer Verlag Berlin (1967). [9] M. Hall, T h e T h e o r y of G r o u p s , Macmillan New York (1954). [10] W. Opechowski, C r y s t a l l o g r a p h i c a n d M e t a c r y s t a l l o g r a p h i c G r o u p s , NorthHolland (1986). [11] V. Kopsk#, Czech. J. Phys. B 88 945 (1988). [12] V. Kopsk~,, Ferroelectrics (1990) in print.
534
S OFTWARE PACKAGES: Space G r o u p s and t h e i r R e p r e s e n t a t i o n s B.L. Davies School of Mathematics, University College of North Wales Bangor, Gwynedd, UK, LL57 1UT. R.. Dirl Institut fiir Theoretische Physik, Technische Universit~.t Wien A-1040 Wien, Wiedner HauptstraBe 8 - 10, Austria
A b s t r a c t : The aim of this contribution is to present universal software packages for space groups. The first set of packages, called IRREP, allows one not only to analyse and to modify consistently the algebraic structure of every space group but also to compute their irreducible characters and irreps, and finally, as special cases, so-called limit.representations. All computed entries, such as characters or matrix elements, are, as in all our packages, in principle coded in analytic and not in numerical form. To run safely all software packages consistent DATA-files have to be created. The creation of the corresponding DATA-files is achieved by tailoring extra menu driven DATA-packages including not only several validation routines but also comprehensive HELP-files to prevent the inexperienced user from generating erroneous DATA-files. The DATA-packages are written in C by using the interface management system CSCAPE being a trademark of Oakland Groupj Inc. All softwate packages are written in P A S C A L and are designed for IBM-compatible PCs under MS-DOS 3.1 upwards, workstations under VMS or UNIX, and for main frames, like VAX, CYBER etc. The computed entries ate already prepared to be used as DATA for other routines. A. Settings of Space Groups: Let g be a space group with 2" as its translation group, "P -" g / T its point group, and fix its setting by choosing an origin and the orientation of its axis [1]. The latter point, even though very often concealed, is crucial when defining space groups. Starting from standard settings g,tan = ~ { T , ' P ; (El0)} of space groups which are tabulated in Ref.[1], non-standard settings are obtained from ~stan by conjugation with arbitrary elements ( W l w ) of the Euclidean Group g(3) respectively. a.o.-.,°.
= a{~', 7.; ( w I w ) } = ( w l w ) • a . , . . • ( w I w ) -1
(a)
To be more strict, a given space group in different settings, say ~ = G { T , 7~; (El0)} and g ' = g { T , "P; (WIw)} where (W[w) must not belong to the Euclidean normalizer o f g , , ~ , , implies that they are isomorphic (~ -~ G t) but not identical space groups. For a given standard setting of ~ its composition law has the following form
(R,In(R) + t) • (SlnCS) + v) = (RSI.(RS) + t(R, S) + t + ~v)
(2)
where t(R, S) = n(R) + Rn(S) - n ( R S ) E T are special primitive lattice translations which only occur if ~ is non.symmo~hic. Let (R']n'(R') + t') E g.on-.~a, be the images of (RIn(R) + t) E g,,an under the conjugation with (W[w) E ~(3), then (RJn(R) + t) ~ ( R ' [ n ' ( R ' ) + t') means in detail R ~ = W R W -x and n'(/~ I) = W n ( R ) + w - W R W - l w and t' = W t respectively, The strategy is to start from space groups in standard settings and to allow the user to carry out any re-setting by specifying the group elements (WIw) E £(3) respectively. The special cases (Elw) E £(3) entail shift of the origin, whereas the special choices (W[0) E £(3) lead to purely re-oriented space groups respectively. This freedom in defining space groups is of particular importance when analysing group-subgroup relations.
535
B. Induced Space Group Irreps: It is well-known from standard literature [2] that every Q-irrep is equivalent to an induced representation ID k,~;x = IB k,~;x T ~ where ]13k,~;x denotes an allowed irrep of the corresponding little group ~(k) C G respectively. The groups ~7(k) are determined by 1-dimensional ~-irreps
(3)
Dk;'~(t) = exp{-I-i)~k, t}
where for convenience we incorporate a so-called A-switch taking the values ~ = 4-1 to be able to start from either sign in the inducLion procedure. This is the reason why space group irreps are provided with an additional label, namely )~ = =1:1respectively. As a matter of preference one can either determine allowed irreps of the little groups g(k) or equivalenty projective irreps of their corresponding little co-groups T'(k) - g ( k ) / T . Preferring the second approach allowed G(k)-irreps are expressed in terms of mk,e;x(RIn(R ) + t) = e +~xk't ~:~k,(~;£(~) ,
(4)
where ( R ] n ( R ) + t) E ~(k) in order to correlate g(k)-irreps /13k,~;~ with 7~(k)-irreps IR.k'~;~ respectively. It holds that ~ ( R ) I R ~ ( S ) = Rk(R, S)~:~(RS) with Rk(R, S) = exp{+ik, t(R, S)} where R , S e ~ ( k ) and R k : 7~(k) x 7)(k) z , C are uniquely determined factor systems. Using in part a matrix notation for allowed g(k)-irreps, induced space group irreps are given by ID~'~'(.RIn(R) + t) = A k ( ~ , .RS) ~+,~,~k.t ,,r,k;~, ,_ a,sj / mJ
mk'e; (R- RS)
(5)
The entries R, S are coset representatives which decompose 7~ with respect to the corresponding little co-group ~ ( k ) "~ G(k)/7". The entries Ak(R, S) = 5({R) *'P(k), {S} *'P(k)) are generMized Kronecker delta's which determine uniquely the generalized permutational structure of space group irreps by fixing a lexicographical order of the cosets {R},'P(k) respectively. The exponential factors • ~x_s(/~) = exp{+i~Rk. [t(/~, S) - t(R, R -1RS)]) occur for non-symmorphic space groups but are umty for symmorph~c ones. The symbols (k,~;)~) define G-lrrep labels where k E RBZ(~T,7 ~) with ~ E A(k), and A = +1 are adopted by convention. Note the A-sign is, from the representation theoretical point of view, immaterial but has to be fixed from the outset. We take advantage of this freedom and implement in all our software packages this A-switch which allows one to deal with either sign. The rules we adopt are {]Dk,e;+(/~ln(R ) + t)}* = ]]3 k,e;- ( R I n ( R ) + t) which provides us with a simple but extremely efficient tool for correlating space group irreps with their complex conjugates in the simplest form. This option is used in another software package called SUB to compute co-irreps for Shubnikov space groups of any type. N O T E : Space group irreps are independent of lattice constants although space groups depend on lattice constants. This allows one to deal rather with space group types, i.e. infinite families, than with single space groups. Thus lattice constants are hidden parameters which never enter into the discussion as long as one deals with a single space group. When dealing with group-subgroup relations only the ratios of lattice constants enter but never their absolute values. Moreover space group irreps neither depend on any shift of the origin nor on any re-orientation if the correlated representation domains are identified correspondingly. C. Standard M i l l e r - L o v e S p a c e G r o u p I r r e p s : Induced space group irreps are called Standard Miller-Love Space Group Irreps and are denoted by ]]3 k,¢ = I D k'l~;'(" if they are defined by Eq.(5) and if in addition the following conditions are satisfied: * Each space group ff must be in standard setting as given in l~eL[3]. Note that almost all settings coincidence with one of the standard settings tabulated in Ref.[1]. • The representation domains R B Z ( T , 7~) have to be identical with those tabulated in Ref.[3] and are called standard ones. Note there exist infinitely many equivalent shapes of representation domains; which one is taken is a matter of preference, but some are more appropriate than others
536
(see Ref.[4]). However, to avoid ambiguities, representation domains have to be fixed to achieve uniqueness as regards orthogonality and completeness relations of space group irreps. • The little co-group irreps :IRk,¢;+ must be identical with Miller-Love irreps as tabulated in P~f.[3]. Initially these irreps were computed by successively applying the induction procedure as a step-by-step procedure to group-chains of the form {E} ,a "P(kt) ,~ "P(k~) . . . . . . ,a "P(km-t) ,~ "P(k) where the index of T'(k$) with respect to its successor "P(kj+l) is two or three exclusively. We call this type of group-subgroup chain a composition series of "P(k). Note for a given "P(k) it is not unique in general. However when taking one composition series it fixes the generators (augmenters) of T'(k) and accordingly of G(k) respectively. The coset representatives decomposing "P into disjoint cosets {~} • "P(k), being necessary to fix space group irreps (5), must be optimized ones. They have to satisfy additional conditions: they have to form a group, if possible, and they have to contain the inversion element, if possible. R E M A R K S : All software packages create by default single and double valued standard MillerLove space group irreps. In fact when talking about space groups we always mean double space groups but keep the notation concise. D. Non-Standard S p a c e G r o u p I r r e p s : Non-standard space group irreps are obtainable in various ways, either by altering the algebraic structure of space groups, or by modifying the irreps themselves. The first aspect is related to the use of new settings of space groups and the second to alteration of the conventions defining standard space groups or, as the most general modification, the combination of both aspects. All software packages, not only Ilq.REP, allow one to carry out these modifications simultaneously as long as they are compatible. Accordingly, apart from having the option to modify consistently the algebraic structure of space groups, one can employ new representation domains T d B Z ( T , 7~)"'~°, one can modify 7~(k)irreps ] R k,e;x either by using new composition series {E} ~'P(kl,),~'P(k2,) . . . . . . ,~'P(km,-1),a'P(k) or by carrying out arbitrary similarity transformations U ~;x n%~;x (R) U ~;'xt or by choosing new sets of coset representatives {_RZ}, or finally combining simultaneously all types of modifications which however, in some instances, may be interrelated. To summarize let us list explicitly for the items (1) new origin, (2) reorientalion, (3) new composition series or similarity transformations, (4) new conventions, and (5) new coset representatives, induced space group irreps by tracing them back to standard ones.
m ,
2(RIn(R)
+ w -
mwk, W R W,- ~, W ~ W - t ( w R w ) + t) =
~_ ]DQk+g,,~;,k
,'~,
," ~ ,
+ t) =
-llwn(n) + wt) -
n_S)
+ e) = En,,s, .
+ ,) _,_
+ t)
~a,s__(R) u ~ ;~n~.~; ~ (_R- 1 RS.)u~ ;~ t "l'~k,~t;,k .
.
k ~"A .
+ t)
ll~k,~';,k ~
$
s_,,QS_Q-
:lDk,¢;~ ~k,¢;~, :ID~,~,C/~InCR ) + t) ~_s,,s__Y ]"l~k'~;'k '[ ax,_szCRInCR) + t) = E_R,,s_, "-"R_x,a' _ _
R.EMAI:LKS: To recapitulate, when changing the setting of space groups non-trivially, one deals with isomorphic but different groups. This is, in contrast to the other options, an essentially different concept as the latter options only refer to a single space group. The most general modifications are achieved when combining both concepts, for instance shifting the origin and taking non-standard representation domains. The application of all options is available simultaneously in all our software packages. E. Package m IRI:tEP: The software package IRREP mainly serves to compute for arbitrary space groups in any setting, standard or non-standard space group irreps and other entries that are correlated to the latter. As a matter of fact, we use in all our software packages
537
fixed data sets as reference data, namely (i) representation domains 2~BZ(T, 7~), (ii) Miller-Love irreps of "P(k), and (iii) "P(k)-irrep label sets A(k) precisely as they are tabulated in Ref.[3]. The package I t L R E P can be used to (1) compute characters of space group irreps, (2) compute space group irreps, (3) compute the complex conjugate of space group irrcps, (4) verify the composition and associative law of every space group G, (5) determine limiUrepresentations [5] for non-generic k-vectors (k , , b where T'(k) C 7~(b)), and finally (6) impose periodic boundary conditions (BOUNDARY INTEGERS = EVEN) and construct corresponding irreps for finite space groups [6]. "lT3k'~;A("~ --
--
k - - * b
- -
Ak(R, RS) e "t'iARb't b;~
]l~k.-.*b,~;A( R - 1 ~ )
--
+ t) =
RS) e
R E M A R K S : To recapitulate, package IRREP allows one to (1) compute single and doublevalued representations of any G, (2) access all settings for any G, like shift of origin and/or reorientation, (3) carry out modifications, like complex conjugation, new representation domains (extended zone scheme), new composition series, or arbitrary similarity transformations to transform T'(k)-irreps, new coset representatives, and finally (4) can pass over to limit-representations depending on the applications one has in mind. For instance option (3) can be used to compare space group irreps being computed independently inside and outside of R B Z ( T , 7~) or of B Z ( T ) . In particular, it allows one to study the periodicity of ~-irreps in reciprocal space when replacing k by k q- g and to compute the corresponding similarity matrices IF k,~ ;~ respectively. In addition a subroutine checks the validity of Schur's Lemma and computes simultaneously Schur matrices.
F. User Friendly Masks: To run safely M1 software packages consistent DATA-files are created. The creation of the corresponding DATA-files is achieved by tailoring extra menu driven DATA-packages (UFM) including not only several validation routines but also comprehensive HELP-files to prevent the inexperienced user from generating erroneous DATA-files. The DATA-packages are written in C by using the interface management system CSCAPE being a trademark of Oakland Group, Inc. The aim of these DATA-packages (UFMs) is to (1) create consistent DATA files for the EXE files comprising the various packages, (2) provide comprehensive HELP files, and (3) help to select options, such as new settings of space groups, or similarity transformations, or coset representatives, etc. A c k n o w l e d g e m e n t s : One of us (RD) is very grateful to Professor Michel for valuable discussions concerning the periodicity behaviour of space group representations. 1. T. Hahn
International Tablesfor Crystallography (Reidel, Dordrecht, Holland, 1983) 2. C.J. Bradley, A.P. Cracknell
The Mathematical Theory o] Symmetry in Solids (Clarendon, Oxford, 1972) 3. A.P. Cracknell, B.L. Davies, S.C. Miller, W.F. Love
Kroneckcr Product Tables, Volume 1 (Plenum, New York, 1979) 4. B.L. Davies, R. Dirl Proceedings of the 15th ICGTMP World Scientific (1987)728 5. B.L. Davies, R. Did Proceedings of the 17th ICGTMP World Scientific (1989) 393 6. K. Did, B. Lipp, B.L. Davies
Finite space groups TUW - UCNW preprint (January 1989)
538
SOFTWARE
PACKAGES:
Transformation Coefficients for Space Groups 1~. Dirl InstituL fiir Theoretische Physik, Technische UniversitKt Wien A-1040 Wien, Wiedner Hauptstral3e 8 - 10, Austria B.L. Davies School of Mathematics, University College of North Wales Bangor, Gwynedd, UK, LL57 1UT.
Abstract: The aim of the second contribution is to present further universal softwaxe pwekages for space groups. The second set of packages is called COEFF consisting of four packages: SUB, CG, SYMCG, and SYMPW. All packages Mlow one not only to treat every'space group in arbitrary settings (shift of origin a~ud/or re-orientation) but also to use space group irreps in arbitrarily modified forms (new representation domains a~nd/or new little co-group irreps and/or new coset representatives). Each COEF-package yields the matrix elements of the corresponding unitary similarity matrices in analytic form. A. Introductory Remarks: For more details concerning the definition of space groups in standard or in non-s~andard settings, or the computer generation of space group irreps in standard or in non-s~andard forms, the reader is referred to Ref.[1]. By definition, we call a space grou p G{q', 7); (WIw)} = (WIw) * G{ ~r, 7); (El0)} * (WIw) -1 a non-standard setting of g{7", 7>; (El0)) if the element (WIw) e ~(3) does not belong to the Euclidean normalizer of g. Induced space group irreps are expressible in the form + t) = A k ( 2 i ,
e
and define standard Miller-Love space group irreps if (1) the space groups G are in standard settings, (2) the sign A is plus one, (3) the representation domains R B Z ( T , 7)) are identical to those that are tabulated in Kef.[2], (4) the little co-group irreps/It k,e;+ are ML-irreps, and (5) the coset representatives {1~} are optimized ones. To transform G-irreps in s~andard form where G is in standard setting, into non-standard forms and/or G into non.standard settings, the reader is again referred to l~ef.[1]. To recall, all software packages, not only IP~REP, create by default single and double-valued standard Miller: Love space group irreps but likewise allow one to carry out any re-setting of space groups and/or any modification of space group irreps as long as they are compatible. B . P a c k a g e - - S U B : One part of our software package called SUB deals with the algebraic check of group-subgroup relations and the second part with the actual computation of tables of multiplicities and corresponding subduction matrices. On some features and computational results of package SUB we have already reported in Ref.[3].
B.1 G r o u p - S u b g r o u p Relations g C 7"& A complete set of certain types of group-subgroup relations is tabulated in Ref.[4] but where all groups are assumed to be in standard settings. To cover all possible types of group-subgroup relations we extend the scheme by admitting that groups, sub-groups, and super-groups may be simultaneously in non-standard settings. Let g be a subgroup of 7/. To be more strict we assume in general ~ { ~ ' a , 7)G; (Vlv)) C ~ { ~ ' ~ , ~'H; (El0))
(2)
where (VIv) e ~(3) that ? t is in standard but g in non-siandard setting. Note iu'paxticular, translational group-subgroup relations TG (V) C TIt ( E) where TG (V) = (VJv)* TG (E) * (VIv)- a
539
involve the possibility of considering Bravais lattices 7"a(V) and ff'H(E) that refer to different lattice constants. The option of entering arbitrary ratios of lattice constants of the corresponding Bravais lattices in conjunction with arbitrary settings of space groups opens up a wide field of applications. B.2 S u b d u e t i o n s - - C o m p l e x C o n j u g a t i o n : The second part of package SUB deals with the actual decomposition ofsubduced 7~-irreps into direct sums of[7-irreps. To distinguish between 7£-irreps and g-irreps we employ the notations IDq,n;x for 7-~-irreps and /D k,~;x for [7-irreps where h ~ 7£ and g ~ [7 respectively. Moreover 7£-irrep labels are denoted by (q, 7/) where q RBZ(~/'H, 'PH) and r} e A(q); and g-irrep labels are denoted by (k, ~) where k e RBZ(qr'a, 7)a) and ~ ~ A(k) respectively. Finally A, A~ = :t:l can be chosen independently. Without going into details let us summarize the possibilities offered by the package SUB. In all the cases described here, we start from a fixed triplet (q, 7}; A) and compute for the fixed q, not only complete tables of multiplicities for all ~ ~ A(q), but also for the given r/the corresponding similarity transformation. Again we stress the fact that A and M can be chosen independently. The results of package SUB are (1) arbitrary subduetions for [7 C 7£, (2) compatibility relations for G = 7-l, (3) generalized compatibility relations for [7 C 7£ by considering limit-representations, and (4) complex conjugation for [7 = 7~ respectively.
w . ,A,A' . t ~D.,.;~(g) w~'~, A,A' w q A,A' . , ~ 1"~'sq*'rl;A(g) wq°'rl"
,v
=
~k,e • re(q, 7; AIk, ~; A') ~D k,e;x' (g)
=
~k.,~o @ m(ko, ~; A[ko, ~o; A') I19k°,e*;x'(g)
=
~k.,~o @ m(qo, r]; Alko, ~o; A') ID k*,~*;x' (g)
=
E k * e . @ re(k, ~; +]k*, ¢*; - ) ] D k " e ' ; - (9)
The last facility especially can be used not only to verify, for arbitrary single and double-valued space group irreps, their reality and degeneracy due to Kramer's degeneracy, but also to construct explicitly co-irreps, in full generality, for all Shubnikov space groups of type II. B.3 A u t o m o r p h i s m s -- Subduetions -- C o m p l e x Conjugation: For m a x i m u m versatility of package S U B the user can also invoke the option of considering automorphisms acting
on ?t exclusively. The most general automorphisms are elements of the Affine Group .A(3). Let a = (Zlz) be an antomorphism of 7£, i.e. a(7£) = (glz) * 7 £ . (glz) -1 = 7£ which means that 7i is mapped onto itself. Now let G C 7£ be a group-subgroup relation, it remains valid on replacing 7£ by a(q-/). This entails G C a(7£) ¢==~ a - l ( G ) C 7£ which reveals the subtle point that one has to distinguish between the cases a-X(g) = g and a-X(g) ¢ g respectively. Starting from a given 7£-irrep, say ]Dq,~;x subduction to G-representatious means IDq'n;A(a(h)) ~ [7 = (IDq'";A(a(g))
,¢==~ a - l ( g ) = [7
iDq,.;~(aCh)) ~ [7 ¢=~ a_1([7) ¢ [7
(3)
Thus when carrying out (3) one can do it either directly or split the procedure into two steps: (1)
aDq,.~(a(h)) ~ m°Cq),°<.);~(h) and (2) ID°Cq),°C.);X(h) I [7 where a(a,O) = (a(q),a(V)) are the images of the 7£-irreps labels (q, 7) under the antomorphisms a respectively. Again without going into details let us summarize the possibilities offered by the package SUB. In all the cases described here, we start from a fixed triplet (q, r]; A) determining a 7£-irrep and compute for the fixed q not only complete tables of multiplicities for all ~1E A(q), but also for the given r} the corresponding similarity transformation. The results of SUB are (1) au~omorphisra mappings for ~ = 7-/, (2) au~omorphism mappings for [7 C 7/, (3) automo~hisms combined with compatibility relations for g = 7"/, (4) automorphisms combined with generalized compatibility relations for g C 7£, and (5) automorphisms combined with complex conjugalion for [7 = 7£. wa(k'e)tA,A']Dk'e;A(a(g)) W ~ ~'e) -~ Ek',¢, • m(a(k), a(~); Zlk', e'; a')
540
mk',¢;z(a)
q,~) w X,,V "(q'~)~ {]Dq'~;~(a(h)) ~. ~} ~..r a (~.~, = ~C.',e ~ m(.(q), ~(~);.~lk, ~); .V) ~k,~;Z(g)
, , ~, (g) W "(R*'e)t ILk-,~;X(a(g)) W~Ck.d) ~,~, = ~ k . , , . @ m(a(ko),a(~);Alko',~$;A') ]Dk*,e*;
wa(.q*,~) n~ w ax,x ( ~ ,~) = Eko,e. @m(a(qo),a(o);alko,~o;a')IDk*'~*"X'(g) x,a' t {]Lq*'°;X(a(h)) ~ ~'J
W~:(~'e)t ]D~'e;+(a(g)) W~:(~'0 = Ek,,¢, • m(a(k'), a(C); +1k', ~';-) ]I3k''¢;- (g) For instance the last facility allows one to construct for every Shubnikov space group of type III and of type IV corresponding co-irreps. To sketch the procedure, let ~ be a subgroup of index two of the space group 7-/, and 7/(G) = ~ +ho*~ a asset decomposition. To arrive at a Shubnikov space group of type III or of type IV, symbolically written as 2v1{7/(~7)} = g + (c; h o ) . G, one considers the extensions h ~ (c; h) for all h ~ 7 / \ g and requires in addition (c; ho) ~ = (e; El0) * (e; h~) where /~ denotes the non-trivial element of the centre of SL/(2). Applying package SUB to this particular situation one arrives at type I: type II: type III:
]Dk'~(ho~)
wk'e;'l'(C; ho) {W",e'~(c; h°)} o =
+ ]D"'e(~10)
wk,e;4"(c;h°) {WR,e;+(e;ho)}* WR'~;±(e;ho) {wk'~;+(e;ho)}?
-- IDk,e(J~10 ) IDk,~(h~) + ]Dk'~(EI0)
= =
which allows one not only to check directly reality and degeneracy of single and of double-valued space group irreps but also to construct explicitly co-irreps of Shubnikov space groups of any type due to the knowledge of the similarity matrices WR'~;±(e; ho) respectively. C . P a c k a g e - - C G : The software package called CG deals with (i) the analysis of socalled Wave Vector Selection Rules, (ii) the computation of multiplicities for Kronecker product decompositions, and (iii) with the actual computation of Clebsch-Gordan matrices. The basic ideas have already been published elsewhere (for further references consult eg. Ref.[3]). By definition, Clebsch-Gordan matrices are unitary matrices that reduce Kroneeker products of g-irreps ID ~AsX" k'~lk''~'l"~ k : / / =/Dk'e;~(g) ® 1Dk"~';X'(g) into direct sums of their irreducible constituents. They are called standard CG-matrices if ~ is in standard setting and if the G-irreps are in standard form respectively. kt
t
c.k,dk',~ '? ]Dk,~l ,~ t.~ (.k,~lk',~' = ~ k " e" @ m(k,~; AIk',~'; A'llk" ,~''; A") mk",e";z'(g) ~ X A ' X" X;X ~ K~/ v A X ~ X " ,
(4)
Note CG-matrices are denoted by ~k'~lk"~' their non-zero entries are located at positions that are determined by so-called Wave Vector Selection Rules (WVSRs). By definition, Leading WVSRs (LWVSRs) are WVSRs of the type S k + S ~ k t = k ' + g where it is assumed that k, k ~, k" E RBZ(~) respectively. Note that depending on the chosen k-vectors k and k ~, more than one LWVSR may exist which implies a natural splitting of the multiplicities. m(k,~; AIk',~'; Yllk",~"; A") = ~(s__,s__,) m(Sk,~; Al~'k',~'; A'llk",~"; A") Note that the sum runs over all pairs of admissible coset representatives ( S , ~ ) that define LWVSRs. This splitting is extensively used in package CG to compute CG-matriees. To achieve the most compact representation of CG-matrices we apply a rearrangement procedure which consists of rearranging rows and columns of CG-matrices in a specific manner. The basic ideas of this rearrangement procedure are described in Ref.[5]. The rearrangement procedure, k ~'k' '~*' ~ "-';~x,xa k,~lk',~' , leads to new unitary matrices that decompose into symbolically written as Cx~Ix,, a direct sum of unitary sub-matrices (Sub-CG-matrices). To summarize "'xXR'X"A k'l~lk"~' = ~ k "
tL.. w.,~ ~(,5",S')_ _ ~./:t''__ ~ B(.....~') ask,~ls_'k',~a "-%~A',~" I , , ~..~../
541
(5)
k elS/k t el
where the matrices A ~ '~ (k", E") are called Leading Sub-CG-matrices. The unitary matrices B ( R " ) are monomial for the vast majority of possible cases. Again without going into details let us summarize the possibilities offered by the package CG. In all the cases described here, we start from two fixed triplets, say (k, ~; A) and (k', ~'; ,V), and compute for the fixed k and k', not only complete tables of multiplicities, but also for the given and ~' the corresponding CG-matrices. The package CG allows one to compute (1) generic CG-matriees (given by (4) if none of the resulting k" coincides with limit-vectors of RBZ(ff)), (2) non-generic CG-matrices (at least one k u coincides with a limit-vector), and (3) limit CG-matrices (at least one constituent of the Kronecker product is a limit-representation of g), for arbitrary settings of g and arbitrary forms of G-irreps.
C k'~clk''(tt ]D k'elk''~'(''~ 6?k'(lk"~'
m(k, ~; mlk', ~'; A'llk", ~o" ; A") m"o,e= .......,~' (g)
....
cko'elk:'e't Tl"k*'elk~*'e'("~' ----~"~k",e" 1~ m(ko, ~; Alk'o,~';A'llk", ~"; A") ]Dk",e";x"(g) AA'A" A;k' kS/ oko'elk'*'e' ~AAIAR E M A R K S : Thus the program can be run for Kronecker products of reducible g-representations. Corresponding results have been discussed in l%ef.[6]. The package SYMCG offers analogous options for decomposing symme$rized Kronecker powers of space group representations. D. Package -- SYMPW: The software package called SYMPW allows one to construct systematically so-called Symmetrized Plane Waves (SPWs). These states transform according to space group irreps and are linear combinations of plane waves. To sketch the procedure, one starts from a given plane wave, say ~k,s where k E RBZ(ff) and g E Tr6¢, and constructs by a two-step procedure SPWs. The first step consists of generating such linear combinations of the PW ~k,g that transform according to allowed g(k)-irreps. The second step consists of applying the induction procedure to the latter states to generate SPWs. The procedure written symbolically '~k'g(x)
'~lg;~'")(x) ~
~(klg;~,t0) (x .a,, ~ ~,
corresponds precisely to the induction procedure {{D k = "/" - irrep} 1' g(k)} T g of g-irreps. To summarize, for uniqueness of the SPWs, one has not only to restrict the k-vectors to RBZ(G), but also to take from each g-star with respect to 7~(k) only one element, as otherwise ambiguities would occur. 1. B.L. Davies, K. Dirl Proceedings of the 18th ICGTMP Lecture Notes in Physics (Springer) (1991) [preceding paper] 2. A.P. Cracknell, B.L. Davies, S.C. Miller, W.F. Love
Kronecker Product Tables, Volume 1 (Plenum, New York, 1979) 3. 'B.L. Davies, R. Did Proceedings of the 17th ICGTMP World Scientific (1989) 393 4, T. Hahn International Tablesfor Crystallography (Reidel, Dordrecht, Holland, 1983) 5. K. Dirl, B.L. Davies Proceedings of the 14th ICGTMP World Scientific 268 (1986) 6. 1%. Dirl, B.L. Davies Proceedings of the 15th ICGTMP World Scientific 722 (1987)
542
ICOSAHEDRAL DISSECTABLE'TILINGS FROM THE ROOT LATTICE Do M. Ba~ke, P. Kramer, Z. Papadopolos and D. Zeidler, Institut fiir Theoretische Physik der Universit/it, D-7400 Tfibingen Hypercubic lattices in 6D with the hyperoetahedral point group f~(6) model icosahedral quasicrystals under the restriction to the icosahedral group A(5). Three such lattices have the typical positions P : (100000), 2 F : (110000), and 2I : (111111). Here 2 F is the sublattice of P with nl + . . . + n~ = even. Moreover 2 F is the root lattice D6 [1] [9.]. We construct and dualize the Voronoi domain Y with the Voronoi vectors (roots) d=el q- el, i ~ j. The extended Coxeter-Dynkin diagram/gs of Ds codes seven "simple" roots and seven vertices v,n of a fundamental simplex S, Table 1. The domain V is generated by acting on S with the Weyl group W, the subgroup of ~2(6) with even number of reflections. Representative boundaries P(n) of V are obtained by applications of subgroups H < 12(6) to subsimplices S, Table 2. Hypercubes are denoted typically by
P(O...Oei+,...~6):x=
j 6 ½(/__~lA/e,÷ ~
i=j+l
eie,), -l_
and the join of a polytope to a point e i by P o ei : z = / z P + (1 - #)ei, 0 _/~ < 1. The boundary P*(6 - n) dual to P(n) is the convex hull of all lattice points from Do whose Voronoi domain contains P(n) [3], these points are given in Table 2. In Table 3 we give the representative boundaries of V under the restricted space group ADo ~ , A(5). A(5) determines the irreducible subspaces E~ and E~.. The inflation transformation of Do given in the orthogonal basis by the matrix
1 M2P = 2
Ii
1 1
I -1
-1 1
1 -1 1 1
1 1
-1 --I
1 1
1 -I
-1 1
-1 1
1 11
-1 -1
1
1
1
1
commutes with A(5) and scales the subspaces E~, E~_ by r and - r -1 respectively. By projection to E~ one obtains from the klo~z construction two tilings T ( 2 F ) and T*(2F) with tiles PII(3) or PI[(3) respectively. The tiling 7"*(2F) is generated from a graph inside V±, a triacontahedron as for the known tiling T*(P). Its tiles are six tetrahedra of different shape. These tiles obtained by projection turn out to be identical to the set given by Mossed and Sadoc 1982 [4]. T*(P) and T*(2F) are related by the properties: (1) the vertices of T*(2F) are all vertices of T*(.P) with nl + . . . + ns = even, (2) to any rhombus face of T*(P) there corresponds one short or long diagonal edge of T*(2F), (3)T*(2F) has additional long edges between even vertices of T*(P). Any additional long edge is generated at and only at an odd vertex of type 6 of T*(P) [5]. This vertex is surrounded by 2 thick and 2 thin rhombohedra. This configuration when
543
rebuilt in T*(2F) contains one long edge which is not the long diagonal of a rhombus face of T*(P). (4) given a tiling T*(2F), there is a unique tiling T*(P) whose even vertices coincide with the vertices of T*(2F). The tiling T*(2F) is dissectable along sets of parallel planes, each orthogonal to a 5-fold axis. Any such plane has the vertex connections of a 2D quasilattice called the triangle pattern [6]. This 2D quasilattice is projected from the root lattice .44. The kinematical diffraction from point atoms in vertex positions is shown in Fig. 1. The tiles of T(2F) are two rhombohedra as in T(P) and four pyramids on the rhombus base. Three types of vertices correspond to two deep and one shallow hole [1] in Ds. Diffraction data are shown in Fig. 1. The Fourier module in both cases corresponds to the lattice I. 1. J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer, New York 1988 2. M. Baake, D. Joseph, P. Kramer and M. Schlottmann, preprint TPT-QC-90-03-1 3. P. Kramer and M. Schlottmarm, J. Phys. A22, L1097 (1989) 4. 1%. Mosseri and J.F. Sadoc, in The Physics of Quasicrystals, eds. P.J. Steinhardt et al., p. 720, Singapore 1987 5. M. Baake, S.I. Ben-Abraham, P. Kramer, and M. Sehlottmann in: Quasicrystals and Incommensurate Structures in Condensed Matter, p. 85, eds.: M.J. Jacam~n et al., Word Scientific, Singapore 1990 6. M. Baake, P. Kramer, M. Schlottmann, and D. Zeidler, Mod. Phys. Lett. B4, 249 (1990); M. Baake, P. Kramer, M. Schlottmarm, and D. Zeidler, Int. J. Mod. Phys. B, to be published
Table 1 "Simple" roots and vertices of the fundamental simplex S for/P6 m
"simple" root
v e r t e x Vm
0
--el
-- e2
½(111111)
1
el -
e2
2
e ~ -- e3
3
e:l - - e 4
4
e 4 -- e 5
5
e5 -- e6
6
(e5 + e6)
½(111111) ½(001111) ½(000111) (oooon) (000001) (oooooo)
544
,-M
~--I
x
X
x
eq x ,m4
x
X
X
x
x
x
x
x
5~
.~
o=
o ,..el w~ O0
o
~
~
~
~
~
CO i.-I
¢0
##
VI vi v
v, v
vi v
""
"
""
I Vl
+ @ c£
.~
LT
Vl ~
~
@ ~
I
+
I
vi +
'"
"
0
C~
CD
0
VI
+
~o
Vl
~
~
~
~
Vl
¢~
¢~
~
VI
,~
+
+
+
VI
~
~" ~
+
~ _y_ .v +
@
@
~'
0
0
0
CD
CD
vi "
I Vl
~" ~" ~" ~" ~" ~" +
0
+ " +
++++++o,.,++
@ _... ~'
@
Vl
~
~
I
Vl
+
0
0
~-~
Vl
~"
,-~
@
545
Table 3 Space group representatives of boundaries under An6 ;~a A(5)
n
P(n)
P*(6 - -
n)
l el 5
P(lOOO10)o
O, e s + e l e5
3
P(1oooio) o
0,-es+el
P(11oolo) o el P(11ooio) o e 1
O, eS"-I- e l , e 2 -J- el
V(lOO!lO) o e l
0, e5 -t- el,e4 + e l
P(lOOiiO)
0,--e5 + e l , - - e 4 -{- el
Oel
0 , - e 5 + el,e2 + e l
P(O001il)
0, ea -- e~, es + e4, --e5 -F e4
P(mooo) P(mOlO) 0 e 1
0, ez + e2, es -I- e l , e 2 We1
P(Iii010) o e 1 P(1i1010) o e 1 P(].llOTO) v(ziloio)
0, e5 + e l , e s + e l , e 2 + el O, e5 + e l , - - e 3 + e l , - - e 2 + el 0, e5 + e l , e 3 -t- e l , - - e 2 -I- el
o e1
0, -e~ + el, es + el, e2 + el
o eI
0,-e5 + el,es + e l , - e 2 + el
t degenerate in Ell and E±
546
0
(o)
0
"0
•
o
0
O"
o° 0
.o o
o...
A
o 0
o
o
o
•
.°.o
.
o
0
*
o
0
o
o
o
"0
0
•
•
•
%
~
•
°
0
o
•
-
o
0
0
--
*
0
0
•
0
o
o
,
o
o
.
.v
o
0
0
0
•
o
0
•
^U
:o
o o. :o 0°
•
o
0
0
0
•
o
o
0 ,.
o
*
• 0
°0
0° o
0
o"
• •
0
0
"0
0
*
0
o
o
o
o 0
.
o.
O"
0
.
• • 0
.
o
0
/
.. Oo o/o ° o . .
'~,"
,F~
0 ° o o " CD " o o ° 0 ~ o . °. ° o O o f l ' . OoO/.o o 0 • 010 . ~'o
o
o',
o
o o " o o . 1•- o
•
o"
"/'/. ' o
o'a
o °.
• o
O.
0
• '0
o
o
--0"
o •
.0.-I..~.
¢
0
/
o°o o
o
o
o" ."
0
0
'J u
• 0
~.o o "o 0
0
o
o.'° 0 . 0
",
. 0-._o
0 o
00
.°.. o
o
o
0 o 0 0"..'0°
0
."
- -
.-
-. 0 0 0
,°
*° 0 0 0 ", 0 0 0 o° o 0 0 O" o o "0 "0"0 o 0 " . . " 0 o o "o" ° 0 o'°. 0"o 0 "-'-° 0 . 0 ° o o o 0 • .. 0 o o .o o 0 °
o o:o'. 0
o:o
o" o 0 0 ° . . °
0o0
o .0 ." 0 ° o o
• .'.
.
o
o
°o
o
0
o_
° o
0
o °.o
O'o'O
.0
• " 0."
•
" ." 0 % 0
. '^°<
o
0
0
./~
'o
/
0
o'/'o
L../ o''~ o
o
"2"
0,.
".[."
0.°'~(-o
o/o
".~
.cp..
o
o
"°
0
"
/'jo...o ~.
0
". ° ~ ° / ' o O o
"o" ° . °
°
o..o ~, " O ° ,,a o . O O.o 0 . •
"o . 0 . 0
o
o
.'O:oo O0
"
O
03'0
" o
o
A
< o.,-o'"
o ... o o o . . . . o o o p % , . , o < . " . :
O.
. 0 ° o o o 0 o o "0" o
000 ..
o
• .'o'..o o .'o .'I'. ~ ' . "o" o + ~ . . o." o o . o " • o . t . ~ . : 5 ~ . U o "'.o
o'° o o.°.
0
o
,
0
o
o''.(b.''o
o o ..'o . ° . 0 ° 0
_ ,/X
-i oj~"~%
c_.
0
o.''o
o
o
o" o
.*0
o ° o o " o " ." o T ¢ ' . ' ~ ' % o .'bO. o °o.'. %Ooo.O L / . . o - o _ _ . . - o
,,
\~/
u .'.'. 0 o.° .=.. u/ o °. )~_o 0 ." ", 0 ^o o o., 0 olo "(~'--. 0"o o u
•
° o 0
°e 0
•
,t
0 0
0
o° 0
0
.
0
• •
e
C~h
'o
*
o. Oo o .o. o oO .;
0
• o
o: :o ; o:-:0
o
•
.
0
o
o
O
o
°
0
o
0
•
o
• ~
•
.
•
0
3
o
•
_
A
0 /
o o
oo+o:olI
X
°. o" % °
o
OO°o. oO< 0
0
"~ 0
,,o . o ^ ' . o .
v
~
*"
".^. o . " 0
'~
0
+.
"
o~'0
oO o ..'o . 0 " o o" "o ° ' 0
o. o " 0 . " 0
0
0
•
e
• 0
0
0
.'0 • O" o o "0 • O" 0 o 0 " .. 0 . O . ' o o ' O 0 " ' ° 0".°.0 """ 0 ° 0 o '=' O . o 0 ." ". O " o o ° = 0 ,= o o .0 _" 0 . 0 • " 0 . " o o o
o OoO o '~ 0 ". o
0
O.o.-O :.v.
G'~-- c o: 'o. .
o o Oo o'°'o •
°.
o
0"..o0o
v
0
."
°
•
0
o
o
o°o" 0 ".o
o
o
.+
o
°'0
O"
Fig.1 Kinematical diffraction with point scatterers at (a) the vertices of T*(P), (b) the vertices of T*(2F), (c) and (d) the vertices of T(2F) corresponding to first and second type of deep holes of Ds. Circles represent diffraction spots on the Ewald plane perpendicular to a twofold axis, areas are proportional to intensities relative to the central peak (cf. P.Kramer and D.Zeidler, Acts Cryst A45, 524 (1989)).
547
GROUPS
WITH G-NUMBER
PARAMETERS
Y. Ohnuki and S. Kamefuchi Department of Physics, Nagoya University, Nagoya 464-01, Japan Institute of Physics9 University of Tsukuba, Ibaraki 305, Japan §I
Introduction
G-numbers, denoted as x I, x Z, ..-, Xn, YI' Y2' '''' Yn' '''' are defined as a generalization of complex and Grassmann numbers. They obey the commutation relations
[xa'
xB]-(a~)
= [Ya'
YB]-{aB)
(a,8 = I, 2,
where ( ~ B ) = ( S G ~
= [xa'
"'', n)
is a relative
YBI-(aB)
= ....
o ,
(1,1)
,
signature
between two i n d i c e s
a and
~
The adjoints xa of x a are assumed to be g-numbers with the same (aB) as those of x a. * The g-numbers x a and x B may be regarded as classical analogues of the operators as(k) and a~(p) respectively, which are defined by [as(k),
a~(P)]_(aB)
aa(k}lO> In this connection
[as(k),
= ~a86(p-k),
we further
[xa,
= 0 .
(].2)
assume
~B(k)]_(aB)
the following:
= 0
where ~a(k)maa(k) or a Ca(k). Genelizing consider g-tensors T~IB2...~ ~ which
~a(k)TBIBZ...B ~ = j ~:l ( a B j
aS(p)]_(aB ) = 0 ,
(1.3) the above satisfy
)TBI82..-B ~ ~ a (k)
we the
shall also relations
,
Tal~Z...aiS~182...~ m = i~l j ~ l ( ~ i ~ j ) S s 1 8 2 . - . ~ m T a l a 2 . . . ~ Algebraic discussed
and analytic in some details
properties of g-numbers elsewhere [I].
548
have
i
.
already
(i.4)
been
The c o m m u t a t i o n r e l a t i o n s (1.2) are g e n e r a l l y those of anomalous case [2]. It is known, however, that under c e r t a i n conditions any anomalous case can be c o n v e r t e d into the normal case with help of Klei n transformations so that given a theory with commutation relations of a n o m a l o u s case then c o r r e s p o n d i n g to it we can always find a t h e o r y with those of normal case, w h e r e the both t h e o r i e s are physically indistinguishable [2]. On the other hand, invariances under transformations of a g r o u p are usually formulated in the f r a m e w o r k of the normal case. H e n c e there m a y a r i s e a q u e s t i o n as to what s y m m e t r y t r a n s f o r m a t i o n s are in the c o r r e s p o n d i n g a n o m a l o u s case. For instance, suppose a s y s t e m of two fermi fields ¢I and ¢2 with 2 equal mass, which are c o u p l e d t h r o u g h the i n t e r a c t i o n g(~2 CGO¢~ ) + In the normal case, where {Ca(x), CB(y)}:aa~8(x-y) and {Ca(x), ¢R(y)}:O~ at equal time, the t h e o r y has U ( 2 ) - i n v a r i a n c e and the energy spectra are c l a s s i f i e d by i r r e d u c i b l e r e p r e s e n t a t i o n s of this group. However, a n o t h e r type of c o m m u t a t i o n relations, c a l l e d a n o m a l o u s case, + is allowed for the a b o v e Hamiltonian, i.e., {¢G(x)¢~(y)}=~(x-y), {¢~(x), ~(y)}=O, and [el(x), ¢2(y)]:[¢l(X), ¢ 2 ( y ) ] = 0 at equal time. Though the group U(2) is no longer a p p l i c a b l e in this case, all p h y s i c a l c o n s e q u e n c e s d e r i v e d here should be the same as those in the normal case [2]. Thus we may expect that there also exists a kind of symmetry which leaves the commutation relations and also the Hamiltonian invariant and enables us to c l a s s i f y all of the energy spectra in place of U(2). In such a s y m m e t r y the g - n u m b e r will be seen to play a crucial role. We shall d i s c u s s this p r o b l e m from a general point of view.
§2
G-Trans format ion
Consider a m a t r i x g=}{gaB{}, c a l l e d g-matrix, w h o s e elements ( a , B = l , 2 , . . . , n ) are c o m p o n e n t s of a g - t e n s o r of order 2. By means it we i n t r o d u c e the t r a n s f o r m a t i o n
a~(k) which
=
immediately
Zpgaoap(k), leads
Thus,
(2.1)
us to
[a~lk), a~lp)l_laB ) taR(k) ,
golf 8
a'~(P)l_laB)
:
o
=
,
E ogaog~ *pS(k-p)
.
(2.2)
iff E #gapgBp *
= 8aB
(2.3)
,
549
the commutation relations for aa(k)'s are kept invariant under the gtransformation 12.1). Moreover, if g and g' satisfy Eq.(2.3), so does the product gg" as well. In what follows we shall assume (2.3). From 12.1) we also have the transformation for a~(k): a'÷lk)a with
g~
= Zpap+(k)gap* = E p g ~ a ~ ( k ) =
(~P)(OP)g*ap
which form a g-matrix g=llg~ll. (p#) is independent of p, since -
12.4)
But it does not satisfy
unless
-~
Epg~g~
= (a~)Ep(op)g~pgap
•
(2.5)
Thus throughout this paper we will also assume the signature (pp) and denote it as
= (pp) which
(2.3)
the p-independence
(p=l,2,...,n)
of
(2.6)
assures
According as ]:+ or -, xa is called a generalized bose number [3] or a generalized Grassmann number [4]. The relation (2.3) may be written as gK=l with K~R=gR~'-~ which ]-I yields det(g) det(K)=l, thereby implying the existence of [det(g) and hence of g-I [3], [4]. Since g-l= K and therefore ~g=l, we are left with pgpagpB
= 8aB
In the same way,
•
owing
(2.8) to the existence
of ~-I we obtain
The m a t r i c e s g - 1 and 2 -1 a r e g - m a t r i c e s and s a t i s f y Now i t i s an e a s y t a s k t o show t h a t
E a':(k)a~(p)
(2,3)
and ( 2 . 8 ) .
= Eaa:(k)aa(p)
(2.10) and,
in addition
~aa~(k)a'a(P)
to these,
=
if g u B = g ~ ,
Eaaa(k)aa(p)
we have
(2.1o')
550
T h u s we m a y d e f i n e , as a g e n e r a l i z a t i o n groups with g-number parameters by U(n)g
: {glEpg~pgBp
O(n)g
= {glg
: 6aB}
of the o r d i n a r y
U(n)
a n d O(n),
,
(2.11)
and
We can f u r t h e r
~ U(n)g
introduce
,
g~B
: gi~}
generalized
SU(n) g
: (glg e U(n)g,
SO(n) g
: {glg
e O(n)g
SU(n)
det(g) ,
(2.12)
'
: I}
In this c o n n e c t i o n it s h o u l d be n o t e d example, does not behave in the t r a n s f o r m a t i o n s of U(n)g, t h a t is,
by
,
: i)
det(g)
a n d SO(n)
(2.13) .
(2.14)
t k)aB(q)aa(p) that E G a 5( same way as ~8(q)
for under
(2.15) where the n o t a t i o n ~ m e a n s the right and left hand following:
~a(aB)a~(k)~B(q)aa(P)
the same of t r a n s f o r m a t i o n properties sides. We e a s i l y see it must be
-
iB(q)
,
(2.16)
which suggests the w a y of m a k i n g the indices contraction tensors o f lower order. For i n s t a n c e s u p p o s e that+ b~(k) antiparticle a n n i h i l a t i o n o p e r a t o r s u c h that b (k) ~ aG(k). d e f i n e the c o n t r a c t i o n of the i n d i c e s of a u a n d b a by
a~
I °Bi
B2" ..Bib
@; ~
of the
E j~l= (~B)aaOBl j B2...B
to is Then
get an we
~ b
contraction
(2.11)
~ OB1B2...~A w h e r e O B 1 B 2 . . . B £ ~ aBlaB2 . . . aBA This method for indices contraction 0
is
easily
generalized
to
~l''Gs-lo as+l''~£ BI''Bt-I® Bt+l"Bm contraction t-I
= E i~s+l (aip)j~l(Bjp)Oal• . a s _ i O ~ s + l =
551
• .~ ~ BI" " B t _ l O B t _ l " "B m
~ 0
,
(2.12)
al''as_l~s+l''~ £ Bl''St_iBt+l''B m a
if Oul..a~ bl 0,8 m ~ aul.. ~A bBl''bBm'
§3
Generalized
Parmutation
Operators
To construct up irreducible representation spaces of generalized groups we have to devise, in additibn to the contraction technique mentioned above, those manipulations of indices which just corresponds to permutation operations in the ordinary theory. On account of the relation aa(k)aB(p)
~ (uS)aB(k)ae(p)
. ~.(~B)aB(k)a~(p)
,
(3.I)
We introduce here a generalized permutation operator ~g(~<~ B) by ~g(a~-~8)[aa(k)as(p)]
~ ~.(aB)as(k)aa(p)
(3.2)
,
which in the special case, where the signature (plP2) is either + or for all @I and @29 just represents a simple exchange of ~ and 8. Generalizing the above we define =g(a) (aeS N) by =g(¢)[a~l(1)a~2(2)'''a~N(N)]
= ~a~01(1)a~2(2)'''a~0N(N)
, (3.3)
where the arguments 1,2,...,N in the round brackets ape abbreviations of kl,k2,...~k N respectively and ~ a product of relative signatures obtained in the following way: The permutation a (1)...a (N) aI aN a (1)...a (n) is performed through a succession of interchanges of
ao2
a¢ 1
two adjacent indices like a (p)a (k)~a.(p)aB(k) and with each of these interchanges we associate aSfaot~r ~'(~T). For example,
~g(¢~'", T)[ae(1)aB(2)aT(3)] = ~.(aB)(BT)(T~)aT(1)aB(2)aa(3) Then,
the
realizing
followings
the
can
be
shown
permutation
[5]:
(3.4) (i)
There
aal(1)..-aaN(N)~a
are
various
(1)...a
ways
of
(N)
by
~¢1 ~¢N applying the above procedure. In spite of this the factor ~ in (3.3) is unique. (ii) ~(o)'s (oeSN) form a group, and obey =g(~)= g (z)==_(ar). (iiz) The commutation relations not including ~functions are compatible with permutations under consideration, that = i (apj).O . I) then is, if OlO2 ..0%a~ ( '
aa(1)OplOl...o~ ~j=l
552
~g(~
-
=
And
8)[s5(i)o
PIP2...p~
=
'
a (z)]
.
j~l (aPJ) Opip z "@£
if gaBOpip2...pA
~g(e
- -
(3.s)
B)[as(1)aB(2)]
= n~ = 1( 5pj )(Bpj).o plpZ...p A ge8 then
mg(51+b~2)[galBiOplpZ...p£g52B ~]
= j~l = (elPJ)(BIPJ)'0 P1P2' . "PA .mg(a1<4 az)[g alB2ga2B2 ] = j~l (eZPJ)(B2PJ)'=g (el +~'a2)[galB2g52B2]'O PlP2"''P~ = where
(3.6)
~g(~l<~ 52 )[galBlg5282 ]~l'(5152)(BIG2)(~IBI)ga2BIga1B2 • Now applying a ' ( j ) = ~Bj g~jpj°aopj(J) we write, by means
a~(j)gBT=(aB)(sT)gsTa~(j) , the monomial a~l(1)a~2(2)'''aLN(N) •
' (2)
... a"
a~l(1)a51
(N) =
aN
.
~
ESI,B 2, '',B N
x aSl(1)aB2(2)'"
aBN(N)
of
as
. . .
galB 1
gaN~ N (3.7)
,
where u is a product of relative signatures which appear through the process of shifting gsB's to the left. Then according to (iii) we obtain mg(a)a'al(1)a'a2(2)
... a'aN(N) = ~a'Sal(1)a'aa2(2)
= EBI ,B2,...,8 Nu m(5)(a)[galBlg52B2g x aB1(1)aS2(2)
... aBN(N)
... a'aoN(N)
... g~NSN ] ,
(3.8)
where the superscript (a) in mg(5) has been added to explicitly show that the permutation under consideration is the one with respect to ~I,~2,''',~N. We shall call mg(a) the g-permutation. It should be noticed here that unlike the ordinary permutations the g-permutation is not applicable to an arbitrary function of 5's but to a quantity in which they are situated in a linear order like a monomial in a's and g's (or
553
in aT's and theorem [5]:
g's).
Then, from (3.8) we
can
derive
the
following
Theorem G-transformations (a) II"
g
(~)
"
are commutable with g-permutations; '
(2)...a'
[aalaa 2
(N)]
aN
EB1,B2,...,BNGal~la2B2..oaNBN
=
x ~g( a ) ( a ) [ a S l ( 1 ) a 8 2 ( 2 )
... s BN (N)] ,
.=
where GalBIa2B2...aNB N
..
UgalBlga2B2.
(3.9)
.
gaNBN
Thus the symmetry properties of indices (of homogeneous plynomial
as usual.
We then immediately obtain from the Theorem
y(a) g
.
.
[aalaa 2
... a '
(N)]
aN
= EB , . . y(a) l)aBz(2)...s (N)] I B2' '''BNGalBIaZB2 "'aNON g [aBl( 8N (3.10) Espeoially, if denoting the antisymmetrization operator d E for n indioes as ~g m ¢~S
sign(a)~g(e)
(3.11)
,
n
we get Mg(a)[a~l(1)aa2' (2) ... a'an(n)] = det(g).~g(a)[a l(1)aa2(2 ) ''' San(n)] Accordingly,
if det (g)=l, ~g "(a)[aal(1)
. . . . aa (n)] becomes
(3.12)
invariant
n
under such g-transformations.
Since mg(~)Mg=sign(r)~g,
~(a)[aal(1)a 2(2) ... a (n)] ~
g
,
an
$ala2' ''an
is regarded as a totally antisymetrized ~a G . , . a is a g-tensor just corresponding 1 1 n in the usual theory.
554
tensor of order n. to the Levi-Civita
(3.13) Here symbol
Thus i n the a b o v e we have b e e n able to introduce, in a c o n s i s t e n t way, the n o t i o n s of the c o n t r a c t i o n s a n d p e r m u t a t i o n s of the indices of g-tensors. A c c o r d i n g l y , as done in the o r d i n a r y cases, we can e x p l i c i t l y c o n s t r u c t up i r r e d u c i b l e r e p r e s e n t a t i o n spaces of a g - g r o u p on the b a s i s of the Fock space. Of c o u r s e there are v a r i o u s kinds of e x a m p l e of a p p l i c a t i o n of ggroups. A hidden symmetry Especially, in the case w h e n [7] the g-group technique approach. The details of t o g e t h e r with r e l a t e d topics.
in p a r a f i e l d t h e o r y is one of them [6]. K l e i n t r a n s f o r m a t i o n s are not applicable would provide us with an advantageous our t h e o r y will be published elsewhere
References [I]
[2] [3] [4] [5] [6]
Y. Ohnuki and S. Kamefuchi: N u o v o Cim. 70A 435 (19821, 73A 328 (1983), 77A 99 (1983), 83A 275 (1984). G. L~ders: Z. N a t u r f o r s c h . 13a 254 (1958). H. A r a k i : J . Math. Phys. 2 267 (1961). Y. Ohnuki and S. Kamefuchi: L e t t . Nuovo Cim. 3__O367 (1981). Y. Ohnukl and S. Kamefuohi: J . Math. Phys. 2_!1 601 (1980). Y. Ohnuki and S, K a m e f u c h i : to be published. Y. Ohnuki and S. Kamefuchi: Prec. of XV Int. Conf. on Differential Geometric Methods in Theoretical Physics, 1986 Clausthal, (ed. by H. D o e b n e r et al.; W o r l d Scientific 1987),
p.41. [7]
Y. Ohnuki
and S. Kamefuohi:
Prog.
555
Theor.
Phys.
7_~5 727
(1986).
REGGE CALCULUS WITH TORSION
Christian Holm Institute for Theoretical Physics A TU Clausthal, D-3392 Clausthal, FRG
and J6rg D. Hennig Arnold Sommerfeld Institute for Mathematical Physics TU Clausthal, D-3392 Clausthal, FRO
1. Introduction to Re~,e Calculus Regge Calculus was introduced by T. Regge [1] almost 30 years ago for Riemannian manifolds without torsion. Regge defined the concept of curvature and metric for a simplicial manifold, giving thus up the differentiable structure, and gave us a "discrete" version of Einstein's theory of gravity. This is done for several reasons: for a compact manifold the triangulation is finite, so one has to deal with only a finite number of simplices; the discreteness makes numerical calculations possible, this is helpful for example in strong fields; and finally, Regge calculus is viewed as a possible road to quantum gravity [2]. In two dimensions a differentiable surface is approximated by triangles, whose interior is assumed to be flat. The metric information of the manifold is encoded in the edgelengths of the simplices. For a n-simplex the number of edges is n(n+l)/2, which matches exactly the number of independent components of the Riemannian metric tensor g~v, so that this information is exactly equivalent to specifying the metric. The curvature is measured by carrying vectors around closed loops, which encircle a hinge, which is a (n2)-simplex. The angle 0 by which the parallel transported vector has been rotated is called the deficit angle associated with that hinge and is a measure of the scalar curvature R:=Rlavg/-tv. Regge gave a discrete version of the Einstein-Hilbert Lagrangian LE_H=
JR/det(-g)d4xas LR=
E0h(l~t)Vh(Ij.t). all hinges h The variation of LR with respect to the edgelengths l~t gives the discrete EinsteinRegge equations, one equation for each edge. It is interesting to note that in the variation
556
we have ~0h(l~)=0. This is completely equivalent to the situation in the continuum where the variation 5R also does not contribute to the field equations. Regge's formalism is a so-called second order formalism, because he varies only the metric. A first order formalism, where one varies the connection and the metric (or tetrad) independently, would be closer to gauge theoretic formulations and also would allow the introduction of torsion. This puts us in the need of talking about the idea of a connection in a simplicial manifold.
2. Simplicial Connections On a pseudo-Riemannian manifold the Levi-Civita connection is the unique connection which is metric and torsion free, and as such is defined by the metric alone through Christoffel's formula. Similarly there is a natural parallel transport defined on a Riemannian simplicial manifold. Because the parallel transport inside a simplex is trivial, one has to define the transport only for crossing the interface o12 , a (n-1)-simplex between two neighboring simplices Ol and 02. For a vector parallel to 012 the transport is the obvious one: two vectors V(Ol) E T(Ol) and v(o2) E T(02) in adjacent simplices are called naturalparalle1 if they can be obtained by parallel transport from the same vector in T(012). Vectors orthogonal to 012 remain orthogonal and preserve their orientation and length with respect to the different metrics in o l and o2. This completes the operational definition of the Levi-Civita parallel transport. That this transport is torsion free can be seen by considering an elementary torsion parallelogram. Since every Riemannian simplicial manifold can be embedded in a higher dimensional Euclidean space [3] we call this connection umklapp-connection for one can think that at o12 the tangent space T(ol) of O1 gets tilted onto T(O2). The umklapp-connection can be defined more abstractly as a linear isometric orientation preserving map between T(ol) and T(o2) which respects natural parallelism.
/I
----...
i j
Figure 1: a frame e(1) is transported from ol to 02 by an umklapp-connection (e(2)), a metric connection with torsion (e(2)'), and a symmetric, nonmetric connection (e(2)").
557
Analogous to the continuum case we generalize this concept and call a linear map between T ( o l ) and T(o2) which is isometric and orientation preserving a metric simplicial connection. Because the natural parallelism is not necessarily respected in this case a frame e(1) transported from O1 to 02 can get rotated by a rotation K when it crosses 012. The rotation K corresponds to the contortion one-form Wk that appears in the decomposition of a general metric connection one-form Wrn into Levi-Civita part wLC and Wk: Wm=wLC +Wk. In the most general case a simplicial connection is a linear non-singular orientation preserving map from T(O1) to T(O2). An arbitrary simplicial connection is called symmetric if it respects natural parallelism. This completes the definition of simplicial analogues of important differential connections.
3. Torsion As we noted before torsion can be described via the contortion matrix K which is naturally defined on the (n-1)-dimensional faces between two adjacent simplices. This kind of torsion has been termed interface torsion in [4] and discarded because it would not contribute to LR. This is not generally true. In the continuum we have the relation R=RLC + Dwk - WkAWk and something similar holds on he simplicial level. If the contortion obeys a cocycle condition, it will define a one-form wk on the whole n-simplex [5] and contribute with a quadratic term in Wk to the Regge action, similar to Drummond's term arising from "body torsion", which in our opinion is just a special case of our notion of torsion. If one builds these torsion degrees of freedom into the Regge action one should obtain the discrete Einstein-Cartan equations upon variation. The exact formalism will be worked out in a subsequent publication.
4,Dual Lattice It turns out to be advantageous to formulate the theory on the dual lattice instead of the simplicial lattice itself [6]. The dual to a simplicial lattice is in general not made out of simplices, but consists of polyhedral cells. These cells are called Voronoi cells [7]. The dual to a vertex P is the set of all points that are closer to P than to any other vertex. In this way a n-simplex is dual to a 0-polyhedron (a vertex), and in general a n-k-simplex is dual to a k-polyhedron. Because the connection and torsion are defined on (n-1)-simplices, they correspond to links L(ij) in the dual lattice. The curvature was defined by going around a hinge, which is a 2-polyhedron, a so-called plaquette, in the dual lattice, whose boundary
558
is made up by the connection. This makes Regge calculus almost look like lattice gauge theory, where the gauge fields are defined on links, and the field strength is a plaquette variable, made up by the product of the link variables around a plaquette. i..(/
L(ml)
1) ~( lk)
n
L(Ln) _
/ L0i)
LCk.i) j
Figure 2: Plaquette in the dual lattice .5. Further Remarks What remains to be done? The formalism of Regge calculus should be closer to differential geometry. In particular one should be able to derive a simplicial analogue of Cartan's structure equations. To achieve this goal we want to use the natural isomorphism of the de'Rham cohomology onto the simplicial eohomology, In de'Rham cohomology we have the set of closed p-forms ZPDR:={wpldwp=0 } and the set of exact p-forms BPDR:=.{Wplwp=d~p_l }. The de'Rham p-th cohomology groups are then defined by HPDR:=ZPDR/BPDR. In simplicial homology we have the p-cycles Zp:={~plC3Ctp=0} and the p-boundaries Bp:={aCtp=0} that define the p-th homology group Hp:=ZP/Bp. De'Rham's theorem makes them naturally isomorphic. Differentiable manifold
Simplicial manifold
p-forms exterior derivative d metric gpv Grassmann product ^
p-cochains coboundary operator a* edge lengths 1/.t Cup product
The motivation is to make use of the simplicial topology as much as possible already for the formulation of the theory in order to concentrate on the algebraic quantities. So we try to use as little as possible of differential geometry, but as much as possible of algebraic topology. The last remark concerns the idea of a space time point. We saw that the connection links tetrads defined on adjacent simplices. On the dual lattice the connection sits on links
559
and the tetrads are defined on points. This suggests to take the idea of a simplicial manifold very seriously and to use the whole simplex as defining a space time point. Because the tetrad is defined on it it possesses by definition local Lorentz invariance. Because a space time point is then an extended object it can possess internal symmetries which can act as a source of gauge fields. If Regge simplices are physical then they can define a fundamental volume and therefore length, which is presumable of the order of the Planck length. All presented material will be published in an expanded and more detailed version elsewhere.
Acknowledgements C. H. would like to thank the DFG for financial support through a postdoctoral grant.
References 1. T. Regge: Nuovo Cimento, 19, 558 , (1961). 2. M. Rocek, R. M. Williams: Z. Phys. C 2.L 371, (1984); R. M. Williams: Class. Quantum Gray. 3, 853, (1986); H. R6mer, M. Z~ihringer: Class. Quantum Gray. 3, 897, (1986); J. B. Hartle: J. Math. Phys. 26, 804, (1985); M. Lehto, H. B. Nielsen, M. Ninomiya: Nucl. Phys. B 272, 228, (1986). 3. R. Friedberg, T. D. Lee: Nucl. Phys. B 242, 145, (1984). 4. I.T. Drummond: Nucl. Phys. B 273. 125, (1986). 5. R. Sorkin: J. Math. Phys. 16, 2432, (1975); 6. M. Caselle, A. D'Adda, L. Magnea: Phys. Lett. B, 232, 457, (1989). 7. T. D. Lee: "Discrete mechanics", in Proc. Intern. School ofsubnuclear physics, Erice, Italy, (1983).
560
QUASIPERIODICITY; LOCAL AND GLOBAL ASPECTS
L, I.)
)
DANZER, Dortmund
I n t r o d u c t i o n a n d d e f i n i t i o n s . This l e c t u r e i s d e v o t e d to d i s c r e t e p a t -
terns, mainly t i l i n g s in the euclidean p l a n e long-range
ort:entational order,
]E2 a n d i n
but lack translational
the most w e l l - k n o w n e x a m p l e s a r e the p l a n a r main interest comes from the q u a
sicry
PENROSE--tilings, but our
stals.
Since physicists a n d m a -
thematicians sometimes seem to speak different languages, ful to define the most relevant terms : A tope ( p o l y g o n ) called
facets.
IE3 , which show
symmetry. Probably
it m a y
be use-
tile is understood to he a poly-
h o m e o m o r p h i c to a ball. Its faces of codimension one are
A global t~ling is a covering of space without overlap.
T w o tiles with a n intersection of codimension one h a v e to share exactly a full facet a n d are called a t i l i n g , whose union is
adjacent. A patch is a collection of m e m b e r s of homeomorphic to a ball. A p~ototi]e i s a represen-
tative of a class of pairwise congruent tiles. W e only consider tilings with a
fin i t e
may
bear
there m a y
set of prototiles, called the
markings
(e.g.
then in the protoset
be two or more congruent prototiles w h i c h differ by their m a r -
kings. If a protoset ~ may
protoset. Sometimes the prototiles
dots ) on their b o u n d a r y ;
is given
(e.g.
PENROSE's
dart and kite ), one
consider the family of a 1 1 tilings, that can be m a d e up using these
prototiles. But usually it m a k e s
more sense, to restrict attention to those
tilings, which also satisfy some conditions, say
C
. The family of all
species defined b y ~ " a n d C ; we denote it periodic, iff it admits at least one translation ( ~/= 0 ); otherwise it is called nonperiodic. A species all of w h o s e m e m b e r s are nonperiodic, we call aperiodic. these tilings we call the
by
sp( ~r, C ) . A tiling is called
2.) Three methods h o w to construct aperiodic species.
a) Matching r u l e s .
Given a protoset
a r e l o o k i n g for t i l i n g s
~
:=
{ S1, S2 . . . . .
Sm I
' we
( k e e p i n mind : t h e y h a v e to be f a c e - t o - f a c e
s a t i s f y i n g a c e r t a i n set of local matching r u l e s .
)
I n p r i n c i p l e such m a t -
c h i n g r u l e s j u s t c o n s i s t of the l i s t of a l l p a t c h e s up to a c e r t a i n s i z e , which a r e p e r m i t t e d I ) . i) S t r i c t l y s p e a k i n g we s h o u l d s a y But for brevity we
,, a l l c o n g r u e n c e - c l a s s e s of p a t c h e s "
shall omit this distinction in the sequel.
) Research supported b y the
DFG
(Projekt
561
.Quasiperiodizittit ")
Iff a species
~@ c a n be defined by some f i n i t e p r o t o s e t
(~'n R ) f g ether with a list of local matching rules, we say ~
~-
to-
satis-
(LMR).
~fies
The matching rules are called sists of p a i r s
strictly local iff the list mentioned only con-
of a d j a c e n t
tiles. We then replace
(LMR) by (SLMR).
£
An important example are the two PENROSE-
triangles shown in figure I. Here the rule reads : ,,Dot to dot and stroke to stroke." A very profound example, that m a y turn out to be of great importance is given in
Figure i
[12] .
b) Inflation -- deflation. Sometimes it is possible to dissect every prototile
S(~ into a patch of smaller tiles, where every piece is congruent to -I some q S v , where ? ( > 1 ) is the same for a l l ~ and 12 .
(I)
i Iff the repeated iteration of this process remains f a c e - t o roto.
factor If
~
?
to
,o,,o ,oo
).
is a p a t c h , we denote its i n f l a t i o n 1 by
t e r n a t i n g l y i n f l a t e d with the f a c t o r
?
infl( 0~
) . If a p a t c h is a l -
and e x p a n d e d by
~ ~it becomes
larger and larger. Denes KONIG's infinity lemma guarantees, that by this procedure one even obtains tilings of the whole of space. Using the triangles of figure 1
with the inflation indicated this inflation-method yields
exactly the same tilings of the plane as the matching rules (cf. [2 ; ). As the example of a square dissected into four squares shows, the inflationmethod m a y produce trivial tilings. This cannot happen, if the species satisfies the following condition: Every tiling ~9 of ~
P = A species s a t i s f y i n g
7
admits a u n i q u e
deflation. I. e.:
(innC Q ( I , D ) is n e c e s s a r i l y a p e r i o d i c .
this principle was discussed in
An e a r l y example of
[I l "
c) De BRUIJN's s t r i p p r o j e c t i o n method.
details cf. [4~, [5~, [6], [8~, [9J
562
In i t s simplest form and [ l l J ) i t
( f o r more
may be described as
permutations of n letters. Applied to the coordinates of the
"
euclidean space
•
IEn
becomes
it
a discrete subgroup of
=
/
G, and
I
%
~
IEd
IE
( 0 <~ d < n ) some invariant subspace,
,
*
~n ), which is
invariant under
I
O(n, ]R )
and even maps 7Zn onto itself. Let p be a sublattice of 7zn (maybe ~
i~ / ~ z F ~
C
lE J" its orthogonal comple-
ment and finally F a polyhedron in tEn which is a fundamental domain of ~'~ (in the strict sense,
C
is the basis of the strip S. Figure 2
hence half open half closed ). Now we consider the strip
S :=
F 4-lE
d
and the orthogonal projection ded, ~ (S d-faces of =
~ (5 r~ ~ ) into IEd . Since F is bound ~ ~ ) is a discrete set of points in lE . Projecting also the F into lEd we obtain a tiling there. Provided lEdo ~ =
~0~ (---- lEln ~
) the tiling will be nonperiodie and even q u a s i p e -
riodic (see section 5 for definition ). If the strip S is replaced by a d translate of S , we get another tiling in IE In this way a whole species of tiiings can be constructed. With G = D= (hence n = 5 ), d =2 and J 5 = 7Z the PENROSE-tilings of sections a) und b) can be reproduced. For later use we define (~P~)'*
{ strip The species projection ~ method, can be defined by de
BRUI]N's
3-) Some properties of species defined by iterated inflation ( cf. [i/~3 ). ]n this section we suppose ~ to be a protoset satisfying ( I ) . The dissections of the inflation give rise to the. following -- n on local ! -- matching rule : I Only those patches are permitted, which occur in the interior (~)
• of ~
k
k(
infl S • )
for some prototile S ~
fold inflated.
563
when it is
k--
It is e a s i l y seen, t h a t the species sp[ ~ , ~ ) defined by the protoset and ( ~ ) coincides with the family of all t i l i n g s t h a t can be obtained from ~ome S~ by i n f i n i t e l y often repeated i n f l a t i o n and e x p a n s i o n . As a l r e a d y mentioned ( 1 ) implies the existence of global t i l i n g s s a t i s f y i n g (:@¢) e v e r y w h e r e ; (I,
D)
and
imply, t h a t a l l such t i l i n g s are nonperiodic.
It seems, t h a t the following is an e s s e n t i a l p r o p e r t y of all t i l i n g s , may help to u n d e r s t a n d q u a s i c r y s t a l s . Definition : A family ~
of tilings in. IEd
is called . _ r e p e t i t i v e , (i)
that
iff
none of its members admits a transl tion ( =~= 0 ),
(ii) there is a c o n - ~ ~
stant
~
/ ~
~
l), ~ 2 ) ~or which the \ following holds :
C~)
~
/ $eca / t i otmall n of ~ .
,f e , belong to ~ x, y are points in
, / TThe patch J4 Us
shadowed,
IEd, ? ~
0
and i f ~
lies in the sectio~ Large section of ~. ,.containing a translate of . ~ .
the patch ~
+ ~dj-
of 6~ then
Figure 3
in the section
61~ Here 6~ With
(y + ~ ? ~ d l
]Bd
=
and ~
~
. We roll
~ k
~
IEd and the
~
of ~4can be found. is independent of
R-- constant
this becomes the definition of a
The hast property we n~ed., is
(~
a translate
denotes the unit--ball in
, x , y ~
of Q
of the family
repetitive
minimality (with respect to
No proper subset of the protoset
~
, ~
.
tiling.
( I ,D ) :
"~" also satisfies
( I , D ) (neither with the same inflation nor with any power of it ) •
Now we can state the following
584
Theorem 1 : a )
Suppose the protoset ~
satisfies
There is a natural number I , and e v e r y V , the prototile t~ Moreover, for e v e r y that
e
( I , D , M ) . then we have :
such that f o r e v e r y ] =~ d , e v e r y S ~ o c c u r s in the patch i n f l ] ( q J S ~ ) . #
there is a
&
.
~
--~1~ is the density of the copies of
(3: , ~ )b ) The species
certain large
0 < ~
S~
in
<=~ 1 ,
e v cry
such global
tiling. sp( ~
It s h o u l d be m e n t i o n e d , local matching rule.
with
, ~F: ) is repetitive.
that in g e n e r a l
(~)
T h e r e a r e many e x a m p l e s ,
c a n n o t be r e p l a c e d b y a n y w h e r e to e v e r y g i v e n r a d i u s
p a t c h e s c a n be u s e d f o r a p e r i o d i c t i l i n g in s u c h a w a y ,
t h a t e v e r y p a t c h of t h e so fQr,med g l o b a l t i l i n g , c o n t a i n s a b a l l of r a d i u ~
greater than
~
which contradicts
(~)
,
.
We c o m p l e t e t h i s s e c t i o n w i t h some r e m a r k s h o l d i n g f o r a l l r e p e t i t i v e t i l i n g s , regardless,
w h e t h e r t h e y a r e d e f i n e d b y i n f l a t i o n or n o t .
Remark 1 :
I f a t i l i n g b e l o n g s to a r e p e t i t i v e f a m i l y i t i s i t s e l f r e p e t i t i v e .
Remark 2 : ( R ) Exception :
~
Remark 3 : ev e ry
implies, that
no p a t c h d e t e r m i n e s a t i l i n g
c o n s i s t s of j u s t one t i l i n g . ( R e m a r k s If the t i l i n g
~
is repetitive
(and
globally.
1 and 2
are trivial.
h e n c e h a s a f i n i t e p r o t o s e t ),
d e c o r a t i o n of t h e p r o t o t i l e s b y f i n i t e l y m a n y p o i n t - - m a s s e s
a discrete massdistr~bution, nerated
the c a r r i e r
)
l e a d s to
of w h i c h i s a s u b s e t of a f i n i t e l y g e -
77.. -- m o d u l e .
We may c a l l two t i l i n g s
p a t c h - e q u i v a l e n t , i f f to e v e r y p a t c h i n one of
them there is a translate in the other. Obviously this is an equivalencerelation. Remark/4 : Every two tilings in a repetitive family are p a t c h - equivalent. Remark 5 :
Let
~
be a repetitive tiling with the
R - constant
cl( 0'9 )
be the class of all tilings being p a t c h - equivalent to
cl(
~
is a repetitive family and its
~.)
A special species in
)
gular icosahedron in with the property
R--constant equals
~
[
and
~ ; then .
]E3 . (cf. ~14, section &2). Let X be a fixed re-
IE3 . We consider the family ~"~ of all tetrahedra
tlhat every plane containing a facet of
to one of the fifteen plane mirrors of every edge of such a T
X
T
T
is parallel
(cf. figures 4 and 5 ). Hence
is parallel to some axis of rotational symmetry of o
X . If the dihedral angle equals
90
, it is parallel to one of the axes
565
3
2
The twofold a x i s i s o r t h e g o n a l to the p l a n e mirror
Figure
4
of twofold symmetry
( red
edges ). Similarly to these fifteen directions there are ten directions for axes with dihedral angles of o
60
or
120° ( g r e e n )
and six directions for the remaining
edges,
(white)
at which the dihedral ano
gle is a multiple of 36 |t turns out, that there are exactly fifteen similarity classes of such tetrahedra. Each of them possesses F~ure
5
at least one edge with a n o n o
primitive
angle
(36
o
, 60
o
and
90
are p r i m i t i v e ) . Through an edge
with a non-primitive angle one m a y cut the tetrahedron in two smaller ones by a plane, which is again parallel to a plane mirror of the new tetrahedra will again belong to
X ; hence
~.
]t turned out, that the four tetrahedra A, B, C, K
given by the table be-
low allow an inflation. Their shapes as well as the inflations are shown in Figure 6. Here tex labelled
A 3a stands for the facet of
A, that is opposite to the ver-
3a • This labellin E is not the same as in the table, but m a d e
up in such a way, that at every vertex of the filings we shall construct, all vertices of the tiles meeting are labelled by the same number. The letters
a
and
e.
and
b
only distinguish the two ends of the red edges of
The inflationfactor ~ equals
q~ : =
save space the small triangles in figure 6
566
1/2 (I + ~ are denoted by
A ,B
) . In order to B2
instead of
Tetrahedron
E d g e I - 2
2 - 3
3 - 1
2 - 4
I - ~
3 -
A
36 0
a
60 ° " ~ b
72 ° -6 a
108 °
a
900
I
60 °
B
36 0
36 ° 2" a
60 ° -6 b
120 0
b
1080
"C ~ a
90 °
1
C
36 0
a -4 11: a
60 ° "6 b
90 °
120 0
b
72 °
a
36 °
a
K
36 0
60° b
72 ° ~ - { a
900
I/2
a
Edgelengths:
a :=
114
~"
900
I0 + 2 ~
"6"/2
~-951057
b
90 ° "6-4/2
, b :=
112
.~-4 B 2. As the labelling of the vertices in the small triangles show, inflation permutes
[%/
I,A.I i~ i 14
V~E
the indices 1, 2, 3 cyclically.
..x
"~'3
Blllt
..-7
' ~ - K ~ ,"'~
-
l-.~I IL z ~ d .
If
~.e
,,,,,a,; ,,-<,-, ,,, given below su~mari~esthe i n f l a t i o n as it ! , o w s . in how many pieces o, ,,hat k i n d s every prototile is d , s sected. For e x a m p l e
the third c o l u m n
567
m e a n s ~C C =
A0 COC~,
10l
.
0 3 2 6
M:=
0 2 1 4
1 0 2 2
0 1 0 1
One crucial fact about the family ~ i : = { A, B, C, K~ is, that in this special case the matching rule (~)
defined by inflation is equivalent to a LMR,
namely : Only
wellorien
ted
tilings are permitted.
That is to say : Not arbitrarely congruent copies of the prototiles are allowed, but only those, which belong to ~
(cf. the beginning of this section ).
(A I) in turn is equivalent to the following rule : , (
) ~'~
i
If none of the three edges of a facet Z~ of a tile
T
is red,
match facet. Else the other side of z~itmaYthe mirroranYimageC°ngruentof T must be placed.°n
Since all entries of some power of the inflation-matrix
M
(in fact of M 3 )
are strictly positive,
~i
(D)
sp ( 7 I, A 1 ) . So we may apply theorem 1 and get
is satisfied by
theorem
is minimal. Finally it can be shown, that also
2 :: a) sp( ~ I ' AI ) satisfies ( I, D, M ) and hence also ( R ). The inflation--factor is "C, the R-factor at most 40 2). In theorem I we can choose
J =3
•
b__Z) Because of the red edges
(cf.
(~122 every member
of this species
splits uniquely into octahedra < A > , < B > , < C > , < K > , where < K > consists of eight copies of K , while the others contain four congruent tetrahedre each. c__~_}The vertices of ~hese tflin6s fall into four classes, as described above ( cf. figure 6 ). d__Z} There are exactly three members of sp( ~ 1 " A1 j which are 61obal]y symmetric with respect to the full icosahedral group of order 120. By inflation and subsequent expansion they are permuted cyclically. e__~} There are exactly 27 vertex- stars, that can be built up by copies of A, B, C, K and satisfy (A I } 3). Only 22 of them can occur in a global ti]ing. There is only one vertex-star in class ~. Every other vertex-star after at most six inflations is turned into one of the three with full icosahedra] s y m m e t r y - g r o u p ( c f . d]). With respect to the well-known channelling effects in physics the folowing theorem due to Th. STEHL]NG may be of some interest. 2) I conjecture the minimal R - factor to lie in the interval [5; 3) A vertex-star is a patch with exactly one interior vertex.
568
7 ~.
Theorem 3 : S u p p o s e ces of
~
dron
63~ sp [ ~ I
" A1)"
Consider the s e t
i n c l a s s 2 a n d choose a w h i t e a x i s
L
V
of all v e r t i -
of the i c o s a h e .
X .
a } Then the set of all lines parallel to point of
V
L
a n d passing through some
is d i s c r e t e ; every such line contains infinitely m a -
ny points of
V . { Hence the six bunches of such lines will show up
in the FOURIERtransform
of
~ .)
b ) Similarly the set of all planes orthogonal to some point of
V
L
and passing through
is discrete.
When the tiles are m a r k e d by their intersections with these planes and the corresponding planes with respect to the other white axes of X, all tiles belonging to the same prototile are m a r k e d equally. (A 1)
is equivalent to the L M R : These markings
shall fit together
as to form full planes. This theorem establishes the threedimensional
analogue
AldMANN -- bars of the planar PENROSE--tilings Probably
the species
following way: gral vectors (rood 2 )
sp ( ~r 1 ' AI ) can also be described by
Consider the lattice (Xl . . . . .
and
x6)T
xI + x2 + •
by the vextors
(0 1 T"
(-I -~;
onto
0 "C
1
t h e r e i s a t i l i n g c o n g r u e n t to 2 6~ IE i ' m /-~ =
C 1 + IE3 , where
C1
regular dodecahedron.
SPM
, . , ~x 6 ~
~
in the
(z)
with
O) T and
in
O
]E3 be s p a n n e d (I" 0 - i
and q~
IE3 . Then to e v e r y t i l i n g
of c l a s s I i s of the form
is unique since
x 2~m
. + x 6 ~--- 0 (rood /4). Let
]E6
I0] ).
2 D 6 which consists of all inte-
IE d" be its orthogonal complement
o r t h o g o n a l p r o j e c t i o n from sp ( ~ r 1 ' A1 )
F' : =
, where x I ~
i 0 "c~)T
(cf. ~8] or [11] ), let
every vertex
to the well-known
(cf. [9, Chapter
IE3
0 r
be the
~
of
such that
z e [" . This v e c t o r
{ 0 3 . These p o i n t s seem to l i e i n a s t r i p
i s some p o l y t o p e i n
IE J', whose c o n v e x h u l l i s a
Similarly the vertices of class 2
q~ ( P + z 2 ), while class 3 is in are appropriate vectors of ZZ 6.
of ~
q'6 ( ~-~ + z 3) , where
are in
z2
and
z3
5 . ) Some more p r o p e r t i e s of a p e r i o d i c s p e c i e s : We b e g i n with a w e a k e r v e r sion of
(R) : f The same as
(W
~
)
(R),
except t h a t
~- may d e p e n d on
~
( b u t not
L o n a n y t h i n g else ) .
The c o n s e q u e n c e s of
(R)
mentioned above remain v a l i d .
569
In case the species
is defined by a LMR, remark
2 implies:
If a tiling is built up adding
step
by step one tile after the other, one has a choice infinitely often. But it may happen
-- and in all known
does happen
-- that a patch
cases of LMRs that force aperiodicity, it
~
, after it was enlarged by a tile in accor-
dance with the L M R , no longer can be extended to a global tiling mind:
sp( ~r , C )
is defined as a family of g l o b a l
non-locality. More
(R) ). This property we call
f
sp( -~ , C )
tilings and so is
precisely :
non-local,
is called
iff to every radius
(keep in
6" there ex-
l ists an extentable patch ~ and l tile T , such that ~4u (T} ~a (NL,
o ~bs tei ly l s ~
]
C,
but is no longer
extendable and there is a patch
[
~
, still obeying
~,tainins ~ u ( T
C
con-
and
. 6" m d Z
Obviously we have
Remark 6 :
If
sp( ~ " ,
C)
is weakly r e p e t i t i v e ,
and the patch
~
vertex of T
, then the ~" in
can be placed in a ball of radius (NL)
We also should mention PENROSEs
'For eve
(P
~LJ
is at most
~
, centered at a
~ • ~(
notion of non-locality
~
).
(cf.
[13, Chapter 2]):
positive o,: there are p tches
,.,
fT,.l"
such that all three are extendable to global tilings of the given species, but . A u { T 1 , T 2 }
is not and
It is still not quite clear, how the atoms in a ted. Therefore I only
(i )
~
is called
q u a s fpe
it satisfies ( R )
i ( ii ) its
( or
quasicrystal
C~.
are loca-
r i odi
c , iff
( w R ) respectively ) and
FOURIERtransform
-- consists of D I R A C -
) ~
d i a m ( T 1 ta T 2 ) >
s u g g e s t the following definition :
A species
(~
Figure 7
~-s only (,, is strictly point " ),
-- these being located exactly at the points of a finitely
I
generated
I
-- the
L
-- M
7Z-module
generators of is everywhere
M
M
What is the relationship between
Q3 :
Is
~I
' A4
~,
) quasiperiodic
570
of generators
d ).
Q i: Does
Q 2 :
sp(
over
dense (hence the number
exceeds the dimension 6.) Open questions and theses.
,
are independ
(R)imply
(NL) ?
and
(NL)? (PNL)
(If not:does ?
Q ?)
n
Q &:
Consider the following property,
(9
~#~
{ The same as a
(D
W
(D),
translate
str~ger than
(D) :
but now the tiling (~ is required to be
of ~
~).
) is met by the two planar PENROSE-tilings
with global
as well as by the three tilings mentioned in theorem 2d. tors are
Z"- and
~3
respectively.
Does
(D)
imply
D 5 - symmetry
The inflation-fac(D*)
in the sense,
m:
that Q5 :
q
is replaced by some power Does
(5PM)
It does not imply VILLE-numbers
imply (R)
may
(wR)
q
.
? Does
1 , k 1 ) satisfy
sp(~
(D)?
?
(because
LIOU-
play a rSle ).
Thesis 1 : Instead of studying t i l i n g s , we should rather consider graphs.
DELONE--
I call a graph in
straight edges of bounded
DELONE-graph (
]Ed with
length a
se~figure 8 ), iff
( i ) its vertices fall into finitely m a n y classes V I , V 2 . . . . Vk , ( i£ ) every V• forms a ( r, R ) system
42
(in the sense of DELONE ).
The r61e of the prototiles may be play-
ed by the rooted subgraphs of graph2 (top of figure 8 ) (Cf. [19] ). theoretical radius Thesis
2 :
If the
might consider
answer
to
Q1
is in the negative
t w o--I e v e 1 matching
not be extended, beyond the point second--level
MR
x
(as
x
to
I assume ),
rules : If e.g. in figure 7
we
~r can-
according to the first--level M R ,
should allow to go on. (Presumedly
back -- tracking from
Figure 8
the
nature will not do
T .)
Thesis 3 : Given a patch of a member o f a repetitive species it is always possible, to decide
(in a finite procedure ), whether it is still extendable,
after some tiles have been added : Compare ches of one known
the new patch with all subpat-
extendable patch of appropriate
radius
(cf. remark 6 ) .
There should be simpler =t-it~ria ! I like to express m y gratitude to Dr, F. GAHLER, ful discussions and to K.P. NISCHKE
Gen~ve
for the program
for several fruit-
for figure 5.
4) In other words: Not only the species, but every particular tiling has to be similar to itself.
571
Correction : and
y
In the definition of (R)
have to be v e r t i c e s
as well as in figura3
the points
x
of the tilings.
REFERENCES [I]
ROBINSON, R.M.: Seven polygons which admit only nonperiodic tilings of the plane. Notices Amer. Math. Soc. 14 (1967), 835.
[2]
GUY, R.K.: The Penrose pieces. Bull. London Math. Soc. 8 (1976), 9 - I0.
[3]
PENROSE, R.: Pentaplexity. Eureka 39 (1978), 16 - 22.
[4)
De BRUI]N, N.G.: Algebraic theory of Penrose's non-periodic tilings. Nederl. Akad. Wetensch., Proc.Ser. A 8~ (1981), 39 - 66.
ES]
KRAMER, P.: Non-periodic central spacefilling with icosahedral symmetry using seven elementary cells. Acta Cryst. A 38 (1982), 257-26~.
[67
KRAMER, P. and R. NERI : On periodic and non-periodic space fillings on IEm obtained by projections. Acta Cryst. A 40 (1984), 580-587.
[7~
SHECHTMAN, D., I. BLECK, D. GRATIAS and J. W. CAHN : Metallic phase with long-range orientational order and no translational symmetry. P h y s . Rev. L e t t e r 53 (198~), 1951 - 1953.
[8]
KATZ, A. a n d A. M. DUNEAU : Q u a s i p e r i o d i d p a t t e r n s s y m m e t r y . ] . P h y s i c s 47 (1986), 181 - 196.
and icosahedral
['9~ GRONBAUM, B. and G.C. SHEPHARD : Tilings and Patterns. (Freeman, New York, 1987). [i0~ 11] [12J
STEINHARDT, P.J. and OSTLUND, S. (Editors) : The Physics of Quasicrystals. (World Scientific, Singapore, New Jersey, Hong l~ong, 1987). WHITTAKER, E. ]. W. and R . M . W H I T T A K E R : Some generalized Penrose patterns of n-dimensional lattices. Acta Cryst. A 4~ (1988), 1 0 5 112. KATZ, A.: Theory of matching rules for the 3-dimensional Penrose tilings. Commun. Math. Phys. 118 (1988), 263 - 288.
13
JARIC, M . V . ( Editor ) : Aperiodicity and order, Vol. 2 : Introduction to the mathematics of quasicrystals. (Academic Press, 1989 ).
14]
DANZER, L.- Three-dimensional Analogs of the Planar PENROSE T i l i n g s a n d Q u a s i c r y s t a l s . D i s c r e t e M a t h . 76 (1989), 1 - 7 .
--[15J GODR~ECHE, C . : The S p h i n x : A l i m i t - p e r i o d i c t i l i n g of t h e p l a n e . P h y s i c s A; M a t h . Gen. 13 (1989), L l 1 6 3 - L I 1 6 6 .
J.
~161
SENECHAL, M. and J. T A Y L O R : Quasicrystals: The view from Les HOUCHES. Math. Intelligencer 12 (1990), 54 - 65.
|17~
BAAKE, M., D. JOSEPH, P. K R A M E R and M. S C H L O T T M A N N : Root lattices and quasicrystals. To appear in ]. Physics A .
18J [19]
KASNER, G., D. W E G E N E R and H~J BOTTGER : Investigation of Danzer's Three-dimensional Quasicrystal'. Preprint. DANZER, L.: A Local--Global Theorem for DELONE--Graphs. ration.
Address of the author :
In Prepa-
M a t h e m a t i s c h e s I n s t i t u t d e r U n i v e r s i t a t Dortmund Postfach 50 05 O0 D - - 46 Dortmund 50 B. R. D.
572
DERIVATIVE MOUFANG TRANSFORMATIONS E.Paal Dept. of Mathematics, T a l l i n n Technical U n i v e r s i t y 1Akadeemia tee, 200108 T a l l i n n , Estonia A Moufang loop [1,2] i t y element
e
i s a quasigroup
G with the two-sided i d e n t -
in which the Moufang i d e n t i t y (ag)(ha) : a(gh)a
holds. The ( f i r s t )
d e r i v a t i v e loop
GaI
of
G
is defined [2] as
the
Moufang loop with the d e r i v a t i v e m u l t i p l i c a t i o n (gh) a' := (g~)(ah) where
~
denotes the inverse element of
transformation group of a set of X be denoted as g -4 Tg of G i n t o
X
, a
in
G. Let
and l e t the i d e n t i t y
Tr(X)
be the
transformation
E. A p a i r (S,T) of the mappings Tr(X) is said [3] to be an action of
g -4 Sg , G on X
if I)
Se = Te = E
2)
SgTgSh = SghTg ,
3)
SgTgTh = ThgSg
and
are s a t i s f i e d f o r a l l b i r e p r e s e n t a t i o n of
g,h G
of
(in
G. The p a i r
(S,T)
is c a l l e d also
x -4 gx := SgX
,
x -4 xg := TgX
(x ~ X; g E G) are c a l l e d G-transformations of X. For a f i x e d element a of G, the ( f i r s t ) d e r i v a t i v e of a b i r e p r e s e n t a t i o n
(S,T)
of
g - 4 (Sg)~ := TaSgT8 G
into x
i g - ~ (Tg) a := SATgSa
,
Tr(X). The transformations -~
( g x } a'
:=
( S )g'
a x
(S,T) aI
G can be defined as the p a i r of the
mappings
of
a
T r ( X ) ) . The transformations
,
x
573
-~
( x g ) a'
:=
(Tg)' a x
(x ~ X; a,g E G)
are c a l l e d the d e r i v a t i v e s of G-transformations of I (gx) a
l ° The d e r i v a t i v e s a,g ~ G)
I (xg) a
and
of
gx
and
xg
X.
(x E X;
can be r e d e f i n e d by (gx)~ = (g~)(ax)
and
(xg) a ' = (x~)(ag) ,
respectively. 2 ° The
( g x ) ai
derivatives
(xg) aI
and
(x E X; a,g E G)
obey the
identities I (ag)x = a(gx) a
I (ax)g = a(xg) a ,
I x(ga) = (xg)Aa
3° (S,T)
is closed under the double d e r i v a t i o n : !
!
((gX)a) ~ = (gX)ab for a l l
x
in
I g(xa) = (gx)aa .
,
X
and
I
and
g,a,b
in
((Xg)a) ~ :
I
(xglab
G. This p r o p e r t y can be f o r m a l l y
expressed as ((S,T)~)~ : !
4 ° The d e r i v a t i v e
(S,T) a
!
(S,T)ab
of a b i r e p r e s e n t a t i o n
(S,T)
of
G
turns out to be a b i r e p r e s e n t a t i o n of the d e r i v a t i v e Moufang loop of
G. 5° Every b i r e p r e s e n t a t i o n of the d e r i v a t i v e Moufang loop
turns out to be the d e r i v a t i v e of some b i r e p r e s e n t a t i o n of In view of G
I Ga
4°
and
are n a t u r a l to c a l l
l ° - 5°
of d e r i v a t i v e
!
Ga
of
5° , the d e r i v a t i v e s of b i r e p r e s e n t a t i o n s of
its
d e r i v a t i v e b i r e p r e s e n t a t i o n s . The p r o p e r t i e s
G-transformations are in good accordance
with
the ideas of V.D.Belousov [ 2 ] .
REFERENCES I.
R.H.Bruck. A Survey of Binary Systems. Third E d i t i o n . B e r l i n Heidelberg-New York,
1971.
2. V.D.Belousov. Foundations of the Theory of Quasigroups and Loops. Moscow, "Nauka", 1967. 3. E.Paal: Trans. I n s t .
G
G.
of Phys. Estonian Acad. Sci.
64 I04 (1989).
574
6__22142(1987);
ON THE EXPLICIT FORM OF CONSISTENT ANOMALIES*
J.A. DE AZCARRAGA AND J.~vl. IZQUIERDO
Depar~amento de Fisica TeSrica and IFIC (CSIC) Universidad de Valencia, ~6100-Burjasot (Valencia), Spain
Abstract We show that the two most frequent expressions for the anomalous commutators can be both derived from quantities associated with the Wess-Zumino-Witten action.
1. I n t r o d u c t i o n It is known that the differential geometric methods are useful in the description of chiral anomalies (see, e.g.,Ill ). In particular, the method of cohomological descent of Stora and Zumino allows us to compute the expression of the Schwinger terms from the form which defines the corresponding two-cocycle and which is obtained by descending from the Chern-Simons form t21 [3]
However, a two-
cocycle is determined only modulo a two-coboundary and, accordingly, the form of the Schwinger terms is not unique. The anomalous commutators may also be obtained by field theoretical methods in a number of ways; in that case, the calculations seem to favour one specific expression for them. The explicit form of the anomalous term in the Gauss commutators (we shall use commutators rather than Poisson brackets and restrict ourselves here , Talk delivered at the XVIII Int. Colloquium on Group Theoretical Methods in Physics, Moscow, June 1990. Work partially supported by CICYT (Spain) under grant # AENg0-0039
575
to D = 4) is usually given by Gab or Gab, where
Gab(x, y) = ~
Tr({Ta, Tb}Tc)e ijk Oi6(x - y)OjACk(y)
,
(1.1)
tz 2
Gab(X, y) =4-~r 2 eijk Tr{[Ta, Tb](CgiAjAk q- AiOjAk q- AiAjAk)
(1.2)
+ Oi(TaAjTbAk)} 63(x - y) Cohomological arguments show that the difference between (1.1) and (1.2) is given by the two-coboundary (Ac°)ab generated by the zero component of ca~(A) = 4--~2e P Tr{Ta(AL, OpAa + O~,ApAv + A,,ApAv)}
(1.3)
Using differential geometric methods, the form (1.1) has been given in t'l ta] ; (1.2) has been derived, e.g. , in tSl
Perturbative and functional theory arguments
however, tend to select I6j (1.2), although (1.1) was given, e.g., in trl In recent years, the Wess-Zumino-Witten (WZW) term ISJ has been proposed as an effective action to describe the anomalies; Faddeev has also suggested that the anomalous theories might be consistenly quantised tgl . It is thus interesting to know what field quantities, obtained from the W Z W action, lead to (1.1), (1.2). Since (.1.2) can be derived from ittl°l
, and the known c o [(1.3)] cannot
be obtained from a local functional of A and U, U being the additional field in the WZW action, it would seem that this action is limited to reproduce the untilded form of the Gauss commutator. Our purpose is to show that this is not
the case and to identify the quantity which relates the above commutators (1.1), (1.2) from the WZW action. In the following, canonical quantisation will be used throughout.
576
2. The Gauged WZW Model Non abelian chiral anomalies [1] arise in theories where a Weyl fermion field is coupled to a YM gauge field; the gauge s y m m e t r y of the action, which is also the expression of a redundancy in the description of the system, is lost in the q u a n t u m theory. This is so because the fermionic path-integral measure in the functional Z[A] is not invariant EllJ under the gauge transformations A~ =
g-lA~g+g-lO~g ' ¢ = g - 1 ¢ , g(x) = expOa(x)Ta, 5A~a =O~Oa +C~cA~b c. In terms of W[A] = -ilogZ[A], the gauge variation Z[Ag] = exp(ial[A, g])Z[A] is written locally as W[A g]-W[A] = al [g, g] (rood 2~'). The variation has to satisfy the WZ consistency condition, al [Agl , g2] - a l [A, glg2]+al [A, gl] = 0 (rood 2~r). If the action of the gauge group on the field U is given by g : U ~ g-lU = U g , $¢b = O=Ya~(¢) , (where Ya = Ya~b (¢)~-~, a [Ya, ]~] = C'~b~) then the WZ condition may be rewritten with gl = g, g2 = g -1U as al[A g, Ug]-al[A, U]=-al[A,g]
(rood 2zr).
Thus, an action defined by al has the same transformation properties as the chiral functional W[A], and it will generate anomalous commutators. We analyse now whether a model based on c~1[A, U] can be used to reproduce the different expressions for the anomalous Gauss commutators (1.1), (1.2). This is the case for (1.2); we now identify in the W Z W action the quantity that leads to (1.1). Let us write the total action for the 'gauged' W Z W model. The first part is the gauge invarlant term _To= ~ fd4xTr{VZU\7zU-1}; VzU = (0~ + Az)U. T h e pure ('ungauged') W Z W term is not s~rictly invariant under G [~21 , and cannot be gauged by simply replacing 0z by ~Yz; moreover, the 'gauged' W Z W term is meant to reproduce the gauge variance of the chiral functional.
The
term I w z w is given by the integral of the form w~(A, U) which is obtained from
dwl(A, U) = ~uw5(A) where ws(A) is the Chern-Simons form[2) : Iwzw[A, U] _ 48~r 2-ih / d4xe~VP~Tr{(O~A, Ap + A~O~Ap + Ai, AvAp)a~M
577
- 2 A~avAvaa - Agavapaz }
ih
/ d ~ T r e , ~ p ~ {a~a~apa~a~ } '
240r 2
(2.1)
B where ag = OgUU -1 and OB = M . By construction, the functional (2.1) has the right transformation properties. The total action I is completed by adding to the previous terms the YM part I y M = ½ f d4xF~,vF #v, where F = dA + [A, A]. The Gauss constraint is the equation of the motion for A°a(X). If the total action I is split into I y M + I ' , the equations of the m o t i o n are
51
G~a - 5A~ - O ~ F : ~ +
r~, ab ~vg ~ab~,'~
5I' _ avFVag + jag = 0 + 5Aag
(2.2)
where c3,F~" oi = cgiE~ and G O = -cgiE~ + J°a . Explicitly,
fvc L-~vtt~=, ab + T2 mzdVg¢cYa d Gga =OvF: # + ~,=b='~ ih ~, 4~rie ~ P Tr[Ta'(2avApa~ + 2avapA~ + A~Apa~ - A~apAa
(2.3)
--t-avApAa-avApaa -- avapaa -- avapAa -- Avapaa)] Although using (2.2) we m a y conclude t h a t j v is a conserved current, it is not the Noether current.
In fact, if under a gauge transformation of p a r a m e t e r s
On(x), 5Z = f d4xt?a(x)Ra(x), where Ra(x) is the anomaly, then we find without using the equations of motion t h a t j $ - Jag - Aa~ = ca~ + Orbt'", where b,v=-b~,t, , and t h a t O~,(j~a - J~)=Ra';
thus, O~,c~=Ra'. If we now c o m p u t e the anomalous
Gauss c o m m u t a t o r s [Ga(x), Gb(y)]~o=,o, the result is t h a t the original algebra is modified by the addition of the t e r m (1.2)
. However, if we now consider the
c o m m u t a t o r s of the zero component of Ga~ =- cOt,F~" + jga
--
cOvb~'~
"~-
Gga + C a", ,
i.e., if we use essentially the conserved Noether current, we obtain precisely the
578
commutator (1.1)
J~
(see(1.3)).
~ abZ-Xux.c
The Noether current is given by
--~
--
48~r2Tr { Ta(-AuOpAa - O~ApA~
.I
- A v A p A a + auapAa - auApaa + Avaoaa - auapaa + AuApaa+ avApAa - AuapAa)}
, (2.4)
so that A~U(A)= ~ 2 e m ' ~ T r { T a ( A ~ O p A ~ + O,,ApAa + AvApA~)}
ih uvp~ +2--~20,{c
Tr[Ta(Apaa
-
apAa)]}
(2.5)
;
the last term in (2.5) corresponds to a trivially conserved superpotential (which, we note in passing, is absent in the D = 2 case) and has been ignored in Gau. We can now evaluate [Ga, Gb] by replacing again the momenta by their canonically associated operators. From Ga -- Ga + co we obtain [Ga(x), Gb(Y)]~o=~o = ihCC=bGc(x)6(x
-
Y) -F Gab(X, y) -F (2XC)ab(X,y)
(2.6)
Since we are ignoring the last term in the expression (2.5), ca0 does not depend on av and then the action of Ga(x) on cb(y) is reduced to the action of Xa(x) = -~hViT-~ . Thus, Gab reproduces (1.1) . Explicitly, •
~2 (2xc)ab(x, y) -- 4Sir 2 eiykTr[2{Ta, Tb}OiAj(y)O~k6(x - y)
- ([Ta, Tb](AiAyAk + OiAiAk + AiOjAk) + Oi(TaAjTbAk))6(x - y)]
(2.7) ,
so that the freedom (1.2) ,(1.1) given by the cohomological descent procedure corresponds in the gauged WZW model to modifying the Gauss law by using the
Noether current; then, the tilded Gauss law generators lead to (1.1).
579
References 1. S.B. Treiman, R. Jackiw, B. Zumino and E. Witten Eds., Current Algebra and Anomalies, World Sci. (1985) 2. R. Stora, in Progress in Gauge Field Theory, G.'t Hooft Ed., Plenum (1964) and in New Perspectives in Quantum Field Theories, J. Abad et al. Eds., World Sci. (1986); B. Zumino, Y.-Shi Wu and A.Zee, NucI. Phys. B239, 477 (1984); J. Mafies, R. Stora and B.Zumino, Commun. Math. Phy, 102, 157 (1985) 3. B. Zumino, Nucl. Phys. B253, 477 (1985); R. Jackiw, in Ref. 1 and in Modek of Field Theory, vol. 2, I.A~ Batalin et al. Eds., Adam Hilger (1987) 4. L. D. Faddeev, Phys. Left. B145, 81 (1984); L.D. Faddeev and S.L. Shatashvili, Theor. Math. Phys 60, 770 (1985) 5. B. Zumino, in Ref. 1; T. Pujiwara, S. Kitalcado and T. Nanoyama, Phys. I, ett. B155, 91 (1985) 6. S. Jo, Nucl. Phys. B259, 616 (1985); Phys. Left. B163, 353 (1985); K.Fujikawa, Phys. Left. B171, 424 (1986); M. Kobayashi, K. Seo and A. Sugamoto, Nuel. Phys B273, 607 (1986); A.Yu. Alekseev, Ya. Madaichik, L.D. Faddeev and S.L. Shatashvili,Theor. Math. Phys. 73, 1149 (1988); F.S. Otto, H.J. Rothe and K.D. Rothe, Phys. Zett. B231, 299 (1989); Y.-Z. Zhang, Phys.tlev.Lett. 62, 2221 (1989); C. Schwiebert, Phys.Lett. B241, 223 (1990) 7. L.D. Faddeev and S.L. Shatashvili, Phys. Left. B167, 225 (1986); H. Sonoda, Phys. Left. B156 ,220 (1985); Nucl. Phys. B266, 410 (1986) (see also A.J.Niemi and G.W. Semenoff, Phys. Rev. Left. 55, 927 (1985)) 8. J. Wess and B. Zumino, Phys.Lett. B37, 95 (1977); J.Wess, Acta Physica Austriaca Suppl. IX, 494 (1972); E. Witten, Nucl. Phys. B223, 422 (1983); Commun. Math. Phys. 92, 455 (1984) 9. L.D. Faddeev in Superstrings, Anomalies and Supergravity, G.W. Gibbons and al. Eds., Cambridge Univ. Press, (1986) 10. R. Percacci and R. Rajaraman, Phys. Left. B201, 256 (1988); Int. J. Mod. Phys. A4, 4177 (1986) 11. K.Fujikawa, Phys. Rev. D21, 2848 (1980) 12. see in this respect J.A. de Azc£rraga, J.M. Izquierdo and A.J. Macfarlane, Ann. Phys. (N. Y.), July/August 1990
580
BERRY PHASES A N D WYCKOFF POSITIONS FOR ENERGY BANDS IN SOLIDS *+ J. ZAK Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel and Materials Science Division, Argonne National Laboratory, Argonne, IL 60439 USA
As was shown by Berry [1] the wave function of a time dependent physical system acquires a geometric phase under an adiabatic and cyclic variation. Soon after its discovery, Berry's phase was measured in numerous experiments in different fields of physics [2]. Two ingredients appear in the definition of Berry's phase: adiabaticity and the existence of a parameter space. Thus, in the elementary example of a spin 1/2 particle in a magnetic field B, one has the following situation: adiabaticity is satisfied when the period T of the B-rotation is much larger than the period z of the Zeeman splitting 'c = 2~ (he0 is the Zeeman splitting); the parameter space is given by the direction angles of the field B. In the band structure of solids, adiabaticity is satisfied, when the time dependent perturbation is slow, e.g., when the frequencies corresponding to the relevant energy gaps are much larger than the frequencies in the Fourier transform of the perturbation. Since in solids, the energy bands form a piecewise continuous spectrum, the k-vector in the Brillouin zone can serve as the parameter space for the definition of Berry's phase [3]. In a periodic solid, k is a conserved quantity, and the Bloch function ~nk( J ) is specified by a band index n and k. By applying a perturbation one can make k vary on a closed path in the Brillouin zone, and ~nk( F ) will acquire a Berry phase. The latter can assume in general a value between 0 and 2~. However, when the solid possesses some symmetry, Berry's phases can become quantized and they assume discrete values when k varies along vectors of the reciprocal lattice K [3]
581
On the other hand, it is well known that energy bands can be generated by using Wannier functions which are defined with respect to discrete centers in the Bravais lattice. These discrete centers are called Wyckoff positions and they are the symmetry centers in the unit cell of the Bravais lattice [4]. The Wyckoff positions are used for labeling band representations of space groups [5,6]. In this talk we show how the Wyckoff positions are connected to Berry phases for energy bands in solids. The discussion is restricted to simple energy bands (in a simple energy band one Bloch function corresponds to each k-vector in the Brillouin zone). Let Gw be an isotropy group of the Wyckoff position -~. An element (3'/~) of Gw when applied to ~ gives
(3,fi)
We
w -- =3'w _. + t
(1)
(7/t)
define a Bravais lattice vector --w
in such a w a y that
= e~/o
.-. + ~
--.
(2)
In the case of a simple band, we have for any element of Gw the following relation [5] (y/~)e ~'I~
Un (3,'lk-*,~ = D(J)(3,)Un (y'lk,q)
(3)
where Un(k, q) is the periodic part of the Bloch function ~nk ( J ), D(J )(3') is a one-dimensional representation of the point group of Gw, and where the subscript and superscript of R were deleted for simplifying notations. Berry's phase in solids, for an energy band n and a K-vector of the reciprocal lattice, is defined in the following way [3]
582
~n(k)= fK Xnn (-k)"dk (4)
where the integration is along the K-vector, and where
Xnn(k)=if(u2 ¢
Un(k,q)4Un(k',q)dq 3k
(5)
In Eq. (5), fl is the volume of the unit cell of the reciprocal lattice, and where the integration is over the unit cell in the Bravais lattice. From Eqs. (3) and (5) we find Ok' ~ (~,)
(k) = -~ + --=
Ok
(6)
where k" = ? "lk- and where the second term is a diadic product. This means that the i-component of both sides of Eq. (6) can be written as follows
Xi(k) = -Ri + __3k'. X (k') 3ki
(6a)
where now the re is a scalar product on the right-hand side. By using the definition of Berry's phase [Eq. (4)], we find
n
-
n
~"
"
583
(7)
This is the main result of the talk: Eq. (7) gives a connection between Berry's phase on the K-vector path in the energy band n, and the Bravais lattice vector R which according to Eq. (2) can be used for determining the Wyckoff position W.
As an example, let us consider the space group F222 (#22). The Wyckoff positions corresponding to maximal isotropy groups [3] are as follows [4] (Ux, UY, U z are rotations by ~ around the x, y, and z-axes correspondingly)
= (0, O, 0); Ga: E, U x, U y, U z
= (o, o, ~ G,: ~, (ox/ooc), (uY/oo~), Tjz ~-C,_ b. :I.G <."E, ( U x / 0 ~, , .,. ( U Y / ~ 0, ~
~4"4" 4T
-
"~,
--
2 2f~
U z -"h0
)
(8)
"2
The F222-space group is face-centered orthorhombic with unit vectors of the Bravais lattice
(9)
and unit vectors of the reciprocal lattice
~=(-~' ~ ' ~,~,~-(~,-~,~,~=(~ ~,-~ b a'b
584
(10)
The band representations for the Wyckoff positions in Eq. (8), all correspond to simple energy bands. Let us apply Eq. (7) to the Wyckoff positions ~ and b in Eq. (8). For ~, the R-vectors [Eq. (2)] are zero for all the symmetry elements. Correspondingly, from Eq. (7), one finds
}#)(~)
For the b-Wyckoff position
=
~#) (~,)
~#) (~,)
=
=
o
(11)
Rb is zero for U z and is -(00c) for U z and UY.
Formula (7) gives a set of equations for determining the Berry phases of the energy bands that are built on the Wyckoff position b. We have cc"1 = U x or U y in [Eq. (7)]
u x ~ =-(~1 + ~,2 + ~3), u XE2 = ~,3, u x ~ = ~-2 (12)
By using the fact that ~n(-K) = -~n(K) we find the following set of equations (we use Eqs. (7), (12) and the value of the Rb vector for U x and U y [Rb = -(0,0,c)])
[~) (K2)- ~n(b) (Y'.,3)= 2~
i3~) (~q)- ~ ) (K3)
=
2~
The solution of these equations is as follows
585
~n(K1) = ]3n (K2) = -[3n(K3) = ~
(13)
In a similar way, one finds the following Berry phases for the Wyckoff positions c and d. We have
2
(14) =
£,2)=
2
Let us remark that the choice of the group F22 was not accidental. In fact, there is a very special reason for choosing this group, and it is as follows. In one-dimensional crystals, it was proven many years ago [7] that there is a oneto-one correspondence between the Wyckoff positions (these are 0 and a/2, where a is the lattice constant) and the symmetries of the Bloch functions in the Brillouin zone. It turns out that such a one-to-one correspondence is, in general, correct also for simple bands in three-dimensional solids. There are, however, exceptions, and one of them is the F222-group. Thus, it was recently found [6] that the a, b, and c, d Wyckoff positions, in pairs, lead to identical symmetries of Bloch functions in the Brillouin zone. These pairs of Wyckoff positions constitute therefore an example, when Wannier functions around different symmetry centers in the unit cell of the Baravais lattice, lead to identical symmetries for their Bloch functions. It was recently shown [8] that despite of them having identical symmetries, the Bloch functions ~(ka) and ~ ) for bands of the type a and b (belonging to Wyckoff positions a and b) cannot be connected by a continuous k-dependent phase factor when the crystal becomes infinite. A similar situation prevails for the Wyckoff positions c and d. What this means is that topologically the bands of type a and b are different (also c
586
and d). In this talk we show that the bands, a and b (also c and d) have also different Berry phases [Eqs. (11), (13), and (14)]. The discussion in this talk was restricted to simple bands. The latter correspond to a single Wyckoff position. In general, for composite energy bands, there is a star of Wyckoff positions that appear in their definition via band representations [5]. Little is known about the topology of composite bands, and it should be of much interest to connect their Wyckoff positions and Berry phases.
References
1. M.V. Berry, Proc. Roy. Soc. London, A392 451 (1984). 2. I.J.R. Aitchison, Phys. Scr. T23 12 (1988), and references therein. 3. J. Zak, Phys. Rev. Lett. 62, 2747 (1989); Europhysics Lett. 9 615 (1989). 4. International Tables for Crystallography, Edited by T. Hahn (Reidel, Dordrecht, 1983). 5. J. Zak, Phys. Rev. Lett. 45 1025 (1980); Phys. Rev. B25 1344 (1982); Phys. Rev. B26 3010 (1982). 6. H. Bacry, L. Michel, and J. Zak, 16th International Colloquium on GroupTheoretical Methods in Physics, (1988). 7. W. Kohn, Phys. Rev. 15 809 (1959). 8. H. Bacry, L. Michel, and J. Zak, Phys. Rev. Lett. 61 1005 (1988).
587
CRYSTALLOGRAPHY OF QUASICRYSTALS: THE PI:tOBLEM OF I:~ESTOlt,ATION OF BROKEN SYMMETI:tY
V. A. Koptsik Dept. of Physics, Moscow State University, 117234, Moscow, USStt,
Abstract. - In the paper, the concept of fuzzy packings of the structure units have been introduced which throw a new light on the crystallography of quasicrystals. With this concept one constructs the sructure of 2D-quasicrystal on a base of pentragrid h'om the rhombi on one kind which constitute a lacy cover of the plane. The space symmetry of such a structure is a positional colour groups isomorphous to wreath product of two generalized groups of the prototype phase ( T 1 x T 5 x T 52 x T 53 x T s ' ) ~ 5 m T1
r 7'1 x T 2,
T 5
= (T1 x T 2 ) C ~ C l o , , = T l w r ~ C , o , ,
,
= T2 x rT3, T 52 = 7'3 x T,l, T 53 = T,t x T s , T 5~ = T5 x T1
describling the symmetry of a long and a short orders of the pentagrid correspondingly. The same is true for the symmetry group of the Penrose pattern which is dual to those pentagrid, its transformation being positional rigid translations and rotations accompanied by the appropriate local transformations of the inflation-deflation types rearranging the internal structures of the "physical points", the smallest domains of the pattern pocessing by the pentagonal colour point symmetry group. It is shown that colour and nDapproaches constitute the isomorphic languages of description of the icosahedral quasicrystal phenomenon in the frame of common Y ( n ) = V(3)E@ Y(d)l
588
construction of nD-crystallography. Some physical consequencies and related topics of the colour approach are considered. 1. Introduction At the XVIII International colloquium on group theoretical method in physics (Moscow, June 4 - 9, 1990) there were delivered some papers on the theory of the quasicrystMline state of the matter, which is the problem of the relations between n-dimensional (nD) and 3D crystallography (n -- 3+d > 3). In [1] the authors use the technique of the group representation theory for the consideration of different possible embeddings into 6D periodicae space. In [2] the direct methods of constructing the nonperiodical tiling of 2D and 3D-space was used. In the present paper we reconstruct the hidden space symmetry of quasicrystals in the terms..of the positional colou,'ed groups which axe nothing else than isomorphous representations of nD-space groups but more suits to the quasicrysta~ structure. 2. Equivalence 9 f parents phase a~ld superspace restoration It is known that the strip projection method is one of the most effective one for constructing the quasiperiodical structures if the orientation and the thickness of the scrip for projecting nD-regular strip lattice structures onto 3D space axe appropriately chosen. (The necessary references may be found in [1,2]). At the same time the changing of ordinary orthogonal projection onto the generalized one [3] allows one to restore the broken periodicity of quasicrystals at the level of coloured positonal groups isomorphous to the nD-space groups in the full correspondence with the principle of conservation (undecreasing) of abstract symmetry for the isolated physic~ systems in the course of their structure transfomrations [4]. (We shall deal with this subject in the next item of the paper). Tha~ the nD supercrystal may be considered as the regular prototype of the parents phase for the quasicryst~ and instead of projection one may consider the appropriate structure phase transition. The p,'ocedure of recovering of the parent phase structure aJad symmetry was developed by Dr. A. Talis in his Thesis [5] at the first time for the polysystem molecular crystals processing by the local symmetry which don't belong to their space symmetry group [6, 7]. The inversion to the parentz phase restoration is a positional
589
phase transition which illustrated there by the schematic examples of Fig's la, b (which are Fig's 2.4 and 2.6 of Dr. A. Talis Thesis). The parents phase structure shown at Fig. la manifests three levels of orderness. At the molecular level one can see pentagonal molecules orderd in triplets by the local axes of the three-fold rotation 3(~) = t(~)3(6')t-~(~) which in turn are ordered according to the transformations of the space symmetry group P4. As the result the parents phase structure maps onto itself by the twisted and iterated wreath product of the Q-type (Wrq), [8, 9] 5' -- [5m(75, +,51)Wrq3(~x)]WrqP4 where ~ + 7~i is the positional address of the first pentagonal molecule and 5m = Cs. is its point symmetry group. Let us admit that the phase transition of the paxents phase is accompanied by the clockwise and counter-clockwise rotations of molecular triplets #(r'3) and #(r'4) correspondingly, by the local similarity deformation of the claster ~ ( ~ ) and by the average homogenous deformation ,~ = (,~(~) of the whole structure which transforms the circle into the ellipse and the tetragonal unit cell (UC) into the monoclinic ones (Fig. lb). Than the space groups S goes into S ' = ~)(r-')S~)-~(~. It may be shown that under the action (×) of si E S the arbitrary point
of the parents phase goes into the point Yj%l)d -- $1~jn,k
or explicitly
•
--,
1
-.
11
..-,
×
11x -~
1
•
t
Ii
-.
-'
*j
1
-,
= "-+
jn
11
= etct(Pl)Clm~(?'ll)P1161"111-- Vj'nGV,
¢i, jn
P~¢~ ,
Cj,
¢~EP4; 11
c~,~, cl,~E3C3rn=C3,~Bci~*j ;
P11~E 5m = Cs~ ;
t(7~),
590
t(~'11) e T(2)
where the multiplication is performed in the natural sequence from the right to the left and the internal structure of the positional operators is shown by the brad, ets [a(... b{[c(.., ded-1...)c-~]c}b-~...)a-~]a, [¢,(...)¢r ~] mea,~s the positional loads of ¢i E P4, E(2) being the Euclidean group in 2D and T(2) its translation subgroup. Fl'om the determination of the action the multiplication law for the twisted and iterated wreath product is followed [6, 7]. 3. Coloured translation symmetry of the Penrose pattern With such a fund of knowledge and examples we can proceed now to the problem of quasicrystals. I start with 2D-quasicrystals (QC). The well known example of 2D-QC is Penrose filings of the plane by a system of two (thid~ and thin) rhombi. Let us take for the starting physical point the regular decagon consisting of 5 thi~k and 5 thin rhombi(Fig. 2a). If we accompany the translation rT, by the reconsruction of internal structure of the decagon it will be the colour translation quite analogous to the case of magnetic symmetry [3]. This reconstruction are performed by the local transformation of the inflation and deflation type [1,2]. If one imagine that the second pattern is superimposed into the first one and shifted at the vector g, than for the coincidence the structures of both physical points it is necessary to eraize some of the lines and to draw the other ones only. If one performe the translation 2g, the list of local transformation will be different. If the algorithm of transition fl'om one physical point to another is established than the list of local transformations of internal structures of physical points will be in the one-to-one correspondece with the so-called "matching rules", which aa'e used for the constructing of the Penrose pattern [1,2]. This list may be enormously big and has no practical use for the determination of the positional colour space group of the Penrose pattern, but for the physical applications it may be limited by the boundary conditions. For the symmetry applications in the macroscopic physics there is no necessity to determine the full list of the local colour transformation. It's enough to determine the list tbr the finite fragment of the Penrose pattern possessing by the constant density of some quasicrystal physical characteristic which we wish to determine. The situation is quite analogous to the case of imperfect crystals which possesses by the constant desity of vacancies [3, 10]. It is more useful to know that such positional colour space group does
591
exist in principle. In the accordance with the principle of abstract symmetry conversation [3,4] we can choose for such a group the junior color group ¢(3) w(d) C P wr q¢(3) ,
q~(3) ~-~ q~(3)~p(d)or~(3) ~q(d) ~-~ q~(3 4- d)
which is isomorphous to the space group ~(3) in 3D and ¢ ( 3 + d ) in nD-space at the same time. In order to determine the colour group ~(~) explicitly we shall go fl'om the Penrose tiling to its dual which is the pentagrid construction. 4. Point groups of the positional rotations for the 2D-pentagrid The procedure of restoration of broken symmetry of quasicrystals has much in common with that of incommensurate crystals. We shall consider that procedure for two-dimensional pentagrid construction, which is the direct product of five 2D t r a n s l a t i o n groups, T 1 x T s x T s2 × T 53 x T 54, T 1 = T1 × T2,..., T s4 = T5 x T1. The 1D lattices Ai = Ti~'l corresponding to the T~(i = 1,... ,5) are situated at the same 2D plane. They a~-e rotated relatively each other at the angles multiple 72 ° and didn't intersect each other in one point in three. We constructed our's pentagrid in accordance with the intersections scheme of J. E. S. Socolar and P. I. Steingardt shown at Fig. 5 in [11]. A fragment of such a pentagrid is presented there at Fig 2b and the immediate vicinity of one of its nodes at Fig. 2c. It is important to note that the nodes of pentagrid are not dimensionless geometrical points as in the case of crystal lattices, but small irregular pentagons of finite dimensions with its own boundary structure and site symmetry. The rotational symmetry of the pentagrid looks ruther unusual. It is described by the systems of the restricted by one operation generalized fivefold axes, each of them being splitted into five separate axes, 5modS~ = {5 I×2, 52×3, 53x4, 54×5 , 5 s×I }, localized at the vertexes of the fixed pentagons. At every vertex it is a11owed to perforin only one power of the positional rotation. 51×2 -- (51×2) -t = 52xI etc., which brings the lattice AI into A2 and A2 into At, etc. So, we define a generalized rotation 5rood52 as the simultaneous rotation by 72° around the local components 51x2~5 2x3, 53×4~ 54x5, 5 5×1 of this operation. The multiplication law will be the component-by-component multiplication
5~,od~ =
{(51×~)~,(5~×~)2,(5~×~)2,(5"×s)~,(5~×~) ~} =
592
= {5 z×3, 52x4, 53×5, 54×z, 55×2 } which gives a new system of the rotation axes localized at the vertexes of the five-pointed star shown at Fig. 2c. Having in mind that the second power of the 5~od52 operation is accompanied by chaalging of points of rotation one can represent the above-mentioned multiplication law by the multiplication of the appropriate substitutions of the pentagrid line numbers 5modS~ ×5rood52 ~
2
4
3
5
4
1
5
2
3
4
5
=
2'
Proceeding in the same way one receives 53od52 = {51×4,52X5 53xl ~4X2~,'j~5X3"tf,'Jmod52~4 __-- {51X5,52X1,53X2,54X3,55X4} 5 and the defining Equation 5roodS2 = 1 for the pentagrid cyclic symmetry group 5mod52 ----- {1,5rood52, 5rood52 2 3 4 , 5rood52,5rood52 } •
Let us notice that the local counter-clockwise rotation at the mutual supplementary angles of 72° and 288 ° are performed around the local axes situated at the vertexes of the pentagrid corresponding to the components of the operations 5rood52 and 5rood52. The same is true for the rotations at 144 ° 2 3 and 216°performed by the components of the operations 5rood52 and 5rood52 correspondingly, but the rotations centers are situated now at the vertexes of the five-pointed star. In splitting of single classic rotation centers into dozens local ones it's manifested themselves the unusual fuzzy (dispersion) character of the pentagrid rotational symmetry. The pentagrid T = T 1 x T 5 x T 52 x T 53 x T 54 does conside with itself as a whole under the dispersion rotations around the generalized axes 5rood52. But the simultanity of rotations around local components of the axes 5rood52 is no more unusual than that one in the case of the exchange magnetic symmetry. It is known that in exchange magnetoorderd crystals the magnetic moments (spins for shortening) of atoms rotate
593
simultaneously with the temperature or they change simultaneously its signs under the action of the time inversion operator 1~ at the every sites where magnetic atoms are situated. In another less exotic but equivalent representation of the pentagrid rotational symmetry to every single 5-fold axes located in the bari-centers of the pentagonal nodes there are assigned the appropriate positional colour loads. This loads are interpreted as local shifts of the pent&grid lines by -4- A ¢'i towards or away from those centers. Thus, under the rotatins by 72° around the central axis of Fig 2c the stright line 1 - 1 standing at the distance I gl I from the ba~'i-center goes to the position 2 ' - 2 ' , which doesn't coincide with the straight line 2 - 2. The coincidence may be achived by a local shift of 1D-lattice 2 ' - 2 ' at a distance - A E~ towards the pentagon center. Under the same rotation the pentagrid line 2 - 2 goes to the position 3' - 3' (not shown at Fig 2c), whirl1 does coincide with the line 3 - 3 after the local shift by + A e'3. The positional symbol of the colour rotation may be written as 5 (-a~2'+a~3'°'°'°} in the sequence of vectors ~'1, e'2, 6"3,E4, ~'5 • The positional action of 5('") at Ei may be defined as follows:
The representation of those operation by the positional substitution of the states of the pentagrid lines will be
5(-A~2'+Z~"3'0'0'0) ~
1
2
3
4
5
-- & e2
0
0
0
0
2
3
4
5
1
~
1
2
3
4
5
2
1
1
1
1
2
3
4
5
1
"
Here the first row shows the initial positions of the pentagrid lines, the second one shows its deviations from the lines of the regular pentagon, the third one shows the same for its final states, and the last one gives the new postions of the pentagrid lines. Having designate the states 0, - A e'2 of the lines by the figures 1, 2 one receives for the second and the fifth powers of the operation
[5('-)]2~
(12111)(12111) (12111) 1
2
3
4
5
1
2
3
4
5
2 2
1 3
1 4
1 5
1 1
2 2
1 3
1 4
1 5
1 1
594
=
1
2
3
4
5
2 3
1 4
1 5
1 1
1 2
etc., ...,[5("')] s = 1 in full correspondence with the previous description. Therfore the both descriptions are equivalent, 5roodS2 ~-* 5 (-h*2'+n¢3'°'°'°). 5. Generalized .space symmetry of the 2D pentagrid The result of our consideration in the previous section states that the pentagrid T = T 1 x T 5 x T 52 x T s3 x T 5~ is coincided with itself as a whole under the automorphic transformations 5modS2T(5mod52) -1 = T or 5(~)T5 (~)-1 = T where (w) stands for the positional colour loads ( - / x ~'2, + / k F3,0,0,0), I/~g3 [=1/xg2 I. But 5,~oa2 or 5 (~) will be only subgroups of the complete point symmetry group 5mod52mor 5(W)m(W) correspondingly because of the inversion i and the mirror reflection m which are the symmetry operations of the pentagrid. Remind that for 2D space i = 2 = C2, the rotation around two fold axis. Then 5 = 5i = i5 is equivalent to 10 = C10, the rotation around ten-fold dispersion axis and one will receive in the equivalent representation 10,~odlo2 or 10(~°)m(~) for the pentagrid point symmetry group. For the receiving of the generalized pentagrid space symmetry group ¢(~') = T(~)G (~) it's sufficient to show that
and
T (:) f~(:~) ~ ° m o d 5 2 ra(::)(6~ ~ /) T <:>-1 --- $~ox)5:m('°:)(r~
having in mind the appropriate colour loads (w) of the combined symmetry operations (... Ckplq~;1... [¢i) (~) e (I)('°) , Ck e (~ C (I) ~ (I)(~) , P i e P , i. e. the positional shifts of the selective systems of the pentagrid lines. Then this space group will be the twisted wreath product of two generalized groups because the conjugated copies of 2D translation subgroups T i = giTlg~ -1 are enumerated by the elements of the subgroup gi E 5rood52 C 5,~od5~m . Let us use now the "empty" pentagrid as the support of the decoration by unit cells in order to pass to the quasicrystal structure. For such UC's we shall take the least rhombi at the intersection of two pairs of adjacent pentagrid lines, the starting rhombus being attached to the vertex 51×2 of the small pentagon at the center of Fig 2b. Using the other components 5 2xa , ... , 55×1 of operation 5rood52 one recieves the central flower consisting of five rhombi, then using the self- similarity property of quasicrystal one goes from the zero pentagon with the embedded central flower to the first
595
enveloped pentagon in which five similar flowers are situated. At the third step one goes to the second enveloped shell of flowers of the decagonal shape by the translations of the starting (zero) flower at five UC's parameters along ten 4-4 directions of the pentagrid lines. As the result of such a process one receives the lacy packing of the rhombi gathered in flower clusters in the 2D plane. The complete structure of decorated pentagrid will be the unity of discrete orbits of the coloured positional subgroup of the generalized space symmetry group ¢(~) = T(~')G (~) which was established for the undecorated pentagrid, each flowers being translationaly and rotatinaly equivalent in such orbits in the colour sence. The fuzzy character of the lacy packing manifests themselves by the overlapping of UC's in some places, in the absense of their strict congruency, in the zigzag-like colour periodicity along the pentagrid strip of the doubled thickness and in the presence of unshadowed holes, the summary amount of which may be diminished by inserting the other "glued" UC's in such holes. Centering at the every UC's the circle of appropriate (fuuzy) radius one may proceed to the atomic decoration of the pentagrid network and after the calculation of Fourier spectru,n go to the comparison of the theory with the diffraction experiment. At the next level of structure one may pass to the tiling of 2D plane by the enlarged unit cells (EUC's) which are the deformed hexagons with the internM angles of 72° and 144° and the equal edges. Because of the quasiregular fuzzy decoration of EUC's by UC's the deformed hexagonal domains doesn't lose the quasiperiodical underlying structure at the periphery of Fig. 2b as it does in the central tilings [12]. In the conclusion of the section let us state that the group theoretical techniques for the symmetry groups of the decorated pentagrid may be developed by analogy with geometro-geometrical interpretation of the twisted and iterated symmetry groups of the polysystem molecular crystals (see section 2). Having no space for demonstration of the multiplicatin law in the terms of cumbersome positional operators let me only refer to the analogy of those groups with the classic cubical and hexagonal space groups which
596
have the structure of twisted wreath products
where G~b = 23, m3, 7~3m, 432, m3m, Ghez = 3, 32, 3 m , . . . , 6 / m , 6 / m m m . Then the (T x x T s x T ~2 x T 5~ x T 54)(~°)R,-l'OVC~(~°) group of the decorated pentagrid is the natural generalization of the constructions as stated above. Conclusion The generalized symmetry of the icosahedron quasicrystals and its decahypergrid duals may be described in (3 + d)D colour space by the color positional subgroups ¢(3) w(d) of twisted and iterated wreath product groups isomorphous to the prototype symmetry group (T 1 x T 3 x T 32 x T 2 x T 23 x T232)X~53m ~ ¢(3 + d) ,
32C53m
in the same way as we deals in the previous sections with the symmetry of Penrose pattern and its dual pentagrid. What of orbital structures will be really preferable it makes no difference for the macroscopic physics of quasicrystals. In accordance with the abstract symmetry conservation principle some or other group of such a type does exist mad by virtue of the damn of isomorphous ( ~ ) , and homomorphous (~---), relations y ~ ~ y ~ ¢(3) ~ ¢ ( 3 F (d~ ~ ¢(3 + d) = ¢(~,),
J => d > 3
the physical properties of quasicrystals may be described by the tensor (V ~) and vector (V) representations of the ordinary space groups ¢(3). At the same time only ¢(3) ~(a) groups will be in full correspondence with the real structure of quasicrystals whereas ¢(3 + d) groups may be considered as the symmetry groups of diffraction patterns or the symmetry groups of prototype crystals fl'om which quasicrystals are originated through the structure phase transitions. The fuzzy character of superspace symmetry elements was discovered by P. M. Zorkii with the colaborators at the first time for the polysystem molecular crystals (see bibliography in [13, 14]). The fuzzy set approach to quasi-symmetric systems in different aspect was recently developed in [15].
597
Acknowledgements This work was completed during nay stay at the Faculty of Mathematics of Dortmund University by the invitation of Prof. Danzer with the finaalcial support of DAAD organization. I wish to express may gratitude to Professors L. Danzer, P. Kramer, A. Dress, F. G&hler, H. Wondratschek, Th. Hahn, H. Jacobs, B. Harbrecht and their colleagues for the fl'uitful discussions azxd the assistaalce. I am much indebted to S. Sdm~idtke for retaping the text and to Klaus-Peter Nischke for computer grafic of Fig. 2b and preparing of the text. Figures Descriptio n Fig. la. The parents phase of molecular clasters /~(r~) maps onto itself by the iterated wreath product S = (5m(r"l + ~'n)wrq3(r~))wrqP4, only indexes of vectors ~, ~j, ~jk being shown. Fig. lb. Superspace molecular crystal is originated from the parents phase of Fig. la through the structure phase transition, the order parameter being the positional field distorsion tensor 7}(7v) (see explanation in the text). Fig. 2a. Fragment of Penrose tiling of the plane as the model of 2D quasiperiodic crystal. Fig 2b. Fragment of 2D pentagrid decorated by the rhombi gathered in pentagonal clusters (flowers), only 1-1 and 2-2 grid lines being shown. Fig. 2c. On the explanation of the global rotational pentagonal symmetry of 2D pentagrid (see the text). REFERENCES
[1] M. Baake, P. Kramer, Z. Papadopolos and D. Zeidler: this volume; [2] L. Danzer: this volume; [3] V. A. Koptsik: Comput. Math. Applic. 16,407 (1988);
598
[4]
V. A. Koptsik: J. Phys. Clfi, 23 (1983);
[5]
A. L. Tails: On the theory of colour (super-Fedorov) symmetry of the space modulated crystals and modulus structure. Thesis. Moscow University (1989);
[6]
A. L. Tails and V. A. Koptsik: In the book "Group theoretical methods in Physics". Eds. A. Markov and V. I. Ma~'ko, vol 1, 527 (1986) (in Russian);
[7]
A. L. Tails and V. A. Koptsik: Soviet Physics Crystallography 35 Nb. 6 (1990);
[s]
B. H. Neumann: Ar&. Math. 14, 1 (1963); Composito Math. 13, Fasc. I, 47 (1957);
[9]
N. S. Zefirov, L. A. Kaluzhnin and S. S. Trach: In the book "Investigations on Algebraic Theory of Combinatorial Objects" (Proceedings of Moscow Colloquium) Moscow (1985) (in Russian);
[10]
V. A. Koptsik: J. Phys. C16, 1 (1983);
[11]
J. E. S. Socolar and P. Steinhardt: Phys. Key. B34, 617 (1986);
[12] V. V. Manzhar: Soviet Physics Crystallography 35 Nb 4 (1990);
[13]
P. M. Zorkii and V. A. Koptsik: In the book "Modern problems of physical chemistr31". Moscow University, Moscow, vol 11, 113 (1979) (in Russian);
[14]
V. A. Koptsik: In the book "Problems of crystallography. On the 100years birthday of Acad. A. V. Shubnikov". Moscow, 69 (1987) (in Russian);
[15] J. Maruani and P. G. Mezey: Molecular Physics 69, 97 (1990)
599
A
t
Fig Fig.
]a
Fig.
2 a
600
1b
I.
4,
2.-.
"2
4
Fig. 2 b
.:2
Fig.
601
2 ¢
