The Conceptual Foundations and the Philosophical Aspects of Renormalization Theory

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THE CONCEPTUAL

SCHWEBER

FOUNDATIONS

THE PHILOSOPHICAL

ASPECTS

AND OF

THEORY*

RENORMALIZATION

1.

S.

INTRODUCTION

a representation for describing structure for the hierarchical

subatomic entities and as a frame that can be built from these entities nuclei, atoms, etc.), quantum field theory (QFT) (mesons, nucl?ons, a reductionist embodies view of science. Serious doubt has often been cast on the whole program, the foundations them particularly when selves were found to be in a state of confusion. During the 1930s and most of the 1940s the infinite results that occurred within the framework calculations made most physicists of QFT in higher-order perturbative of QFT.1 The doubts were dis doubt the stability of the foundations a a renormalization and for while after procedure was proposed pelled As

work

carried out in 1947-48 by Kramers (in Schweber, 1985), Bethe (1947), Lewis (1948), Schwinger (1948a, 1948b), Tomonaga (1946), and Feyn man (1948a, 1948b, 1948c), and spectacular successes were achieved to electromagnetic in explaining and predicting radiative corrections in QFT got a further boost confidence processes. More specifically, a proof of the renormalizability when in quantum of the S-matrix - the - was electrodynamics (QED) suggested simplest case of a QFT 1986a, 1986b). However, by Dyson (1949a, 1949b; cf. Schweber, Dy son's proof was not conclusive. Some loopholes such as those related to the overlapping and the lack of a rigorous proof of the divergences in each order - were of the renormalization convergence procedure

later closed by Ward (1950, 1951), Salam (1951a, 1951b), Weinberg (1960), and Mills and Yang (1966). Others have persisted, most notably, the nonconvergence of the power series used in the perturbative expan sion of the 5-matrix in terms of which Dyson's and Weinberg's proof - as are formulated (1952), Hurst (1952), and pointed out by Dyson Thirring (1953). Soon after Dyson's ments the stability of challenging

?

1993. 97: 33-108, Synthese 1993 Kluwer Academic Publishers.

Printed

serious argu proof was published, renormalization theory at various

in the Netherlands.

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- were advanced and conceptual mathematical, by physical, 1973a, 1973, 1969b, 1973b, 1983), Schwinger (1969a, (1970, (1953), Landau et al. (1954a, 1954b, 1954c, 1954d, 1956), 1983), K?llen and Pomeranchuck Landau and (1955), and others, (1955), Landau even by Dyson were various to himself. There the responses (1952) one of renormalization At end of the difficulties theory. conceptual were to who the axiomatic field theorists, spectrum sought clarify the a stable renormalized structure of QFT and to construct theoretical

levels Dirac

-

1964; Velo and theory of quantum fields (cf. Streater and Wightman, were Landau, At the other end 1973; Wightman, 1986). Wightman, who denounced the whole frame and other S-matrix theorists, Chew, not because its renormalized work of QFT, version, merely including the strong and weak interactions, of its empirical failure in describing but principally because of its conceptual 1990; instability (cf. Cushing, Cao, 1991). can one The basic question that had to be answered was: How account for the great empirical success of renormalized QED, a concep this paradox logician confronting tually unstable theory!2 A stubborn or renormalization field would probably either theory (or reject theory and Chew did such as Dirac, and some physicists Landau, both!), so. But most field theorists reasoned differently. Ignoring the stability that if meaningful cal in renormalization theory, they argued problem the framework of renor culations could only be carried out within should be malized theory, then in fact renormalizability perturbative on theory construction. It is a historical taken as a crucial constraint of QFT beyond the scope of QED fact that the further developments as a of the have been accomplished renormalizability using principle case most in is unified field The Weinberg's guideline. convincing point interactions. As Weinberg (1980a) remarked theory of the electroweak not he of in his Nobel if had been lecture, guided by the principle have his theory of electroweak interactions would renormalizability, vector not only from SU(2) x t/(l)-invariant received contributions - which were believed to be renormalizable, boson exchanges though not proven to be so until few years later by 't Hooft (1971a, 1971b) and Veltman, and others (Lee and Zinn-Justin, 1972; 'tHooft 1972a, - but also from et Becchi al., 1974) 1972b, 1972c; SU(2) x ?/(l) to be nonrenor invariant four-fermion couplings, which were known malizable,

and the theory would

have

lost most

of its predictive

power.

RENORMALIZATION

THEORY

35

this historical fact deserves reflection, Certainly philosophical parti to the criteria of theory appraisal, cularly with reference theory accep tance, and theory choice. nature and essential character the mid 1970s the fundamental During of the renormalizability As a result principle began to be challenged. of two decades of fruitful interactions between and statistical QFT of the theoretical certain foun mechanics, by understanding physicists a dational aspects of renormalization underwent radical transfor theory was the emergence mation. At the heart of the transformation of the new concept of 'broken scale invariance' and the related renormaliz ation group approach. Weinberg (1978) was one of the first to assimilate the physical insights developed by K. G. Wilson (Wilson principally context and Kogut, in of critical phenomena for 1974; Wilson, 1975) the existence of the fixed solutions of renormalization example, point in coupling-constant for trajectories group equations and the conditions - and to fixed the space passing through points apply them within context of QFT. His intention was to explain or even replace the a more with fundamental renormalizability principle guiding principle, which he labeled "asymptotical safety". Yet this program was soon to be overshadowed that of 'effective field theory' (EFT), also by another, initiated by Weinberg (1979, 1980b). At first, EFT was a less ambitious than that safe th?ories, be program by asymptotically encompassed as its conceptual cause EFT still takes renormalizability basis. EFT, a a thor to has led radical of with however, outlook, change together a examination of the of and clari very concept ough renormalizability fication of the ontological basis of QFT (cf. Schweber, 1993). to a comprehensive The present paper is an introduction account of the history of renormalization which the authors have under theory, taken and is still in progress. After the necessary background giving (Section 2), we examine the foundations (Section 3) and the philosoph ical aspects (Section 4) of renormalization the help of a theory. With of the nature of physical theories in general, and of the the hypothetico-deductive in method by QFT adopted case on more the this in the of bearings study general topics particular, of science, such as the criteria for theory acceptance and philosophy are versus and the of issue 'realism instrumentalism', theory choice, brief analysis structure of

discussed

in some detail

in the last section.

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BACKGROUND

Renormalization It can system. theory is a complicated conceptual be understood from various perspectives. the renormalization First, can be viewed as a technical device for circumventing procedure the infinite results that occur in QFT i.e., isolating and discarding the concept of renormalization calculations.3 Second, perturbative helps to clarify the conceptual to establish basis of QFT and, it is hoped, its can to status also be elevated the of stability. Third, renormalizability a regulative principle, and theory selection guiding theory construction within the general framework of QFT. of renormalization the emergence theory in the late Historically, difficulties of QFT. In its original 1940s was a response to the divergence renormalization in formulation, theory was technical and conservative to and it is these initial characteristics in character, important keep mind when trying to understand its further developments. It was techni nu cal because it involved a series of algorithmic steps for obtaining to that could be compared the theory, numbers the Lamb shift and for data, Retherford, (Lamb example, experimental moment of the electron 1947) and the anomalous magnetic (Nafe et was it took the framework It conservative because of QFT al., 1947). as given, and made no attempts to alter its foundations. In fact, Dyson as one of its endearing features its conservative character took a to introduction the frame Thus brief 1986a). (Schweber, conceptual work of QFT is in order. that obey equations QFT is a system consisting of local field operators re and anticommutation commutation of motion, certain canonical a and Hilbert lations (for bosons and fermions, space of respectively),

merical

results

from

state vectors

that is obtained by the successive of the field application vacuum to to which is assumed be the state, operators unique. Let us that are involved in greater detail. look at three of the assumptions (i) The locality assumption. a spacelike surface commute each other. The assumption

this asserts that field operators on (bosons) or anticommute (fermions) with is a legacy of the point model of particles

In QFT

of interactions and its description among them. At first glance, locality seems merely of the possibility to be a statement of the rejection to keep and a means the representation in of action-at-a-distance, an construe of the with But examination compliance special relativity.4

RENORMALIZATION

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37

tion of the point model of the electron reveals that it is also an attempt to resolve a difficulty in Lorentz's theory of the electron (1904a, 1904b). to in the field J. J. Thomson (1881), the energy contained According to e2l2a. Thus, when of a spherical charge of radius a is proportional the radius of a Lorentzian electron goes to zero, the energy diverges if is given a finite radius, then the repulsive But the electron linearly. the sphere of the electron makes the configur Coulomb force within was to the the sug ation unstable. Poincar?'s response (1906) paradox a force in that there exist cohesive gestion might nonelectromagnetic so to balance side the electron the Coulomb that the electron force, would not be unstable. Two elements of the model have exercised great that the mass of the influence on later generations: (a) the notion electron has, at least partly, a nonelectromagnetic origin; and (b) the notion that the nonelectromagnetic when interaction, compensative to combined with the electromagnetic would lead the ob interaction, servable mass of the electron. Thus, the point of departure of Stueckel

berg (1938), Bopp (1940), Pais (1945), Sakata (1947), andmany others in their studies of the problem

of the electron's

self-energy

is Poincar?'s

ideas.

The equilibrium of the Poincar? electron is not stable against defor out by Fermi in 1922 (cf. Rohrlich, mations. This was first pointed to the and this elicited another kind of response observation 1973), first stated Frenkel Frenkel that since the difficulty, by argued (1925). electron is elementary and has no substructure, the inner equilibrium of an extended is a meaningless electron the classical problem within framework. By adopting the point model, Frenkel eliminated interaction' the parts of an electron - and thus the between - but the 'self-interaction' he could not eliminate problem the point-electron and the electromagnetic field it produces The Maxwell's Frenkel left open theory. abandoning problem more acute when QFT came into being.

the 'self stability between without became

was quickly accepted by physi idea of the point-electron5 Frenkel's cists and became the conceptual basis for QFT. The idea of looking as Dirac for a structure of the electron was given up because, (1938, a too "the electron is for the question p. 950) suggested, simple thing of the laws governing its structure to arise". It is clear, therefore, that what is hidden in the locality assumption is an acknowledgment of our ignorance of the structure of the electron and that of other elementary entities described by QFT. The justification given for the point model

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the consequent

is that they constitute good, locality assumption at the in present experi approximate representations energies available that are too low to explore the inner structure of the ments, energies particles. When Jordan (in Born et al., 1926) (ii) The operator field assumption. and Dirac (1927a, 1927b) extended the methods of quantum mechanics to electromagnetism, were pro the electromagnetic field components moted from classical commuting variables to quantum mechanical oper ators. The same procedure to fields describing could also be applied fermions

1927; Jordan and Wigner, 1928; cf. Darri (Jordan and Klein, a These local field have direct operators gol, 1986). physical interpreta tion in terms of the emission and absorption and the creation and annihilation of the quanta associated with the particles. The creation of a particle is realized as a localized excitation of the vacuum. Accord a localized excitation ing to the uncertainty principle, implies that arbi are available amounts momentum of and for the creation energy trary to the vacuum of particles. Thus the result of applying a field operator a single particle, state is not a state containing but rather results in a of states containing that is arbitrary numbers of particles superposition of the relevant quantum numbers.6 constrained only by the conservation An operator field is defined by the totality of its matrix elements, it should be clear that an overwhelming of these refer hence, proportion to energies and momenta that are far outside experimental experience. status (iii) The plenum assumption of the bare vacuum. The ontological It of the 'bare' vacuum has been a widely discussed may be subject. Pauli recalled that Furry and Oppenheimer and (1934), Weisskopf to Dirac's idea (1943), and others raised objections (1934), Wentzel a vacuum state in which all the of the (1930) being (one-particle) energy states were filled. The strongest argument put forth negative was to special the the against following: according plenum assumption a vacuum zero must state be Lorentz-invariant of the energy, relativity, zero

momentum,

zero

angular

momentum,

zero

charge,

zero

whatever,

that is, a state of nothingness 1983, p. 69). However, (cf. Weisskopf, to be caused by the vacuum fluctu when certain phenomena supposed to the ations were analyzed, the very same physicists who objected vacuum as the took substantial, tacitly something plenum assumption or assumed it to be an underlying medium, namely as a polarizable

RENORMALIZATION

THEORY

39

In other words, the scene of wild activities. substratum, they actually adopted the plenum assumption. The local coupling among quanta and the fact that the application of field operators on the vacuum results in strictly local excitations imply one has to consider virtual processes that in QFT calculations involving except for the consequences arbitrarily high energy. However, imposed as unitarity, no em there exists essentially by such general constraints for believing the correctness of the theory at these pirical evidence at the inclusion of these virtual processes energies. Mathematically, that are obviously arbitrarily high energy results in infinite quantities are not external. They are undefinable. Thus the divergence difficulties are internal to the very nature of QFT: within the constitutive they In this sense the occurrence of QFT. canonical formulation of the struc divergences clearly pointed to a deep instability in the conceptual ture of QFT. In addition to various proposals for radically altering the foundations of QFT, two different responses were advanced to overcome this insta bility. The first one was developed by Pais and by Sakata, independently and was in the spirit of Poincar?'s solution to the stability problem of It put forth the idea of compensation: the Lorentz electron. fields of in such a way as to cancel the unknown introduced particles were divergences produced by the known interactions. The second response was the renormalization is the central subject of the program, which present paper. In the 1930s, Dirac (1934), Heisenberg (1934), Weisskopf (1936), Kramers (1938), and others had already put forth the idea of renor in terms of subtractions. malization But it required the precise experi mental findings on the spectrum of hydrogen and deuterium that were obtained and instruments using techniques during World developed War II to stimulate the further elaboration of this idea. The explanation of the accurate and reliable data obtained by Lamb and Retherford in their measurements of the fine structure of hydrogen and of Rabi's structure of hydrogen results on the hyperfine and deuterium became an outstanding In for theoretical the process they challenge physicists. for obtaining finite numbers for the measured developed algorithms in their and put forth suggestive QFT-based calculations, quantities ideas for justifying the algorithms (Schweber, 1986a). The ideas and algorithms Bethe, Lewis, by Kramers, developed can be summarized as follows: and Tomonaga Schwinger,

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terms that occur in the QED The divergent calculations in a Lorentzidentifiable and gauge-invariant manner, as modifying can be interpreted the mass and charge ameters that are introduced in the original Lagrangian. or renormalized, mass the modified, By identifying masses with the observable charge parameters physically

(i)

(ii)

are and par and and

are absorbed charges of physical particles, all the divergences into the mass and charge renormalization factors, and finite are obtained.7 results in good agreement with experiments Thus the measurements of Lamb and of Rabi can be ex QED. plained within the framework of (renormalized) The

crucial assumption was first expressed gram

the whole

underlying succinctly

by Lewis

renormalization

(1948,

mass of the electron is a small effect and electromagnetic from a failure of present day quantum electrodynamics cies. . . ,

The

...

arises

and somewhat on QED: Electrodynamics ably accurate those

aspects to modification

energy

more

fully by Schwinger

(1948a,

pro

p. 173): its apparent divergence above certain frequen

p. 416)

in his first paper

at ultra-high but is presum revision requires energies, unquestionably at moderate to isolate It would be desirable, relativistic therefore, energies. and are subject of the current involve high energies, theory that essentially

and are

theory, by a more satisfactory thus relatively trustworthy.

from

aspects

that

involve

only moderate

It is clear that only when a physical parameter (which when calculated is actually finite in perturbation in QFT may turn out to be divergent) and amalgamation into the 'bare' par and small can its separation ameters be regarded as mathematically justifiable. The failure of QFT as at ultra-relativistic in per indicated energies, by the divergences turbation theory, implied that the region in which the existing frame from the region in which it work of QFT is valid should be separated It is is not valid and in which new physics would become manifest. to determine where the boundary is, and one does not know impossible can to the that are not be used calculate small effects what theory from unknow in QFT. However, this separation of knowable calculable of a cutoff, is realized mathematically able, which by the introduction can be schematized that the parameters by using phenomenological must

include

these

small effects.

RENORMALIZATION

THEORY

41

nor Tomonaga made explicit use of a nor Schwinger to terms with corrections cutoff. They directly identified the divergent them from the expressions and removed and the charge, the mass and charges. By contrast, the masses for real processes by redefining cf. calculational efficient 1948c; algorithm (1948b, Feynman's a use on cutoff. relativistic of the explicit Schweber, 1986b) is based it which makes The latter consists of a set of rules for regularization, a and in to calculate physical quantities gauge relativistically possible in the limit but still results in divergent invariant manner, expressions a finite cutoff, this artifice as the cutoff mass goes to infinity. With of transforms essentially divergent quanti purely formal manipulations into quasi respectable mathe of parameters, ties, i.e., the redefinition of mass and charge, other matical operations. If, after the redefinition are insensitive to the value of the cutoff, then a renormalized processes Neither

Lewis

theory can be defined by letting the cutoff go to infinity. A theory is are sufficient to if a finite number of parameters called renormalizable one. a as renormalized define it to introducing cutoff is equivalent relativistic Feynman's Physically, to the infinite cancel an auxiliary field (and its associated particle) field. of the due to the ('real') particles contributions Feynman's original or compen approach is different from realistic theories of regularization and positive sation. In the latter, auxiliary particles with finite masses are are and described in to be observable assumed principle, energies that enter the Hamiltonian Feynman's explicitly. by field operators in the sense that: (i) the auxiliary masses theory of a cutoff is formalistic as mathematical are used merely which finally tend to parameters, are and in nonobservable and (ii) the coupling con principle; infinity stant associated with the auxiliary particle would be imaginary.8 Repre are the papers of Sakata (1947, sentative of the 'realistic' approach see and Sakata also Hara, 1947; Sakata and Umezawa, 1950; 1950), et and Kawabe, Umezawa 1949a, al., 1948; Umezawa (Umezawa as well as that of Rayski (1948). 1949b), and other Japanese physicists, to Feynman, Rivier and we in addition the 'formalists' find, Among

Stueckelberg (1948) and Pauli and Villars (1949). results showed that Feynman's It was Dyson who, as a synthesizer, formula were and from derivable and insights Schwinger's Tomonaga's a proof of was able to outline tion of QED. Furthermore, Dyson mass and meant that of QED. Renormalizability the renormalizability charge

renormalization

removed

all the divergences

from the 5-matrix

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to all orders of perturbation of QED that theory. He also suggested a well be renormalized QED might quasi-stable theory. Empirically, version of QED has enjoyed great success because of the renormalized the anomalous its astonishing both in calculating power, predictive moment and the Lamb shift in of the electron and magnetic hydrogen, to high energy electron-electron in estimating the radiative corrections In the late 1940s and early 1950s it was and electron-positron scattering. a the divergence difficulties, circumventing hoped that, by successfully and moreover, framework of QFT could be constructed, quasi-stable as Pauli suggested, fix the masses that it might and charges of the particles that appear in the theory. This turned out to be too optimistic, some advances toward a rigorous proof of the renormaliz although were of made. Crucial among these were: QED ability (i) (ii)

the solutions to the overlapping divergences given by Salam, and Mills and Yang; and Ward, which is necessary theorem of Weinberg, the convergence theories all the ultra for the proof that in renormalizable do cancel to all orders of perturbation violet divergences occurrence the of complicated theory, despite divergent subgraphs.

as a even though renormalizability had been accepted in renormalization works the of QED, why program actually property can This be divided unclear. remained QED quite question conceptually into two parts: However,

(i)

(ii)

Why do the apparent divergences arising from a failure of above certain energies unrenormalized QED actually give rise to small effects? of nature by renormalized theo Why are the representations ries stable, and more specifically why are they so very insen sitive

to whatever

happens

at very high energy?

some progress toward an answer to the second part of the no real insight made has been during the past two decades,9 question toward being able to give an answer to question has been obtained (i) during the more than four decades since Lewis first stated the smallness But perhaps an even more fundamental question regarding assumption. all the interactions in nature are renormalization theory was whether While

renormalizable.

RENORMALIZATION

THEORY

43

and Dyson was aware that the answer to the question was negative, so reported to the Olstone conference 1986a). Detailed, (Schweber, after Dyson's classic explicit, negative examples were given immediately For example, Feldman papers on renormalization appeared. (1949) are interactions of the vector meson observed that the electromagnetic out nonrenormalizable. Kamefuchi that the Fermi four (1951) pointed are also nonrenormalizable, and Peterman fermion direct interactions a that the interaction of and Stueckelberg noted magnetic mo (1951) ment term field with the electromagnetic Pauli of the form (a Thus the unavoidable is likewise nonrenormalizable. ques fif/a^i/jF^) tion arose: Should nature be described only by renormalizable theories? For physicists, such as Bethe, who had elevated from renormalizability a property of QED to a regulative principle guiding theory selection, et al., 1955). They justified the answer was affirmative (see Schweber their position in terms the aim of fundamental

of predictive power. They argued that since to is formulate theories that possess physics must laws" considerable "fundamental contain power, only predictive a finite number of parameters. theories are consis Only renormalizable tent with this requirement. While the divergences of nonrenormalizable them into appro by absorbing possibly be eliminated an number of infinite parameters would priately specified parameters, be required and such theories would initially be defined with an infinite in the Lagrangian. number of parameters appearing to the interaction Lagrang the renormalizability According principle, ian of a charged spin 1/2 particle interacting with the electromagnetic field cannot contain a Pauli moment. Similarly, a pseudovector coupling of the pion to the nucle?n was excluded. By the same reasoning, Fermi's interaction lost its status as a fundamental theory of weak theory. A more complicated was the of the renormalization constraint application theories

could

in the strong rejection of the pseudoscalar coupling of pions to nucl?ons was interactions. the renormalizable. Formally, pseudoscalar coupling was not realizable because the radiative correc Yet its renormalizability are too large to justify the use of perturbation tions it produces theory which is the only framework within which the renormalization proce re to the popularity dure works. This contributed of the dispersion and to the adoption of Chew's lations approach 5-matrix theory ap proach, which rejected the whole framework of QFT, by a considerable number The

of theorists.10 case of the Yang-Mills

field

(1954a,

1954b)

deserves

special

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in the original Yang-Mills attention. Physicists were interested theory even though to be renormalizable, it was conjectured in part because the massless bosons invariance could not be required by the gauge for nuclear forces. The massive version of short-range responsible was because the massive bosons gauge Yang-Mills theory unacceptable of the spoiled not only the gauge invariance but the renormalizability was as attracted the to well. Gell-Mann and tried find theory by theory a "soft-mass" mechanism that would allow the renormalizability of the of gauge boson masses, massless but theory to persist in the presence he did not succeed Where Gell-Mann 1987, 1989). (cf. Gell-Mann, failed, Weinberg (1967), Salam (1968), and Gross and Wilczek (1973a, and of renor 1973b) succeeded with the help of the Higgs mechanism After the apparent proof of equations, respectively. a the of latter theory became the renormalizability Yang-Mills theory,11 case of QFT and constituted an extension the paradigmatic of Dyson's a area. new into original program to claim that the most It would not be too great an exaggeration in QFT that have been achieved in the past four advances substantial the have been guided and constrained decades by renormalizability true that renormalization It is certainly theory saved QFT, principle. and allowed one to calculate with it, and thus made it manipulable, revived the faith of theorists in QFT (Schweber, 1986a). Be that as it as ever no consensus to been reached whether renormaliz has may, or a an universal is essential characteristic of QFT ability principle of nature. As a matter of fact, all the possible descriptions constraining since the early 1950s serious arguments have been advanced challenging the stability of renormalization theory and casting doubts on the foun of the dations of QFT. The debate has led to a deeper understanding to and has the of and renormalization, clarify helped philosophy physics of QFT. This can be viewed as another way in which the foundations renormalization theory has advanced QFT into a new phase.

malization

group

theory is certainly unstable due to the presence and by virtue of the infinities that stem from divergences the infinite volume of space time. The latter have to be disentangled as a candidate the Fock representation from the former and excludes relations. The occur form of the canonical commutation for the Weyl rence of these two kinds of infinities makes to define a it impossible An

unrenormalized

of ultraviolet

Hamiltonian operator, of QFT collapses.

and the whole

scheme

of canonical

quantization

RENORMALIZATION

THEORY

45

in renormalized theories is very different from The stability problem ones. The ultraviolet are supposed that of unrenormalized divergences to be circumventable the renormalization Some of the by procedure. as to such how define local fields and their equiv remaining difficulties, conditions and the associ alence class, and how to specify asymptotic in a rigorous fashion by axio ated reduction formula, were analyzed matic

field

theorists

with

the help of distribution theory and normed new Yet created 1989). problems by renormaliz on three that were advanced serious criticisms

algebra (cf. Wightman, ation theory invited different levels. At the mathematical level, Dirac (1969b) criticized renormalization a procedure of infinitesimals, for infinities instead theory neglecting at custom in the odds with usual mathematics. Lewis's small radically ness assumption seems to and Dirac's invalidate criticism, anticipated

but the assumption itself has to be justified in the first place.12 At the physical level, Heitler (1961) and others noted that the mass as of particles the differences (such pions and the nucl?ons), which are identical except for their electric charge, could not be calculated using that if the mass renormalization theory. It is not difficult to establish are of electromagnetic then the differences electro origin, divergent mass to will lead infinite differences. This difficulty magnetic self-energy that renormalization theory could not fulfill Pauli's clearly indicated theory to account for the mass hope that it would provide a general In addition, renormalization theory a framework to accommodate the as of the weak such important phenomena CP-violating representations interactions and the gravitational interactions. But the gravest defect of renormalization around 1970, when it was theory was made manifest

ratios of the "elementary was criticized as being

particles". too narrow

that it is in direct and irreconcilable conflict with the chiral recognized and the trace anomalies that occur in high orders of QFT, ironically, as a consequence of the demand of renormalization (cf. Jackiw, 1972). At the conceptual level, the stability of renormalization theory was and others. In 1953, K?llen Landau, K?llen, by Dyson, challenged claimed to be able to show that, starting with the assumption that all constants are finite, at least one of the renormalization renormalization constants in QED must be infinite. For several years this contradictory result was accepted by most physicists as evidence for the inconsistency was as out of QED. later However, by some critics (e.g., pointed on some notoriously et al., 1959), his results depended Gasiorowicz

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treacherous of the orders of integra arguments involving interchanges over an infinite number of states, and was thus tion and summation inconclusive. K?llen himself later acknowledged this ambiguity (1966). More serious arguments the stability of renormalization challenging in terms of the breakdown of perturbation the theory were expressed As is well renormalization known, Dyson's ory. theory was only formu lated within the framework of perturbation theory. The output of per a set turbative renormalization is of well-defined formal power theory series for the Green functions of a field theory. However, it was soon realized that these series - and in particular the one for the S-matrix - were most Thus theorists were thrown into a state likely divergent. of confusion and could not give an answer to the question: In what sense does the perturbative series of a field theory define a solution? to be disillusioned the first theorist enough, Interestingly by per was himself. In 1952, Dyson turbative renormalization theory Dyson that suggested that after renormalization gave an ingenious argument were divergent. The subsequent all the power series expansions dis cussion by Hurst (1952), Thirring (1953), Peterman 1953b), (1953a, field theorists added Jaffe (1965), and other axiomatic and constructive to the assertion that the perturbative further weight series of most even is field there still no com theories renormalized diverge, though cases. most in plete proof A divergent series for a Green function may still be perturbative to a solution of the theory. In the mid 1970s the existence asymptotic of solutions for some field theoretical models was established by con that the solution structive field theorists and these indicated a posteriori is uniquely determined by its perturbative (cf. Wightman, expansion were exhibited only for field-theoretic Yet solutions models these 1976). in space time continua of two or three dimensions. As for the more in 1952 Hurst had realistic four-dimensional QED, already suggested indicate that the excellent of QED with experiments may agreement series is an asymptotic the that the pertubative However, expansion. the behavior of of QED by K?llen, Landau, investigations high energy and especially and Low (1954), showed that the per by Gell-Mann in breaks down, ironically, as a turbative approach QED unavoidably and of charge renormalization. Landau of the necessity consequence his collaborators argued further that remaining within the perturbative to no interaction framework lead either would (zero renormalized

RENORMALIZATION

THEORY

47

of ghost states rendering the theory charge)13 or to the occurrence Both the results demonstrated inconsistent (Landau, 1955). apparently in renormalized of QED. theory perturbative inapplicability freedom in a wide class of non After the discovery of asymptotic in quantum Abelian theories, gauge chromodynamics especially was that the QCD would get rid expressed hope perturbative (QCD), thus eliminate most doubts as to the of the Landau ghost and would this did not last long. It was of QFT. However, consistency expectation soon realized that the ghost that disappeared at high energy reappeared were reminded at low energy (cf. Collins, Thus field theorists 1984). of the limits of the applicability of per forcefully and persistently turbative theory. As a result, the stability problem of QFT in general, and the consistency renormalization theory in problem of perturbative a was state in of uncertainty. particular, The attitude of theoretical physicists toward this issue differed sharp most For ly. stability is just a pedantic problem. practicing physicists, As pragmatists, and they are only guided by their scientific experiences have little interest in speculating about the ultimate stability of a theory. For Dirac the (1963, 1969a, 1969b, 1973a, 1973b, 1983), however, was to with renormalization the cutoff existing theory going infinity In his opinion, what was required illogical and nonsensical. physically were new forms of interaction and new mathematics, such as the pos or of sible use of an indefinite metric nonassociative (1942), algebra or perhaps even more The positions esoteric. (1973a), something and by Chew were more radical and drastic (cf. adopted by Landau were not What forms of Cao, 1991). they rejected merely particular and perturbative interactions versions of QFT, but also the paradigm set by QFT, the style of reasoning exemplified by QFT, and the general framework established For them the very concept of a local by QFT. field operator and the postulation of any detailed mechanism for inter were in a microscopic actions region spacetime totally unacceptable even in principle. to be observable, because these were too speculative Their position was supported by the presence of divergences in QFT and by the instability or even the inconsistency of the perturbative zero charge argument renormalization theory,14 even though Landau's could not claim to be conclusive. The most positive attitude was taken by the axiomatic field theorists, who

later called

themselves

constructive

field

theorists.15

In the spirit

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they tried to settle the question of the stability of and took this as the only way to give clear QFT by axiomatization, answers to conceptual problems. While Hilbert tried to legitimize the use of mathematical entities with a proof of the consistency of a formal system consisting of these enti field theorists went the other way round. They ties,16 the axiomatic tried to prove the internal consistency of QFT by constructing nontrivial a is of whose existence the axioms With alone. consequence examples overcome out radically altering the foundations to of QFT, tried they of Hubert's

tradition

its consistency, the apparent difficulties with step by step. Although remain of did they find any many open, nowhere important problems indication that QFT contained basic inconsistencies. The axiomatic field theorists took the fields to be operator-valued test functions of fast with distributions defined infinitely differentiable or at test decrease with functions infinity having compact support. of the physical this was a mathematical idea of Essentially, expression a thus defined theory may the exact point model. However, modifying still be nonrenormalizable in the sense of perturbation theory. That is, even no there is itmay still be an unstable though inconsistency theory involved in it. Since the mid 1970s, there have been major efforts using the approach of constructive field theory to understand the structure of nonrenor malizable theories and to establish the conditions under which a nonre normalizable theory can make sense. One of the striking results of this that the solutions of some nonrenormalizable theories have is enterprise a is number of This finite contrary to their arbitrary parameters. only series. It has been speculated in terms of the perturbative description to be renor that the necessity for an infinite number of parameters an come in perturbation from malized power theory may illegitimate the case that in series expansion (cf. Wightman, 1986). It is certainly field theorists have and the constructive the axiomatic these efforts and a flexible frame of mind. Yet future develop exhibited openness ments and proving the consistency in understanding the foundations in some involve and stability of renormalization changes theory may and that have not been that have not yet been challenged, assumptions axiomatization of the theory. In any case, the present captured by any a soluble four-dimensional field theory, despite failure to construct indicates that the axiomatic intensive efforts for nearly four decades, and the constructive field theorists are meeting considerable difficulty

RENORMALIZATION

THEORY

49

in solving the consistency sense, let alone problem of QFT in Hilbert's its stability. It also has dampened their initial optimism somewhat. The finitists, among whom are Salam (1973; Salam and Strathdee, of supergravity 1970) and the various advocates (e.g., Hawking, 1980) et and superstrings Green than al., 1987), are more optimistic (e.g., the axiomatic and the constructive field theorists. Their hope is that by in the existing formulations interactions of QFT including gravitational a to construct it will be finite without any systems, theory possible infinite renormalizations. They thus hope to solve the stability problem of QFT without It should, however, be involving any renormalization. recalled that hopes come and go, and moreover that they seem to be short-lived.

advanced contrary views on renormalization, respec Sakata and Sakata Umezawa, 1950; Sakata et tively by (1950, 1956; were expressed in terms al., 1952) and by Schwinger (1970,1973,1983), of their concerns regarding the structure of the "elementary" particles. For Sakata, renormalization theory was only an abstract formalism, behind which lay hidden the concrete structure of elementary particles. a His position was that when renormalization theory would encounter Two

additional

it would become neces (and its limitations would be exposed), more to to structure and the look for of the elemen sary analyze closely more were Under Sakata's efforts invested in influence, tary particles. in of the constituents of model-building elementary Japan particles than structure of QFT. As a result, little in the analysis of the theoretical theory as an essential concep emphasis was placed on renormalization tual ingredient of QFT (cf. Takabayasi, 1983; Aramaki, 1989). are of particular views of renormalization interest, not Schwinger's one is of the because he founders of renormalization merely theory, but principally because he has given penetrating analyses of the philos and is one of its most incisive program ophy of the renormalization to Schwinger, critics. According the unrenormalized which description, as its field local contains basis, operators adopts conceptual speculative defect

about the dynamic structure of the physical particles that assumptions are sensitive to details at high energy. However, we have no reason to believe that the theory is correct in that domain. In accordance with a Kramers's that should have QFT precept structure-independent the renor character, which Schwinger accepted as a guiding principle, malization that to very he elaborated removed reference any procedure high energy processes

and the related

small distance

and inner structure

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He thus shifted the focus from the hypothetical world of assumptions. to the observed world of physical localized excitations and interactions to proceed found it unacceptable in this tortu particles. But Schwinger ous manner of first introducing physically extraneous structural assump them at the end in order to obtain physically tions, only to delete a rejection of the philosophy results. This constitutes of meaningful But renormalization renormalization. is essential and unavoidable in a local operator field theory if the latter is to make any sense. In order to bring his criticism to its logical conclusion, introduced Schwinger sources and numerical valued fields to re numerically (nonoperator) sources symbolize These the interven place the local field operators. tions that constitute measurements of the physical system. Furthermore, all the matrix

elements of the associated fields, the operator field equa relations can be expressed in terms of the tions, and the commutation sources. An action principle gives succinct expression to the formalism. as to Schwinger, his source theory takes finite quantities According and it is thus free of divergences. This theory is also sufficiently new experimental to be able to incorporate results, and to it can do them in a reasonable manner. Most important, extrapolate so without to of extend the theory to falling into the trap having - which constitute domains where arbitrarily high energies unexplored sure to is be encountered. unknown new, physics Thus in Schwinger's the ultimate fate of renormalization approach to excluded from of is for it be eliminated and any description theory nature. He tries to implement the concept of a local this by abandoning a drastic alteration of the foundations operator field, which constitutes of QFT. The radical character of Schwinger's foun approach, whose in the 1960s and dations were laid in his 1951 paper and elaborated until the mid 1970s when the renormaliz 1970s, was not recognized was first that time in new, important By ability principle challenged.

primary, malleable

and renormalizability had been gleaned from sights into renormalization studies using renormalization group methods, resulting in a new under and of and also in novel attitudes of renormalization QFT, standing interest in nonre toward scientific theories in general. Thus a renewed theories manifested, and the 'effective field theory' ap to In this changed conceptual its context, gain popularity. proach began most theorists the e.g., Weinberg (1979) began to perspicacious in the radical shift of that Schwinger's ideas were essential realize in fundamental outlook physics.

normalizable

RENORMALIZATION

3.

THEORY

OF

TRANSFORMATIONS

51

FOUNDATIONS

of renormalization The conceptual foundations theory have undergone a radical transformation the last four decades. These changes during are the result both of attempts to solve conceptual anomalies within the theory itself and of fruitful interactions between QFT and statistical as regulariz such concepts concerning Implicit assumptions physics. and have ation, cutoff, dimensionality, symmetry, renormalizability of these concepts is being been clarified, and the original understanding transformed. New concepts of symmetry-breaking, either spontaneous or anomalous, of decoupling of renormalization group transformations, from low energy phenomena, of sensible processes field have been and of effective theories theories, on in dramatic statistical progress drawing heavily physics. developed, As a result of these advances there has emerged a new understanding a clarification structure of QFT of the theoretical of renormalization, a most crucial shift of out and its ontological basis, and, importantly, Section 3.1 examines look in fundamental these foundational physics. in whose philosophical transformations, implications will be discussed of high energy nonrenormalizable

3.2.

3.1. As

Cutoff noted

essentially

in the last section, of two steps:

the renormalization

procedure

consists

(i)

for a given theory (e.g., QED or the Weinberg-Salam the an is ory of the electroweak interactions), algorithm specified low en for an unambiguous separation of the ascertainable from the high energy processes that are not ergy processes known, the latter being describable only by new future theo ries; and

(ii)

the incorporation of the effect of the neglected high energy on the that is described processes by the theory physics a the is accomplished redefinition of number of finite by of the parameters theory.

The

redefined

be determined

parameters by experiments

are not

by the theory but can 1948a, 1948b). The implicit

calculable

(Schwinger,

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to be possible will and redefinition for the incorporation requirements in Section 3.2. Here our focus is on the separation. be examined For Schwinger and Tomonaga, who directly the infinite separated were terms by contact transformations, the 'not known' contributions terms that have by divergent proper gauge and simply represented were Lorentz and that transformation identified by properties properly an appropriate first constructing of the vacuum state and of description states for the given theory. There were, how the physical one-particle as to where in their formulations the boun ever, no clues whatsoever the the knowable from unknowable energy region lies. dary separating It is buried and hidden in the divergent somewhere integrals. Thus can only be viewed as a species of the incorporation and redefinition essentially extremely

formalistic manipulations of divergent with an quantities, tenuous Dirac's criticism, 1969a, logical justification (see

1969b). and most other physicists, Pauli and Villars, took an ap Feynman, differed and that from Schwinger's Tomonaga's. They tempor proach the theory with the help of a regularization arily modified procedure so to make the integrals finite. In the momentum cutoff regularization scheme introduced by Feynman (1948c) and by Pauli and Villars (1949), line separating the knowable the boundary region from the unknowable is clearly indicated by the momentum cutoff introduced.17 Below the to is and the be the cutoff, trustworthy, theory integrals supposed can be justifiably manipulated and for the higher-order corrections occur The unknown that above the calculated. high energy processes as they have to be. Up to this from consideration cutoff are excluded seems to Schwinger's scheme in im Feynman's superior point, were ideas of which first the basic renormalization, clearly plementing It also seems to be more respectable stated by Schwinger. logically and mathematically. the following difficult question must be answered However, by the are the effects of the excluded schemes: How various regularization on the low energy phenomena taken into ac high energy processes is specific to local field theories and is unavoidable count? This question the ruling that framework. solution, which became Feynman's at to to the is take the end the calculation. cutoff of orthodoxy, infinity are taken into consideration, In this way, all the high energy processes can be incorporated and their effects on the low energy phenomena by in of the theory's that the the appear parameters redefining specification

within

RENORMALIZATION

THEORY

53

The price for accomplishing in the manner of Schwinger. Lagrangian we can no longer take the cutoff as the threshold energy at this is that which the theory stops being valid and where new theories are required we would face a serious for the correct physical description. Otherwise, to the cutoff that the anomaly: taking infinity would mean conceptual are not is and trustworthy theory high energy processes everywhere This is in contradiction with the basic idea of renormaliz unknowable. ation, and the divergent integrals that result when taking the cutoff to are clear indications that this is not the case.18 infinity The implications of taking the cutoff to infinity are very significant. First of all, the boundary line separating the ascertainable and verifiable domain from the unknowable becomes buried It and hidden. region also changes the status of the cutoff from a tentative, and tantalizing, threshold and thus essentially device, energy to a purely formalistic reduces the whole Feynman-Pauli-Villars scheme to Schwinger's orig inal formalistic one. Physically, cutoff regulariz Feynman's momentum ation can be regarded as another, more efficient, formalistic algorithm for manipulating the divergent which replaces Schwinger's quantities, canonical transformations. direct identi Or, equivalently, Schwinger's can as fication of the divergent be viewed integrals combining Feyn two steps of introducing a finite cutoff, followed by taking it to man's infinity. More significantly, taking the cutoff to infinity also reinforces a prevailing formalistic claim that the 'physics' should be cutoff-inde to the cutoff should be removed on and all explicit reference pendent seems compelling the The claim because the parameters. redefining the of any physical meaning. step deprives quantities cutoff-dependent the claim in turn allows one to take a purely formalistic Conversely, of the cutoff, and forces its removal from real physics. interpretation But what if the cutoff is taken seriously and interpreted realistically as the threshold energy for new physics? Then the orthodox formalistic cannot scheme collapses and the entire perspective the cutoff changes: be taken to infinity and the obverse side of this same coin is that the In fact, the impor physics cannot be claimed to be cutoff-independent. tant advances since the mid 1970s in understanding the physics and the have come from such a realist interpreta of renormalization philosophy tion (see Polchinski, several in 1984; Lepage, 1989). There were to tertwined strands of physical that led foundational this reasoning in turn, was reinforced by philosophical and practical change, which, considerations.

The

rest of the present

section

is devoted

to disentang

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ling these strands of physical reasoning, leaving the other considerations to be examined in the next section. to take a To begin with, let us examine the reason why it is possible realist position on the cutoff. As we noted above, the motivation for the effects of the taking the cutoff to infinity is to take into account - which a finite are excluded by introducing high energy processes - on If we can find other ways of cutoff low energy phenomena. these effects while keeping the cutoff finite, then there is no reason to infinity. In fact, the realist for the cutoff taking compelling has adherents since the late 1970s precisely because gained position theorists have gradually come to realize, using detailed power-counting arguments and careful dimensional analysis, that the high energy effects can be retained without can the cutoff to infinity. This objective taking be achieved by adding a finite number of new, local, nonrenormalizable as the original Lagrangian, interactions that have the same symmetries of the parameters with a redefinition of the theory (see combined Wilson, 1983; Symanzik, 1983; Polchinski, 1984; and esp. Lepage, to It is be noted that the introduction of nonrenormalizable 1989). causes no difficulty because interactions the theory has a finite cutoff. There is a price to be paid for taking this realist position. First, the formalism becomes more complicated by adding the new compensating The cost is not very high since there are only a finite interactions. that need to be added, as these are subject number of new interactions retaining

to various constraints. Moreover, this position is conceptually simpler the realist formalism than the formalistic one. Second, is valid only up can only probe a to the cutoff energy. However, since any experiment limited range of energies, this limitation of the realist formalism has actually not caused any real loss in accuracy. Thus the apparent cost is illusionary. The next question cutoff so that ways

is how to articulate the physical realization of the can be found to determine its energy scale. The cutoff in realist theory is no longer a formalistic device or an arbitrary as the embodiment but acquires physical significance of the parameter, structure of QFT, and as a boundary hierarchical energy separating sets of parameters by different regions that are separately describable with different and different laws (interactions) symmetries. physical of spontaneous The discovery of the mechanism symmetry breaking to be discussed and of the decoupling theorems, below, suggests that of heavy bosons, the value of the cutoff is connected with the masses

RENORMALIZATION

THEORY

55

Since which are associated with the spontaneous symmetry breaking. the otherwise nonrenormaliz the symmetry breaking makes negligible due to the absence of all other interactions able interactions19 detectable that are forbidden by the symmetry, the energy scale of the cutoff can be established the by measuring strength of the nonrenormalizable in a theory. interactions fashion that a realist The above discussion has shown in a preliminary a an not the is untenable of cutoff However, conception position. is possible only when this conception convincing proof of its viability is integrated into a new conceptual network that provides new foun interac dation for understanding nonrenormalizable renormalizability, us turn to in in other strands this network. and Let QFT tions, general. 3.2.

Symmetry

and Symmetry

Breaking

for having a renormalization The essential motivation from the necessity of dealing with the divergences solutions of a quantum field theory. perturbative

comes procedure that occur in the In the traditional

after separating the invalid (divergent) parts (formalistic) procedure, the effects of the inaccess from the valid (finite) parts of the solutions, on accessible and knowable ible and unknown high energy processes are absorbed by modifying the parameters that low energy phenomena enter in the definition of the theory in terms of its Lagrangian. For this to be possible, the structure of the amplitudes however, amalgamation that simulate the unknown and inaccessible high energy dynamics has to be the same as the structure of the amplitudes for the responsible low energy processes. the renormalizations Otherwise, multiplicative would be impossible. To guarantee the required structural similarity, a crucial assumption about the unknown high energy dynamics has to be made that is implicitly built into the very scheme of multiplicative renormalization. This is the assumption that the high energy dynamics same as the low is constrained the those that constrain by symmetries a a the of solutions constitute energy dynamics. Now, representa theory tion of

the symmetry group of the transformations the under which were if is invariant. different Therefore, theory symmetries displayed in different energy regions, this would by the dynamics imply different structure for the solutions constraints and a different group-theoretical in the differing renormalizability

If this were the case, pieces of the dynamics. of the theory would definitely be spoiled.

then the

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one of the simplest cases, the renormalizability In the case of QED, is guaranteed the somewhat of the U(l) by mysterious universality with the of gauge symmetry. However, symmetry breaking, discovery the situation became more First, in the early 1960s the complicated. mechanism of spontaneous symmetry breaking (SSB) was introduced and then in the late 1960s the phenomenon of anomalous was These encountered.20 breaking required that the (ASB)

and studied,

symmetry about the relationship between above general consideration symmetry and renormalizability be refined and be made more sophisticated. at the of SSB was first noticed SSB. The phenomenon Consider was of the and and rediscovered Cao, 1991), century (Brown beginning It was explained in the 1950s in investigations of superconductivity. context and integrated within the field-theoretical into the theoretical structure of QFT by Heisenberg, Nambu, Goldstone, Anderson, Higgs, matter in the early 1960s.21 In condensed and statistical and others the properties of the solutions concerning physics, SSB is a statement of a dynamical that some asymmetrical system, namely, configurations more stable than symmetrical ones. Essentially, are energetically SSB is concerned with the low energy behavior of the solutions and asserts that some low energy solutions exhibit less symmetry than the symmetry of the system, while others possess the full exhibited by the Lagrangian to its foundation, of the Traced SSB is an inherent system. symmetry of because the existence and the deter the system property dynamical mination of the asymmetrical solutions are completely determined by to the dynamics and the parameters of the system. They are connected structure of the solution, which, in statistical physics, the hierarchical is manifested

in the phenomena

of continuous

(second-order)

phase

transitions.

sense only in gauge theories when physical one are of its mathematical involved. Otherwise, symmetries of massless Goldstone bosons the existence namely, prediction the framework of gauge Within would contradict physical observations. statements listed in the previous para all the SSB theories, concerning are to in valid. addition another There these, is, very important graph to our discussion. In a gauge theory, as assertion that is of relevance In QFT, continuous

SSB makes

in the case of the electroweak for example theory, in contradistinction case to the diverse low energy phenom of explicit symmetry breaking, ena can be accommodated in a hierarchy with the help of SSB, without of the theory. The reason for this is that spoiling the renormalizability

RENORMALIZATION

THEORY

57

lower than the SSB affects the structure of physics only at energies is broken, and thus does not affect the the symmetry scale at which a statement of a theory, which is essentially of the renormalizability of the The of behavior theory. high energy understanding profound these implications the strong impetus to search of SSB has provided of nature, in which natural laws with for an ultimate unified description invariance properties, theories, and asymmetrical symmetrical states all from the emerge highest symmetry that characterizes physical the in the early universe, under conditions present physics passing a as the temperature of transitions decreases sequence through phase while the universe expands until it reaches the state described by QCD and the electroweak theory. Such an enterprise has to meet several stringent constraints. One of them arises due to the occurrence of ASB. Generally speaking, ASB is the breakdown of a classical symmetry caused by quantum mechanical effects. It is possible that some symmetries the system possessed in its in its quantified because classical formulation may disappear version, In QFT the latter may introduce some symmetry-violating processes. these arise because of loop corrections, and it is related to the renor malization and the absence of an invariant regulator. procedure an In particular, ASB role in QFT. the desire to important plays a can from broken symmetry safeguard being anomalously place a very on are If constraint the model concerned strong symmetries building. as occur and then the such local, gauge symmetries general covariance, rence of ASB, which is unavoidable in chiral theories, is fatal because the renormalizability of the theory is spoiled and unitarity is violated.22 Since any realistic model must contain some chiral sector(s), there is no way of avoiding the presence of ASB. The only way out, then, is to make some ad hoc arrangements for canceling the anomalies.23 This severe on to also leads restrictions the choice of spacetime requirement different

dimensions (10 or 26) and of symmetry groups (50(32) or E8 x ES) in the context of superstring theories (cf. Green et al., 1987). While it is are a great success or a crushing debatable whether such restrictions a of ASB there is no doubt that the investigation defeat, occupies central place in the research on the foundations of QFT. are global, then the occurrence of ASB as in the case of global y5 invariance for (cf. Bell and Jackiw, 1969), or of scale 7r??>yy decay in QCD with massless for hadrons quarks obtaining massive

If the symmetries concerned is harmless or even desirable, explaining invariance

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as bound states. But the implications of the anomalous symmetry break so down of scale invariance are extremely that they demand profound separate discussions. 3.3.

Scale

Invariance

and Renormalization

Group

Approach

The idea of scale dependence within the framework of QFT appeared earlier than the idea of scale invariance, and it can be traced to Dyson's work on the smoothed interaction (1951). In this repre representation can be treated the low of the interaction sentation, frequency part was to be the from which high frequency thought part, separately in producing renormalization effects. To this end except a of the the guidelines adiabatic hypothesis, defined, Dyson adopting a of the and electron smoothly varying interac smoothly varying charge tion with the help of a smoothly varying parameter g. He then argued that when g is varied, some modification in the defi had to be made to of in nition the g-dependent order for the interaction, compensate effect caused by the change of the g-dependent charge. In line with this a similar con idea of Dyson, Landau and his collaborators developed a out in of smeared interaction series of influential cept papers (Landau et al., 1954a, 1954b, 1954c, 1954d, 1956). In accordance with this a as not of the the interaction should be regarded concept, magnitude constant but as a function of the radius of interaction that must fall off ~ exceeds a critical value P \la, where a rapidly when the momentum is the range of the interaction. As a decreases, all the physical results ineffective,

tend

to finite limits. Correspondingly, the electron's charge must be an as as function of the of interaction. unknown radius yet regarded studied the short distance be With the help of this concept, Landau some significant results, which were re havior of QED and obtained and Landau had the idea that ferred to in the last section. Both Dyson to the parameter the charge of the electron was scale corresponding In that the addition, Dyson hinted, though only implicitly, dependent. more be of should QED Landau, physics scale-independent; explicitly, that the interactions in QED might be asymptotically scale suggested invariant.

in particular and Petermann In later works, those of Stueckelberg Low and (1953) and of Gell-Mann (1954), Dyson's varying parameter g and Landau's range of interaction were further specified as the sliding and Low, the renormalization scale, or subtraction point. In Gell-Mann

RENORMALIZATION

THEORY

59

character of parameters and the connection between scale-dependent at different renormalization scales were elaborated in terms parameters of renormalization and the scale-independent group transformations, character of the physics was embodied in renormalization group equa were not appreciated tions. However, these elaborations until much later the late 1960s and early 1970s - when a deeper understanding of the ideas of scale invariance and of renormalization group equations was gained, mainly the result through the researches of K. G. Wilson, of fruitful interactions between QFT and statistical physics. The idea of the scale invariance of a theory is more complicated and different from the idea of the of the very independence physics on the renormalization scale as expressed group by the renormalization a The scale invariance of theory refers to its invariance equations. under the group of scale transformations. The latter are only defined for dynamical not variables but for the dimensional par (the fields), a as for otherwise scale such transformation would masses, ameters, result in a different physical theory. While the physics should be inde of the choice of the renormalization scale, a theory may not pendent be scale-invariant if there are any dimensional parameters. In Gell-Mann

and Low's treatment of the short distance behavior of when the electric is QED, theory is not scale-invariant charge in terms of its value at very large distances. renormalized The scale invariance would be expected in this case because the electron mass can be neglected and there seems to be no other dimensional parameter in the theory. The reason for the unexpected failure of scale appearing the

invariance is due entirely to the necessity for charge renormalization: there is a singularity when the electron mass goes to zero. However, when at a relevant energy scale the electric charge is renormalized a to suppress effectively renormalization scale by introducing sliding irrelevant low energy degrees of freedom, there seems to occur an scale invariance. This scale invariance' is ex asymptotic 'asymptotic a terms Gell-Mann and in Low of law for the effective pressed by scaling for the bare charge, that is, by condition charge and by the eigenvalue the statement that there is a "fixed" value for the bare charge indepen dent of the value

of the measured charge.24 was a suggestion by Johnson (1961) in the early 1960s there Although that the Thirring model might be scale-invariant, the real advance in was nature the of scale invariance made in mid the 1960s understanding as a result of developments in statistical physics. Research in this area

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was

also stimulated of field-theoretical in anomalies by the discovery the study of current algebra and in the short distance of expansion that products of quantum field operators. Here we want to emphasize the interaction between QFT and statistical mechanics, which can read in the shaping of Wilson's ideas, played an important ily be discerned role in the development. the interaction is very interest Conceptually, but In also Widom 1965, ing, quite complicated. (1965a, 1965b) pro posed a scaling law for the equation of state near the critical point that

earlier results obtained generalized by Fisher the relations (1964) concerning was puzzled by Widom's Wilson work was Wilson familiar with justification. a he had found natural Moreover, just

Essam

and Fisher (1963) and the critical among exponents. it lacked a theoretical because Gell-Mann and Low's work. basis for the renormalization a lattice field theory, by to develop group analysis, while working one momentum scale for the problem and solving eliminating (Wilson, At Wilson realized that there the should be applications time, 1965). of Gell-Mann and Low's idea to critical phenomena. One year later, Kadanoff derived law the idea which Widom's (1966) scaling using - that embodied the renormalization transformation group essentially a fixed point of the transformations on the the critical point becomes Wilson assimilated Kadanoff's parameters. quickly scale-dependent it into his thinking about field theories idea and amalgamated and critical phenomena, the of broken scale invariance.25 concept exploiting on Wilson had also done some seminal work in 1964 (unpublished) operator product expansions (OPE), but had failed in the strong coup After about the implications of the scaling theory domain. thinking ling of Widom and Kadanoff when applied to QFT, and after having investi of Johnson's the gated the consequences (1961) suggestion concerning concern scale invariance of the Thirring model and that of Mack (1968)

at short distances, ing the scale invariance of the strong interactions Wilson his theory of OPE basing it on the new idea of reformulated at scale invariance (1969). He found that QFT might be scale-invariant if the scale dimensions of the field operators, which are short distances that the canonical defined by the requirement commutation relations are scale invariant, were treated as new degrees of freedom.26 These can be changed by the interactions between the fields scale dimensions which and can acquire anomalous values,27 mathematically correspond to the nontrivial in critical phenomena. exponents for the foundational The most important implications

transformations

THEORY

RENORMALIZATION

61

in statistical physics advances of QFT from the dramatic stemming can be summarized two Wilson stressed: by concepts (i) the statistical limit of a local theory, and (ii) the fixed points of renormaliz continuum ation group transformations. noticed that systems described by statistical physics and First, Wilson embodied various scales. If functions of a continuous variable, by QFT such as the electric field defined on space time, are themselves indepen so that functional to form a continuum dent variables and assumed can be defined, then one can define a statistical integrals and derivatives continuum limit that is characterized by the absence of a characteristic to each in all scales are coupled scale. This means that fluctuations to a process. In QED other and make equal contributions calculations this typically leads to logarithmic divergences. Thus renormalization is for the study of these systems. That this concept of statistical necessary a central position in Wilson's continuum limit occupies thinking on on in in and renormalization is in reflected QFT general, particular, his claim that "the worst feature of the standard renormalization is that it gives no insight into the physics of the statistical procedure continuum

limit" (1975, p. 775). the various parameters the physics at various Second, characterizing renormalization scales reflect the scale dependence of the renormaliz are related to each other by the renor ation effects. These parameters malization that are described by the renormaliz group transformations In ation group this the renormalization sense, group equations. the behavior of QFT by following the equations study high energy variation of the effective parameters of the theory caused by the anom of scale invariance of the theory. alous breakdown Since the late 1960s it has been recognized that the scale invariance of any quantum field theory is unavoidably broken anomalously because of the necessity of renormalization. This insight was foreshadowed by Adler (1969), Bell and Jackiw (1969), and others in 1969 in their studies of current algebra. They found that in perturbation theory the equal time commutators of field operators were affected by renormalization. This

the existence of chiral anomalies and discovery helped ascertain the anomalous dimension of quantum fields, and led to the idea of ASB. A more convincing is based on the concept of dimensional argument The scale invariance of a theory is equivalent to the transmutation. conservation

of

the scale

current

in the

theory.

To

define

the scale

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is required, current, a renormalization for, as a product of procedure at a same point, the scale current implicitly contains an two operators even in a theory without ultraviolet any dimen singularity. However, to introduce a dimensional it is sional parameter, still necessary par ameter as a subtraction in order to avoid renormalizing, point when and in order to define the coupling constant. the infrared divergences to And this breaks the scale invariance of the theory. The necessity was transmu introduce a dimensional called "dimensional parameter tation"

and Weinberg (1973). Precisely because of dimen by Coleman the in a renormalized sional transmutation, scale invariance theory is broken anomalously, though the effects of this breakdown unavoidably can be taken care of by the renormalization group equations. In statistical physics, the renormalization group approach effects con scale levels. By scaling out the nections between physics at different and by locating stable infrared fixed irrelevant short-range correlations it has made possible the conceptual unification of various de points, - such as those of excitations elementary (quasi-particles) scriptions - the and collective ones (phonons, plasmons, spin-waves) explanation critical and the calculation the of various of of behavior, universality same In and critical the order parameters QFT, components. approach can be used to suppress the irrelevant low energy degrees of freedom, fixed point. In both cases, the essence and to find a stable ultraviolet as Weinberg is to concentrate of the approach, (1981) has indicated, on the relevant degrees for a particular of freedom and problem,28 the goal

is to find fixed 2Q

point

solutions

of

the renormalization

group

equations.

to Wilson, the fixed point in QFT is just a generalization According condition for the bare charge in of Gell-Mann and Low's eigenvalue a law At the fixed QED. holds, either in Gell-Mann scaling point sense or in Bjorken's, and the theory is asymptotically Low-Wilson's is broken at nonfixed points, and The scale invariance scale-invariant. can be traced by the renormalization the breakdown group equations. of a given field that if the renormalization group equations a fixed then in that field stable ultraviolet theory possess point solution, no can causes and it thus be trouble, theory the high energy behavior an toWeinberg safe theory". called, according (1978), "asymptotically safe theory may be a renormalizable An asymptotically theory if the It is clear

fixed point.30 Weinberg, how is the Gaussian fixed point it possesses ever, argued, and supported his position with a concrete example of a

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scalar theory, that the concept of 'asymptotic five-dimensional safety' and thus can than the concept of renormalizability, is more general explain and even replace it. There may in fact be cases in which theories are asymptotically safe but not renormalizable in the usual sense, if a are with Wilson-Fisher fixed associated they point. The conceptual developments described in this section can be summa rized as follows: in systems with many scales that are coupled to each a characteristic other and without scale, such as those described by the scale invariance is always anomalously broken due to the of This renormalization. breakdown manifests itself in the necessity anomalous of fields in the framework of OPE, or in scale dimensions at different the variation of parameters renormalization scales that is If have no charted by the renormalization these equations. equations fixed point solution, then they are not asymptotically scale-invariant

QFT,

and

if they nonrenormalizable;31 theory is, rigorously speaking, a are then fixed scale-in possess they point solution, asymptotically variant and the theory is asymptotically safe. If the fixed point is Gaus But there may be some asymp sian, then the theory is renormalizable. are safe theories that nonrenormalizable if the fixed point they totically a occurrence is Wilson-Fisher fixed With the of the more possess point. one of the of is fundamental which safety, guiding principle asymptotic of the renormalization the fundamental consequences group approach, ly of the renormalizability principle began to be seriously challenged. the

3.4. Decoupling

Theorem

and Effective

Field

Theories

to the renormalization group approach, different renormaliz According to ation prescriptions lead different of a theory. only parameterizations a An of this freedom in convenient important choosing application renormalization is embodied in the decoupling theorem, prescription first formulated by Symanzik and then and Car by Appelquist (1973), razzone (1975). The theorem is concerned with renormalizable theories in which some fields have masses much than the others and is larger states The based on power-counting theorem in that such arguments. can theories a renormalization be found the such that heavy prescription from the low energy physics, except particles can be shown to decouple for producing renormalization effects a the of power pressed by experimental mass.

and corrections that are sup momentum divided by a heavy

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important corollary to this theorem is that the low energy physics is describable field theory (EFT), which by an effective incorporates that are actually only those particles important at the energy being studied: there is no need to solve the complete all theory describing the light and the heavy particles (cf. Weinberg, 1980b). The EFT can be obtained by deleting all heavy fields from the complete renormalizable the coupling constants, masses, and the theory and suitably redefining scale of the Green's functions, group equa using the renormalization a description tions. Clearly, of the physics by an EFT is context-depen An

dent. It is delimited and thus energy available, by the experimental able to keep close track of the experimental situation. This context of an EFT is embodied in an effective cutoff that is repre dependence mass a sented by associated with SSB. Thus, with the decoupling heavy a hierarchical theorem and the concept of EFT emerges picture of one nature offered by QFT, that explains why the description at any one level is so stable and is not disturbed by whatever happens at higher and thus justifies the use of such descriptions. energies, seems to be an apparent contradiction There between the idea under group approach and the idea underlying EFT. lying the renormalization on the absence of a characteristic While the former is predicated scale in the system under consideration, the latter takes seriously the mass in which mass scale of the heavy particles, scale plays the role of a or a cutoff characteristic scale in the low energy physics that physical involves only the light particles. The contradiction imme disappears if we remember that the heavy particles still make diately, however, to the renormalization contributions in EFT. Thus effects the mass scale of the heavy particles in an EFT actually plays only the role of a of such scale, not a genuine one. The existence pseudocharacteristic a reflects scales hierarchical of pseudocharacteristic ordering couplings at different energy scales, but it does not change the essential feature a of systems described the absence of characteristic by QFT, namely, at various energy scales. While scale and the coupling of fluctuations some couplings between fluctuations at high and low energy scales exist and manifest in the renormalization themselves in effects universally are low energy physics, others and reveal no observable suppressed clues in low energy physics. is not absolute The above assertion that the decoupling is reinforced that the influence of the heavy particles by the important observation on the low energy physics is directly detectable in some circumstances:

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if there are processes that are exactly (e.g., those of weak interactions) forbidden by symmetries conservation, etc.) (e.g., parity, strangeness in the absence of the heavy particles that are involved in symmetry interactions that lead to these processes and breaking (e.g., WZ-bosons in the weak then the influence of the interactions), heavy are observable, due particles on the low energy phenomena although, a to the decoupling of divided theorem, energy power by suppressed these effects can be described by the heavy mass. Typically, by an effective

nonrenormalizable theory (e.g., Fermi's theory of weak inter as a to a renormalizable low energy approximation actions), which, a the electroweak cutoff or theory (e.g., theory), possesses physical set characteristic scale the 300 Gev for energy by heavy particles (e.g., the Fermi

the experimental the cutoff energy approaches theory). When nonrenormalizable and new the becomes energy, theory inapplicable, a that for its either renormaliz (i) physics appears requires description able theory or (ii) a new, effective theory with a higher cutoff energy. The first choice represents the orthodoxy. The second choice presents a serious challenge to the fundamentality of the renormalizability prin and popularity. ciple, and is presently gaining momentum

3.5. A Challenge

to Renormalizability

The concept of an EFT clarifies how QFT different forms and allows two different ways (i)

at different scales takes to look at the situation:

If a renormalizable then theory at high energy is available, at can the effective lower in be obtained any energy theory a totally systematic way by integrating out the heavy fields of the theory. In this way, the renormalizable electroweak can as and be understood effective theories at QCD theory low energy of some grand unified theory. They thus lose their status of being fundamental theories. Another presumed also with this way of looking at the compatible possibility, a tower of effective assume to is that there exists situation, theories that contains nonrenormalizable each interactions, with fewer numbers of particles and with more small nonre terms than the previous. When normalizable interaction the mass cutoff than is much M) (heavy particle physical larger the experimental

energy

E,

the effective

theory

is approxi

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terms being the nonrenormalizable of E/M. The second way corresponds more closely to what high en ergy theorists actually do when studying physics. Since no knows what the renormalizable body theory at the unattain or even it exists at all, we able higher energies whether is, have to probe the accessible low energy first and design that fit this energy range. We then extend representations our theory to higher energies only when it becomes relevant an to our understanding We thus obtain of physics. endless renormalizable, suppressed by a power

mately (ii)

in which each theory is a particular theories,32 a to situation and none can response particular experimental as be the fundamental ultimately regarded theory. In this can be re the requirement of renormalizability approach, a on the nonrenormalizable condition interactions placed by in the effective theories: all the nonrenormalizable interac tions in an effective theory describing physics at a scale m must be produced by heavy particles with a mass scale M are and thus (> m), suppressed by powers of m/M. Further in the renormalizable effective the more, theory including mass inter with these nonrenormalizable M, heavy particles actions must disappear. tower

of

These clarifications, group equa together with the renormalization to come to a new understanding of renor have tions, helped physicists As David Gross put it, renormalization "is an expression malization. of the variation of the structure of physical interactions with changes in the scale of the phenomena being probed" (1985, p. 153). Notice is very different from the old one, which that this new understanding on the high energy behavior and on ways of circum focused exclusively It shows a more general concern with the finite venting the divergences. with finite changes of interactions variations of the various physical non and thus energy scales, enough leeway for considering provides renormalizable interactions. A significant change in the attitude of physicists with respect to what in theory construction has taken should be taken as guiding principles of EFT. On place in recent years in the context of the development was taken for many classic work, renormalizability the heels of Dyson's for acceptable years as a necessary quantum field theories. requirement

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can probe only a limited range Now, reflecting the fact that experiments it seems natural to take EFT as a general framework for of energies, interactions results. Since nonrenormalizable analyzing experimental occur quite naturally within this framework, there is no a priori reason to describe cur to exclude them when constructing theoretical models accessible rently physics. In addition to being compatible and congenial with the new under nonrenormalizable interactions seri of renormalization, standing taking some nonrenormaliz is also other arguments. First, ously supported by are malleable

to accommodate and enough experiments area in of the observations, gravity. Second, they possess especially power and are able to improve this power by taking higher predictive and higher cutoffs. Third, because of their phenomenological nature, theories, which, they are conceptually simpler than the renormalizable as stressed by Schwinger, extraneous involve physically speculations structure of the physical particles. The traditional about the dynamic theories was that they are unstable argument against nonrenormalizable able

theories

at energies cutoff). A more (undefinable higher than their physical extended discussion of the question of the stability of such theories will be given in the next section. Here we focus on the high energy behavior of nonrenormalizable interactions, which many physicists have regarded as one of the most fundamental in QFT. questions As noted above, within the original framework of EFT which takes as its conceptual theories as basis, nonrenormalizable renormalizability ones secure standings to renormalizable low energy approximations the experimentally available energy only as auxiliary devices. When new and to their cutoffs appear, they become approaches physics begins incorrect and have to be replaced by renormalizable theories. Within the framework of Weinberg's safe theories, nonrenor asymptotically status. Neverthe malizable theories have acquired a more fundamental a common still share feature with all the less, they EFT, namely, discussion of them is based on taking the cutoff to infinity, and thus of the cutoff. falls into the category of formalistic interpretations we if EFT to take the idea its However, underlying logical conclusion, then a radical change in outlook takes place and a new perspective of QFT can be developed and a new appears; a new interpretation theoretical structure of QFT waits to be explored. Thoroughgoing advo cates of EFT, such as Georgi would and argue (1989b) (1989), Lepage that when

the experimentally

available

energy

approaches

the cutoff of

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alizable effective theory, we can replace it with another nonrenormaliz able effective theory of much higher cutoff. In this way, the high energy interaction above the cutoff will be behavior of the nonrenormalizable care of taken by: properly (i)

(ii)

caused by the the variation of the renormalization effects, of the the cutoff and calculable renormalization change by and group equations; counter terms.33 additional nonrenormalizable

the cutoff is always finite and can Thus, at any stage of development, as was argued in Section 3.1. In be given a realist interpretation, to the finite cutoff, we also find two new ingredients that are addition structure of QFT but are legit in the traditional absent or forbidden structure of the new formula in the theoretical imate and indispensable of renormalization tion of QFT. These are the variations effects with counterterms and the that of cutoff nonrenormalizable specific changes are legitimatized of finite cutoff. the introduction by in the conceptualization transformations of renor These foundational an stem internal which from evolution malization, conceptual partly in statisti and were inspired to a large extent by the significant progress and the further fertile soil for the acceptance cal physics, have provided of Schwinger's insightful ideas. These had been advanced development formula in his criticism of renormalization theory and of the operator in the presentation of his source theory. tion of QFT, and were detailed in his work on the views strongly influenced Weinberg Schwinger's to chiral and on dynamics Lagrangian approach phenomenological EFT. We can easily find three theory and the new formulation (i) (ii) (iii)

features of QFT:

shared

by Schwinger's

source

that they are fundamental theories; new particles in being able to incorporate their flexibility and new interactions into the existing schemes; and the possibility of each of them to consider nonrenormalizable the denial

interactions. a fundamental difference exists between the two schemes. However, of QFT is still a local operator field theory and The new formulation contains no characteristic scale, and thus has to deal with the contribu at arbitrarily high energy. On the other hand, tions from fluctuations one, in which the theory is a thoroughly phenomenological Schwinger's

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is only from the operator is different field, field, which at low energy. There is thus excitation for the one-particle responsible no question in Schwinger's of renormalization theory. On the other has taken a renormalization of QFT, hand, in the new formulation more and more sophisticated in has form, gained predictive power, and tool. calculational has become an evermore powerful numerical

4.

PHILOSOPHICAL

RAMIFICATIONS

examined both the cognitive content of renormalization theory Having of its structure over the last four decades, we and the transformations of are in a better position to clarify the implications and ramifications we a found Most the theory from notably, perspective. philosophical on in theoretical that the recent developments support a pluralism an an antireductionism in and antifoundationalism epistemology tology, are in sharp contrast with the neo These in methodology. implications Platonism implicit in the traditional pursuit of quantum field theorists, entities as the ontological foundation of physi which took mathematical rational and which assumed cal theories that, through (mainly mathe an one at stable theory arrive ultimate could human activities, matical) scientific theories of of everything. Also, contrary to the previous image new was structure the of in the mathematical that QFT, image implicit is that scientific theories are not to be fostered by the EFT approach as necessary but rather of scientific conceived rationality, products in the of r?visable should be seen as contingent nature, descriptions course of changing circumstances. have These important implications bearing on the debate about realism versus instrumentalism. 4.1. Atomism

and Pluralism

in Theoretical

Ontology

within the framework of QFT for de various models developed are in nature: the atomistic world the subatomic particles de scribing are to be regarded scribed by the fields that appear in the Lagrangians as elementary the atomism, or the constituents of the world. However, theories have a halfway notion of atomicity, adopted by unrenormalized

The

the reason for this is As clearly pointed out by Schwinger, an implicit assump theories that unrenormalized field contain operator to that is sensitive tion about the inner structure of physical particles at this the details of dynamic processes high energy. Mathematically,

character.

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has manifested itself in the divergent assumption integrals.34 Metaphys constitu ically, the assumption implies that there exist more elementary ents than the physical particles described in a by the fields appearing and thus status or the contradicts of the particles, that of Lagrangian, the fields, as basic building blocks of the world. The fundamental of the renormalization in significance procedure this regard can be described in the following way: by removing any to inaccessible, reference and the related very high energy domains structural

the renormalization reinforces the assumption, procedure commitment of QFT. This is because the particles or the fields in the renormalized theories do act as the basic building appearing blocks of the world. But note that atomicity here no longer refers to the exact point model. To the extent that one removes the reference to the inaccessible very high energy domain - which arises conceptually - renor in the exact point model by virtue of the uncertainty principle malization blurs any point-like character. This spatially extended yet structureless model, (seemingly) quasi-point adopted or produced by renormalized in two ways. On the one hand, is justified it is QFT, success. It is also justified philosophically supported by its empirical by as as the experimental energy is not high enough to arguing that, long so that all the statements detect the inner structure of the particles, about their (small distance) structure are essentially the conjectures, model is not only an effective for experi quasi-point approximation mental purposes, but it also reflects a necessary that stage of cognition we must go through. atomistic

to the 'true' theory that Dirac Compared (1983) and Tomonaga once to and the for, (1965) yearned compared 'theory of everything' that superstring theorists are still searching for,35 the quasi-point model seems to be merely a mathematical device needed and useful in a transition period. A disciple of Dirac would argue that the quasi-point model should be discarded once the structure of the elementary part icles is known. This is true. But those committed to atomism would and practiced) formulated the struc argue that in physics (as presently tural analysis of objects at any level is always based on (seemingly) - at structureless the next level. objects genuine or quasi-point-like For example, the analysis of deep inelastic lepton-hadron scattering would be impossible if there were no analysis of elastic scattering of as a basis. Thus the adoption the partons of the quasi-point model seems to be unavoidable in QFT. This is probably what Feynman meant

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that "[w]e will start by supposing that there are (quasi we no field theory at otherwise would have because point particles) all" (1973, p. 775). as can be summarized The dialectics of atomism in this context follows: on the one hand, the structureless character of particles as we

when

he stated

at any

is not

and context absolute but contingent low energy experimental justified only by relatively dependent, probing. some in experiments When the energy available becomes high enough, sooner or later is revealed, of the inner structure of the particles and an turns out to the notion of the absolute be illusion. On indivisibility the other hand, with the unraveling of the structure of particles at one - as a for the un level, there emerge at the same time precondition at next structureless the level. And thus objects raveling (seemingly) in terms of the original pattern of 'structured objects being expressed structureless objects' remains, and will remain as long as (seemingly) know

them

level

Thus the idea expressed QFT remains the mode of representation. by the bootstrap hypothesis in 5-matrix theory that everything is divisible is incompatible with the atomistic paradigm adopted by QFT. extends the atomistic The EFT further, and approach paradigm within that framework is given a more the domain under investigation structure. The hier and a more sharply defined hierarchical discernible mass a chain of spon is delimited scales associated with archy by and is the broken theorem. taneously symmetries justified by decoupling structure from a metaphysical The examination of the hierarchical per that are worth spective yields two seemingly contradictory implications seems to lend structure the hierarchical noting. On the one hand, to the of in a reduc support possibility interpreting physical phenomena at tionist or even reconstructionist to extent least the that SSB fashion, works. Most of the efforts expended during the past two decades by the mainstream of the high energy physics community, from the stan can as to superstring dard model be this theories, regarded exploring other On the the and EFT theorem hand, taking possibility. decoupling the reductionist seriously would entail considering (and a fortiori the an to and would lead its rejection and illusion, program constructivist) to a point of view that accepts emergence, hence to a pluralist view of theoretical ontologies.36 possible The decoupling theorem does not reject the general idea of causal connections between different hierarchical levels. In fact, such connec tions are assumed to exist and to be describable by the renormalization

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basis of group equations: they are thus built into the very conceptual the theorem. It is the attempt to give the connections universal signifi cance and the stipulation of their direct relevance to scientific inquiry that is rejected. More precisely, what is to be rejected is the suggestion to infer that it is possible simply by means of these kinds of connections the complexity and the novelty that emerge at the lower energy from the simplicity at higher energy scales, without any empirical as required by the decoupling The necessity, theorem and EFT,

scales input. of an

at the lower ontologies empirical input into the theoretical applicable - scales to which at the the scales energy ontologies higher energy - is in scientific investigations scales have no direct relevance fostering a particular of the physical world. In this picture the representation as layered into quasi-autonomous latter can be considered domains, and associated each layer having its own ontology 'fundamental' laws. to in theoretical This hierarchical does not mean ontology pluralism at between ontologies the different levels. reject the causal connections it merely So in a weak sense it still falls into the category of atomism: of deducing various entities from some basic rejects the possibility ontology. commitment It is precisely the strong antireductionist the concerning levels that the different hierarchical differentiates between relationship is EFT of atomism that nurtured from the cruder version by pluralistic the constitu that is adopted by conventional version of atomism QFT, are reductionism and constructionism. In tive components of which on an the that is contin addition, historically empirical input emphasis gent also sharply contrasts the hierarchically pluralistic version of atom atomism mathematical ism with the neo-Platonic implicit in the tra ditional pursuit of quantum field theorists. The latter takes ahistorical that foundation and assumes mathematical entities for its ontological can be deduced in all empirical phenomena from it. These distinctions to the epistemological have direct relevance commitment ontological of renormalization and methodological components theory. We turn to a discussion in the next few sections. of these matters

4.2. Antifoundationalism

in Epistemology

of renormalization In our discussion of the foundational transformations we out the and statistical between that interactions QFT theory, pointed - which a For the of SSB crucial role. concept example, physics played

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of massive is indispensable both in the proof of the renormalizability structure for theories and the hierarchical non-Abelian gauge delimiting in the study of supercon of effective field theories - was first developed the concept of broken scale invariance and the ductivity. Similarly, to related renormalization which led the EFT group approach, approach were the result of the and to a new understanding of renormalization, of ideas and techniques in investigations of amalgamation developed the short distance behavior of QED, with some arising in the study of a question critical phenomena. of fundamental However, significance was left unanswered: What is it that makes the exchange possible and so extremely fruitful? seem to be quite mysterious. At first sight the exchanges Nor is the a resolved historical of the analysis by Why are puzzle developments. the physical insights obtained from one phenomenological domain (e.g., to another and applicable spins in crystal lattices) relevant, translatable, different domain continuous More fields)? entirely (e.g., specifically, even after rewriting the lattice formalism in a continuum language, which is mathematically it is still very difficult to understand possible, to QFT and vice is applicable why the formalism of critical phenomena versa. In particular, the effective Hamiltonians of statistical physics involve terms of arbitrary complexity and are thus quite different from the formalism of renormalizable field theories that involves only a finite terms. Similarly, number of interaction the concepts employed in the as of critical such lattice and block theory (QFT), phenomena spacings are to and field deal spins (coupling-constant renormalization) designed a to with infrared (ultraviolet) that has little do divergences, problem with QFT (critical phenomena). the puzzle can be solved in a realist-essentialist way. The Assuredly, interactions between different physical theories can be explained by, or taken to argue for, the transcendental unity of physical phenomena on the ontological of physical truths on plane, and/or the universality a the epistemological One of commitment plane. prominent example to realism-essentialism is neo-Platonism, which frequently finds many adherents mathematical In the last three decades, among physicists. on neo-Platonists fundamental the context of working physics, within have taken mathematical entities ahistorical and QFT, relations, parti and supersymmetries, and their representa cularly gauge symmetries as true and/or the hidden essence tions, expressing reality manifesting overt terms beneath in both of entities and their existing phenomena,

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in the real nature of various By revealing themselves are appropriated to constitute the universal foun to this view, all physical theories. According (i.e., insights lose their fundamental non-pure-mathematical) importance and survive only as suggestive and all theoretical activities in heuristics, reason physics can be reduced to the search, by means of mathematical that would group structure) ing, for the one key factor (mathematical in a consistent fashion the complexity of the universe explain (cf. patterns.

phenomena, they dation of physical

Radicati, 1984). The latest example of such an overly grandiose and totalizing concep tion of physical theory is the search for the theory of everything by the pursuit has not been successful, theorists.37 Although superstring it has deep roots that may be traced back to the ideas of Felix Klein

(1872, 1918), Sophus Lie (Lie and Engel, 1893), Minkowski

(1908,

and Heisen 1909), Weyl (1918a, 1918b, 1929), Cartan (1922), Einstein, effectiveness of berg (1960, 1966). It is nurtured by "the unreasonable as Wigner in the natural sciences", once put it (1960). mathematics successes of this approach we find Dirac's pre the spectacular Among of the diction of atomic and (1930), Wigner's positron descriptions nuclear spectroscopy and Ne?man's classifica (1931, 1937), Gell-Mann tions of hadrons and Salam's predictions of (1964), and Weinberg's intermediate bosons and neutral currents. Thus it should not come as a surprise that at present a great many of the theorists in fundamental the sway of the successes of the Weinberg directly under physics, are preoccupied Salam model, this mathematical by pursuit.38 as this mathematical As prevalent foundationalist is, the tendency recent developments in the theory of renormalization, from the renor to malization method the EFT have in the group approach, pointed the of direction: theoretical in dif opposite empirical input ontologies ferent domains of investigation have been emphasized and have pro vided a strong argument for the fundamental importance of phenomen a contrast to In of physical ological approaches. totalizing conception a a view that localist theory, supports phenomenological approach theories as histori physical (or more generally, scientific) and context-dependent.39 This view applies not only to cally the phenomenological laws describing natural phenomena, but also to laws that explain phenomena the fundamental and give them mean

characterizes

situated

ing.40 By accepting

the existence

of fundamental

laws,

the localist view

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distances

itself from another foundationalist tendency that takes sensory as the foundation of human knowledge. experience The application of a localist view to fundamental laws needs to be a and of the has to be justification explicated philosophical application we stems have As the localist view of from the seen, theory provided. in QFT with infinities, and finds its justification difficulties encountered in an empiricist of the natural world, regarding knowledge position to which knowledge is constitutive rather than given. Lo according calists believe that our knowledge of the natural world is constructed in which our sensory input and feedback process through a complex our theoretical models are assimilated and accommodated in a self of sequence modifying learning, hypothesizing, deducing, predicting, the entire process being constrained testing, and correcting by spatio in producing reality. The limited nature of our experience temporal of the world undermines the universal claim of physical knowledge laws: it only allows ascertaining in resemblance family (regularities) local region of space and time. From local regularities we cannot con struct physical theories that are unique and necessary. On the contrary, all theories are context-dependent, culturally relative, and historically changeable. the concept of fundamental laws which explain phe Furthermore, nomena and give them meaning in terms of abstract concepts raises three questions that need to be answered: (i) (ii) (iii)

What makes abstract concepts possible within framework? to phenomena? What constitutes giving meaning What constitutes scientific explanation?

the

localist

The answer to (i) is shifting metaphors. connect the most Metaphors abstract of concepts with the reality of everyday life via a kind of in structures, thus ascribing meaning to abstract concepts. similarity a to this the of in is metaphorical view, According concept meaning the typical instance, and literal (truth-functional) in the only limiting case. Thus, all theoretical in the sense that they concepts are dynamic are subject to metaphorical and transformation. expansion Concerning to believe that nature of the learning localists due the feedback (ii), cannot be constituted process, meaning by any atomistic fact or refer as claimed Russell and the early Wittgenstein ence, by (1914) (1922),

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and their dispositions but rather is conferred networks, by holistic contexts and transformations. And since the distinct metaphorical that and interactions of different parts of the net define the intersections can have fixed meaning no phenomena across work do so differently, contexts. This means different metaphorical that all understanding of a of the assimilation network of dynamic phenomena requires meanings that is controlled and operated by metaphorical associations and meta a a answer to shifts. is that scientific Thus, (iii) phorical provisional as a can be understood of the explanation metaphorical redescription domain of phenomena.41 a new view of scientific The metaphoric has created explanation different physical the interactions between for understanding potential the theories. Roughly themselves either manifest exchanges speaking, or in A in mathematical mathematical analogies. analogies physical and fixed points in such as renormalization group equations analogy, statistical physics and in QFT, works principally because different physi cal interpretations formalisms in different domains of the mathematical are connected of phenomena transformations of con by metaphorical in the formalisms. A physical analogy, such as the sup of freedom irrelevant by using renormalization degrees pression in is itself a meta in and statistical both QFT group equations physics, one of the in obtained domain of insights physical phorical expansion to to another. apply phenomena The epistemological position supported by the recent developments is empiricist in renormalization theory, especially by the EFT approach, in nature. Physical theories are justified by empirical data from which instruments for organiz the theories are abstracted. They are effective and order and the local data coherence, by imposing they conceive ing can theories be seminal and express local causal regularities. Physical cepts

involved of

the boundary delimited by the data in a domain beyond and suggestive are expressed in an apparently universal because the local regularities are ex to be and thus able mathematical formalism, metaphorically a cannot be But the in coherent way. fully justified panded extrapolation formalism itself and awaits an empirical justifi by the mathematical is most in these recent cation. It seems to us that what significant of the claim of is the rejection by the EFT approach developments in and its emphasis the mathematical formalism for QFT, universality on the local nature of theoretical structures and on their variation with changes

in the scale of the phenomena

being

probed.

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the idea successively ad rejects uncompromisingly position super during the last fifteen years by grand unified theorists, and superstring theorists that the development of gravity theorists, will end with fundamental the defini discovery of an ultimate, physics formalism. Rather, the development tive, and conclusive mathematical is taken as a process of successive that is assumed not extrapolations to have an end, with every step of the extrapolation being justified by a collective of theory and observation before and after reinterpretation the extrapolation, and through debates within the physics community about the meaning of theoretical It is also taken to be a concepts.

This vanced

in which the ahistorical quest for logical process of social construction coherence and stability is but one component, and even this component has to be defined historically because the understanding of coherence and stability differs from time to time.42 The social character of the an ideological process is highlighted by the fact that it also incorporates cultural and aesthetic consider component by expressed preference43 are to of which the unification and the criteria ations, examples appeal a of simplicity. Most the is characterized process by important, socially defined instrumentalist and progress: physics becomes more powerful is judged to be so by the pragmatic criterion of empirical adequacy in an ever expanding local domain in (success explanation, prediction, and control). this instrumentalist neither However, pre progress truth for theories that supposes nor implies any universal or permanent far from the data. extrapolate In summary, the empiricist position in epistemology that is supported the recent in renormalization by developments theory is characterized its and its antiessentialism its rejection of a antifoundationalism, by fixed underlying natural ontology entities, by mathematical expressed and its denial of universal, mathematical truths in the physical purely world. This position is also characterized by its constructivism, emphas nature the constructive of their local theories, socially izing physical character, and their dynamics realized through a social process of meta phorical

expansion

4.3. A Reappraisal

and transformations.

of Criteria for Theory Acceptance

QFT, as other branches of mathematical physics, aims to find laws of nature that are fundamental. to phenomenological In contradistinction or laws that can be induced from a limited number of observations

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fundamental laws can (usually) only be formulated experiments, by a within mathematical framework and are frequently working suggested are introduced. The parti in which unobservables by thought processes cular aim of mathematical has determined its specific procedures physics to be hypothetico-deductive. In this approach, hypotheses, including in experiments. idealized models, do not (necessarily) Most originate are are mathematical and often, directly they symbolism inspired by as in the case of Dirac and Gell suggested by symbolic reasoning, or come as in the from the the free creation of mind, Mann,44 simply are case of Einstein new to able ideas that generate (cf. 1936). They notions far deductions from This fact, encompass beyond experiment. in mathematical does not mean that hypotheses however, physics are autonomous and are not constrained by the external world. On the an to test in order to establish have external contrary, undergo they are impos their validity. Since direct tests for highly abstract hypotheses conse to have mathematical deduce certain factual sible, physicists no for of their their confirmation. But quences hypotheses particular can be thought of as being unambiguously refuted or con hypothesis firmed - as Duhem (1954), Milne (1929), and Hesse (1974) have force the nature of network of virtue theories in fully argued by physical is only one thread. For this reason, some refinements which a hypothesis are required: criteria have to be of the hypothetico-deductive method can be judged so that theories constructed with this method formulated a can or and rational choice made should be acceptable unacceptable,45 theories. there be several acceptable are the empirical Paramount among the criteria for theory acceptance and the of the stability theory. Dyson's adequacy renormalizability to be a regulative principle because the requirement was soon elevated was some to renormalization have thought by physicists procedure and empirically and (ii) (i) calculable, adequate, predictive, we as stable. have indicated in Section However, 2, later theoretically showed that renormalizable did theories, although inquiries they enjoy great success in their empirical adequacy, were far from being stable. can readily be asked: It is clear from this example that three questions

made

(i) (ii) (iii)

QFT

What

of stability? is the precise meaning and stability? is the relationship between adequacy is the primary determinant of theory appraisal? Is it or some other criterion? adequacy, stability,

What What

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us take the third question first. the criteria for theory Among such as cultural prefer those that are based on ideology, appraisal, and aesthetic considerations unification, ence,47 (e.g., simplicity), non-ad-hoc-ness and possess only heuris plausibility naturalness), (e.g., in (nonempiri tic value without Those grounded logical compulsion.48 as such constitutive cal) metaphysics, (i.e., unexplained metaphysics and modes their qualities of interactions), and that prefer explainers, consilient with ideas, assign the metaphys types accepted explanation ical intelligibility and explanatory value to the theories under consider ation. Those from the constitutive level (with respect to theory, not to the object of knowledge), such as empirical adequacy and stability in Let

are thought to be necessary at least for the final acceptance theorizing, of a theory. The actual appraisal of a scientific theory is a complicated and intricate business that takes place on several levels simultaneously. on one level may be compensated Weakness for or neutralized by no can be taken as the at another and criterion level, strengths single in the appraisal of scientific theories. In valid determinant universally a the appraisal of a particular scientific the theory by community, determination of which criterion should act as the determinant and which criteria should recede into the background ismore than a matter on the concrete of logic and depends the historical back situation, and the cultural Leibniz's and context.49 ground, Berkeley's critiques on the metaphysical level of Newton's substance50 concept of space without in the eighteenth ceased to bother natural philosophers century because of the empirical success that Newtonian mechanics enjoyed.51 The same a became crucial consideration in Mach's and Ein critiques, however, suc stein's assessments of Newtonian mechanics its despite empirical cesses. One of the factors that contributed to the new weight of meta was the general crisis faced by the mechanical physical consideration world

view

toward is nating example been retained by grounds in the face

the end of the nineteenth illumi century. Another the following: unified theories grand (GUTs) have on rational the high energy physics community of contrary empirical evidence (the seeming failure to find any event of proton decay despite intensive search) due to the faith in renormalizable theories.52 prevailing It iswidely held that the renormalization to remove infinit procedure ies has established the stability of QFT. Another idea widely accepted is that nonrenormalizable theories are unacceptable because of their even though they are empirically in the energy instability, adequate

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a global and universal to us. Both accessible ideas presume to of that refers all energy ranges and to all possible stability conception kinds of interactions, But it would be too narrow-minded respectively. to exclude nonrenormalizable theories from consideration, while impor tant interactions such as gravity steadfastly resist description by renor range

malizable

theories. Also, it would be expecting too much to assume to renormalizable be stable in this theories sense, global and universal while of these theories out in ignoring the inconsistency (as pointed Section 2) and the incomprehensibility of some features of these theo ries in terms of accepted ideas. We have in mind such things as the states and the cancelation of infinities, which presume ghost decoupled

the physical reality of ghost states and infinite totalities. A proper assessment of renormalizable theories and nonrenormalizable theories of the concept of stability, and this leads us to requires a clarification the first question. In mathematical the concept of stability comprises three physics, related but distinguishable components, namely, logical consistency, mathematical and decidability, and conceptual definability stability. as a requirement Defined of noncontradiction, is logical consistency in formal thinking. Yet mathematical is necessary certainly physics more than a formal system. Its understanding requires a semantic and a pragmatic in the system, and this analysis of the concepts employed we shall refer to the situation. For the sake of distinction, complicates with all the and semantic logical consistency pragmatic complications as conceptual that is introduced stability. One by the complication of is the following: conceptual stability stability requirement conceptual that a new concept and its implications always involves the requirement be consilient with the network of accepted ideas in the system. In this a is not the criterion for theory sense, necessary stability conceptual For of Newton's is action-at-a-distance acceptance. concept example, not consilient with the idea of contact that people had interactions and acquired from everyday experience, is not consilient with the spatiotemporal had underlain the classical conception cases mathematical lies stability, which

the concept of quantum jumps picture of physical events that of nature, in both although between the logical and the is not disturbed. That concep of the stability, components conceptual tual stability is not at the heart of mathematical physics can also be an nature justified from epistemological point of view: the metaphoric and scientific of meaning that (see Section 3.2) entails explanation

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shifts, systems are dynamic in nature, subject to metaphoric conceptual and transformations, and obey rules different from those associations, consilience recedes while of formal logic. The importance of conceptual the metaphoric network tries to assimilate apparent contradictions. This once at to few serious attention the present people pay explains why situation of wave-particle duality in quantum physics. paradoxical the The mathematical of concept of stability needs more component reason The this is at the core of the component why explanation. not that the in mathematical is theoretical concepts only concept physics are expressed in terms of a mathematical and hypotheses formalism but also that the logical validity of the conceptual is realized system a to formalism. and of stable greater importance through Additionally, our concern here, is the fact that the universality of theoretical claims of conceptual and the global character stability are largely derived from and justified by the formal character of the symbolic system of mathematics that does not refer to any concrete meaning. For example, the most striking and persistent symptom of the conceptual instability is the occurrence of infinite integrals, the lack of of QFT indicating at high energy and thus challenging of QFT its global definability a limitation to the global validity of a mathe character. Traditionally, matical formulation is taken as a decisive criterion for its rejection. Illustrative include Fermi's and interactions theory of weak examples A serious that contain chiral and/or gravitational models anomalies. to this negative attitude toward the limits of the globalism challenge comes from the EFT ap and universality formalism of mathematical as It the limit struc takes proach. positively indicating the hierarchical ture of the phenomena under investigation, and as delineating the valid to the traditional domain of an effective Lagrangian. This challenge of the concept of mathematical understanding stability requires justifi cation and invites a new understanding of the concept. However, before an new some of the presenting understanding, exposition preliminary comments

are

in order.

The

connection between the problem of mathematical stability and in the context of QFT is not peculiar the existence of infinite quantities to QFT. Rather, it is universal and the whole problem of mathematical to understand from attempts stability actually originates properly and to treat the infinite totalities that occur inmathematics. If mathematics to the description is to be restricted of observable objects, which are no finite in then number and from antinomies size, always generated

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to ob arise. However, infinite totalities would unavoidably adjoined to servable objects are theoretical infinite objects (various totalities), in mathematics ideal concepts, which there correspond and statements, inferences involving infinite totalities. The introduction of infinite total ities is inevitable even in classical mathematics. For example, the central is defined in terms that of a real number, notion of classical analysis, com infinite totalities. The situation became much more was transfinite mathematics first introduced when by Cantor, plicated and Russell into their Principia and then incorporated by Whitehead It was quickly discovered that transfinite math Mathematica (1910-13). as to class of all cardinal num contradictions such "the ematics led of actual

bers" and "the class of all those classes which do not contain as member", etc. Three responses were generated: (i)

(ii)

themselves

and others was to devise ad logicist response by Russell for avoiding antinomies while accepting remedies the actual infinite totalities. of concept The intuitionist response by Brouwer (1913) and others was to eliminate the notion of actual infinite totalities from math The hoc

ematics.

(iii)

The formalist response by Hilbert (1918, 1930) and others was to try to accommodate transfinite mathematics within a as concerned with observable that is conceived mathematics objects.

is the most transfinite relevant Hilbert's response methodological one to our concern with the stability problem. According to Hilbert's into mathematical theories transfinite concepts are admitted formalism, as and usefulness for such of their because purposes only simplification to them. status is going to be accorded unification. No full ontological The amplification of the mathematical system can only be justified by the corner proving its stability.53 Thus the concept of stability became the whole edifice stone of Hilbert's formalist program of reconstructing is crucial in order to legit In particular, this concept of mathematics. infinities. imate the use of ideal concepts Any unstable system involving as and has to be re is regarded by the formalist-finitist meaningless in the formalist-finitist jected. This attitude, which originated philos to haunt later became very popular and continues ophy of mathematics, field theo axiomatic-constructive mathematical physicists, especially rists.

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the collapse of the formalist-finitist However, program in mathemat attitude ics, in the wake of G?del's work, has suggested an alternative toward mathematical takes ordinary stability. The new understanding mathematical and acknowledges that in formal seriously experience are there of the mathematical in which we en systems parts practice counter statements decidable within the system, by a finitist method but that there are also statements whose truth value can be convincingly and consistently but usually only by semantic analysis, or by decided, A supporting considerations. external, informal, and empirical is the G?del by (1938, 1939, 1940, 1947), which suggestion example was later proven by P. J. Cohen (1963), that the continuum hypothesis set theory. Thus the meaning within axiomatic is formally undecidable is independent of the hypothesis of formal axioms, and the stability of set theory loses its formal character. The stability is estab axiomatic lished by semantic analysis, which is certainly not as pure as the formal 1976, (cf. Kreisel, approach, and depends on context and interpretation other

1980). even in pure mathematics Thus, according to this new understanding, line separating mathematical there is no sharp boundary stability, which is pure, formal, and universal, and conceptual which is impure, stability, to and revisions with local, subject changes in the context-dependent, context. Notice that we are not suggesting a thoroughgoing intuitionist in mathematical that rejects the constructibility practice in of nonconstructive mathematical stability significance practice. that both mathematical and conceptual Rather, we want to emphazise to certain local characteristics. stabilities are committed to answer the second question We are now in a position concerning the relationship between stability in theorizing and empirical adequacy. as to why a globally inconsistent theory, such Implicit in the question as QFT, can be empirically a is view of adequate global stability that our legi in the failed formalist-finitist program. However, originated room for a globally unstable timation of a nonglobal view has provided concept

of

and only required that local stability theory to be empirically adequate, - for in theorizing is necessary ad though not sufficient empirical an in local the latter character. We have thus equacy, always being the question. of local adequacy The priority and dominance is in general agreement with the methodological recent developments in renormalization theory. swered

over

global stability of the implications It implies a rejection

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- a of any attempt to mathematize tendency that origin physics totally ated with Descartes, and and which has led to Leibniz, Eddington, to trying to and necessity, striving for global stability, universality, derive all the observational facts of nature from hypothetic overarching to the formulation of the theories of and, more concretely, principles, and superstrings. While to the general frame committed supergravity work of the hypothetico-deductive local method, adequacy also implies the of and em stressing indispensability phenomenological approaches their increasing phasizing importance. Phenomenological approaches are inductive in nature. They pay more attention to the empirical input from observations and experiments. theories from a They extrapolate to a larger but still local domain, local domain and verify that every step of the extrapolation 4.4. Realism Steven

is empirically

confirmed.54

versus Instrumentalism in his Nobel lecture stated that the constraint of one to the true the that way "may point theory" and might be the key criterion which would help us to true physical theory of the infinite variety of conceiv

Weinberg

renormalizability "renormalizability pick out the one able quantum field theories" (1980a, p. 516). His strong belief in the existence of a "one true theory" shows that he is well within the to which scientific theories are tradition of scientific realism, according true descriptions structures of of real entities that are the underlying the world and that reveal its essential physical features. If the word to be used in the sense of literally true, that is if the 'true' is meant and statements of the true theory are thought to be in a categories detailed naive

Naive

isomorphism

with

the world,

then this position

is usually

called

realism.

realism

determination

faces an insurmountable of

theoretical

difficulty, which is the under by data. As Duhem (1906) models may all fit a given set

ontology of theoretical pointed out, a multiplicity of data well, yet presuppose different ontologies entities (fundamental and properties). in the context of quantum mechanics, For example, who is able to tell which is the literally true description of the quantum

the canonical quantization world: scheme in which the probability of an electron's to is ascribed its wave function that, according presence to Born (1926b), is a "ghost field" that carries no energy and momen tum, and thus has no ontological status; or Feynman's path-integral

RENORMALIZATION

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85

and Hibbs, of quantization 1948a, 1949b; Feynman (Feynman, an in is which the electron itself entity 1965), ontologically probabilistic electron can take possess and all the possible paths that this probabilistic or reality? In the context of relativistic quantum me equal possibility

method

the chanics, which theory is the literally true theory: Dirac's positron a a sea in is which hole in the filled of ory, energy negative positron or Feynman's is an elec electrons, theory (1949a), in which a positron tron evolving along the inverse arrow of time?55 In the context of is to be taken as literally renormalization theory, which procedure true:

scheme the Pauli-Villars (in which unobservable regularization are or 't Hooft's auxiliary particles introduced), (1971a, 1971b) method is of dimensional which the dimension of spacetime (in regularization taken to be a complex variable subject to analytic continuation)? A much stronger argument against naive realism is provided by the the frequent occurrence of concep history of science, which chronicles tual revolutions in which one theory is replaced by another postulating a radically different ontology. This entails that the history of science is and supports the following metainduction: essentially discontinuous just as no fundamental theory of two hundred years ago is regarded as true no present today, similarly, day fundamental theory can have any chance to be regarded as true two hundred years later.56 realism about entities ismore sophisticated and emphasizes Hacking's the existence of unobservable entities. The application of this kind of to in realism to mathematical and renormalization physics, particular Just think about the auxiliary particles in Pauli theory, is questionable. and Villars's scheme and the regularization ghost in the Fadeev-Popov on of fields. his gauge quantization put Although Hacking emphasis in experiments those entities that can be manipulated such (including entities as fractionally the boundary line separating charged quarks), and unmanipulable is quite fuzzy. unobservable entities are a long chain and these involve manipulations possible a given theoretical that presupposes framework. Thus the the unobservable entities is far from clear cut. The defects about entities lie in:

manipulable Only indirect of inference reference to of realism (i) (ii)

the acceptance of Aristotle's realist ontology of fixed natural or even justifiable; not is which kinds, justified the neglect of Duhem's underdetermination thesis: no data can unambiguously unobservable theoretical enti determine

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so that which unobservable entities exist is always a unsolvable and philosophically question; the underestimation of the peculiar features of mathematical work in this discipline is done by physics. The predominant and symbolic and its aim is thought processes reasoning, a one-to-one not to formulate between the isomorphism and unobservable in a theory and those observable entities a physical in the world, but to establish analogy by which the structural relations of the world can be grasped within ties,

(iii)

the symbolic system that consists principally inmathematical formalisms. This is well illustrated by what Maxwell did with a his mechanical model of the aether. With this model, was established and dynamic theory of electromagnetism in a set of equations. Just as an existential claim expressed so is extremely for Maxwell's idle-wheel dubious, particles a claim for is Hacking's number of unobservable probably entities.57

Instrumentalists argue that if the truth claim for theories and the claim for unobservable entities are untenable, existence then scientific as or instruments can mental theories constructions, only be regarded for reasoning about the world. A scientific theory may well be coherent and in its internal structure, be useful in organizing empirical materials, in terms of prediction, and be very valuable Yet control, explanation. the instrumental these traits have nothing to do with truth. Essentially, a via logic, for a is ist argues, a scientific shorthand, theory merely to that reference observed makes only phenomena. complex expression the progress of science, but only in The instrumentalist acknowledges an increase in power as judged by the pragmatic the sense of manifesting or of convergence conver instrumentalists deny approximation realist ontology gence to truth because reject Aristotle's they generally to converge of fixed natural kinds, so that there is nothing to.58 must face the following difficult questions: How can Instrumentalists criteria

better

of prediction

and control, not to truth. Moreover,

in the sense

constructions theories, as mental mainly consisting of symbolic reason it Or in the same vein: What makes adequate? ing, be empirically possible for theories to achieve instrumental progress? These questions of the relationship between symbols and reality. require a clarification it must have some For a symbolic system to be empirically adequate,

RENORMALIZATION

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contact with reality. But mere contact is not enough. A symbolic system based on myths also has contact with reality. The essential difference to realism about structural re science and myth, between according structure in lies fact that the of symbolic system in science the lations, in reality so that there is a is constrained the structural relations by see correspon between instrumentalists Since them.59 correspondence dence as taboo and so reject it, they are unable to answer adequately the questions raised above. For realists about structural relations, how answer is simple: the correspondence the exists but it is not a ever, one-to-one it is in nature, as those Rather, isomorphism. metaphoric that exist in models and in analogies. So the symbolic system cannot in be taken as literally true, nor is the existence of any ingredient the system guaranteed. the loose correspondence exists only Rather, as scientific laws in science, embodied between the structural relations and principles, and those in reality. As to the unobservable entities, as the carriers of the constellation function of structural relations they in a scientific theory. The possibility of rearranging the relations has status. the their entities of Illuminating deprived primary ontological examples can be found in the formalistic treatment of auxiliary particles and in the mathematical of the Higgs multiplet. manipulation to Realism about structural relations has an interesting application are theories. As well is there often several theories known, equivalent with different unobservable entities that explain data equally well. Yet the equivalence is not absolute. One theory may provide new insights and new directions for development that people would not have thought once referred to of had they worked with the other theory. Feynman and stressed the different psychological this phenomenon implications our explanation would refer to the differ of different theories. Rather, ential surplus of structural relations of 'equivalent' theories: differing theories numbers of relations other equivalent usually possess differing than those required for explaining the available data, and these are carried

entities. Another by different unobservable interesting appli cation of this viewpoint is to the explanation of how a symbol, such as in the 1930s), can be transformed 'quark' in the 1960s (or the neutrino as we now believe in it. into a real entity, the quark (or the neutrino) a The transformation is achieved of the of through change meaning term from metaphoric to literal, which in turn resulted from theoretical the fact that a stable constellation of relations that had been found could be ascribed to the term.

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Remarks

as any other history, has to be rewritten The history of renormalization, from time to time, due to changes of interests and circumstances. For on was to in the late 1940s and when the focus how 1950s, example, a of deal with ultraviolet renormalization would divergences, history about when and how infinite results were derived and Pauli, 1929; Oppenheimer, QFT (Heisenberg 1930; Waller, and 1930; Dirac, 1934; others), when and how the nature of the diver - was determined - whether linear, or logarithmic gences quadratic, were 1934; Weisskopf, 1934, 1939), how these divergences (Peierls, mass with shown to be identifiable, and renormaliz respectively, charge

narrate

the events

within

and how effec ation factors (Kramers, Bethe, Lewis, and Schwinger), to separate unambiguously infinities tive techniques were developed mass of and and to renormalize Tomon parameters charge (Schwinger, and, finally, how these techniques were elevated aga, and Feynman); to deal systematically to a research program with all kinds of diver a were to arbitrary in calculation that encountered gences perturbative and others). (Dyson, Salam, Ward, Mills and Yang, Weinberg, would into the story Gell-Mann and Low be K?llen, Landau, brought advanced in the early 1950s, when giving an account of the arguments, scheme. In the indicating the internal instability of the renormalization when renormalizable Wein the the mood of early 1970s, triumphant order

electroweak success, great empirical theory enjoyed berg-Salam be a record of events would history leading to this success: SSB

the and

dimensional mechanism, for non-Abelian gauge

quantiz regularization, path-integral and theories 1963; Fadeev (Feynman, use of renormalization 1967; Boulware, gauges ('t 1970), the Popov, turned their attention 1971a, 1971b), etc. After that, physicists Hooft, a final grand unified theory (Georgi to the ambitious task of constructing and Glashow, 1974; Fritzsch and Minkowski, 1975). Since how to deal is an essential with the relevant degrees of freedom part of that en Higgs ation

that led to the renormalization terprise, the various lines of researches Cal and Kadanoff; Wilson; Low; Fisher; group equations (Gell-Mann an of the would become and lan, 1970; 1970) Symanzik, important part an were of SSB of the various grand integral part history. Applications of the hierarchical unified theories and with them came a realization It would take a great deal structure of the domains under investigation. of guts

for a historian

of renormalization

to ignore

the evolution

of

RENORMALIZATION

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89

ideas and techniques So in some sense, leading to the EFT approach. as the history of renormalization, any other internal history of science, cannot simply escape being somewhat Whiggish. it is still However, in the pejorative sense, if critical aspects possible not to be Whiggish and perspectives of the theory are provided and put into a proper to be solved, when and the problematics intellectual context, they are a can serve two functions. The out. Such narrative emerge, pointed first is to provide the story of the evolution leading to the present stage in freeing physicists of knowledge, and the second consists from the of current ideas that present only one of the possible domination rami fications. Such a history would in the field the dangers show workers to current ideas. As our own presentation of confining themselves has made is linear or predetermined clear that no development and an of a different perspective might be fruitful. examination 4.5.1.

Change

of Outlook

We have pointed out that the presence of divergences results from too as literally true and globally naive a realism, which takes point models and that the search for renormalizable theories to establish applicable; the global stability of QFT is too high an expectation. With the develop ment of the renormalization with the discov group approach, especially structure of the hierarchical of the domains ery investigated by QFT, and the development of the EFT approach for dealing practically and with each of the the localist view of effectively layer hierarchy, physical theories gained momentum, and the whole outlook about renormaliz seems to cease to be a necessary ability has changed. Renormalizability for and nonrenormalizable theories can QFT construction, requirement be legitimately

employed.

4.5.2.

Experience

Scientific

and Foundational

Problems

on the internal consequences they can have long-term of the in the of the foun subject, changes developments understanding dations of a scientific theory do not normally have a great impact on the of the theory by the scientists in scientific practice and on the conception

Even

though

the field. This is because conflicting schemes are not usu foundational in with conflict the scientific and the silent majority ally existing practice, on of scientists would normally rather than draw their experience rely

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from a new perspective by a new under suggested inspiration as we of foundational have tried to make But, standing problems.60 a on to decide choice foundational schemes clear, among requires some - in other than scientific in experience particular, aptitude expertise their

acquired mainly through the study of logic and of the and the philosophy of history philosophy, philosophy science - and the availability of historical obtained from the insights study of the history of science. conceptual

analysis and of

NOTES * B. Dreben, R. Jackiw, J. Polchinski, H. Putnam, We would like to thank R. S. Cohen, A. Shimony, and S. Weinberg for critically the manuscript J. Renn, D. Robinson, reading and suggestions. the course of this work, we have and giving us their comments During L. M. Brown, R. S. conversations with S. Adler, W. Bardeen, benefited from helpful D. Gross, R. Jackiw, M. Fisher, H. Georgi, S. Glashow, S. Coleman, B. Dreben, Cohen, E. Lieb, F. Low, D. Nelson, J. Polchinski, H. L. Kadanoff, G. Lepage, Johnson, 'tHooft, A. Shimony, S. Saunders, M. Veltman, S. J. Renn, H. Schnitzer, G. K. Wilson, T. T. Wu, and Y. Yao. Our work F. Wilczek, A. S. Wightman, Weinberg, K.

Putnam, has

been

in part

supported

a grant

by

from

the National

Science

Foundation

(DIR

9014412 (4-59070)). 1

to small perturbations A stable theory is a valid theory that is insensitive that occur in that is entailed of the its presumed domain, i.e., the domain by the basic assumptions in any whatever taking into consideration theory. A stable theory can withstand happens to be defined The theory is assumed formalism by a mathematical part of its domain. and valid

the perturbations in a domain D, be might of stability

turbation' definition

do not the will

of stability will inconsistent theory

discussion An

statements domain

ously cutoff

initially extension

be refined during in Section be given is unstable because

the course

of

the paper.

A

more

general

4.3.

to be valid when it ceases contradictory if in some of its presumed still be unstable appear. A consistent theory may an unrenormalized In the case of QFT, it is undefinable and undecidable. theory it entails no self-contradictory because consistent yet unstable statements) (i.e.,

may be the occurrence region,

of the formalism. The is assumed theory changes on only a subdomain D\ could be defined A 'per D" of D. This to a larger subdomain initial of D'

include

and

and the

of

infinities

the procedure truth value of

indicates

that

it is not definable

of subtraction some

provides its statements.

of

in the ultra-relativistic

no effective In contrast

energy way to decide unambigu with this, a nonrelativistic

is stable. domain theory with a restricted one if the effects into a quasi-stable of the An unstable theory may be transformed can be absorbed In the case into adjustable 'instabilities' parameters. phenomenological a nonrenormalizable a renormalizable of QFT, one, while theory seems to be quasi-stable turns out to be much more than it appears the situation is not. Yet theory complicated to be: rigorous investigations or even inconsistent undefinable

have

shown

(see Section

that

a renormalized

2), while

a cutoff

perturbative nonrenormalizable

theory theory

is

RENORMALIZATION

may

be quasi-stable

with

a domain,

presumed

THEORY

by the usual

91

local field

theory

(see Section

3.5). The

of a stable theory has some affinity with the notion of a "closed theory" concept in the late 1920s and that he published in Dial?ctica in 1948. that Heisenberg advanced in Modern Science' in Heisenberg of a "Closed Theory" See The Notion (1974). For a of these ideas, see Chevalley of the formulation (1988). The notion of there (1989). But the conceptualization theory was also used in Schweber in the formalism. Thus it the stability of the theory against small changes emphasizes seems extremely to make in the formalism difficult small changes of quantum mechanics and the usual notions of causality. that maintain validity, empirical logical consistency,

historical

account

a quasi-stable

an interesting of quantum nonlinear (1989a, 1989b) has outlined generalization in subjecting the basic framework that can be used as a guide of the theory an investigation to experimental tests. Weinberg's formalism of the validity of permits that is at the core of the usual linear version of the the linear superposition principle

Weinberg mechanics

et al., 1989). But Polchinski (see Bollinger (1991) has shown that inWeinberg's an isolated of quantum could receive information via EPR mechanics system and allow EPR and that, in a Stern-Gerlach correlations communication, experiment, of the wave be possible. communication between branches function would 2 success and conceptual the tension between is not unique Certainly empirical instability to QFT. The case study presented in this paper serves a dual purpose: for (i) to provide

theory version

a contemporary of the tension in one of the most mature example in turn would and (ii) that the analysis have the physical sciences; an impact on the understanding of QFT and the direction of its future development. 3 exact: Note that this statement, is by no means in some cases, although quite popular,

analysis philosophical and sophisticated of

also require renormalization. We thank L. M. Brown for bringing theories would An is Landau's this point to our attention. illuminating example quasi-particle picture of at low temperatures Fermi by a dilute liquids. In this picture, liquid 3He can be described near a spherical excitations Fermi and the quasi-particles surface; gas of quasi-particle finite

can be viewed

as

of the constituents 3He fermions, with altered masses idea here, as that adopted is to replace a by field theorists, are the effects of interactions system with a simpler one in which interacting complicated constants. absorbed into redefinitions of masses and coupling 4 We shall not explore this aspect of the locality assumption, it is one of the most although and

interactions.

'dressed'

The

versions

basic

the foundations of quantum and comprehensive concerning problems physics, from both and have been investigations, physical philosophical intensively perspectives, undertaken since Bell's classic papers and further references, 1966). For details (1964, compare Bell (1987). 5 cannot claim originality Frenkel in this matter. The idea of point particles has a long us of this point. For tradition. We thank one of the referees of Synthese for reminding profound

a brief

exposition

(1982a, 1982b). 6

of

the

tradition

within

the context

of

classical

physics,

see Harman

we refer to in the text is that in the Heisenberg Note that the field operator picture, which has different from that in the interaction physical interpretation picture. 7 a (multiplicative) In addition, of the field operators is also necessary. renormalization 8 We thank L. M. Brown for bringing (ii) to our attention. point 9 This will be discussed in the next section.

TIAN

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10 For more of detailed descriptions see Cushing Chew's 5-matrix theory, 11 In the late 1960s and early 1970s, 'tHooft 'tHooft and 1971b; (1971a,

SILVAN

S.

SCHWEBER

what

the way for the acceptance of exactly paved (1990) and Cao (1991). Veltman 1968b, 1969a, 1969b, (1968a, 1970) and a series Veltman, 1972a, 1972b, 1972c) published for proving of the Yang-Mills the renormalizability theory

of papers that paved the way in general, and that of the Weinberg-Salam regularization using dimensional particular, ances.

The

and

of Veltman

work

'tHooft

at the one-loop is less convincing; to be cultivated

is convincing proof feel that the result soil

model and quantum in chromodynamics that preserved and Poincar? invari gauge was et al. (1974). Their by Becchi improved some people at the two-loop level, however,

level; as for the higher-loop a remark made to us

remaining conversation 1989). (Nov., 12 on renormalization in 1969, Dirac At a meeting theory to the Heisenberg for solutions malization when looking to the cutoff. insensitive

it seems levels, by T. T. Wu

to be virgin in a private

accepted

the need

equations

of motion

for renor that are

to be logical, that 8mlm and ?e/e shall be small, It is necessary, for the calculations so that the terms involving not make But we must them shall be small.... [the that by doing so they like to make g?? ?? and believe physicists cutoff] g?? oo.Many get

a relativistic

1969b,

theory.

But

13 This, however, The divergence malization theory 14

calculations

cannot

be made

relativistic.

(Dirac,

(1981) and Gell-Mann (1987). series shows the instability of renor perturbative a bearing on its consistency. Yet its undefinability, without as shown by Landau at ultra-relativistic and by energy,

is controversial. of

See Weinberg

the renormalized

in terms

of

of perturbation the breaking indicates Gell-Mann and Low, its negation: entails perturbative turbative. 15 Wightman field theory,

such

p. 2)

its inconsistency. at some energy

That region

an expansion is, assuming being becomes the expansion nonper

field theory as an offspring of axiomatic quantum (1978) takes constructive some difference the concern of axiomatic field between them. While with the constructive fields, theory of quantum models and constructs solutions satisfying see field theory, of axiomatic development

theory

is the general

specific former.

Lagrangian For early

field

theory

starts

the

requirements Jost (1965) and

from of

the

Streater

field theory, compare Velo and Wightman and Wightman (1964); for constructive (1973). 16 a proof of consistency that in Hilbert's formalist and finitist program Notice guarantees its stability. 17 can be stated for other regularization schemes: the same claim as for The following can be made for the (essentially regularization equivalent) not for dimensional which is more formalistic regularization, to the point discussed here. and irrelevant 18 that "it now appears further explanation likely (1989, p. 3) asserts without Lepage that this last step (taking the cutoff to infinity) is also a wrong step in the nonperturbative theories, including QED". analysis of many 19 simulate the low energy evidence of the inaccess interactions The nonrenormalizable the Pauli-Villars-Feynman lattice cutoff scheme,

but

ible high energy dynamics and are thus suppressed divided by the mass of the heavy boson.

by a power

of the experimental

energy

THEORY

RENORMALIZATION

93

20 symmetry

Explicit

review

(1991). 22 As

unitarity

out by pointed can be saved

and

Steven

that spoils Lorentz integrals 23 An illustration important renormalizable

when

only

(1983) in the mid

physics his thinking 26 (a) Wilson

a constraint

the number

operators (Wilson, degrees

and Cao

is the following: the standard model so that and leptons is the same, lepton sector cancel each other.

of quarks and by the

in the renormalization points of QED. its asymptotic Thus scale how the progress in statistical and of Kadanoff, had influenced

described vividly the works of Widom

especially

is the

of fixed space

only a conjecture. in his Nobel lecture

1960s, in theoretical

see Brown

in a private (22 Sept. 1991), correspondence a definition of the measure for the path

has

remains

subject,

to a pure

invariance. of such

caused

group invariance 25 Wilson

this

terms

to take

by the quark sector been no proof for the existence constant of the coupling transformations

anomalies 24 There

of

analysis

conceptual Weinberg is willing

if one

mass

e.g., adding non-gauge-invariant to our discussion here.

breaking, is irrelevant

theory, Yang-Mills 21 For a historical

physics. corroborated

his dynamical view of the scale dimension of field an analysis of the Thirring and of \4 theory model (Wilson, 1970a) new the dimensions of spacetime have become then even 1970b). (b) Since sense. at least in an instrumentalist of freedom, In statistical the e physics, further

with

was in QFT, dimensional tech Both introduced; expansion technique regularization. are based on this new conception of spacetime dimensions. While realistic field niques can be have to be four-dimensional, theoretical models and only toy models ultimately are of great in statistical two-dimensional models however, two-dimensional, physics, in the real world. relevance 27 It is worth that there noting the asymptotic scale invariance

is an important difference between Wilson's of concept at short distances of QFT and that of Bjorken (1969). While about the form factors in deep inelastic Bjorken's scaling hypothesis lepton-hadron seem to turn off at very short distances, that the strong interactions suggests scattering Wilson's effects way

formulation of

of

Bjorken's

of OPE

interactions

into

reestablishes

the scale

the anomalous

dimensions

corrections logarithmic expressing soon fitted ideas were into the

to the

invariance

scale

framework

the only after absorbing is just another the fields. This

of

invariance

of

the

an non-Abelian

of

theory.

Thus,

gauge theory found its appli

as asymptotic and reexpressed while Wilson's idea has freedom, (QCD) cations in other areas 28 that the relevance is sometimes (1983) also noticed Weinberg problem an appropriate than simply choosing energy scale: it involves turning plicated

more

com

on collective

ones and turning off the elementary of freedom degrees (e.g., hadrons) (e.g., quarks and gluons). 29 are: (i) The most widely known of the renormalization group equations applications a basis for a perturbative in QCD, which provides the observation of asymptotic freedom and Wilczeck, that is renormalizable QCD 1973; Gross 1973a, 1973b); and (ii) (Politzer, the calculations electroweak

of

the variation

interactions,

al., 1974). (Georgiet 30 The Gaussian fixed field

distributions

which

of

the coupling

lent

support

point corresponds are Gaussians.

constants

with

to the proposal

to a free massless

in the strong and energy of grand unified theories field

theory

for which

the

TIAN

94

YU

CAO

AND

SILVAN

S.

SCHWEBER

31

as perturbatively is regarded renormalizable the breakdown of QED only because as pointed out by Gell-Mann and Low, energy, theory at ultra-relativistic perturbation is ignored. 32 is entailed That the tower of EFTs would be endless formulation by the local operator our earlier discussion of QFT. Compare 33 This means that the nonrenormalizable the extension

of

the subdomain

within

on

the operator field assumption in Section 2. effective the effects of theory can also absorb as presumed its domain, by a local field theory

and thus qualifies itself as a quasi-stable theory. 34 that the root of Mary Hesse (1962, p. 262) asserts to the basic notion traced of atomicity. If the notion

the divergence of atomicity

difficulties

can be

to the point if as However,

refers

or to local excitations and local couplings, the assertion is correct. then the root of the divergence usual, it refers to the structureless constituents, elementary is the structure in unrenormalized difficulties field theories, operator implicit assumption rather than the notion of atomicity. 35 It is worth of string theory that the point of departure is the possibility of stressing or rather the substratum, not as the basic building of the universe blocks, considering as they are in QFT, zero-dimensional but as one-dimensional point particles, objects either tiny closed loops or open curves with endpoints. 36 Karl Popper the implications about of the emergence argues convincingly viewpoint for pluralism in theoretical 1970, esp., pp. 6-9). (see Popper, ontology

model

in arguing Abner for a general reductionist the program, Shimony (1993), rejects as a idea of objective and suggests instead a notion of epistemic emergence emergence 1987). (see also Shimony, compromise 37 Part of the initial attraction of string theory was the fact that any consistent relativistic is highly constrained. Thus a quantum objects theory of extended a quantum extra sional strings contains of supersymmetry, theory that agree with those found in and matter multiplets gauge groups, scheme of the standard model. 38 results until the new colliders The dearth of new experimental another 39 The due 40

Some

lie. Part

of one-dimen

dimensions, gravity, the grand unification go

into operation

is

factor. localist

to Mary

maintain

theory

view

Hesse.

of scientific

theory

See Hesse

(1974,

in the next few paragraphs is principally expounded and Hesse 1980) and Arbib (1986). Ian Hacking (1983) and Nancy (1983), Cartwright

in particular philosophers, laws are possibly that, while phenomenological of the reason for their rejection of fundamental

laws certainly true, fundamental laws is that they play down the the function of fundamental laws,

in science, which is largely of explanation importance on predictive success. This, in turn, seems to have its roots in their and focus principally are defined of rejecting universals, which in terms 'naive' empiricist traditionally position of an Aristotelian of fixed natural kinds. realist ontology

such as late Wittgenstein (1953) and Hesse (1974), empiricists, 'sophisticated' an alternative Aristotelian have developed also rejecting universals, theory of is based on Wittgenstein's of "family which resemblance". thus universals, concept They a localist view of fundamental a framework within which laws and scientific provided More

while

can be developed and Hesse, 1986, esp. Chap 8). (cf. Arbib explanation 41 was first elaborated Its starting This position by Hesse (1965). point was Black's in the light of Wittgenstein's interaction theory of metaphors (1962), which was modified

RENORMALIZATION

THEORY

95

It was further developed and Hess See also resemblance". by Arbib "family (1986). Hesse (1963). 42 4.3. This point will be discussed in Section further 43 in Japan during the 1950s and the 1960s was characterized Fundamental by physics rather than the study of dynamics. The influence of Taketani model-building strong (1942)

and

Sakata

1950;

Sakata

et

(1947, al.,

1952)

1950, made

1956;

Sakata

ones. over dynamical investigations 44 Dirac's work on the relativistic theory work on the Pauli algebra; Gell-Mann's

and Hara, physicists

Japanese of

the electron

the quark and Gell-Mann

the SU(3) A-matrix. Cf. Dirac (1977) 45 In choosing criteria for theory appraisal, when to a program. have actually committed themselves

from accepted inevitability, derivability principles, his theory construction. See Weinberg (1983). 46 on the criteria of theory For a detailed discussion

1947; Sakata so-called

prefer

started

model

with

started

with

and Umezawa, 'substantialistic'

his "playing" his playing

with with

(1987). constructing For example, and simplicity

some

theories

physicists takes logical in principles

Weinberg as guiding

compare among numerous appraisal, recent discussion A widely acclaimed of the role of in theory appraisal is that given by van Fraassen (1980). empirical adequacy 47 For example, aether theories century British physicists during the nineteenth preferred favored while continental action-at-a-distance theories. physicists 48 In Reichenbach's (1938, 1951), they are out of the context of justification. terminology 49 introduced the context of the well-defined between Thus, boundary by Reichenbach references,

Buchdahl

(1970,

1980).

and the context of justification is to be blurred discovery again. 50 Note that this criticism did not challenge the logical consistency of Newton's theory. 51 This assertion in the literature thank is not undisputed 1986). We (see Freudenthal, one of the referees to our attention. for bringing this point 52 cases can be easily in the history Similar found of twentieth-century One physics. is the faith that physicists had in Pauli's hypothesis of the neutrino, example prior to its in 1952 (see Reines, detection albeit the faith was based on the more entrenched 1989) of energy conservation. principle 53 Cf. note 15. 54 is initiated Often the extrapolation

by what

Schnitzer

(1988)

has

called

a "crucial

calculation". 55 As was pointed

out to us by one of the referees, note that these two theories are not as well as fer to bosons is applicable While Feynman's theory completely equivalent. fails in the case of bosons. Yet this difference carries no weight in asserting mions, Dirac's that Feynman's is truer than Dirac's. We shall return theory of the positron discussion in the last paragraph of this section. of equivalent theories 56 was first pointed out by Putnam This metainduction (1977). As we have refers to "fundamental" theories. In a letter

to us

(22 Sept.

1991),

Steven

Weinberg,

referring

to a further stated

it, it

to the metainduction,

commented: to is true, as it may well be, there is no reason a permanent to be true, and that we will never discover After all, a nineteenth say that "Just as century explorer might as the source of the Nile two decades of the Nile ago is regarded Even

continue

if this

that it will suppose fundamental theory. no supposed source today,

so no present

TIAN

96

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AND

SILVAN

SCHWEBER

S.

as the true source two decades be regarded from now." to the Nile was discovered, and it stopped being possible statements. It seems to me entirely that some variant of today's possible as a fundamen theories will be recognized 'overly grandiose' superstring permanently tal theory. source

day

He

the Nile

of

Eventually make such

the source

also

"I was

added:

will

of

saddened

by your

retreat

from

'naive

realism'

at the end

of

the

paper". 57 The

is an unobservable 'stable' or 'quasi-stable' issue is: How entity rendered enough to justify Hacking's claim as to their reality? 58 realist ontology of fixed natural The of an Aristotelian kinds has various rejection to this, it is only one of them. In addition ramifications. Instrumentalist epistemology a way to a sophisticated In the domain of ontology, realist epistemology. also prepares idealisms and for dialectical materialism. the rejection also opens a door both for various 59 to Schlick and Russell discussed the reference of theoretical (1918) (1927) ontology issue has recently been discussed characteristics of reality. The the structural by Maxwell and Friedman and Cao (1986). (1985), (1970), Demopoulos 60 to an novel may be happening in high energy that is related Something physics in of science: Do social factors the affect the cognitive sociology interesting question content and structure of a scientific in the physical sciences? The transition from theory

to the EFT approach content theories the cognitive significantly changes involves nonrenormalizable and of QFT. The EFT approach interactions, of the limits of the applicability of some version of it justifies the determination (in part) such as the superconducting of multi-Tev accelerators the building (SSC). supercollider to high energy experimenters connected with the SSC project. This is certainly attractive renormalizable

and

structure

a large sum of funding to high energy theorists because attractive in the 1980s a is important Thus for the support of their research. to operate seemed that certainly had some impact of mutual reinforcement mechanism on the development of EFT approach. A social study of this episode would be interesting, The

is also

SSC project and

this

is involved

but we

have

nothing

more

to report

on

the matter

here.

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