New measures for nonrenormalizable quantum field theory

ANNALS

OF PHYSICS

117, 19-55 (1979)

New Measures for Nonrenormalizable Quantum Field Theory JOHN R. KLAUDER Bell Laboratories, Murray Hill, New Jersqv 07974 Received December 22, 1978

Generic interactions characteristic of so-called nonrenormalizable scalar and spinor quantum field theories are interpreted as discontinuous perturbations in the sense that the theory does not return to the unperturbed theory as the interaction coupling vanishes. To proceed beyond this interpretation specific alternatives to conventional quantization schemes are developed. Solution of a highly idealized (independent-value), nonrenormalizable scalar field theory automatically entails a formally scale-invariant measure (rather than the conventional translation-invariant measure) in a functional integral formulation, and the success of this measure suggests its use more generally. Such a measure can be motivated (by augmented field theory) on heuristic grounds as taking into account the partial hardcore nature of the interaction responsible for its behavior as a discontinuous perturbation. This modification leads generally to what we call scale-covariant quantization, which can be formulated in terms of unconventional functional differential equations, coupled Green’s function equations and operator field equations. Use of affine fields establishes equivalence of these various approaches and enables analogous coupled Green’s function equations for models with fermions to be most easily obtained. The basic concepts of this program are illustrated with elementary wave-mechanical examples.

1. INTRODUCTION

AND OVERVIEW

One of the greatest challenges that faces contemporary quantum field theory is the formulation and analysis of theories conventionally regarded as nonrenormahzable. Several arguments suggestthat such a challenge exists and that it must be faced. On the ore hand, nonrenormalizable models [e.g., (+4)1L, a quartic self-interaction in n space-time dimensions where n > 41 have well-behaved, nontrivial solutions when regarded classically [l]: quantum mechanically it is a different story. As is well known a perturbation analysis of the quantum theory leads to an infinite number of distinct counterterms [2], while a renormalization group approach leads to a trivial (free) behavior [3]. Both results are unacceptable. On the other hand, there is as yet simply no guarantee that nature has restricted herself to renormalizable theories. It would be foolhardy to seek all answers within the narrow confines of renormalizability when the answer-possibly even economically formulated-may lie outside that limited set. Contrary to the self-serving attitude that dismissesany possiblephysical relevance to nonrenormalizable theories simply because they are nonrenormalizable, we argue that only experiment can decide the relevance of any theory and for that purpose methods to attack these presently insoluble problems need to be found.

19 0003-4916/79/010019-37$05.00/O All

Copyright 0 1979 by Academic Press, Inc. rights of reproduction in any form reserved.

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Of course, nonrenormalizable theories have been the subject of various investigations and approaches. In dealing with the nonlinear spinor theory Heisenberg and collaborators introduced an indefinite metric in the Hilbert space at the outset to soften the singular nature of an otherwise nonrenormalizable theory [4]. Various attempts have been made to sum selected graphs with leading order divergences on the assumption that lack of expandability in the coupling constant was the main source of difficulty [5]. In fact, comparison with the nonexpandability that arises for infrared divergences of super-renormalizable theories has led to the widely held view that ultraviolet divergences typical of nonrenormalizable theories can be understood in terms of superrenormalizable theories [6]. Consequently, heavy reliance has been placed on renormalization-group analyses which have generally proved successful for the calculation of critical exponents and other nonanalytic, infrared-dominated properties [7]. In another approach special properties of certain interactions are invoked to argue for the mathematical existence of some form of “minimality” in the singular nature of the omnipresent ambiguities in the Green’s functions [8]. In still another approach it is argued that certain theories (e.g., asymptotically-free gauge theories) that by naive power counting ought to be renormalizable but instead through cancellations turn out to be super-renormalizable should admit a slight extension (say of space-time dimensionality) to theories that are superficially nonrenormalizable before the real limit of renormalizability sets in [9]. And in yet another approach it is argued that nonrenormalizable theories become defined as alternative, renormalizable one through selected graphical resummations [lo]. In our view these ideas, generally speaing, are either incorrect or inadequate to handle nonrenormalizable models. As traditionally encountered, infrared and ultraviolet divergences are fundamentally different since the former arise from nonlinear interactions which are short range in configuration space while the latter arise from nonlinear interactions which are long (not short) range in momentum space [ll]. Thus there is no contradiction in accepting the results of renormalization-group analyses for critical phenomena and rejecting them for nonrenormalizable models. What goes wrong with the conventional renormalization-group approach-or stated positively, what should be done differently-is a subject treated in this paper. Of course, no one can exclude that special interactions may yield to special’ methods, but such cases are by their very nature highly specific. (Even the present author has discussed a soluble, covariant nonrenormalizable model that has relatively few surprises except when an operator realization is attempted [12].) What is ultimately needed, however, is not just a collection of tricks for a few specific models, but a generic understanding of just what nonrenormalizable models are all about. Such an understanding, we believe, is at hand, and elucidating that viewpoint and some of its consequences are the main goals of this paper. Much of our viewpoint and approach regarding nonrenormalizable models have direct and intuitive analogs in simple Schrijdinger wave-mechanical problems. These analogs provide an overview of our basic concepts, and a discussion of them is a good way to begin.

NONRENORMALIZABLE

The Key Concepts Continuous

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21

THEORY

in Wave-Mechanical

Problems

Perturbations

The traditional viewpoint regarding nonrenormalizable models can be illustrated by the example of a singular radial potential with Hamiltonian H = -4jV

+ $9 + hlrw

(1.1)

in an s-dimensional space, s 3 2, and where h > 0. We have included a harmonic potential r2 to make the spectrum discrete, but that is not essential to the argument. Now, whenever (Y> 3 it follows by simple scaling that the potential dominates the kinetic term at small distances, and furthermore that the deviation of the energy levels from the free (harmonic oscillator) levels behaves for small h as h1/(a-2). Even though for small X the correction to the free energy levels is also small, such a dependence on h of the energy levels is continuous but nondifferentiable at the origin. In the range 2 < 01< 3, where the potential still dominates the kinetic energy at small distances, there is a somewhat different behavior with respect to X. For CY-= 3 the deviation of the energy levels from their free values behaves for small h as --hlnh. For 2 < 01< 3, on the other hand, the deviation behaves for small h as h itself with the characteristic nonanalytic behavior h1/(a-2) appearing now as a higher-order term. By analogy, one commonly supposes that nonrenormalizable models have energy levels (Green’s functions, scattering amplitudes, etc.) that deviate from their free behavior in a continuous but (at some order) nondifferentiable manner as the coupling constant increases from zero. Efforts to sum leading-order divergences are aimed at determining the dominant nondifferentiable (or even nonanalytic) behavior. Clearly, this type of behavior exists and it may indeed apply to certain special nonrenormalizable models. However, in the general case we believe that this description actually understates the real state of affairs. An elementary example of that kind of situation is given next. Discontinuous

Perturbations

Consider the same example as above except now in a one-dimensional This leads to the Schrodinger problem with Hamiltonian H =

-+ayax2

+

4x2

+

x/l

x 1~

space (s = 1). (1.2)

with the same parameters 01and X > 0 as before. Evidently this system has discrete energy levels and eigenstates, and undoubtedly as X ---f O+ these energy levels and eigenfunctions pass continuously to those of the free system, the harmonic oscillator. Such continuity seems obvious enough, but it is fake if OL> 2! [13] The levels and eigenfunctions pass continuously all right to something as X ---fO+, but not to the free system; instead they pass to a “pseudofree” system characterized most simply in this exampIe by Dirichlet boundary conditions [i.e., all eigenstates satisfy #(O) = 0] that are freely implied by the singular interaction whenever 01> 2. (Indeed, how to avoid this situation in the range 1 < 01< 2 is a fascinating story in itself complete

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JOHN

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with viable analogs of normal ordering and of renormalization counterterms that necessarily involve a polynomial or even a nonpolynomial function of the coupling constant h [14].) Since for a! > 2 the Hamiltonian in (1.2) is not even continuously connected to the free theory, then the free theory certainly cannot in any sense serve as the start of a meaningful perturbation calculation. After all, perturbation theory cannot reproduce a discontinuous function. But that criticism does not apply to the pseudofree theory which may serve as the start of a perturbation calculation since it is by definition continuously connected to the interacting theories. In particular, for “soft” perturbations, where 2 < 01< 3, the deviation from the pseudofree behavior is in fact both continuous and differentiable at the origin at least once, while for a: > 3 only continuity remains. But given continuity one can at least hope that some sort of leadingorder-divergence resummation scheme may be used. And for this to work it must entail the pseudofree theory and not the free theory. By analogy, and based on arguments presented below, we expect the generic nonrenormalizable theory to be a discontinuous perturbation, that is, not continuously connected to the usual free theory, but instead continuously connected to a (nonfree) pseudofree theory. Soluble examples of just this kind of behavior will be discussed, and, as it turns out in most cases of interest, the interactions are “soft” in the sense that the interacting theories admit power series expansions in the coupling constant -about the pseudofree solution, of course. Path Integral Viewpoint

The examples we have discussed above can be analyzed in alternative ways as well. In particular, path integral techniques are especially instructive, notably for the diffusion equation where real probabilities and true measures are involved. In this picture the interaction acts as a partial hard core projecting out certain histories that would otherwise be allowed by the free term. As the coupling goes to zero these paths remain absent and their absence ultimately leads to the pseudofree theory and not to the free one. For the one-dimensional example discussed above, it is intuitively clear that the singular interaction retains only those histories for which either x(t) > 0 or x(t) < 0 for all t and suppresses (projects out) any contribution from the remaining paths. Heuristically, the path integral involved may be taken to have the form C(S) = JV s exp Ii s sx dt - s [&(a? + x”) + X/l x I”] dr) 9x,

(1.3)

where s = s(t) is a smooth function, formally 9x = nt dx(t), and M is a formal normalization constant chosen to normalize the characteristic functional C(s). The suppression of certain paths as h + 0+ implies that C(s) does not pass to the characteristic functional of the free theory (which necessarily includes contributions from the suppressed paths). This issue is discussed further in Section IV. In the context of Brownian motion a suppression of paths that reach a certain level leads to the familiar absorbing Brownian motion, and the notion of a hard core has

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real significance [14]. However, in the context of a stationary Ornstein-Uhlenbeck process [15] suppression of paths that reach a certain level leaves only a set of measure zero remaining and so the concept of a hard core has only heuristic significance. This fact does not prevent the existence of a measure with the proper constraint on paths; it only asserts that the two types of measure are mutually singular. This is the typical behavior we expect to encounter in field theory.

Equations of Motion Still focusing on the same one-dimensional example, let us remark on one additional approach to this problem. Take as action principle the conventional expression

z = s [3(3i*:! - x’) - A/;x ‘“1dr

(1.4)

and derive the equation of motion by stationary variation. Conventionally we make an arbitrary variation of the form 6x(t) = &l(t), but instead let us make a variation of the form 8x(t) = &S(t) . x(t), namely a variation proportional to the displacement itself. The former variation is the infinitesimal form of the transformation x(t) + x(t) + cl(t), while the latter variation is the infinitesimal form of a transformation of the type x(t) * S(t)x(t), where S(t) > 0. For the problem at hand, with the singular potential 1x I-&, the first kind of variation may take an allowed path for which x(t) > 0 into a nonallowed path, while the second kind of variation can never do that. The equation of motion given by the latter variation reads x(2 + x) = cd/l x p,

(I .5)

and evidently leads to a solution (say) where x(t) > 0. For this purpose the extra s is immaterial, but when h --f Of, the equation of motion reads x(2 + x) = 0

(1.6)

which admits the solution Alcos(t + B)I, A and B arbitrary, as well as the here unphysical one A cos(t + B). In other words, the equation of motion obtained through variations consistent with boundary conditions implied by the potential admits even after X --+ Of the pseudofree solution otherwise obtained as the limit as h + 0+ of the interacting solutions.

Survey This brief discussion of wave-mechanical problems serves as a prelude to the remaining sections. Quantization of field theories by means of functional integration is discussed in Section II and partial hard-core behavior is attributed to nonrenormalizable interactions for scalar models and (implicitly) for models with spinor fields. An idealized, soluble, nonrenormalizable scalar model (independent-value model) is treated in Section III by functional integration methods, and the lessons learned from

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JOHN R. KLAUDER

that exercise are applied to covariant models in Section IV stressing the importance of scale-invariant formal measures in functional integration. Models so characterized may be called scale couariant, and in Section V unconventional functional differential equations, coupled Green’s function equations and operator field equations for scalecovariant scalar and spinor models are discussed. Section VI is devoted to a brief conclusion.

II. HARD-CORE PICTURE FOR NONRENORMALIZABLE MODELS Scalar Fie1d.v

We first focus on scalar field theories and approach the quantization via Euclideanspace functional integration in an n-dimensional space time. The expression of interest is formally given by (dx = d”x) S(h) = JV J exp (i 1 h@ dx - f Mv@)z

+ m2@2] + A@~} dx) .9@

where h = h(x) is a smooth function (say, Corn), Qi = Q(x) a random formally mD = rj d@(x). a!

(2.1)

field, and (2.2)

The formal constant H is chosen so that S(0) = 1. Such a formal prescription to define the generating functional for Schwinger functions is common [16]. It would seem evident that the limit h -+ 0+ of this expression should reduce to the expression for the free theory, namely

It is this assumption that underlies the program of renormalized perturbation theory. This assumption is surely valid if the interaction is first regularized with a high momentum cutoff. But it is an act offaith that the solution of the problem is that given by the limit of regularized solutions as the regularization is removed. This distinction is not merely a “mathematical technicality”; quite the contrary, for in certain cases no solution exists (other than free) when the problem is approached through a standard cutoff procedure, although an acceptable nontrivial solution can be found otherwise. A concrete example of this behavior will be given in Section III. Under what conditions could the functional integral (2.1) not pass to the free solution as X -+ 0+ ? Such behavior may arise whenever random fields are included for which the free term is jinite while the interaction term is infinite [17]. This simple

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2.5

characterization is very general, and it applied to a wide variety of systems. For the special case of scalar fields under present consideration this question can be examined with the aid of Sobolev-type inequalities. We recall [17] the fact that a finite constant K independent of @ exists such that

il

j CDj’ dnx

2/P< 1

K

s

[(Vo)” + mZQi2] d”x = K /i 0 [If,

(2.4)

provided p < 2n/(n - 2). If p > 2n/(n - 2), no such finite constant K exists. Topological Properties

Let us examine several consequences of the inequality metric on fields given by d(Q) 3 11Sp11where

(2.4). We first consider the

II @l!2 = II 0 IIf + II Qi IIf>.

(2.5)

Here /j @ 11:is defined in (2.4), and (2.6)

If p > 2n/(n - 2) it follows from (2.4) that the metric d is equivalent to the free metric dF[dF(@) = // @ II,] since in that case

II @lit < II 0 II2 < (1 + K) II @I~“,.

(2.7)

Consequently, the topology of the fields is the same as in the free case so long as p < 2n/(n - 2). On the other hand, if p > 2n/(n - 2) this conclusion no longer holds; in that case, the topology on fields induced by /( @ 11is vastly different than that induced by 11@ jjw . Certain fields previously included (for which /j @ /lw < a~) are now excluded (since jl @ ll2, = cc for them), although the remaining set is dense in the original set. An example of an exluded field is one that has a local singularity of the form I x I-” under conditions such that n/p ,< y < (n - 2)/2. The topological considerations referred to above are entirely analogous to those that arise when considering

Here we have replaced )/ @ llu) by I/ @ II2 , and whenever p > 2 we can expect that the topological structure defined by /j @ II is vastly different than that defined by II Q, Ii2 . Again certain fields previously included (for which /j @ II2 < co) are now excluded (since I/ @ IID = cc for them), alth ough the remaining set is dense in the original set. In this case, an example of an excluded field is one that has a local singularity of the form I x I--Yunder conditions such that n/p < y -C n/2.

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Relevance for Field Theory

To say that one set is dense in another may give the impression that no peculiarities in defining functional integrals may arise. Nothing could be further from the truth! And the simple reason for this stems from the fact that the measure given to one set may be nonzero, while that given to a dense subset may be identically zero. This happens with the ordinary Lebesgue measure dx on the real line in relation to the dense subset of rational points. When a topology change occurs in the field case the interaction projects out certain fields leaving a dense set to carry the measure, which is then resealed to a probability measure. Once they are absent these fields might never reappear, which automatically leads to a discontinuous perturbation. More specifically, the set of fields on which the measure of the interacting theory is concentrated just may not converge to the set of fields on which the measure of the free theory is concentrated as h -+ O+. And such a situation may arise whenever the set of fields of finite action for the interacting theory is a dense, proper subset of the set of fields of finite action for the free theory. On the basis of Sobolev-type inequalities, we have argued that this situation may arise for covariant scalar fields whenever p > 2n/(n - 2). It is of basic significance that this very same criterion, namelyp > 2n/(n - 2), is exactly the one which characterizes nonrenormalizable covariant self-interacting scalar models according to conventional renormalized perturbation theory [18]. Thus we are led to identify discontinuous perturbations as giving rise to the behavior normally ascribed to nonrenormalizable fields. Spinor Fields

Once having committed ourselves to this point of view, the argument is not limited in its application just to covariant self-interacting scalar fields but should apply much more generally. One important class of examples pertains to covariant spinor fields for which a typical Lagrangian density in Minkowski space is given by

(2.9) or even with more general nonderivative interaction terms involving p fermion fields. To discuss this situation it is useful to appeal to a more general form of Sobolev inequality [19] given by (j- I @P(41pd’$”

< K j- (I k [2E + m”) I 6(k)12 d”k

(2.10)

which holds with finite, field-independent K provided that p < 2n/(n - 25). On the other hand, if p > 2n/(n - 2l), then no such K exists. For boson fields E = 1 while for spinor. fields f = + since the free Lagrangian contains only one derivative. By analogy, therefore, we expect the topological changes that arise when the interactionterm power p > 2n/(n - 1) to lead to a discontinuous perturbation in a functional integration. The criterion p > 2n/(n - 1) is exactly the one which characterizes

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nonrenormalizable covariant self-interacting spinor models according to conventional renormalized perturbation theory. In particular a local Fermi coupling as given in (2.9) has p = 4 and is thus nonrenormalizable whenever n > 2. This conclusion applies whatever the tensor character of the interaction since the y-matrices disappear in the inequality. Consequently, any delicate balance of cancellations that may occur among various components is not apparent at this level of argument: in any case. such cancellations are not generic. Yukawa

Models

We should also comment on the typical Yukawa model with Lagrangian given by &iy”2, - M)$ + &[(Q$)z - m2+2] - h&j+.

density (2.1 I)

To discuss this situation we note that (2.12) for any pl, ps ) 0 such that pyl + p;l z 1. The same topology as the free case arises provided p1 < 2n/(n - 2) and p2 < n/(n - 1). And a satisfactory solution exists, therefore, so long as (n - 2)/2n + (n - 1)/n < 1, or in other words provided n < 4. Conversely, if n > 4 the topology changes and a discontinuous perturbation is expected, all of which agrees with the onset of nonrenormalizability for this model. Extended Model Fields The inequality (2.10) immediately suggests the possibility of considering extended or generalized models for which 4 is not restricted to either 1 or %. For this generalization the Euclidean action is taken as l f (i k [2c + m2) I 6(k)12 dnk + h f j @(s)~J’ CPX. Y

2

(2.13)

For general ,$ the free term exhibits long-range coupling and although the model has O(n) and translation invariance in Euclidean space it does not lead to a local relativistic theory in Minkowski space. But since the goal is gaining a general understanding of singular interactions in field theory we can also profitably study models that do not exhibit full relativistic covariance. Equation (2.10) implies that the metric d [d(Q) rr ‘1@ I], where 81 @ ii2

c

iI @ i t.,:

is equivalent to the free metric d,,.,,[d,,,(@)

-+-

1’ @

(2.14)

f,

= ‘/ @ I’i,,,E-],where

I/ @ IIf/.,, =-= f (I k != -t m2) 1 @k)l”

t/“h-,

(2.15)

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JOHN R. KLAUDER

wheneverp < 2n/(n - 26). Thus the topology of fields is the same as in the free case. Whenp > 2n/(n - 2.5, on the other hand, the topology is vastly different than in the free case. The general picture for 5 < 1 is qualitatively the same as for 4 = 1, with the principal difference being that for smaller 5 the onset of topological change occurs for smaller p values. The discontinuous nature of the perturbation and the hard-core picture of nonrenormalizability must apply to the generalized models where .$ < 1 just as much as it applies to the covariant case [ = 1, and it must arise at the onset of the topological change, namely at 2n/(n - 28). It is evident that n and f do not enter independently but only in the ratio t/n; that is, nonrenormalizability arises for p > 2/(1 - 25/n). Thus one can study the features of nonrenormalizability by varying II at fixed f (e.g., .$ = 1, which is the usual case), or by varying [ at fixed n. (In fact, all the basic divergence features appear for space-time dimension n = 1, i.e., time alone, when all 5 are considered; thus the basic features of field theory can also be found in noise theory

PI*)

For any it the severity of nonrenormalizability increases as 5 decreases. This can be readily understood. As .$ decreases, the space of fields with finite free action enlarges, i.e., additional functions are added, and so for the interaction term to be finite on the enlarged space the power p must, in general, be correspondingly reduced. Clearly the most severe test of nonrenormalizability arises for the smallest 5, namely for .$ = 0. Here nonrenormalizability sets in for any p > 2 independently of n, which just reflects the fact that a function that is locally L2 is generally not locally Lp whenever p > 2. This is just the idealized example discussed earlier from a topological point of view. Here we see it as a candidate for a nonrenormalizable model, indeed the most severe test of nonrenormalizability there is in this class of extended models. If this most severe test is successfully analyzed, then there is hope that the less severe cases (e.g., 5 = 1) can also be understood. In the next section we shall analyze the idealized model for .$ = 0 and show that a fully consistent solution exists which, incidentally, illustrates all the main features we have ascribed to nonrenormalizable theories.

III.

INDEPENDENT-VALUE THEORY AS A NONRENORMALIZABLE MODEL General Remarks

In the framework of the extended models discussed in the previous section we are led to consider a model formally expressed as (dx E d”x) S(h) = JV 1 exp (i 1 h@ dx -

&m2 f a2 dx - h 1 CD dx) 9@,

(3.1)

where JV is chosen so that S(0) = 1. On the surface, this is just (2.1) with the gradient terms deleted. Without gradients there is no mechanism for the field at one point to make its influence felt at another point. No physics should be ascribed to such a model; its only virtue lies in its solubility as an example of a nonrenormalizable model.

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Symmetry The absence of any propagation leads to independent behavior at every point of space time, and such models may be called “independent-value” models [21]. This symmetry feature is reflected in the fact that S(h) necessarily has the structure S(h) = exp I-- 1 f.[h(.~)] L/X/

(3.2)

for some function L, where L[O] = 0. Next we assert that S(h) is formally the Fourier transform of a positive integrand and that whatever else they do, renormalizations of the P and @’ terms preserve that positivity. Hence S(h) is a positive-definite functional -in fact, a characteristic functional-and in that case all the functions L can be represented in a standard canonical form. Since this canonical form is fairly central to our argument we shall outline its derivation. Let h(x) = hxa(x), where h here is a number, and xd is an indicator function, xA(x) = 1, x E d, and x4(x) = 0, x 4 A. In that special case S(hxJ = exp(--dL[hJ), but this expression is still a characteristic function with respect to the variable h and so exp{--dL[h]f where ~4(u) is a probability

= J eihs dp.(u)

measure for each A > 0. Consequently.

L[h] = 1,1q + d-‘(1

(3.3) it follows that

- ePJLth]) (3.4)

= lii

s

(1 - eihzl) d-l dpJz4).

The canonical form for L[h] referred to above comes from the general form for this limit, and that form is given by [22] L[h] = ihA + Bh2 + J (1 - eihu + ihu/(l -- u2)) do(u)

(3.5)

where A is real, B > 0, and u is a positive measure that satisfies 1 (u2/(1 + 24”)) do(u)

< cc.

(3.6)

For enen functions L[h], as suggested by even interactions, the canonical form becomes L[h] = Bh” + f [I - cos(hu)] da(u).

(3.7)

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JOHN

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KLAUDER

Inequalities

Before proceeding further let us extract one basic result from this canonical form. The functional S(h) is the generating functional of the correlation functions, while lnS(h) is the generating functional of the truncated (connected) correlation functions (denoted by a superscript “Y). In particular, provided the requisite moments exist, it follows for q > 1 that

((J h@dx)“)’ = I;~ J w-1 dpd(u)J I+dx,

(3.8)

and thus all even order truncated correlation functions are nonnegative. Specifically,

((J hdi dx)4)r Conventional Lattice-Space

=

J u4do(u) J h4 dx

2 o.

(3.9)

Approach

At this point it is useful to introduce a lattice-space regularization to the formal functional integral (3.1), and we also specialize to the case p = 4. The expression of interest is obtained as the limit as the cell volume d -+ 0 of a lattice-space-regularized calculation. Guided by conventional wisdom we assume the lattice-space form of interest is given by

(3.10)

:~J...Jexp!i~h,~~-~rn,e~~,~A-h,~~,’A~nd~~ k

where k = (k, ,..., k,) labek a lattice site, d is the cell volume, hk and @, are lattice fields, and mo2 and X, > 0 denote bare parameters (chosen to achieve consistency as A --+ 0). But this simple expression is one of a class that satisfies the Lebowitz inequality [23] which asserts that <@kl@k2@k3@k4)T

<

O,

(3.11)

namely, that all fourth-order truncated correlation functions are nonpositive. In the limit A + 0 this inequality still remains true, and the only way in which this inequality can be compatible with the one in (3.9) is for the lattice-space computed truncated four-point function to vanish, which means the ultimate limit is free, i.e., L[h] = Bh2. In other words, the conventional lattice-space formulation has automatically led to the trivial free result no matter how clever we try to be in choosing mo2 and h, as functions of A. This result for the lattice-space approach is the same as would be calculated by the renormalization group method which asserts that nonrenormalizable theories limit to free theories [3]. Although our argument has been limited to a Q4 interaction because of the Lebowitz inequality, it illustrates some sort of profound and inherent inadequacy in the conventional lattice-space approach for these models.

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Solution and Its Derivation

As seen above the conventional lattice-space approach can be of no use to us in the search for nontrivial solutions. Elsewhere [21] we have shown that the generating functional for the Euclidean action ~(&.&P + h@) dx, p even, formally given in (3.1), is evaluated as exp 1-h j d-y s [l - cos(hu)] e--tbn,‘u2-Ab’-‘u’ du/i u 11,

(3.12)

where b is an arbitrary, positive constant with dimensions (length)-“. If units are chosen such that b -= 1, as was the case in previous studies, then (3.12) simply becomes exp [- s dx s [l - cos(hu)] e-f~~l’u’-~uY du// u :).

(3.13)

A few words regarding (3.12) are in order. Clearly, it is well defined and nontrivial, i.e., h enters in a nontrivial fashion. As X + 0 +, this expression passes to thepseudofrer functional given by exp 1-b j dx j [l - cos(hu)] e-+bn2u2 du/’ u /;,

(3.14)

which is the samefunctional regardless of the interaction; moreover, it is not the free functional (defined as exp[-I3 J P(X) dx], for some B, although for large b it can be brought arbitrarily close to this expression). Next note that the interacting solution possesses a power series expansion (an asymptotic expansion, to be sure) in which the first term is the pseudofree solution and not the free solution. In short, the interacting solution exists and is as well behaved as could be expected with the only difference being that it is completely disconnected from the free solution. We conclude this subsection with a sketch of the derivation of the solution given in (3.12). Derivation of the Solution

To initiate the derivation of (3.12) suppose that L is even, B = 0, and do(u) == P(U) du. Then it follows that the field Q(X) can be given a very simple realization [21]. Let / 0) be a normalized vector, and A(x, U) and A+(x, U) denote conventional Fock annihilation and creation operators for which A(x, U) 10) = 0 and [A(x, u), A+(x’, u’)] = 6(x - x’)6(u - u’).

(3.15)

As usual repeated application of the creation operators A+ on 10) span the Hilbert space. For convenience, let B(x, U) = A(x, U) + C(U); then the field G(x) is defined by Q(x) = j B+(x, u) uB(x, u) du.

In particular,

it follows from the bilinear nature of Q, that

(0 I exp (i j h@ dx) I O? = exp I595/117/l-3

(3.16)

1 dx j [I - cos(hu)] F(u) du/ .

(3.17)

32

JOHN

R.

KLAUJIER

This realization of the operator readily leads to the proper prescription products. Note that @(x)@(y) = b-16(x - y) d-y(x) + :@(x)@(y):

for local (3.18)

where : : denotes normal order of At and A, and QiR2(x) = b j- B+(x, u) u2B(x, u) du.

(3.19)

An arbitrary, positive constant b with dimensions (length)-” has been introduced so that the engineering dimension of QiR2 is twice that of @. As the leading term in an operator product expansion tie take QR2(x) as the proper renormalized local square. More generally, it is easily seen that the distribution limit djRQ(x) = lim A-lb~-l A+0

[s,,,

@(x

+

4

dz]”

(3.20) = bg-l j- B+(x, u) ugB(x, u) du

defines the proper renormalized local qth-power. For suitable C this rule can even be extended to absolute values and their powers. Note that this general prescription to define local products is vastly d&wt than normal ordering appropriate to Gaussian random variables. Let q be even, g(x) be smooth, nonnegative and have compact support, and consider (0 1exp (i I h@ dx - j gqbRq dx) / 0) = exp I-

/ dx 1 (1 - eihu-gb”-‘ua) C2(u) dul

(3.21)

as follows from the bilinear form of both Q, and @a~. If this expression is normalized to unity for h = 0, then the infinite-volume limit g(x) + g > 0 can be taken to yield exp I-

1 dx j [I - cos(hu)] e-gb’-1u”C2(u) du/ ,

(3.22)

and this result formally corresponds to adding to the Euclidean action the extra term g J @* dx. Consequently, it follows that C?(u) = e-Ab’-1w’Co2(u),

(3.23)

where Co2(u) corresponds to the weight in the absence of the @P interaction. To find C,“(U), let q = 2 in the added interaction, namely make a muss insertion. In that case the relevant integrand in (3.22) reads

5 [1-

cos(hu)] e--DbuaC02(u)du.

(3.24)

NONRENORMALIZABLE

QUANTUM

THEORY

33

But the two mass-like terms can also be combined into one in the original functional integral, and through a resealing of the field the relevant integrand can be brought to the standard form i

[l - cos(hu/(l + 2g/m”)‘/“)]

C,Z(U) C/U.

(3.25)

Equality of these expressions implies that C,“(u) = (b/i u i) e-+m2bu”,

(3.26)

where we have identified an overall undetermined scale as the factor b, which itself was arbitrary. This completes our sketch of the derivation of (3.12). Further details can be found in Ref. [21]. The Correct Lattice-Space

Approach

As we have seen, a completely consistent, nontrivial solution exists to the model formally defined by (3.1) in spite of the fact that the conventional lattice-space approach utterly fails. To penetrate deeper into the cause of this failure and to gain further understanding of the solution (3.12) let us specialize the test-function argument of that particular solution to lattice-space test functions which are piece-wise constant on cells of volume d, and (at least approximately) take the Fourier transform to learn the true nature of the lattice-space integrand. For convenience, attention is confined to a quartic interaction (p = 4). Following this procedure we are led to a proper lattice-space form for the generating functional which is given by

S((h,)) 57 N J . .. J

Z h,@,A - +nPbA C Qk2A -- WA3 1 Qk4A/ .

;: fl Lmk/(l tDk 11-y

and We that u=

(3.27)

in particular we assert that as A --+ 0 this expression converges to (3.12) for p :-= 4. now proceed to prove this assertion. In this expression [in which we already see mo2 = m2(bA) and h, = h(bA)3] each ok is independent of all the others, and with @,A, it follows that S({h,\) = n N,, 1 exp(ih,u - $m2bu2 - hb3u”) du/i u ~1-2b3 k

The normalization

factor N,, is easily estimated for small A from the fact that bA .rp, du/j u j1-2bd = B2b” ---f 1

(3.29)

34

JOHN

R.

KLAUDER

as d + 0. Even if B varies as fir, - co < r < 00, the limit as d + 0 is still unity. Thus, to first-order N,, = bd; as it turns out this is the only order needed for the result. To this accuracy the factor 1 u I--(1--2bLl) may be replaced by 1 u 1-l in the second form of (3.28) since convergence of the integral is maintained and the d terms only contribute to a higher order. Thus we are led to the form, accurate to first order in d, given by S((h,}) = n 11 - bd 1 [l - cos(h,u)] e--tmebua-~b3u4 du/] 24]I, Is

(3.30)

which in the limit d -+ 0 becomes S(h) = exp (4

1 dx 1 (1 - cos[h(x) u]} e-*mabue-Ab3u4 du/l u I)

(3.31)

as desired. Besides the evident fact that this derivation is not limited to a quartic interaction, a close inspection reveals several important features. In the proper lattice-space form (3.27), the bare parameters, mo2 and X, , are related to the final parameters, m2 and h, according to the relations rno2 = m2(bA) and h, = h(b@, the need for which is evident from the derivation of (3.31). Alternatively stated these relations automatically incorporate the prescription for the renormalized local-field product as given in (3.20) since @,A = I,,

@(x + z) dz

(3.32)

when x lies on the lattice site k. Multiplicative renormalizations of one kind or another of the bare parameters are not uncommon, and so their appearance in this situation is no cause for surprise. What is unusual, however, is the all-important modification of the measure from the conventional form lJIk d@, to I& dQk/ ] c#Q]1--2bd. The translation-invariant form is replaced with another form which is formally scale invariant in the limit A -+ 0. This is a profound change from the conventional approach, and is, ultimately, completely responsible for making the new form work. So different is this prescription that it is not the A in the exponent (representing the cell volume in the usual integral which gives the action) that becomes the dx in the final expression; rather it is the specific A appearing in the modified measure du/ 1u j1-2bd which fixes the normalization factor IV,, as proportional to A, a term which, ultimately, becomes the dx in the final expression. This fundamental difference is the difference between Gaussian random variables (and functions thereof) and Poisson random variables (and functions thereof); mathematically, recall that the canonical form (3.7) contains Gaussian and Poisson contributions [22]. The distinction between the conventional and alternative approach can be said to lie in the measure chosen for the fields. The translation-invariant form is conventional and is chosen to conform with canonical commutation relations. But for highly singular theories, such as nonrenormalizable ones, the existence of canonical commutation relations is suspect, and therefore there is no compelling reason to choose the

NONRENORMALIZABLE

QUANTUM

THEORY

33

translation-invariant form. Alternative forms should be considered until consistency is achieved, which is exactly the freedom that was necessaryto approach the independent-value models. The conventional lattice-space approach with the translationallyinvariant measurepassedin the limit to the free theory. The so-called renormalizationgroup method (without approximations) is essentially just a special order for carrying out the integrals, and it clearly limits in this case to the free theory as well. But by choosing the alternative measure, the limit is nontrivial and nonfree, and clearly the renormalization-group method applied to the casewith the alternatiae measure would yield the samenontrivial and nonfree results. Clearly then we have no quarrel with the renormalization-group method as a calculational tool. What we object to is the unstated and unjustified assumption that all field theories including those conventionally regarded as nonrenormalizable are regularized by lattice-space techniques invariably involving the translationally invariant measure. Conceivably. one could adopt the viewpoint that the change of the translation-invariant formal measuren.,, c/@(.r) to the scale-invariant formal measure IJ73.d@(x)/ ! @(.u) is a “renormalization effect,” which may be one way of interpreting the fact that the proper evaIuation of (3.1) is given by (3.12). But such a “renormalization” is independent of the coupling and remains present as X --) 0’. If the nonlinear coupling is then reintroduced, the integral already contains the scale-invariant measure. and there is simply no basis for considering any other starting point. Consequently, it is more correct and far more useful if the formal measure on fields appearing in (3.1) is interpreted ab initio as the scale-invariant form rather than the conventional translation-invariant form. In essence,this is the only “trick” needed to find the solution (3.12). If the simple trick of changing the measure for fields was the key to successin understanding the most severe of the nonrenormalizable problems it stands to reason that it may be useful in dealing with lesssevere nonrenormalizable problems. By all means, the successof choosing a new measure in one case strongly suggests its consideration for other cases.Therefore in the next section we initiate the formulation and study of covariant quantum field theories with alternative, scale-invariant measures on a more formal basis. To give it a name we shall generally call such an approach scale-covariant quantization.

IV. SCALE-C• VARIANT FUNCTIONAL

INTECRATIOV

Guided by the solution of the independent-value models we propose, for a classical theory with action functional I =

s

{-i[(z”c$)2 - ?722q52] - h$“i C/-Y.

(4.1)

to choose the unconventional functional integral quantization formally given by Z‘(h) 1~ .k-’ ~exp(i~h$dx+i~

($[(P,c$)” - n&p] - A&j dYy)L?‘qi

(4.2)

36

JOHN

where A”’ is a formal normalization

R. KLAUDER

factor and

9’4 = nz 4&)/I m.

(4.3)

Clearly this measure is formally invariant under the scale transformation d(x) + S(x)#(x), S(x) > 0. The change of measure from the translation-invariant form to the scale-invariant form in no way affects the formal Poincare covariance of (4.2). It is proposed that this formal quantization, supplemented with suitable regularization and renormalization, should apply to nonrenormalizable field theories. It may also be considered to provide an alternative quantization scheme for models normally regarded as (super) renormalizable. It is apparent that as h + 0+ such a theory is formally connected to a pseudofree theory defined formally by

which differs fundamentally from the free theory. These models may be defined in Euclidean space as well (we effectively change the “external field” h by a factor i in the same operation). In this case the generating functional of interest reads S’(h)

= dV’ j exp (i 1 h@ dx - 1 {$[(V@)2 + m2Q2] + XP} dx W@,

1

where A’*’ is (another) formal normalization measure is given by

factor, and the formally

(4.5)

scale-invariant

ST@ = fl d@(x)/1 Q(x)\.

e

(4.6)

The Euclidean form of the pseudofree theory is evidently h@ dx - $

s

[(V@)2 + m2G2] dx

and differs from the Euclidean free theory. Before discussing these relations any further let us offer some additional for them.

(4.7)

motivation

Augmented Field Theory

Recall that with regard to the classical theory it is the classical equation of motion that is sacred and not the classical action. Therefore we are free to introduce an alternative classical action as long as it leads to the sameequation of motion. To this end consider the action I’ = j {+[(a,~$)~ - rn”+“] - X4”} dx - Q 1 x2$2 d.y

(4.8)

NONRENORMALIZABLE

QUANTUM

THEORY

3.1

in which the conventional action has been augmented by an additional term in which x is an auxiliary dynamical field [24]. Stationary variation of I’ leads to two equations of motion, namely (0

+ n?) 4 = -4X$”

- yG#J,

x+* = 0.

(4.9)

But the latter equation implies that either x or I$ is zero, hence x2+ = 0, and so the field 4 still obeys the original equation of motion. The classical theory has not been changed, but the quantum theory will be fundamentally different. The quantum theory is formally given by

where s f “ is a normalization

factor, and (4. I I )

The simple manner in which x appears allows us to integrate out the x field, using, for example, the formal relation that (4.12) which follows from a simple change of variables. When this relation is introduced into (4.10), it is clear that (4.2) is recovered. A similar analysis applies to the pseudofree theories. The same kind of argument may be used for the Euclidean formulation. The Euclidean formulation for the action (4.8) is given by

where

Y’ is a normalization

factor, and formally

9m = n &#qx): .z Again elimination

55x = fl tfX(s). J’

of the Euclidean field X is straightforward

(4.14) using the formal relation

38

JOHN

R.

KLAUDER

Incorporation of this relation into (4.13) evidently yields (4.5), and a similar analysis applies to the pseudofree theories. We have stressed the equivalence between the augmented approach and scalecovariant quantization to show its relation to a fairly conventional approach (albeit with nonexistent dynamics for X) and because we want to emphasize that any additional potential over and above the conventional approach has a density &X”(x) C+“(X) and not, as one might be tempted to assume, something like &(O)ln [ @(x)1. Such a term [in which the weight factor in the measure in (4.5) is put into the exponent] fails to lead to a proper classical potential for three reasons. First, it necessitates a nonvariable, infinite coefficient; second, it does not enter (4.2)-which is the relation where the physics lies-preceded by a factor i as it must to be a potential; and third, it must be proportional to A-which is another way of saying that the classical dynamics has remained unchanged. These three difficulties are not present in the augmented form &y2+2 or iX%D2. While we are still discussing the augmented formulation let us consider two generalizations of this approach. First, we suppose that the interaction term has a different power, i.e.

I’ = 1 {&[(a,+)2 - my] - x I 4 I”}dx - + / x2+2dx.

(4.16)

Evidently no fundamental changes arise in the foregoing discussion on this account. Second, let us generalize the augmented term as well so that

1’ = J M%~~2 - m2421- h I 9 I”) dx - & j- I x Ip I d lp d-x.

(4.17)

Only a short reflection is needed to convince oneself that the classical theory still satisfies the original equation of motion, and that the quantum theory still involves the scale-invariant measure since

To make yet another point let us examine the Euclidean-space expression S’(h) = N

J exp (i J” h@ dx - J {+[(VCD)~ + m2a2] + h 10 I”} dx) .2&B

x exp (-

4 s I X 1~ 1di 12,dx) 9X.

(4.19)

According to the viewpoint that nonrenormalizable interactions act as partial hard cores-namely, that J 1 CD12,dx projects out certain field histories-we can heuristically characterize the role of the factor exp(-+ J I X /2, I @ Ip dx) .9X as preparing the conventional measure BD for the oncoming interaction.

NONRENORMALIZABLE

QUANTUM

39

THEORY

Preparing the Measure in One Dimension The concept of preparing the measure to receive a singular interaction has a useful analog for the one-dimensional systems discussed in Section 1. Equation (1.3) dealt with an anharmonic oscillator with an 1.X 1~;)) a: > 3, interaction, and we noted that all paths were projected out except those that were either always positive or always negative. We can also arrange for this projection to occur with an augmented classical action, taken for convenience still with imaginary time. Consider the action

where 4’ is an auxiliary dynamical variable and p is a positive parameter at our disposal. As in the field case it is easy to see that the (real time) equations of motion are identical to the original one. In carrying out the functional quantization we are led to consider first

When the .v integration is carried out a factor nt 1x(t)18 appears in the integrand, which can aIways be incorporated into 5%‘~. This factor has the desired property of giving zero weight to any path that reaches or crosses the origin, but it gives an unwanted weight to other paths. What is needed, in this case, is to take the limit /I + Of as a final step, which then gives unit weight to any path such that x(t) ;-- 0 or x(t) < 0. and eliminates all other paths. If we formally let 9.x 1~: lim JJ R--0+ ,

s(t)!@ &(f),

(4.23)

then C(s) = Jlr s exp Ii s sx dt - I [t.(3

--- xp) L- A]

dt/ 9’s

(4.24)

holds, and for h = 0 it even correctly characterizes the pseudofree theory. As we have seen, the use of a suitable augmented action can implement path projection in particle mechanics and lends further credence to their use in field theory. Lest the reader worry that the augmented interaction in (4.20) seems to become insignificant as fs + 0+, he or she should imagine rearguing the case with the equally good augmented action (4.25)

40

JOHN

R.

KLAUDER

which leads to the same prescription for S”x as /I + Of. We add the remark that in the case of field quantization the degree of freedom represented by p does not seem to play a role. Lattice-Space Formulation We conclude this section with a discussion of the approach to scale-covariant scalar fields by means of a lattice-space regularization [24]. Based on an analogy with (3.27) we expect that the lattice-space form of (4.5) should read S({k,)) = N 1 ... J” exp [i C h,@&l - &Z C ($* -

- C#Q2•~4 (4.26)

&no2

Here E is a lattice dimension, d s en, mo2, A, , and Z are parameters chosen for consistency, k* represents one of the n next “larger” sites to the site k, and b is a parameter required for dimensional reasons. Hopefully, as d -+ 0 this expression converges to a meaningful quantity for sufficiently many functions h, and this will surely entail choosing mo2 and h, as functions of d; it may even be necessary to choose Z as a function of d (and not unity as might be expected), and perhaps also to choose the parameter b itself as a function of d (in such a way that bd + 0 as d --+ 0 still). An evaluation of (4.26) may be attempted by renormalization-group methods, by one or another “high-temperature” expansion or by any other convenient approach. For completeness we sketch the ingredients of one fairly simple high-temperature expansion. Let J = Z 1 @,,@,E-~A (4.27) denote the “ferromagnetic”

coupling term in the exponent of (4.26), and introduce

or equally well for the correlation

functions

where (.) denotes the average implicit in (4.26), and ( .)O denotes the same average except that the factor J is deleted. Next expand eJ in a power series in J in both the numerator and denominator. For X, > 0 or m02 > 0, and for any large but finite lattice volume, it is evident from the structure of (4.26) that the order of summation and integration in each such expression may rigorously be interchanged. The result for the numerator in (4.29) then is a positive sum of terms of the form

NONRENORMALIZABLE

QUANTUM

41

THEORY

which in view of the absenceof correlations is evaluated as (4.31) where each mq is distinct, np denotes the number of occurrences of 4Tm,, and the II,, obey the sum rule Cn, = s f 2r. Unless n, is even for all 9 this term vanishes. A graphical representation of a term such as (4.30) may be given by denoting each of the external variables Gk:in (4.30) by a “point” at one of the lattice sitesk, ,..., k, . and by denoting each of the r pairs @$BI in (4.30) by a “link” between the two indicated nearest neighbor (hyper-cubic) lattice sites. Just as in the Gell-Mann-Low equation [25], “vacuum” graphs which have no direct link to an external variable sum up to cancel the denominator in (4.29), and so (4.32) in which all vacuum graphs are omitted. Such expansionshave a great deal in common with high-temperature expansions for statistical mechanics, such as for the partition function of an Ising model [26]. Techniques developed in such studiesmay prove useful here as well.

V. ALTERNATIVE

FORMULATIONS

FOR SCALE-C• VARIANT

QUANTIZATION

Scale-covariant quantization of relativistic fields, formulated as functional integrals in the preceding section, may be characterized in alternative ways as well. In so doing we shall also be led to convenient formulations for models with spinor fields. Initially let us continue to exploit functional techniques. Functional DifSerential Equations The Green’s functional Z’(h) in (4.2)-so-called because it is the generating functional for time-ordered Green’s functions-can be written in the form SF’(S+} 9’4 making use of the fact that B’S+ = 94. This is an expression that superficially depends on S - S(x) > 0, but which is in fact independent of S. Consequently. the relation (5.1) holds as an identity. Applied to the specific form of Z’(h) this identity formally leads to the equation

42

JOHN

where K, = 0, + m2. Evidently differential equation [27]

i 6h(x) Ih(x)-L

(&&-)

R.

KLAUDER

this expression can be recast into the functional

Kc (&)

- 4h (&,,I

Z’(h)

= 0.

(5.3)

Here is a formal difirential characterization of the functional Z’(h). Solution of this differential equation, supplemented with boundary conditions and suitable regularization and renormalization, provides an alternative way to find Z’(h). As we shall see this approach may ultimately offer some advantages, both technical and conceptual, relative to the functional integral approach. It is useful to compare the functional differential equation (5.3) with the conventional one, commonly referred to as the Schwinger functional differential equation [28]. The conventional Green’s functional is defined as in (4.2) with the translationinvariant form .9$ replacing 9’4. Thus it has the structure SF{+ + il} 94 but in fact it is completely independent of L’I = /l(x). Consequently, the relation (5.4)

holds as an identity, which in this case eventually leads to the Schwinger equation

Differentiation of this relation once more with G/i&h(x) and comparison with (5.3) shows that there is a real distinction in the scale-invariant and translation-invariant forms since the leading terms are unequal. Further discussion of the relation between (5.3) and (5.5) can be found in Ref. [27]. The pseudofree Green’s functional Z;(h) evidently satisfies the functional differential equation

!

h(x) L i ah(x)

and the functional

-

Z’(h) is related to Z;(h) according to Z’(h)

= N’ exp 1--IA s [S/i 6h(x)J4 dxl Z;(h)

where N’ corrects for normalization.

(5.7)

This relation is exactly parallel to the relation

Z(h) = N exp 1-ih

f [S/i ah(

dx/ Z,(h)

(5.8)

that connects the functional Z(h) with the free Green’s functional, Z,,(h) = exp [i g 1 h(K, 2 iO)-l h dx],

(5.9)

NONRENORMALIZABLE

QUANTUM

43

THEORY

which is the solution of (5.5) for h = 0. Comparison of (5.7) and (5.8) again shows that the pseudofree functional plays an important and central role in the alternative formulation, indeed as important and central a role as that played by the free functional in the conventional theory. A few remarks regarding (5.3) are in order. To make sense of this functional differential equation regularization and renormalization may be expected. One form of renormalization so far neglected asserts that the value at h = 0 should be subtracted off from (5.3), a correction which really arises because the normalization factor ,+“ in Z’(h) formally contributes to the derivation of (5.3) as well. Thus (5.3) is more correctly given by

where. typically, 6 : ---)I’: i i 6/1(.x)

we use the shorthand Z’(h)

= (T&J

Z’(h)

-

(Y&J)

Z’(h)l*_o

. Z’(h).

(5.11)

Also multiplicative renormalizations may be expected for the mass and coupling parameters and probably even for the coefficient of the gradients. Independent- Value Models To indicate the kind of multiplicative renormalizations that can be expected let us briefly specialize to the independent-value models for which (5.10) becomes 6 ___ ht.4 i 8h(.u) I

- nz: :(A)‘:

-

A, :(&f:/

Z’(h) = 0.

(5.12)

Here we have inserted bare parameters in anticipation of renormalization effects. As before, there are no gradient terms, no mechanism for coupling between one point and another. Therefore symmetry requirements dictate that the solution have the form Z’(h)

= exp i II

W[h(x)]

d-x/

(5.13)

for the same reasons that led to (3.2). With this expression in mind it is easy to find what kind of multiplicative renormalizations are required. For example, since

=- f-i it follows

formally

6(x - y) w”[h(x)]

f

W’[h(y)]

W’[h(.u)]}

Z’(h),

(5.14)

that

&i&)

Z’(h)

= -iW”[h(x)]

Z’(h),

(5.15)

where W’[h] = dW/dh, etc. Note that the last term in (5.14) is scaled to zero in the effort to make a well-defined quantity. To achieve this scaling we set PZ,,~= &P/6(0),

44

JOHN

R.

KLAUDER

where m2 is a finite (mass)2 parameter and b an arbitrary but finite, positive constant with dimensions (length)-“. A similar argument shows that we must set h, = b3A/ [S(O)]“. Observe that these are just the (unregularized) form of multiplicative renormalizations already encountered in (3.27). When these arguments are put together it follows that (5.12) becomes h(x) W’[h(x)] + ibm2{W”[h(x)] - W”[O]) - 4ib3h{W”“[h(x)] - W”“[O]} = 0,

(5.16)

where the subtracted terms just represent the effect of the operation : : defined in (5.11). This is a linear, ordinary differential equation for the function W[Iz], and the desired solution must satisfy W[-h] = W[h] and W[O] = 0. Such a solution is readily found and is given by W[h] = ib 1 [l -

cos(hu)] exp{-i[Qb(m2 - i0) u2 + b3hU4])dull u /

(5.17)

which is nothing but the real-time form of the solution (3.12) forp = 4 supplemented with a convergence prescription. Note that (5.16) does not set the overall scale for W, and that scale has (again) been taken as the arbitrary parameter b. The factor i is chosen to equilibrate the phase of the two-point function of the pseudofree theory with that of the free theory. The derivation of this result has entailed only very commonplace renormalizations of the functional differential equation (5.12)-multiplication and subtraction-and it would be easy to supplement our derivation with a conventional regularization (e.g., a lattice-space regularization) so that m,2, h, and (5.12) were all well defined at each stage. No trace of the uncommon regularization of the scale-invariant measure (4.3) in the form that appears in (3.27) is apparent and that was needed in the functional integral approach. In that sense characterization of such models by the functional differential equation seems more familiar, and thus is possibly to be preferred. While the calculation sketched above is quite specific to the independent-value models we wish to draw attention to the important fact that the basic equation (5.16) is linear and homogeneousin the variable W which (effectively) is the generator of the truncated Green’s functions. This came about, in turn, due to the infinite multiplicative renormalizations that were required. Coupled Green’s Function Equations

Guided by the previous discussion let us derive the coupled Green’s function equations implied by (5.10). We recall, by definition, that Z’(h) = so (m!)-1 i” S **+s GA(x, ,..., x3 h(xJ .*. h(xm>dx, ***dx,

(5.18)

with Gi = 1, and that In Z’(h) = $, (m!)-1 i” 1 **. I Gz(x, ,..., xm) h(x,) a**h(x,,J dx, *.* dx, .

(5.19)

NONRENORMALIZABLE

QUANTUM

45

THEORY

Normally one would insert (5.18) into (5.10) and find a linear, inhomogeneous set of coupled Green’s function equations. However, as in (5.16), we expect multiplicative renormalizations to lead to a linear, homogeneous set of coupled truncated Green’s function equations. Making allowance for that expectation, but not explicitly including the multiplicative renormalizations themselves leads to the set of equations

1

GkT(xix, ,..., x,)

+ K,G;:,(x,

x*, x1 ,..., x,) + 4hG:+&,,(.u, x, x, x, x1 .. . .. x,)

= 0 (5.20)

which holds for all even m 3 2 (the odd terms vanish by symmetry). Here x* denotes a variable set equal to x (from a spacelike direction) after the action of KX . These equations also take into account the subtraction at h = 0 of the operation : :. If for simplicity we set g, = Gz for h = 0, then the coupled Green’s function equations for the pseudofree solution are given by

1

g,(x,

,..., x,) + K,g,,L+p(x. s*, x1 ,..., x,,,) = 0.

(5.21)

There is every reason to believe that a perturbative solution of (5.20) could be generated iteratively based on a solution to (5.21), which again stresses the importance of the pseudofree theory. We shall arrive at the set of equations (5.20) from a different point of view below. We also remark that the discussion of Green’s function equations may be carried out for the augmented formulation of scale-covariant theories. Further details may be found in Ref. [27]. Equations of Motion Let us temporarily set aside the coupled Green’s function equations and discuss in a formal and heuristic fashion the derivation of operator field equations on the basis of unconventional variations of the conventional action functional. Scalar Fields Consider

the conventional 1 =

scalar operator action functional

s

{~[(~,~)z

- m2+2] - @4} dx

(5.22)

chosen with a quartic interaction for convenience. Traditionally, the operator equation of motion is derived from stationarity under variations of the form &$ = S/1, A an arbitrary c-number function, which leads to the equation (Cl

+

m2)$

=

-4X43.

(5.23)

46

JOHN

R.

KLAUDER

The variation in question here is an in6nitesimal form of the transformation 4(x) -+ 4(x) + n(x). As noted in Section II, significant changes in the relation between the free and interaction terms in (5.22) occur in the c-number theory for n > 4. Such changes may cast doubt on the q-number derivation of (5.23) for the following reason. If the local products appearing in (5.22) are defined through an operator product expansion instead of normal ordered products-an occurrence rather likely in the presence of highly singular interactions-then the conventional c-number variation indicated above is unsuitable since the definition of local products scales such a variation to zero [29]. Heuristically speaking, one may say that conventional. c-number variations do not respect the additional singularities introduced by the interaction term. Now, this situation may be remedied [29] with an alternative, non-c-number variation, a variation of the form 84 =6S .+, S an arbitrary positive c-number function, namely one which is everywhere proportional to the field C#itself. We call such a variation a scale variation. This variation is an infinitesimal form of the transformation 4(x) + S(X)+(X), 5’(x) > 0, which evidently survives the rule for local products and formally has the feature that it preserves the class of fields for which the interaction dominates the free term and vice versa. This argument is analogous to that presented in Section I to derive (1.5) except of course in that case trouble arose at zero (“field”) amplitude while in the present case trouble arises at infinite field amplitude. With a variation of the scale variety the equation of motion becomes C(o

+

m2)$

=

-4hy54.

(5.24)

If we were dealing with the classical theory this equation would be as satisfactory as the usual one since we can divide both sides by (b(x) save when 4(x) vanishes. If #I vanishes at a point, continuity of the solution fills the gap; if C#vanishes on an open set, then the field still satisfies the conventional equation on that set. But we are not dealing with the classical theory; we are dealing with operator field equations. As such it is by no means a simple matter to “chip off” one # to recover the conventional equation (5.23). The local operator is a distribution, and finding the proper prescription for defining local operator products is a principal point (and without much exaggeration, it is the main point). The prescription necessary to define (5.23) and (5.24) simply may not permit an equivalence of the two to be established. Indeed, we expect that the two equations are fundamentally different, and that such a difference extends to the case h = 0 between the conventional free theory and the pseudofree theory which formally satisfies the equation of motion

$(I3 + mz>$= 0.

(5.25)

This characterization of the pseudofree theory no longer depends on any specifics of the interaction save the essence of its singularity structure, viz. compatibility with scaling transformations. Subsequently, we shall establish the equivalence of (5.24) and (5.20).

NONRENORMALIZABLE

QUANTUM

47

THEORY

Spinor Fields The characterization of scale-covariant theories through equations of motion is not limited to scalar fields. For a self-interacting, nonlinear spinor field theory with action functional

for example, it is traditional to consider variations of the form S$ = S+j, where +j is an arbitrary anticommuting c-number function. Such a variation is an infinitesimal form of the transformation 6 --+ $ + 17.The resultant equation of motion is &A?, -

M)~ - 2hy’3L@y&

= 0.

(5.27)

One the other hand, let us consider scale variations of the form S$ == @U, where U is an arbitrary c-number, nonsingular matrix function. This is just an infinitesimal form of the transformation 4 ---f $U. In this case the resultant equation of motion reads Qmy%

- M)# - 2~~$q4GJyu~~

(5.28)

= 0,

where, for example, in a four-dimensional space-time (rz = 4) I’ stands for any one of the 16 independent, nonsingular 4 x 4 matrices conveniently taken as I, y”, &, yUys, and y5 . Tn other words, this relation should be interpreted as 16 different equations. For c-number fields (or even anticommuting c-number fields) the extra $r could be factored off to recover (5.27) but not so for operator fields since some particular prescription for local operator products is implicit. This distinction also holds for the pseudofree theory formally characterized by J;T(iy”2,

- M)JG = 0,

and asserts that this theory is not equivalent theory. Yukawa

to the conventional

(5.29) free spinor

field

Models

Scale variations

of the action

may also be considered. Under stationary variations of the form S$ = 6s. C$ and S$ := 4 SU, described above, it follows that the two equations of motion become

48

JOHN

The corresponding

R. KLAUDER

pseudofree equations are

Kl

+ m2>4= 0,

fjr(iy@a,

- M)$

(5.33)

= 0.

(5.34)

As in the previous cases these quantum equations are not equivalent to the conventional ones, but they are the same as the uncoupled pseudofree equations (5.25) and (5.29). Afine Fields

It is well known that the (equal-time)

canonical commutation

[4x), WI = -Wx

relation

- 39,

(5.35)

where Y = 1, is closely connected with the translation-invariant measure 94 in a functional quantization. But when Y = ~13,as in the conventional case of divergent field strength renormalization, the canonical commutation relations are clearly illdefined. Proceeding formally, we note that [H44

$4x*> + 9Xx*> 4x)),

4(Y)] = [e)

9Xx*), $(Y)l

= -iY#x*)

If we introduce the Hermitian

(5.36)

6(x - y).

field

4x) = ~W+Mx)

+ doMx)>,

(5.37)

then it follows that b44, +(y)l

= -WG(x

(5.38)

- Y)-

The fields K and 4 are affine fields, and (5.38) is called the affine commutation relation [30]. There are representations of the canonical fields (e.g., the free field), and there are representations of the affine fields. Almost surely a representation for fields cannot fulfill both types of commutation relations. If Y = 1, say, then K is generally a form; if K is a local operator, then Y = co and rr is at best a form. Evidently, rr is the generator of field translation since (5.39)

eifgndx+(y) e+Jgndx = 4(y) + g(y), where g = g(x) is a suitable function (e.g., Cam). On the other hand, of field scaling since eiSSKdX4(y)ecibdx = es(y)+(y) E S(y) 4(y), where s = s(x) is a suitable function (e.g., Corn).

K

is the generator (5.40)

NONRENORMALIZABLE

QUANTUM

THEORY

39

Coupled Green’s Function Eqtfations (Again) Let G:,h,

,. .. xv,) Em(0

T$(x,)

. &.Y,,,) 0, ‘.

(5.41)

denote the time-ordered, truncated mth-order field operator vacuum expectation value for the scale-covariant theory, and consider the quantity G:~~+,(x,x*, .x1,..., s,,,) where the vector difference x’ - .Y* is spacelike. The first time derivative of the variable .Ybecomes

while the second time derivative yields [31]

‘<4(-d .‘. &z, ... $(xm)I oy,

(5.43)

where the “hat” denotes omission of that term. These relations follow from the multilinear nature of Gz. Consequently, 2~G:,',2(x, s*, x1 ,..., x,) = (0 / IT&$(x)

4(x*, +(x1, ... c$(s,,,)/ O‘\T

+ f e, - X0?)(0 I l‘b$w $(-y*),41.G)l r=1 Y &x1) ... +gj ..’ yqx,,,,I O)T.

(5.44)

Consider the last term on the right-hand side in the limit that x* ---f x. We divide this expression by Y, and recall that 4 = rr, at least as a form. lf Y diverges, as expected, then the only terms that survive are those in which 4(x), 4(x*), and 4(x?) are all in one and the samefactor in a decomposition of truncated Green’s functions into (sums and products of) Green’s functions. Thus we may invoke the affine commutation relation (5.38). Ignoring the factor Y itself, which we use only implicitly, we learn that

1

G~(x,

,..., s,,J.

(5.45)

50

JOHN

R.

KLAUDER

In fact, just this kind of reduction of truncated functions due to an implicit infinite multiplicative renormalization occurs for the other terms needed to make up K, , and it follows that

,...,x,) + K,G$+A$, x*, x1,...,x,J I GzCxl = (0 I nK4 &ml

4C&>*-. ~~-&n)I w,

(5.46)

where the limit x* + x is now understood. The upshot of the remarks on truncation made above is that the term 4(x) K,&(x) acts as one field in performing the truncation on the right side of (5.46). Finally, we add the interaction term to learn that

+ KzGi,%x, x*,x1,...,x,> = (0 I TW)

&d(x)

+ 4~G'a:&,x,

+ 4+4(x)1

d(xJ

x, x, x,,...,xm)

.a* KGAW,

(5.47)

and again we expect that an implicit infinite multiplicative renormalization will eliminate any terms in a decomposition of the truncated Green’s function into (sums and products of) Green’s functions in which all four fields 4(x) of the interaction do not appear in a single factor. This means, in particular, that the term d(4&w

+ 4+4(x)

(5.48)

preserves its integrity and acts as one field in performing the truncation on the right side of (5.47). Hence if we invoke the operator equation of motion (5.24) derived by scale variations the right side of (5.47) vanishes and we recover the coupled Green’s function equations (5.20). This demonstration completes the circle and provides the sought for equivalence of: (1) the original functional integral approach with a scale-invariant measure; (2) the functional differential equation for the Green’s functional; (3) the coupled Green’s function equations; and (4) the operator field equation derived by scale variations. It is also now clear that the initial conditions to those field equations should obey the affine commutation relations and not the canonical commutation relations. Coupled Green’s Function Equations for Spinor Fields Guided by the derivation of the Green’s function equations for scalar fields we can readily derive a set of equations when spinor fields are involved. We first note that Ty(X) = Y;l#+(x) r+(x)

(5.49)

NONRENORMALIZABLE

formally

plays the role of the affine field

QUANTUM

K

since

h-(x), $(Y)l = -r$Kmx as based on the (equal-time)

51

THEORY

(5.50)

- y)

canonical anticommutation

relation

l$J’W, #(Y)) = Y,S(x - Y).

(5.51)

Even if Yz == XI it is possible that (5.50) still makes sense. The truncated Green’s functions, defined as usua1 [32]. are denoted by r,r,=(.~yl 1.. , s,,,) = (‘0 j T$!qs,)

.‘. tp(.Y,,l)

0‘ 7

(5.52)

where # is either t or nothing. and an m-fold spinor index is suppressed. Consider the quantity &+,(x*. .Y, x1 ,..., x,,) where the vector difference s - .Y* is spacelike. Then we note that

P,,,,,.‘O1 Tp(s*)

t/J”(x) yP(x,) .‘. $qY-m) 0, 7

Now choose #+(x*) = $(x*)r (including a matrix ,‘r) and ##(x) -3 #(x); add spatial derivatives as necessary without additional cost, and also add an interaction term: let x* -+ X, adopt the commutation relation (5.50), and argue as before regarding the truncated terms. Taking (5.28) as the relevant operator equation, we then find the coupled Green’s function equations

for all even m 2 2, where r is an arbitrary matrix, the limit s* + x’ is understood, and the disposition of the m free spinor indices is left unspecified but is nevertheless fairly obvious. The superscript r serves as a reminder that a factor I’ has been left implicit. If X = 0, then (5.54) reduces to the coupled equations for the pseudofree spinor theory. Yukuwa Models The coupled set of Green’s functions appropriate to (5.31) and (5.32) are readily obtained based on our previous discussion. Let

52

JOHN R. KLAUDER

then it follows that

1

’ Tm,n

(Xl

,...,

x’,

; -Y, x*,

+

&Ln+&

,..‘,

+

x+2,n+l(.~,

-y, Xl ,...,

-%I

x,

; Yl

,...,

Yl ,...,

YJ

Yn)

; x, 4’1 ,...,

YJ

=

(5.56)

0

and i r
6(x

-

x,)(r),

+

(V%

-

-

~T%i,n+l(X,

GL(xl

w

~~$2.n(X*,

,...,

-y,

x,

; .h

,..a,

-Yl ,***,

Yn)

xnt

; Yl ,-..,

X, Xl T..., Xm ; -x, YI ,..., Yn) = 0

Yn)

(5.57)

are suitable transcriptions of (5.31) and (5.32). Here again x* -+ x is understood, and the disposition of the arbitrary matrix r and the m free spinor indices is again left unspecified but is fairly evident.

VI. CONCLUSION In this article we have presented our interpretation of nonrenormalizable interactions as discontinuous perturbations and discussed this point of view in some detail. The partial hard-core behavior of nonrenormalizable interactions in the context of functional integration provides a clear understanding of why conventional perturbation theory fails and suggests that an alternative approach should exist. The clue to introduce scale-invariant measures in functional integrals, rather than the conventional translation-invariant measures, arose from studying the solution to the idealized independent-value model, the most severe nonrenormalizable model of the general type displayed in (2.13). Relativistic models involving scale-invariant measures can also be formulated through functional differential equations and coupled Green’s function equations to some advantage since only traditional types of renormalization are encountered. The identity of operator field equations, derived on the basis of scale variations, to the coupled Green’s function equations was established by using the affine commutation relation (5.38) instead of the conventional canonical commutation relation. The connection to such a scale-covariant quantization that is provided by the operator equations of motion enabled corresponding coupled Green’s function equations for systems with spinor fields to be established most easily. The set of coupled Green’s function equations for scalar and spinor field systems, separately or combined, deserve further study to seek consistent solutions. This is particularly the case for the set of coupled Green’s function equations for the pseudofree (A = 0) models, e.g., (5.21). It is important to emphasize that establishing consis-

NONRENORMALIZABLE

QUANTUM

THEORY

53

tency of such equations is of primary significance to the program presented in this article, and at the present it remains an open problem (although there are some initial studies of such questions [33, 341).Evidently, if consistency can be demonstrated for such equations then more realistic models should be studied. For example, it would be important to have alternative formulations of the weak interactions involving local (Fermi) couplings to compare with experiment. One may worry that the electron field that enters conventional QED satisfiesthe standard canonical anticommutation relation while to be scalecovariant it should fulfill the affine anticommutation relation (5.50). This should be no cause for concern on at least three counts: First, deviations from unitarity for the conventional formulation of a local four-fermion coupling weak interaction theory begin about 300 GeV, and compared to that energy level the highly successfulverifications of QED are extremely low energy; second, the infinite electron field-strength renormalization encountered in QED may even make the af?ineanticommutation relation have greater validity than the canonical anticommutation relation; and third, we expect that suitable parameters exist to make the pseudofree theory arbitrarily close to the free theory, as was already remarked with regard to (3.14). Gravitational theory is another candidate for a nonrenormalizable theory, and a discussion of gravity in combination with various sources from the viewpoint of the hard-core picture of nonrenormalizability has already been given [35]. Besides applicability to generic nonrenormalizable models it is entirely possible that the hard-core picture applies to certain strictly renormalizable models as well. Motivation for this possibility can be seen in the need for infinite field-strength renormalization in order that the affine commutation relation holds. There is even a hint of such a behavior in the analog one-dimensional example of (1.2) when 01:-= 2 where the properties of the solution for h I,> 3/8 are closer to those for C~> 2 than to those for CCC. 2; m particular, the solution for N = 2 and A >, 3/8 can be approached continuously as CY- 2- and not as N + 2. Moreover, the solution for cx.==2 and h ‘173/S is not given as the analytic extension of the solution for 01== 2 and X < 3,s. By analogy, a critical value of coupling constant may exist for strictly renormalizable models above which the behavior is not determined by analytic continuation from below. Thus it becomesimportant to remain open minded about strictly renormalizable models, especially when it is appreciated that, in principle, a hard-core picture can certainly coexist with spontaneous symmetry breaking. soliton or instanton behavior. and other current concepts.

ACKNOWLEDGMENTS This article is a revised version of lecture notes prepared for the IIIrd School on Elementary Particles and High Energy Physics, Primorsko, Bulgaria, October 1977. Gratitude is expressed to the organizers, Professors Ch. Christov (Chairman), and I. Todorov and I. Zlatev (Vice-Chairmen), for the opportunity to participate in that school. Thanks are also expressed to the National Academy of Sciences (USA) and the Bulgarian Academy of Sciences for sponsoring an interacademy scientific exchange visit.

54

JOHN R. KLAUDER REFERENCES

1. See, e.g., M. C. REED, “Abstract Nonlinear Wave Equations” Springer, New York, 1976. See, e.g., N. N. BOGOLUBOV AND D. V. SHIRKOV, “Introduction to the Theory of Quantized Fields” Interscience, New York, 1959. 3. See, e.g., K. G. WI-ON AND J. B. KOGUT, Phys. Rep. 1ZC (1974), 75. 4. See, e.g., W. HEISENBERG,“Introduction to the Unified Field Theory of Elementary Particles,” Interscience, New York, 1966. 5. See, e.g., G. FEINBERG AND A. PAIS, Phys. Rev. 131 (1963), 2724; 133 (1964), B477; N. N. KHURI AND A. PAIS, Rev. Mod. Phys. 36 (1964), 590; A. PAIS AND T. T. Wu, Phys. Rev, 134 (1964), B1303; B. A. AR~UZOV AND A. T. FILIPPOV, Nuovo Cimento 38 (1965), 796; W. G~~TTINGER, Fortschr. Phys. 14 (1966), 483. 6. See, e.g., K. G. WILSON AND J. B. K~GUT, Phys. Rep. 12C (1974), 75; K. G. WILSON, Rev. Mod. Phys. 47 (1975), 773. 7. K. SYMANZIK, Commun. Math. Phys. 45 (1975), 79; see also D. I. BLOKHINTSEV, A. V. EFREMOV, AND D. V. SHIRKOV, Zzv. Vuz. 2. Fiz. 12 (1974), 23 [Sou. Phys. J. 17 (1974), 16501. 8. H. LEHMANN AND K. POHLMEYER, Commun. Math. Phys. 20 (1971), 101. 9. G. PARISI, Some considerations on nonrenormalizable interactions, in “The Proceedings of the Colloquium on Lagrangian Field Theory, Marseille, 1974;” Lectures given at the Cargese Summer School, July 1973. 10. T. EGUCHI, Phys. Rev. D 14 (1976), 2755. 11. J. R. KLAUDER, in “Quantum Dynamics: Mathematics and Models” (L. Streit, Ed.), Springer, New York, 1976, p. 185 [Acta Phys. Austr. Suppl. 161. 12. J. R. KLAUDER, Phys. Rev. D 12 (1975), 1590. 13. J. R. KLAUDER, Acta Phys. Austr. Suppl. 11 (1973), 341; B. SIMON, J. Functional Anal. 14 (1973), 295; B. DE FACIO AND C. L. HAMMER, J. Math. Phys. 15 (1974), 1071. 14. H. EZAWA, J. R. KLAUDER, AND L. A. SHEPP, J. Math. Phys. 16 (1975), 783. 15. See, e.g., the articles reprinted in “Selected Papers in Noise and Stochastic Processes” (N. Wax, Ed.), Dover, New York, 1954. 16. See, e.g., D. LUR& “Particles and Fields,” Interscience, New York, 1968; J. R~EWUSKI, “Field Theory,” Iliffe, London, 1969, Vol. II; H. M. FRIED, “Functional Methods and Models in Quantum Field Theory,” MIT Press, Cambridge, Massachusetts, 1972. 17. J. R. KLAUDER, in “Recent Developments in Mathematical Physics,” Proceedings of the XII Schladming Conference on Nuclear Physics (P. Urban, Ed.,) Springer, Berlin, 1973, p. 341 [Acta Phys. Austr. Supp/. 11 (1973)]. 18. See, e.g., H. UMEZAWA, “Quantum Field Theory,” Interscience, New York, 1956. 19. V. P. IL’IN AND V. A. SOLONNIKOV, Tr. Moscow Zzv. Akad. Nauk SSSR 66 (1962), 205. We thank Prof. Solonnikov for helpful correspondence regarding this work. For a general discussion of such inequalities see 0. A. LADYZENSKAJA, V. A. SOLONNIKOV, AND N. N. URAL’CEVA, “Linear and Quasi-Linear Equations of Parabolic Type,” Trans. of Math. Mono., Vol. 23 (Amer. Math. Sot., Providence, Rhode Island, 1968). 20. J. R. KLAUDER, Phys. Lett. 56B (1975), 93; in “Electromagnetic Interactions and Field Theory,” Proceedings of the XIV Schladming Conf. on Nuclear Physics (P. Urban, Ed.), Springer, Berlin, 1975, p. 581. [Acta Phys. Austr. Suppl. 14, (1975)]. 21. J. R. KLAUDER, Acta Phys. Austr. 41 (1975), 237. 22. E. LUKACS, “Characteristic Functions,” Hafner, New York, 1970, 2nd ed.; I. M. GEL’FAND AND N. YA. VILENKIN, “Generalized Functions. Vol. 4: Applications of Harmonic Analysis,” translated by A. Feinstein, Academic Press, New York, 1964. 23. J.LEBOWITZ, Commun. Math. Phys.35 (1974), 87; B. SIMON AND R. GRIFFITHS, ibid. 33 (1973), 145. 24. J. R. KLAUDER, Phys. Rev. D 14 (1976), 1952. New 25. See, e.g., J. D. BJORKEN AND S. D. DRELL, “Relativistic Quantum Fields,” McGraw-Hill, York, 1965. 2.

NONRENORMALIZABLE

QUANTUM

THEORY

53

26. See, e.g., “Phase Transitions and Critical Phenomena,” Vol. 3 (C. Domb and M. S. Green, Eds.), Academic Press, New York, 1974); in this regard see also E. R. CAIANIELLO, M. MARINARO, AND G. SCARPETTA, Nuovo Cimento 44B (1978), 299. 27. J. R. KLAUDER, Phys. Rev. D 15 (1977), 2830. 28. J. SCHWINOER, Proc. Nat. Acad. Sci. 37 (1951), 452; K. SYMANZIK, Z. Naturforsch. 9a (1954), 809; J. M. JAUCH, Helv. Phys. Acta 29 (1954), 287; J. G. VALATJN, Proc. Roy. Sot. London, Ser. A 229 (1955), 221. 29. J. R. KLAUDER, Phys. Lett. 73B (1978), 152. 30. J. R. KLALJDER, .7. Math. Phys. 18 (1977), 1711. 3 1. See, e.g., K. NISHIJIMA, “Fields and Particles” Benjamin, New York, 1969. 32. See, e.g., D. MATHON AND R. F. STREATER, Z. Wahrscheinlichkeitstheor~e and vet-w. Gebiere 20 (1971), 308. 33. T. YOSHIMURA, Phys. Rev. D 17 (1978), 3284. 34. J. R. KLAUDER, “Scale-Covariant Quantum Field Theory,” 15th Annual Winter School for Theoretical Physics, Karpacz, Poland, February ‘&March 5, 1978, to be published. 35. J. R. KLAUDER, Gen. Relativ. Graoit. 6 (1975). 13.