Review of Invariant Time Formulations of Relativistic

Foundations of Physics, 1Iol. 23, No. 3, 1993

Review of Invariant Time Formulations of Relativistic Quantum Theories

J. R. Fanchi 1

Received January 22, 1992

The purpose of this paper is to review relativistic quantum theories with an invariant evolution parameter. Parametrized relativist& quantum theories (PRQT) have appeared under such names as constraint Hamiltonian dynamics, four-space formalism, indefinite mass, micrononcausal quantum theory, parametrized path integral formalism, relativistic dynamics, Schwinger proper time method, stochastic interpretation of quantum mechanics and stochastic quantization. The review focuses on the fundamental concepts underlying the theories. Similarities as well as differences are highlighted, and an extensive bibliography is provided.

C O N T E N T S

I. I n t roduc t i on A. Out l ine of Pape r . . . . . . . . . . . . . . . . . . . . . 488

B. N o t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . 489

II. Ea r ly Research

A, Te t rode (1922) . . . . . . . . . . . . . . . . . . . . . . 490

B. Fock (1937) . . . . . . . . . . . . . . . . . . . . . . . 491

C. S tueckelberg (1941-1942) . . . . . . . . . . . . . . . . . 494

D. F e y n m a n (1948-1950) . . . . . . . . . . . . . . . . . . . 497

III. Research P r o g r a m s A. P a t h In tegra l F o r m u l a t i o n . . . . . . . . . . . . . . . . . 498

Baru t . . . . . . . . . . . . . . . . . . . . . . . . . 500

B. S c h w i n g e r - D e W i t t P r o p e r Time M e t h o d . . . . . . . . . . . 502

C. D a v i d o n . . . . . . . . . . . . . . . . . . . . . . . . 504

D. M i c r o n o n c a u s a l i t y . . . . . . . . . . . . . . . . . . . . 505

E. Indefini te Mass . . . . . . . . . . . . . . . . . . . . . . 509 F. Algebra ic F o r m u l a t i o n s . . . . . . . . . . . . . . . . . . 512

i 1078 Eas t O te ro Avenue, Li t t le ton, C o l o r a d o 80122.

487

825/23/3-10 0015-9018/93/0300-0487507.00/0 © 1993 Plenum Publishing Corporation

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G. Covariant Hamiltonian Dynamics . . . . . . . . . . . . . . 519 Relativistic Dynamics . . . . . . . . . . . . . . . . . . 519 Constraint Hamiltonian Dynamics . . . . . . . . . . . . 526

H. Probabilistic Formulations . . . . . . . . . . . . . . . . . 528 Four-Space Formalism . . . . . . . . . . . . . . . . . 528 Hostler . . . . . . . . . . . . . . . . . . . . . . . . 532 Stochastic Formalism . . . . . . . . . . . . . . . . . . 333 Parisi and Wu . . . . . . . . . . . . . . . . . . . . . 537

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 538 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

I. I N T R O D U C T I O N

Fock appears to be the first to use an invariant evolution parameter in relativistic quantum theory (Fock, 1937). By 1942, Stueckelberg had presented a new basis for parametrized relativistic quantum theory (PRQT). P R Q T was part of the physics mainstream in the late 1940's and early 1950's when quantum field theory was being formulated. I t helped guide Feynman (1950), Schwinger (1951), and others in their treatment of propagators and divergences. By the end of the 1950's P R Q T was diverging from the mainstream as researchers tried to develop a basis for parametrized theories that could be widely accepted as a paradigm. With a few notable exceptions, P R Q T was neglected during most of the 1960's. Research in P R Q T enjoyed a resurgence based on fundamentally new formulations appearing in the late 1960's and continuing through the 1970's. The 1980's saw an extension of these formulations as many seemingly disparate theories began to find a common ground. The 1990's promise to be a crucial period for establishing the credibility of PRQT.

I.A. Outline of Paper

The primary emphasis of this article is to review the fundamental concepts underlying parametrized theories. To control the volume of literature reviewed, I found it necessary to concentrate on lines of research which view the parameter as physically significant. Lines of research which view the parameter as a mathematical convenience with negligible physical content are also considered, but I limit the discussion to their foundations. In some cases, I review only the founding article(s) and the most recent articles of a particular line of research. Such a choice was made when earlier articles could be traced from the references in the listed article and results of earlier articles did not need to be explicitly stated in the review.

Articles are reviewed in chronological order based on the date of

Parametrized Relativistic Quantum Theories 4 8 9

appearance of the founding paper(s) of a research program. It turns out that a chronological presentation lets us follow the evolution of fundamental concepts as people tried to resolve problems and extend concepts. A research program is defined by its fundamental set of concepts. Articles are classified by their relevance to or consistency with other ideas in a research program. Some articles fit in more than one program, and I have attempted to draw attention to them at an appropriate place. Section II of the review discusses the early work on the subject (pre-1950). The concept of an invariant evolution parameter was introduced in these articles. Section III discusses articles appearing after 1950. An alphabetical listing of articles by author and date of publication is provided as a bibliography. Titles are included to provide the reader with a little more information about the article.

I have designed this review to help researchers learn more about where parametrized theories overlap and where they differ. This article should help people locate other articles that may be relevant to their work. Perhaps the most important conclusion I can draw from this review is that there are many parallels between and areas of compatibility among different research programs.

Most of the modern research programs discussed in Section III are proceeding along theoretical lines, and are not accompanied by experimental programs. In general, researchers have developed formalisms that do not conflict with observation but have not yet generated decisive experimental tests. Scientific acceptance of PRQT still needs to be experimentally justified. Several of the reviewed articles suggest experimental tests (e.g., see the discussion at the end of Section III.F). Hopefully this review will help motivate efforts to design and perform experimental tests.

I.B. Notation

Rather than using the original notation of each paper, I have chosen to use a consistent and modern notation. This choice helps clarify the literature by highlighting areas of agreement and disagreement which may have been obscured by arbitrary notational differences.

Nonzero elements of the fundamental metric tensor are

g o 0 = 1 = - - g l l = - - g 2 2 = - - g 3 3 (I.B.1)

Greek indices run from 0 to 3 and Latin indices run from 1 to 3 unless indicated otherwise. Index 0 denotes the temporal component, while indices 1 to 3 denote spatial components. The scalar product is defined by

A . B = A ° B ° - 7k . g (I.B.2)

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where the four-vector A has the elements

{A"} = (A °, A),

I adopt the notation

{A.} = (A °, - A ) (I.B.3)

and

O ~= , , - V =Ox. ' =ax" (I.B.4)

0 2

for the usual differential operators. The antisymmetric defined as

(I.B.5)

field tensor is

F~v = OuA~ - c~A. (I.B.6)

and the antisymmetric spin tensor is defined in terms of Dirac gamma matrices such that

i cr.~ = ~ [7., 7v] = i(7/,7~-g.v) (I.B.7)

For an explicit representation of the gamma matrices, see a text such as Scadron (1979) or Bjorken and Drell (1964). Natural units h = c = 1 and Einstein's summation convention are used unless otherwise noted.

1I. EARLY RESEARCH

II.A. Tetrode (1922)

Tetrode (1922) appears to he the first researcher to publish a covariant action defined in terms of an integral over proper time:

ATet = m ; d~ (II.A.1)

Equation (II.A.1) shows the action function for an electron of mass m and a proper time defined by

dz = x/dt 2 - dY i d£ (II.A.2)

Parametrized Relativistic Quantum Theories 491

with (Y, t) denoting space-time coordinates. The integral is performed over the entire world line. Tetrode performed a variation of the action using Hamilton's principle to derive classical equations of motion.

II.B. Foek (1937)

Fock (1937) defined a classical action as an integral of a Lagrangian function with respect to proper time:

mFo c = f LFo c d s (II.B.I)

Proper time was treated as an independent invariant parameter. The classical Lagrangian for an electron interacting with an electromagnetic field is given by

m LFo e = - - mYc ,,~ ~' - ~ -- e.f , A ~' (II.B.2)

where

dx~ (II.B.3) You= ds

Fock constructed a Hamilton-Jacobi equation of the form

(OAvoc_ e~)23 + m 2 = 0 (II.B.4) OA~Vs°~ + ~ - ~ [ ( V A v ° ~ + e A ) Z - \ Ot

subject to the condition

OAv°c = 0 (II.B.5) 0s

Equation (II.B.5) was considered necessary for a system to be realistic. Given a classical basis, Fock presented the quantum analog of the

Hamilton-Jacobi equation (II.B.4):

OF 1 ~s 2m

[ if ] - - - - - - F ] ~ v ( i~--eA')( iO~ eAv)--m2 2 7,?v F

1 -- 2m KFooF (II.B.6)

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The function F satisfies a partial differential equation that is second order in space-time and first order in proper time. The condition

gF ds ~-~s = 0 (II.B.7)

is the quantum analog of the constraint [Eq. (II.B.5)] on the classical action. If the operator does not depend on s, the function F(s) evolves from an initial distribution F(0) according to the relation

F(s) = F(0) exp( - iSKFoo/Zm) (II.B.8)

where the dependence on space-time has been suppressed. Denoting the eigenvalues of the operator KFoo by kvoo and substituting Eq. (II.B.8) into (II.B.7) shows that

KvooF = kvocF = 0 (II.B.9)

Thus, solutions of Eq. (II.B.6) had to be stationary states with zero eigenvalues in Fock's formulation. Fock's theory is outlined in English by Nambu (1950).

Nambu (1950) viewed Fock's (1937) parameter as a mathematical convenience for characterizing the space-time evolution of a system. He tried to determine if the invariant scalar parameter should be incorporated into a wave equation as a first derivative in a Dirac-like equation

y5 ~ss-- (7,,0'* + m ) I/I(xO, xI, x2, x3, s)=O, 75 ~ i'Y0~1"~2'~3 (II.B.10)

or as a first derivative as in Eq. (II.B.6). Iterating Eq. (II.B.10) yields

[~@~2 q- ~uOu - m2] ~b = 0 (II.B.ll)

Nambu did not find a good reason to select one approach over the other, so he stayed with Fock's procedure. Katayama et al. (1950), Bakri (1971), and Omote et al. (1989) are three more examples of attempts to express the evolution parameter as a fifth coordinate. These attempts have generally been abandoned, although Barut and Pavsic (1988) have recently found a Kaluza-Klein (five-dimensional) approach useful in describing the interaction of particle spin with the electromagnetic field. It is important to note that Barut and Pavsic introduce a fifth coordinate in addition to the invariant evolution parameter. Some authors have kept the evolution parameter derivative the same order as the space-time derivative--first


order as in Eq. (II.B.10) (Katayama et al., 1950), second order as in Eq. (II.B.11) (Greenberger, 1974a; Caldirola, 1978), first order in the mass unit (Moses, 1969; Johnson, 1969; Garrod, 1966), or first order in the form

a~ i ~ s = V ~ pU ~ , p~ =- - ia ~ (II.B.12)

Equation (II.B.12) is the most promising Dirac-like equation. It has been studied in such papers as Hamaguchi (1954), Enatsu (1954a), Corben (1961), Johnson (1970), Ellis (1981), Lopez and Perez (1981), Herdegen (1982), Pavsic (1984), Kubo (1985), Barut and Duru (1989), and Evans (1989). Wong (1972) used Eq. (II.B.12) with a mass term added to the right-hand side. Many of these papers are discussed in more detail later.

Returning to Nambu's 1950 paper, we note that Nambu derived Schwinger, Feynman, and Dyson propagators from Fock's theory by using the proper time parameter as the parameter in the integral representations of the propagators. He used Fock's theory to calculate the self-energy of an electron and vacuum polarization. Other applications of Fock's theory include Katayama (1951), Mano (1955), Rafanelli and Schiller (1964), Yasue (1977), and Ichiyanagi (1983). Ichiyanagi postulated that wave mechanics is more fundamental than matrix mechanics, then proceeded to reconstruct Nambu's modified Schr6dinger equation. Of these papers, the most interesting from a modern perspective is Yasue's because Yasue coupled a stochastic derivation of Fock's equation with a Hilbert space norm. The role of stochastic processes and the importance of the normalization condition are examined later.

Szamosi (1961) followed a path similar to Fock (1937) and Nambu (1950) by quantizing a classical Hamilton-Jacobi description of a free relativistic particle with spin 0 or spin 1/2. His quantization method yielded the wave equations for a free scalar particle and a free spin-l/2 particle as special cases of a total differential equation describing the particle motion. Szamosi's quantization method differed from Fock's and Nambu's because it led to an independent description of both a scalar particle and a spin-l/2 particle. Szamosi was unable to find a simple formulation of relativistic equations for other stable particles using his quantization method.

As a footnote, Fock (1950) and later Schwinger (1951) introduced the Lorentz covariant gauge condition

( x - x') ~ A~ = 0 (II.B.13)

where x' is an arbitrary space-time point. This gauge was rediscovered by Cronstrom (1980), Shifman (1980), and Dubovikov and Smilka (1981). It is being used extensively in the modern literature because it induces a

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gauge field dependence on x'. For a succinct introduction to the modern uses of Eq. (II.B.13), see Zuk (1986).

II.C. Stueckeiberg (1941-1942)

Stueckelberg was the first to develop a parametrized quantum theory that included a new normalization condition and a new interpretation of the probability function in addition to a parametrized wave equation. Stueckelberg began using proper time as part of a new classical electron model. He derived classical equations of motion from an action defined as the integral of the Lagrangian with respect to proper time (1941a). The problem of interpreting the evolution parameter was addressed in two later papers (1941b, c).

In 194tb, Stueckelberg introduced a space-time normalization of the parameter-dependent wave function

fDO*(x, s) ~b(x, s) d4x = 1, d4x = dx 0 dx 1 d,Y 2 dx 3 (II.C.1)

where the wave function has the explicit dependence

@(X, S) ~--- ~ / (X O, X 1, X 2 , X 3 , S) (II.C.2)

and the integration is over a space-time four-volume D. We shall subse- quently refer to Eq. (II.C.1) as the Stueckelberg normalization.

Stueckelberg (1941b) recognized that interpretation problems existed in his new formulation. This led him to the notion of a charged particle (electron) and its antiparticle (positron) propagating in opposite spatial and temporal directions (Stueckelberg, 1941c). He arrived at this notion by examining the classical relativistic equation of motion for a particle in an electromagnetic field

d2x ~ dx~ rn dz---/- = e ~ F ~" (II.C.3)

where proper time is given by Eq. (II.A.2). Equation (II.C.3) is invariant with respect to a simultaneous change in the sign of the charge and the direction of the space-time four-vector. Feynman (1948a) presented similar arguments for a classical action to arrive at an equivalent interpretation. It should be noted that Feynman attributed this interpretation to a suggestion he received from J. A. Wheeler in 1941. Feynman first referenced Stueckelberg's interpretation, notably Stueckelberg's 1942 paper, in 1949 (Feynman, 1949a).

Parametrized Relativistic Quantum Theories 4 9 5

A 1942 paper by Stueckelberg brought together all of his ideas in a reformulation of quantum theory. Stueckelberg examined the consequences of replacing a classical action of the form

with the action

dq/z Aold = - f m , / G 4 " d:*, (ILC.4)

(,

Ane w = - ½ J 0 ~ ~ d2 (II.C.5)

He introduced the invariant parameter into classical mechanics via the Lagrangian density

~t~Stu ~- ~ern "~ ~¢9m -~- ~I (II.C.6)

Lagrangian densities for the electromagnetic field, matter, and their interaction are, respectively,

1 ~(~Oem - - 16z~ F'"FU'~ (II.C.7)

~ e = _ f ~ 1 -o~ 2 g,,,(x) ?l~'q 3 ( x - q(2)) d2 (II.C.8)

~q]= _ JsAu JU~=_e gF~f(x-q(2))d2 (II.C.9)

The Dirac delta function is defined in the usual way by

f 6(x) f (x) d4x = f ( 0 ) (II.C.10)

Integrating the Lagrangian density over space-time yields the Lagrangian

Lst u = - ½0pO ~ - eA~,(q)(t u (II.C.11)

where only the matter and interaction terms have been retained. A Hamiltonian of the form

1 6 3 L s t u R(p, q)=-~ rc,~", lr~,(p, q)=pu--eA~,(q), p~,- c3i1~, (II.C.12)

was then constructed from the Lagrangian. Neither Eq. (II.C.11) nor (II.C.12) contains the mass. As discussed in more detail in Section Ill.A,

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mass enters the theory as a constant of integration. Poisson brackets were used to derive a set of canonical equations of motion. Cawley (1973) demonstrated the Galilei invariance of Stueckelberg's Lagrangian and action.

By analogy with Schr6dinger's development of his nonrelativistic wave equation, Stueckelberg introduced a probability amplitude which satisfied the equation

k(/~, q)~h = t-~- (II.C.13)

where the momentum operator is defined by

p~ = - ion, (II.C.14)

Equation (II.C.13) is the Klein-Gordon equation with the mass squared term replaced by the term on the right-hand side. Eigenvalues of the Hermitian operator R are proportional to the square of the mass of the particle. We shall call Eq. (II.C.13) Stueckelberg's equation. Fock's equation [Eq. (II.B.6)] differs by the addition of spin terms and the auxiliary requirement equation (II.B.7). Stueckelberg defined the expectation value of a Hermitian operator as

(P(2)>=(tp, Ptp)=f ~p*(q, 2)(Ptp(q,)O)d4q (II.C.15)

The probability amplitude is subject to the normalization condition

f I~L 2 d*q = 1 (II.C.16)

Stueckelberg derived a correspondence principle between classical and quantum mechanics by first finding the expectation values of a four-vector current and a stress-energy tensor, then integrating the expectation values with respect to the invariant scalar parameter. A similar procedure has been used more recently by Horwitz and co-workers. They call their integration procedure a concatenation (Arshansky, Horwitz, and Lavie, 1983).

The derivative of the expectation value of the coordinate four-vector with respect to the invariant parameter yielded the relationship

d ( q ' ~ ) - ( k " ) , {k" } = (k °, Ic) = (co, ~c) (II.C.17) ,/2

where the right-hand side of Eq. (II.C.17) is the energy-momentum four- vector expressed in terms of wave numbers. By analyzing the zeroth (tern-


poral) component of Eq. (II.C. 17) and noting that the invariant parameter is a monotonically increasing function, Stueckelberg recognized the relation between the sign of the energy of a particle and its direction of temporal motion. From Eq. (II.C.17), he arrived at the concept of a positive- frequency wave propagating forward in time and a negative-frequency wave propagating backward in time. This interpretation was independently surmised and popularized by Feynman (1949a, b; 1950). We shall refer to it as the Stueckelberg-Feynman interpretation.

l l .D. Feynman (1948-1950)

Feynman first used a parameter in relativistic quantum theory in 1948 (Feynman, 1948b). This paper introduced his path integral formulation of nonrelativistic quantum theory. In the last section of the paper, Feynman suggested that a relativistic path of a spinless particle could be described by four functions of an invariant scalar parameter. His functions were the space-time four-vector, and the parameter played the role of proper time. The Klein-Gordon equation could then be derived using the path integral formalism for a Lagrangian of the form

dx ~ L F e y : - - 2112~ - - eA~A,, 2 ~ - (II.D. t )

ds

The action for his path integral was the integral of Eq. (II.D. 1 ) with respect to the invariant parameter. Feynman outlined a procedure for deriving a wave equation with the square of the Dirac operator. It is worth noting that Feynman (1948a) used an expression like L w y to justify interpreting positrons as electrons with the proper time reversed. This interpretation follows by observing that LF e y is invariant if the sign of the charge and the direction of proper time are simultaneously reversed.

Feynman returned to a parametrized relativistic formulation in 1950. In Appendix A of one of his classic papers on quantum electrodynamics (QED), Feynman (1950) described "a formulation of the equations for a particle of spin zero which was first used to obtain the rules" he gave for making quantum electrodynamic calculations involving spinless particles (1949b). Appendix A contained a derivation of the Stueckelberg-type equation

i ~-~ OVey(X, s) 1

= ~ (i(?, - eA,)(i~? ~ - eA u) ¢'wy(X, s) (II.D.2)

using Feynman's path integral formulation. Feynman pointed out that a separate scalar parameter must be used for each particle when several

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particles are present. He imposed a stationary state condition on Eq. (II.D.2), i.e., final solutions must satisfy the Klein-Gordon equation. Thus a solution of the Klein-Gordon equation is obtained by calculating

f oo

7tVey(X) = exp(-½im2s) ~/Fey(X, S) ds --OO

(II.D.3)

Feynman did not adopt Stueckelberg's norm [Eq. (II.C.16)]. Instead, he preferred to average over s as in Eq. (II.D.3) and use the resultilag wave function in the conventional norm

i f [- ~['/~ey(X) 0 0 ~Y/Fey(X ) -- (0 0 ~F~ey(X)) ~[/Fey(X)] N3X = 2p0 ( I I .D.4)

Problems with the conventional norm have been discussed by Fanchi (1981a) and Kyprianidis (1987).

Feynman used Eq. (II.D.2) to obtain rules for solving the problem of an electron interacting with an electromagnetic field. His results agreed with the second quantized theory of the Klein-Gordon equation except for the absence of vacuum loops. Feynman suggested that this formulation could be used as an alternative to second quantization, but he remarked that he had not thoroughly analyzed the physical significance of the parameter-dependent formulation.

IlL RESEARCH P R O G R A M S

III.A. Path Integral Formulation

Feynman outlined a procedure to extend his path integral formulation of nonrelativistic quantum mechanics to quantum electrodynamics (QED) in 1950. Morette (1951) succinctly presented the relativistic path integral formalism by writing the probability amplitude K for a particle to go from space-time point x A to space-time point x s as

K(x B, x A) = f exp(iS[x]) d(paths) (III.A.1)

The integral is over all paths x(s) from x A to x B. The action functional Six] is given by

~ L Six] = {x(~), : t (s)} as (III.A.2)


where L is the Lagrangian. Feynman (1950, Appendix A) worked with the Lagrangian of a spinless particle given in Eq. (II.D.1). The corresponding parametrized wave equation is Eq. (II.D.2), where the "physical" wave function was found by integrating over s as in Eq. (II.D.3).

Following a suggestion made by Feynman (1948b, Section i4), Morette (1951) applied Feynman's path integral approach to a Dirac particle in a constant, external electromagnetic field. She used a Lagrangian of the form

LMo r = -~ (III.A.3)

with the corresponding wave equation

0 2im -~s ~/M°r(X' S) = {])#[(~# -- ieA ~] }2 ~/MortX, S) (III.A.4)

Morette recovered a Klein-Gordon type of equation by making the variable substitution

0Mot(X, S) = exp[ ims/2 ] ~JMor(X) (III.A.5)

where the argument of the exponent contains a first-order m rather than m 2 as in Eq. (II.D.3).

Feynman (1951, Appendix D) used a path integral formulation to derive Fock's parametrized spin-l/2 wave equation. His Lagrangian had the form

LFF =- -- ~ dU +e-ffffuAu-'4a. vF"v (III.A.6)

where u denotes a scalar evolution parameter. The interaction term is assumed as a result of "minimal coupling." Miura (1979) studied the geometrical features of the path integral approach including the introduction of interaction terms by requiring gauge invariance. The wave equation with interactions corresponding to Eq. (III.A.6) is

11 e j i ~vF(X, U)=~ (iO,-eA,)(iO"-eAU)-~cruvF ~v ~FF(X, U) (III.A.7)

Feynman derived a Klein-Gordon equation from Eq. (III.A.7) by making the variable substitution

~Fv(X, U) = exp( - - imZu/2) ~r ' tFF(X ) (III.A.8)

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Equation (III.A.6) yields the classical equation of motion

dZx ~ dx v du 2 - e ~ u FVU (III.A.9)

Unlike the usual classical Lorentz equation of motion [Eq. (II.C.3)], Eq. (III.A.9) does not contain m. Neither Eqs. (III.A.6) nor (III.A.7) contain the rest mass m of the particle.

Rest mass enters the theory here as well as in Stueckelberg's work (Section II.Ci as a scale parameter linking the increment du of the parameter u with the increment ds of proper time:

ds = m du (III.A.10)

Feynman derived Eq. (III.A.10) by specifying the minimum value of the action. Other bases for Eq. (III.A.10) have been published. For example, Stueckelberg (1941b, c) and Davidon (1955a) arrived at Eq. (III.A.10) by imposing an initial condition on the magnitude of the proper four-velocity; and Fanchi and Wilson (1983, Section 6.4) made a change of variable using Eq. (III.A.10) to construct mass-independent parametrized relativistic wave equations for a many-body system.

A relationship between Feynman's (1950) parametrized path integral formulation and Schwinger's propagator formalism (1951) was provided by Ravndal (1980). He established the relationship by considering a Lagrangian for a relativistic particle with spin in both an external electromagnetic field and a gravitational field. His work was an extension of Di Vecchia and Ravndal's (1979) construction of a supersymmetric Lagrangian for a massive, spinning particle. Shirafuji (1970, 1971) generalized the path integral approach to describe a particle with two proper times. Henneaux and Teitelboim (1982) used Hamiltonian path integral methods to study a spinless particle in an external field and an electron in an external electromagnetic field. They calculated propagators and scattering amplitudes from an S-matrix. Lebedev (1985) used Eq. (III.A.7) as well as Stueckelberg's scalar particle wave equation to find the WKB asymptotic form of the causal Green's function and calculate a mass shift associated with the self-interaction of a classical particle.

Barut. An interesting modern application of path integrals is the development of a relativistic electron model by Barut and co-workers. They used path integrals to construct both classical and quantum models of spin-0 and spin-l/2 particles. The result is the parameter-dependent theory outlined below.


Path integrals are formulated in terms of classical actions. Following Barut and Duru (1989), we write the action for a single particle as

(III.A. 11 )

where L ° is the free particle Lagrangian, j e is the current, and A" is the four-vector potential. Let z(s) be a parametrized c-number four-component spinor. The Lagrangian for a spinor particle is (Barut and Zanghi, 1984)

L°=-N sz-eN 2=2+70 (llI.A.12)

where z + is the Dirac adjoint of z . The current has the form

j ~ ( x ) = e£7~zS(x ~ - x~(s)) (III.A.13)

There are two fundamental constants in this formulation: 2 with units of action, and e. The mass of the particle appears as a constant of the motion. There are two sets of canonically conjugate variables. One set is the parameter-dependent space-time and energy-momentum four-vectors {x ~, p~}, while the other set is the pair of spin variables {z, ~}. In an N-body system, each particle has its own evolution parameter and constant 2. The equations of motion for the spinor particle can be used to derive the Lorentz equation

zt~ = eF~v2 v, ~z~ = p ~ - eA~ (III.A.14)

where mass does not explicitly appear. The wave equation corresponding to the classical formulation is

i ~=-qk(z, x, s ) = H B ( ~ ( z , x, s) (III.A.15) Us

where the Hamiltonian operator for flat space-time is

HB = ZyUzn~ (III.A.16)

A Hamiltonian for curved space has been derived by Barut and Pavsic (1987). The Dirac-like equation

i ~s ~ -- 7"( - i?~, - eA~)tp (III.A.17)

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can be derived by making the change of variable

¢ = q)z (III.A.18)

where ~ is the Dirac adjoint spinor. Equation (III.A.17) appears in several papers and is discussed in more detail later.

One of the most physically significant applications of the above formulation is to the description of electron Zitterbewegung (Barut and Thacker, 1985a, b). Their work is built on a classical model (Barut and Zanghi, 1984) in which electron spin is viewed as the orbital angular momentum of Zitterbewegung. The spin variables introduced in the action are needed for representing the "internal motion" of the electron. Several other applications of the formalism have been made.

Specific examples include derivation of QED from classical particle trajectories (Barut and Duru, 1989); generalization of the Lorentz-Dirac equation to include spin (Barut and Unal, 1989); derivation of higher-spin equations from the classical model (Barut, 1990); and covariant quantum many-body theory (Barut, 1991). The interested reader should consult Barut, Onem, and Unal (1990) or the articles mentioned above for additional references.

III.B. Schwinger-DeWitt Proper Time Method

Schwinger (1951) was interested in developing a gauge invariant calculation procedure for isolating the divergent parts of integrals involving propagators. He postulated the expectation value of a vacuum current of a charged Dirac field to be

(j~(x) ) = ie tr ~,G(x, x')[x,~x (III.B.1)

where tr denotes trace, and the Green's function for the Dirac field is found from

[7,H~ + rn] G(x, x ' )=6(x -x ' ) , H ~'--- (-iOP-eAU(x)) (III.B.2)

The Green's function is

G = [ - T v H ~ + m ] I-rn 2 - (7~H~) 2 ] -1 (III.B.3)

which has an integral representation

G=i(--7vH~+m) foeXp[--i(rn2--(yj-I")2)s]ds (III.B.4)


in terms of an invariant parameter s. The system evolves with s according to an operator of the form

U(s) = exp( - iHschS) (III.B.5)

with the Hamiltonian

~,~,, p~v (III.B.6) Hsch = -- (7~//") 2 = H~H" - 2~v~v~

and the corresponding wave equation for a spin-l/2 particle is

i ~ss OSch ----- Hsoh ~kS~h (III.B.7)

Schwinger interpreted s as a parameter like proper time. The wave equation does not include the rest mass of the particle. The rest mass appears in the interaction Lagrangian density

i I v exp(-im2s) tr(x] U(s) i x ) ds (III.B.8) Lsoh(x) = 2 J o s

where the kernel contains the trace of the matrix element of the operator U(s). Schwinger's Hamiltonian differs from Fock's Hamiltonian by a factor of 2, which is accounted for in the exponential term of Eq. (II1.B.8).

Schwinger applied his gauge-independent formulation to the problem of vacuum polarization and gamma decay of neutral mesons. He then developed a perturbation theory that was not gauge invariant for solving problems with arbitrarily varying electromagnetic fields. Schwinger showed that the Pauli and Villars (1949) invariant regularization procedure was a "partial realization" of his proper time method. Karplus and Klein (1952) and Karplus, Klein, and Schwinger (1952) calculated the hyperfine structure and Lamb shift of atomic energy levels, respectively, using Schwinger's approach.

Valatin (1954) used Schwinger's approach to study propagators in a parametric integral representation. He transformed the wave equation to attain a kernel that did not contain the rest mass constant. He used gauge- independent techniques to develop a perturbation theory. The perturbation theory was applied to evaluating singularities of an electron propagator. Miyamoto (1970) and Itoh et al. (1971) used Schwinger's proper time method to study the duality between s-channel and t-channel scattering. For a more recent discussion of the Schwinger proper time representation of propagators, see Rumpf (1983).

DeWitt (1975) extended Schwinger's regularization scheme to the problem of handling divergences in the quantum field theory of curved space-time. DeWitt showed that different procedures had to be used to demonstrate coordinate invariance and conformal invariance. DeWitt's

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modification of Schwinger's scheme is called the Schwinger-DeWitt procedure. It preserves conformal invariance as well as gauge invariance. The Schwinger-DeWitt procedure uses the parameter s as a mathematical device that must be integrated over at the end of a calculation. Several authors have used the Schwinger-DeWitt procedure to calculate physically interesting quantities in curved space-time, e.g., Hartle and Hawking (1976), Brown (1977), Lee et al. (1989), and Cai and Papini (1990).

I do not present a thorough review of modem applications of the Schwinger-DeWitt procedure because the applications are viewed as mathematical procedures that do not ascribe any special physical significance to the evolution parameter. Rather, I focus on parametrized relativistic quantum theories which attribute a physical significance to the evolution parameter. Readers interested in more modern applications of the Schwinger-DeWitt procedure should consult papers such as Stephens (1988, 1989), Camporesi (1990), Corns and Osborn (1990), and Molzahn et aL (1990).

III.C. Davidon

Davidon (1955) used classical theory to justify a parameter-dependent wave equation of the form

8 HDav~bDav(X, S) = i~s s ffDa~(X, S) (III.C,1)

where I/ /Da v is a 2-component spinor and the Hamiltonian is

'E ] i e - = + -- F,~ cry. ~ 17, = - i~, - e2~ HD~v -2 H"HU 2 ' (III.C.2)

The field tensor Fu~ is multiplying the 2 x 2 Pauli matrices ap. The average of the field tensor and the 4-vector potential are defined in terms of a shape function F ( x - q) by the equation

Pu~(q) = f F(x - q) Fu~(x ) d4x

A~,(q) -- f F(x - q) A~,(x) d4x

(IH.C.3)

The shape function describes the charge distribution. It is a Dirac delta function for a point particle. Davidon defined an inner product of the form

($b, $ , ) = f $ ~ H~aP$a d4x (III.C.4)

Parametrized Relativistic Quantum Theories S05

Equation (III.C.4) is an indefinite metric since it can be either positive or negative. Davidon ignored negative probabilities because of his assumption that measurements at a given s were not physically significant.

Perhaps the most interesting contribution of Davidon's work at the time was the conversion of the two second-order wave equations with 2-component spinors to four first order wave equations with a 4-component spinor; thus,

where

(i~/y ~ + ~3)¢(+)= o, ¢(4) = F u'2) ] (III.C.5) LV(2)J

- 1 0

0 1

0 0

0 0

0 0 -

0 0

-2iV o

o -2 iN.

(III.C.6)

Davidon viewed Eq. (III.C.6) as a "trivial" mass operator because it did not account for electromagnetic effects. The 2-component spinors u(z), v(2) satisfy the coupled equations

- i l I~a~ut2) = v~z~ (III.C.7)

i I Iu cv~'v(2) = i ~s u(2) (III.C.8)

Substituting Eq. (III.C.7) into Eq. (IILC.8) gives Eq. (IILC.I). Davidon worked with Eq. (III.C.5) to study QED. In 1955a, he found

a free-particle solution and constructed Green's functions for Feynman graphs. In 1955b, Davidon quantized the photon field, calculated the electron self-mass, and discussed a many-electron wave equation, a multiple-parameter formalism, and vacuum polarization.

III.D. Micrononcausality

The proliferation of experimentally observed particles in the post- World War II years stimulated theoretical efforts to understand mass spectra. Enatsu (1954a, b) attempted to calculate quantized masses, or

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mass spectra, using parametrized theories. He applied the proper time formulation of Schwinger (1951) and, to a lesser extent, Feynman (1950). The Schwinger proper time formulation was selected because it was gauge invariant and could remove divergences such as vacuum polarization. Enatsu presented a theory of mass quantization for spin-l/2 particles interacting with a scalar photon starting from a Dirac equation of the form

-iT~a.~, o(x) + f ME(x, x') t~ D(X' ) d4x ' = 0 (III.D.1)

The mass operator in the Schwinger proper time formulation is

ME(x, x') =m6(x--x') + ig2Ge(x, x') D+(x, x') (III.D.2)

where g is the coupling constant and m is the rest mass. Green's functions for the spin-l/2 particle and the scalar photon are calculated from the wave equation

i~s s 0E(x, s )= - 0~O~0E(x, s) (III.D.3)

and have the respective forms

1 GE(x, x')= (-~n)2 to s-2 exp(--im2s) [

× exp k J

and

7"(XU2s-X'u) 4-ml

ds (III.D.4)

Enatsu solved the mass quantization problem for a Coulomb-type self- potential (1954a) and a Yukawa-type self-potential (1954b). His work provided a relativistically invariant formalism for studying a system interacting with itself.

Based on his 1954 papers, Enatsu suggested that the self-interaction of a system was related to the quantization of particle masses. Changes in the bare mass of a particle were calculated by assuming an internal particle structure. These ideas were extended to a relativistic quantum mechanical treatment of pseudoscalar mesons interacting with nucleons (Enatsu, 1956) and to the calculation of the hyperon mass spectrum (Enatsu and Ihara, 1955). Ambiquities in the theory, such as in the decomposition of the

S [ 1 -2 l(xu--x~)(x -ds (III.D.5) D+(x, x')= (-~n)2 s exp 4s


observed meson mass and an assumed relationship between proper times of distinct particles, motivated the search for a more general approach to the quantization of particle masses.

In 1963, Enatsu tried to determine if the classical relativistic Hamiltonian formalism could be used as a basis for quantum field theory in place of the Lagrangian techniques developed previously by researchers such as Feynman and Schwinger. Enatsu began with a variational action principle for a free, spinless particle

6 f"- ~f 5f E d4x ds = 0 (III.D.6) Sl

with a Lagrangian density

~ e = i~*(x, s) ~s OE(x, s) - O,O}(x, s) Ou~ke(x, s) (III.D.7)

where * denotes complex conjugation. The free-particle wave equation corresponding to Eq. (III.D.7) is Eq. (III.D.3). From this Lagrangian basis Enatsu constructed a Hamiltonian density and derived particular solutions.

Enatsu recognized that the invariant parameter was conjugate to the square of the mass. He regarded mass as an "observable whose eigenstates form a complete set" (p. 238). The fixed mass-dependent commutation relation of nonparametrized field theory is

['~(X : F/12), ~ * ( X t :m2) ] = iAs(x - x' :m 2) (III.D.8)

where the right-hand side is Schwinger's invariant delta function [see Eqs. (III.D.t0) and (III.D.11) below]. By contrast, Enatsu assumed the parameter-dependent commutation relation

[tpe(x , s), O*(x', s')] = iJE(x -- x', S -- s') (III.D.9)

where the delta function has the explicit representation

. t ,u p t

A e ( x - x ' , s - s ' ) = ~ 5 exp[_ -4(-~ -----~ j J

1 2 x (s ---Jof-~) ' ~(x) = ~ + i ' ( --1, Xo>;}Xo < (III.D. 10)

The delta function in Eq. (III.D.10) is related to Schwinger's invariant delta function [Eq. (III.D,5)] by the inverse Fourier transform

A , ( x - - x ' : m 2) = A e ( x - x ' , s) exp(- im2s) ds (III.D.11) - - o o

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In the limiting case when s'-+ s, Eq. (III.D.10) becomes

[~k e ( x , s), t~* (x ' , s)] = 6 ( x -- x ' ) (III.D.12)

which is the relativistic analog of the nonrelativistic expression for a Schr6dinger field.

By working with an invariant parameter, Enatsu developed the formalism for both a parametrized Schr6dinger picture and a parametrized Heisenberg picture. The evolution of a system in the parametrized Schr6dinger picture is described by the parameter dependence of the state vector, while operators corresponding to physical observables are parameter independent. The parametrized Heisenberg picture is characterized by parameter-independent state vectors and parameter-dependent operators.

Prior to Enatsu's 1963 paper, quantum field theory postulated the local commutativity of field operators, or microcausality. Enatsu introduced the notion of micrononcausality, or microscopic action at a distance. Following Schwinger (1959) and Nakano (1959), Enatsu chose to work with a Euclidean metric

x 2 + x21 + x 2 + x 2 ~> 0 (III.D.13) and the complex time

t ~ - ixo (III.D.14)

The resulting formalism did not contain any symbolic differences between space and time. Space-time coordinates were considered parameters, and the only "time" with a direct physical meaning was the invariant time.

Enatsu's invariant time was not necessarily identified with classical proper time. Invariant time had a physical significance similar to the "absolute time" of Isaac Newton and Dirac (1953). Dirac introduced an absolute time as a means of parametrizing surfaces of absolute simultaneity. Given an invariant time, Enatsu also resurrected the concept of absolute simultaneity of two spacelike separated points x and x' relative to the invariant time. Enatsu (1963) defined the domain of invariant time as "a four-dimensional spacelike region through which a kind of action at a distance is propagated instantaneously" (p. 260). He argued that micrononcausality should apply "in the microscopic world where Heisenberg's uncertainty principle plays a dominating part" (p. 252). These assumptions were especially useful in solving the two-body problem.

Application of the formalism led Enatsu to a solution of the relativistic two-body problem using the spin-0 wave equation

8 I ¢32 82 1 • - - - + VE ~UE(x, y, s) (III.D.15) y' & Sx ay.


where Ve is the interaction term, and {x, y} denotes the space-time coordinates of the two particles. Equation (III.D.15) is a differential equation whose order is lower than that of the Bethe-Salpeter equation for two- body systems. Enatsu was therefore able to avoid the problem of redundant solutions. A spin-l/2 wave equation was proposed at the end of the 1963 paper.

Enatsu concluded that a relativistic analog of the nonrelativistic Hamiltonian formalism of quantum field theory could be developed when an invariant time is defined. He argued (p. 260) that the "essential virtue of the new formulation" was the introduction of a completely covariant Schr6dinger representation. This representation allowed the distinct separation of the dynamical and kinematical aspects of a system.

In later work, Enatsu (1971) emphasized that his principle of micrononcausality implied the existence of virtual particles with non-negative real mass that propagated at tachyon speeds, if not instantaneously, from one source to another. Enatsu used a spin-l/2 wave equation in 1971 and 1975 to describe the relativistic two-body hydrogen atom. His work with Kawaguchi (1975) led to the calculation of nuclear corrections to the hyperfine mass levels of hydrogen. Enatsu and Kawaguchi (1975) also noted that several of the formulas in two earlier papers (Enatsu, 1963, 1968) left out a factor representing the classical electron radius. Enatsu, Takenaka, and Okazaki (1978) studied the internal structure of the deuteron. They proposed a model based on micrononcausality for hadrons that included a detailed description of virtual mesons and baryons. Enatsu's method of mass quantization was applied to the study of neutral scalar mesons (Enatsu, 1986), mesons with spin and charge (Takenaka, 1986), W-bosons (Fujii and Enatsu, 1988), and Z-bosons (Fujii, 1988). The latter two applications were within the context of electroweak theory (Weinberg, 1967; Salam, 1968). Takenaka has coupled quark model concepts with micrononcausality and Euclidean wave functions to describe hadrons (Takenaka, 1989), leptons, and quarks (Takenaka, 1990).

III.E. Indefinite Mass

Like Enatsu, Greenberger was concerned with devising a theory in which mass could be created or destroyed. He arrived at a parametrized relativistic quantum theory from a different direction than previous researchers. Originally, Greenberger (1963) was attempting to construct a scale-invariant quantum theory of spin-l/2 particles. He noticed that the scale transformation had a gaugelike structure if he made an independent scale transformation at every point in space-time. To preserve gauge invariance, he introduced a vector field in analogy with the Yang-Mills

510 Fanchi

method (1954). The vector field was a function of space-time coordinates and an independent continuous parameter. The continuous parameter in Greenberger's scale transformation theory (1963) was related to the Compton wavelength of a spin-l/2 particle. Greenberger recognized (1963, p. 308) that "there is no evidence that nature has any use" for his scale transformation theory, but he suggested that the procedure may be valuable. By 1970, Greenberger proposed some specific applications.

Greenberger argued (1970a) that the equivalence principle justified the treatment of proper time as an independent degree of freedom with mass as its canonically conjugate variable. According to Greenberger, the most important consequence of this view was the development of a dynamical theory that could accommodate variable mass models. Proper time was taken in its classical context to represent time measured on a clock in the rest frame of the particle. Any relationship between proper time and coordinate time was established through dynamical equations of motion. Greenberger developed a theory based on the proper time transformation

s ~ s' = s + As (III.E.1)

Greenberger insisted on space-time point-by-point invariance of the theory with respect to a proper time transformation. In this case the incremental change in proper time had the form

As = - g a ( x ) (III.E.2)

where g is a coupling constant and the proper time increment is a function of space-time coordinates. Following Yang and Mills (1954), Greenberger's invariance requirement implied the transforms

~G(x, s) --, 7%(x, s') = ~ d x , s) (III.E.3)

Bu(x , s) --* B'~(x, s ' ) = B . ( x , s) - ~ . ~ (III.E.4)

0~ ~ ~ , - g B , ~ s - D ~ , (III.E.5)

Equation (III.E.3) is the transform of the matter field with respect to a proper time transformation. The four-vector field in Eqs. (III.E.4) and (III.E.5) is a new field that was needed to preserve gauge invariance. It has mathematical properties that are analogous to the electromagnetic four- vector potential, but Greenberger argued that it was physically similar to a gravitational potential. He identified Bo as the gravitational potential. For


this reason I refer to B~ as the gravity four-vector. In the nonrelativistic limit the elements of the gravity four-vector are given by

B~ = {~b,/~} ~ {~b, 0}NR (III.E.6)

where the subscript NR denotes the nonrelativistic limit. It is important to observe that the spatial components have vanished in Eq. (III.E.6).

Greenberger (1970a) attempted to identify the physical significance of the gravity four-vector by assuming a simple proper time dependence

~ga(x, s) = e;mSg;Go(X ) (III.E.7)

so that the covariant derivative in Eq, (III.E.5) became

O , gta(x, s) = e i m s ( o l~ - igmB u) g%o(X) (III.E.8)

Equation (III.E.8) shows that the proper time derivative, in the covariant derivative leads to a coupling between the matter field and the gravity four- vector that is proportional to mass. The explicit proper time dependence of the gravity four-vector is responsible in Greenberger's theory for changes in the rest mass of a particle, i.e., Greenberger linked particle decay to the interaction between the matter field and the gravity four-vector (1974a). All of Greenberger's examples (1963, 1970a, b, 1974a, b) assumed the nonrelativistic limit of the gravity four-vector [Eq. (III.E.6)]. Consequently the physical meaning, if any, of the spatial components of the gravity four-vector was not addressed.

Most of Greenberger's work dealt with classical formalism and applications. He presented a quantum mechanical formulation (1974b) based on a mass operator

0 mop = -- i ~ss (llI.E.9)

Replacing physical variables in the classical expression

m2 =PUP"

with quantum mechanical operators led to the wave equation

t?z~a D,DUgto = ~ s 2

(III.E.IO)

(III.E. 11 )

for a matter field interacting with a gravity four-vector. As usual, Greenberger took the nonrelativistic limit of Eq. (III.E.11) before using the theory to describe a decaying particle at rest.

512 Fanchi

As a follow-up on this work, Greenberger (1988) argued that modern attempts to unify gravity with other fundamental forces were heading in the wrong direction. He favored a different approach with the equivalence principle and the uncertainty principle as its basis. He pointed out, for example, that many lifetime-width uncertainty relations should be viewed as mass-proper time relations.

I I I . F . A l g e b r a i c F o r m u l a t i o n s

Three algebraic treatments of parametrized relativistic quantum mechanics (PRQM) appeared in 1969 and 1970 (Johnson, 1969; Moses, 1969; Aghassi et aL, 1970a). It is easiest to put these algebraic treatments in perspective by first reviewing the simplest algebraic formulation of PRQM (Fanchi, 1979).

All parametrized relativistic quantum theories assume that free-particle wave equations are invariant with respect to the linear Poincar6 transformation

' - ~ ( I I I . F . 1 ) x~ - A~xv + a.

where {A~} represent a homogeneous Lorentz transformation and {a.} represent translations along the {x.} axes. In the parametrized theory with an independent variable s, the theory must be invariant with respect to the linear transformation

s' = s + As (III.F.2)

where As represents translations along the s axis. Except for Greenberger (e.g., Greenberger, 1963), all parainetrized theories assume the s translation is independent of space-time coordinates. The effect of allowing a space- time dependence of the s translation was discussed previously. Equations (III.F.1) and (III.F.2) can be denoted by {a, A, As}.

The set of transformations {a, A, As} is an element of an ll-parameter continuous group Fg with the identity element {0, 1, 0}. Elements of the group obey the binary operation

{a~,A~,AsI}{az, Az, As2}= {al + A~a2, A~Az, As~ + As2} (III.F.3)

which expresses the product of two transformations. The symmetry group Fg is the direct product group of the translation group along s (T~) and the Poincar6 group (Pg); thus

Fg = Ts ® Pg (III.F.4)


For an infinitesimal transformation the set {a,, A~,As} becomes y {~,, g~+ e~, e} where (e,, ~,, e) are first-order infinitesimals. The unitary

operator representing an infinitesimal transformation is

U ( l l ) ~--- 1 ~- i U ( l l ) (III.F.5)

where

K(II) = 1J~Veuv + P%# + Hope (III.F.6)

Ten of the eleven generators of Fg are the six independent elements of the antisymmetric angular momentum tensor ju~ and the four elements of the energy-momentum four-vector P~. The eleventh generator Hop generates infinitesimal translations along s.

The generators of the Poincar6 group satisfy the 45 usual commutation relations

]-p,, pv] = 0

[Juv, P~] = i(Pv g~ -- Pu g~)

[J,.v, J;.,~] = i(g..,~Jv~ + g~;.Jv,~ + gv,~J.;~ + g;.~,Jo~)

(III.F.7)

(III.F.8)

(III.F.9)

The remaining ten commutators are evaluated by taking the commutator of the eleventh generator Hop with each of the ten generators of the Poincar6 group. Since Hop is the generator of the group Ts, which is a subgroup of Fg as in Eq. (III.F.4), the generator Hop must commute with all of the generators of the Poincar6 group:

[Ju,, Hop] = 0 (III.F.10)

[Pu, Hop] = 0 (III.F.11)

Equations (III.F.7) through (III.F.ll) completely define the Lie algebra of Fg. In accordance with Lie's theorem, the commutator of any two generators Gr, Gs of a Lie group is a linear combination of the generators such that

Ear, G,] = Z,c'~,G, (HLF.12)

t where Crs are the structure constants of the group. We note that the structure constants of the Lie algebra of Fg are the same as those of the Lie algebra of the Poincar6 group.

Johnson (1969) and Aghassi et al. (1970a) began with an algebraic formulation such as the one described in Eqs. (III.F.7) through (III.F.11). They wanted to develop a Lie algebra that contained a relativistic position

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operator and a mass operator. For a discussion of position operators in nonparametrized theories, see Broyles (1970). Both Johnson and Aghassi et al. presented an algebraic structure with 15 generators. The additional four generators were obtained by adding space-time position operators that satisfy the commutators

IX ~, X v ] = 0 (III.F.13)

[P~, X ~ ] = ig ~v (III.F.14)

[j~v, X;~] = i[g~VX~ _ g~UXV ] (III.F.15)

The XPM algebra of Johnson (1969, 1971) is defined by Eqs. (III.F.7) through (III.F.9) and (III.F. 13) through (III.F. 15). The relativistic Galilean group of Aghassi et al. (1970a) contains Johnson's XPM algebra, although Aghassi et al. preferred to express Eq. (III.F.14) as

[p , , QV] = _ ig,Vl-1, x ~ - - 1Q" (III.F.16)

where Q is a dimensionless operator and I is a "universal length." Aghassi et al. (1970b) showed that their group could be obtained from the Lie algebra of the inhomogeneous de Sitter group IS0(3 , 2), and they studied the representations of their group [Aghassi et aL, 1971]. For comparison, Moylan (1983) determined the unitary representations of the (4+1)- de Sitter group S0(4 , 1).

Following Appendix C of Aghassi et al. (1970a), the transformation properties of the 15-parameter group include Eq. (III.F.2) and

X'~ = A~xv + a~ + b~s (III.F.17)

Equation (III.F.17) replaces Eq. (III.F.1). The group element is denoted by {a, , b , , A ; , As}, the identity element is {0, 0, 1, 0}, and the space-time position translation is characterized by the four parameters {b,}. The composition law of two group elements is

{a2, b2, A2, As2 } { a,, b,, A ,, As~ }

= {a2 + A2a~ + (As~)b2, b2 + A2b~, A2A~, As~ + zls2} (III.F.18)

The infinitesimal generator is

U(15) = 1 + iK(15) (III.F.19)

where

K~15) = ½JUVe,v + Pt'(gp)# + XU(~x),u + Hopg (III.F.20)


and {ep, G} represent infinitesimal energy-momentum and space-time translations, respectively. At this point the formulations begin to differ.

The choice of the form of the generator of s translations is not unique. Johnson (1969) required

[Hj, X ~ ] = iP ~ (III.F.21)

where

H, = x f ~ . P u (III.F.22)

for a free particle. Garrod (1966, 1968) used a similar Hamiltonian in his classical formulation. By contrast, the corresponding commutator for Aghassi et al. (1970a) was

[-HARs, X u ] = 2iP" (III.V.23)

with

HARS = P~P~ (III.F.24)

Schwinger (1951) used a parametrized wave equation with a Hamiltonian like Eq. (III.F.24), while Cooke (1968), Davidon (1955), Feynman (1950), and Stueckelberg (1942) used HARS/2. In addition to Eqs. (III.F.22) and (III.F.24), Johnson considered the Hamiltonian operator

1 Hp = ~m P" P" (III.F.25)

where m is the rest mass of the free particle. Equation (III.F.25) was introduced by Pearle (1968) in the context of a parametrized classical theory.

Johnson rejected Eq. (III.F.25) because he viewed the appearance of mass in the denominator as a restriction on the generality of the dynamics. Other workers, e.g., Horwitz and Piron (1973) and Fanchi and Collins (1978), have taken a different view. Their work is discussed later. Johnson argued that HARS/2 and Hp were deficient because they did not reduce to the "standard" Hamiltonian Po in the rest frame of the particle. This reasoning presumed the existence of a standard, yet much of the work in parametrized theories has been motivated by the search for a covariant Hamiltonian theory that could become a standard. Lastly, Johnson argued that Eq. (III.F.22) could be used to eliminate certain unphysical representations by requiring that the Hamiltonian be Hermitian and positive-definite. These representations correspond to spacelike (tachyonic) states that may

516 Fanchi

be physical, but their existence has not been established (Fanchi, 1990). By presenting a physical rationale for eliminating these mathematically admissible states, Johnson had to explain why exact localization was not possible for a physical state. Furthermore, the positive-definite requirement applied to the Hamiltonian conflicts with the allowed existence of negative energy states in the rest frame of a particle.

Given Johnson's Hamiltonian Hj, he obtained

i~s ~j(x, s )= HjOs(x, s) (III.F.26)

as the parametrized wave equation for free, spin-0 particles. Similar wave equations may be written for the other Hamiltonians. For example, Aghassi et al. obtained

0 1 i t = - i (III.V.27)

for a free particle. Notice the appearance of their universal length l. Aghassi et al. view Eq. (III.F.27) as "the analog of the nonrelativistic SchrSdinger equation" (p. 2759).

Johnson interpreted the parameter s as "the expectation value of the time interval on a clock at rest with respect to the system" (p. 1760). He adopted a scalar product in the four-momentum representation with the form

(~J I¢, > = f q;~k, s) Cj(k, s) d4k (III.F.28)

Equation (III.F.28) corresponds to the Stueckelberg normalization. Johnson defined the probability of observing a particle in a finite space- time volume VST during a measurement performed at a given event s as

PVsT(S) = fVsT ~lj~y, S) @j(y, S) 642 (III.F.29)

The probability of finding a particle at space-time location y for any value of s required that the ds integration be performed after computing probabilities. Johnson noted that reasonable results were not obtained if the integration over ds was performed on the amplitudes rather than the probabilities. He discussed an uncertainty principle between invariant mass and s of the form

Am As >i. h/2 (III.F.30)


Equations (III.F.26) through (III.F.28) apply to massive, spin-0 particles. Johnson applied his formalism to massless particles in anticipation of treating photons. His attempts to describe massive, spin-l/2 particles led to an ambiguity in the definition of the Hamiltonian. In particular, he found that

H1/2 = Y~P~' (III.F.31)

has the same commutator as Eq. (III.F.22) when taken with the Poincar6 group operators J ~ and PC. He argued that Eq. (III.F.22) worked better with other physical observables such as position and spin than did Eq. (III.F.31). Johnson was unable to use Eq. (III.F.22) to directly connect his formulation with the standard Dirac theory of spin-l/2 particles. In later work, Johnson (1971) found a basis in which his difficulties with Eq. (III.F.31) disappeared. Johnson and Chang (1974) used a transformation like the Foldy-Wouthuysen transformation to solve the problem of a Dirac particle in an external electromagnetic field for a Hamiltonian with the form of Eq. (III.F.31). They noted that Eq. (III.F.22) was difficult to transform because of the asymmetrical treatment of its zeroth component.

Relationships between the Hamiltonian in Eq. (III.F.3 I) and an evolution parameter have been studied by Corben (1961, 1968), Wong (1972), Ellis (1981), Lopez and Perez (1981), Herdegen (1982), Barut and Thacker (1985), and Evans (1989, 1990). Corben compared helical solutions of classical equations for a free particle with spin to Zitterbewegung. He defined an evolution parameter derivative of an operator g2 as i - 1 times the commutator of f2 with the Hamiltonian H1/2. Wong (1972) added a mass term to the right-hand side and studied the equation within the context of general relativity. Ellis (1981) sought to extend Corben's (1961) treatment of Eq. (III.F.3t) as the generator of parameter translations. Lopez and Perez (1981) showed that Eq. (III.F.31) was consistent with the probabilistic interpretation of the relativistic wave function proposed by Collins and Fanchi (1978). We have discussed Barut and Thacker's work previously. Evans (1989, 1990) sought to develop a parameter-dependent wave equation that did not include mass. A more detailed treatment of Eq. (III.F.31) was given by Herdegen (1982).

Herdegen (1982) studied both spin-0 and spin-l/2 particles and their classical correspondence. He independently derived many of Johnson's results. Unlike Johnson, Herdegen developed a Hilbert space formalism including a dynamical postulate requiring that quantum states belong to a Hilbert space with a unit norm. Herdegen did not view the evolution parameter s as an observable. Herdegen presented an explicit example of a Foldy-Wouthuysen type of transformation that preserved translation

518 Fanchi

invariance. He found that a free particle exhibited Z i n e r b e w e g u n g only when positive and negative frequency states mixed.

I have dwelled primarily on the treatments presented by Johnson (1969) and Aghassi et aL (1970a) because their work is rich in physical concepts that are directly related to PRQT. Moses (1969) focused on the mathematics of an infinite-component relativistic wavefunction. He .defined a mass operator as

HM = I ( M + M + ) (IlI.V.32)

where

M = { [P~ - eA ~] [P= - eA=] + LM }1/2 (III.F.33)

and L M w a s a poorly defined Hermitian operator. Referring to a classical correspondence in terms of Poisson brackets, Moses constructed a wave equation in the Schr6dinger picture similar to Eq. (III.F.26) but with Hj replaced by HM. Moses' algebra, particularly his definition of the mass operator, led to a commutator that depended on LM. The operator LM appeared to affect the way the relativistic wavefunctions change in proper time, but the physical interpretation of LM was not pursued. Aghassi et al.

(1970b) later related their mass operator with the evolution operator of translations in proper time.

Equation (III.F.14) was used by several researchers, including Falk (1952), Castell (1967), Johnson (1969), Moses (1969), and Aghassi et al.

(1970a), to achieve the transition from classical to quantum mechanics. In the classical theory, the commutator [-X ~, P~] vanishes. Noga (1970) challenged the validity of parametrized algebraic formulations by stating that they led to "completely unphysical results" (p. 310). Noga was concerned about the existence of spacelike states (negative m2), the continuous character of the mass spectrum, and the existence of only infinite spin representations. Johnson (1971) responded that the spin problem was associated with the requirement proposed by Aghassi et al. (1970a) that homogeneous Lorentz transformations be unitary. Johnson's work included both unitary and nonunitary homogeneous Lorentz transformations, thereby allowing finite and infinite spin representations. Neither Johnson (1971) nor Aghassi et al. (1971) were able to counter Noga's con- cerns about spacelike states; however the possible existence of tachyons is a little more palatable today than it was 20 years ago, in part because of possible observations in particle scattering experiments (Chodos, Hauser, and Kostelecky, 1985; Wall, 1989) and superluminal motion associated with some astronomical objects like the jet of particles emitted by quasar


3C273 (Davis et al., 1991 and references therein). For a succinct review of the tachyon literature and a discussion of tachyon kinematics in the context of parametrized quantum theory, see Fanchi (1990).

III.G. Covariant Hamiitonian Dynamics

Pearle (1968) first introduced Eq. (III.F.25) in the context of a parametrized classical theory. He sought a Hamiltonian that was a function of Lorentz invariant scalar products of four-vectors and would yield the usual results of Galilean invariant classical mechanics in the nonrelativistic limit. He obtained the classical form of Eq. (III.F.25) "by replacing the rotationally invariant scalar product of three-vectors with Lorentz invariant scalar products of four-vectors" in the Hamiltonian of the Galilean invariant theory. Pearle did not believe his classical theory could be the basis for a realistic quantum theory because particle masses were not conserved quantities. He was concerned that a quantum theory based on his classical mechanics would describe a universe in which a continuum of masses, including imaginary masses, exists. Other researchers have developed a quantum theory based on Eq. (III.F.25).

Arunasalam (1970) argued from analogy with classical mechanics that a Hamiltonian like Eq. (III.F.25) should be used in the parametrized, relativistic quantum description of spin-0 and spin-l/2 particles. His parametrized wave equation for a spin-l/2 particle interacting with a four-vector potential was

0 1 i~stpA=~m {[7u(P~-eA~)][Tv(PV-eA~)]}OA-HAtpA (III.G.1)

For a mass eigenstate, Arunasalam showed that Eq. (III.G.1) reduced to the usual Dirac theory.

Relativistic Dynamics. A more comprehensive study was undertaken by Horwitz and Piron (1973). They formulated a classical dynamical principle that was a generalization of the Hamilton principle. Suppose an N-body system is characterized by the differential 1-form

N

co = ~, [p,~, d q ] - K(pi, q~) ds] (III.G.2) i = 1

where K is a function of the 8N variables (Pi, qi) of each particle, i= 1 ..... N. The dynamical principle may be stated as follows (Reuse, 1979): Given a closed curve C in

F = {Pi, qi, s} ([I[.Ol3)

825/23/3-12

520 Fanchi

the integral ~cCO is invariant for any (continuous) deformations of C generated by arbitrary displacements of its points along the trajectories corresponding to the evolution of the system. This principle is equivalent to the canonical equations

dq____~_~ = O K (III.G.4)

ds Opi.

dp , , O K (III.G.5)

as Oqf/

where s denotes an evolution parameter. Horwitz and Piron (1973) called s historical time and interpreted s as a parameter analogous to time in nonrelativistic Galilean invariant mechanics.

Equations (III.G.4) and (III.G.5) are covariant when the differential 1-form is invariant. Covariance is satisfied when K is a scalar with respect to the Lorentz group and the parameter s is invariant. Horwitz and Piron proposed expressions for K for a free particle, a particle in an electromagnetic field, and the relativistic 2-body problem. As an example, K for a particle of rest mass m in an external electromagnetic field is

1 K = ~ m [ p ~ - e A ~ ] [ p " - e A ~ ] (III.G.6)

Steeb and Miller (1982) presented a more detailed analysis of the classical theory in terms of differential forms, vector fields, and the Lie derivative. Their work included a brief derivation of constants of motion. We note that the form of K is not necessarily unique. Salisbury and Pollot (1989) adopted a free particle Hamiltonian of the form Ko - (m/2), where Ko is the free particle form of K. They introduced a potential term to their modified Hamiltonian and used it to solve the relativistic harmonic oscillator problem.

Horwitz and Piron transformed to a quantum mechanical system by making the usual identifications

a q~ = x ~', pU = - i = - ia u (III.G.7)

0x~

which satisfy the commutation relations

v i[p~,, qV] = 3 ~ (III.G.8)

The commutator in Eq. (III.G.8) corresponds to a Poisson bracket in Horwitz and Piron's relativistic classical mechanics. Equation (III.G.8) has


been used as the basis for discussing an uncertainty relation such as Eq. (III.F.30) (Arshansky and Horwitz, 1985) and causal propagation (Horwitz and Usher, 1991). The subsequent wave equation is

i ~ O.p(x, s)= KOnp(x, s) (III.G.9)

where K is a self-adjoint operator. The states of the system are described in the Hilbert space L2(R 4, d3xdt) with the Lorentz invariant scalar product

(~HP, ~HP) = f ~IP( x, S) ~Hp(X, S) d4x (III.G.IO)

Solutions of the two-body bound-state problem for spin-0 particles were presented by Arshansky and Horwitz (1989a, b). They calculated the mass spectra associated with a harmonic oscillator potential, a relativistic square well, and Coulomb-like binding. Solving the harmonic oscillator problem by the method of Arshansky and Horwitz (1989a) eliminates the subsidiary condition needed to suppress time excitations used by other authors (e.g., Feynman, Kislinger, and Ravndal, 1971; Blaha, 1975; Kim and Noz, 1986).

The basic formalism of Horwitz and Piton (1973) is very similar to Stueckelberg's (1942). Horwitz and Piron use a similar wave equation and normalization condition. Unlike Stueckelberg, Horwitz and Piron explicitly include m in the function K in a manner that is essentially the same as Pearle's (1968) and Arunasalam's (1970) treatment. Horwitz and Piron also attribute a physical interpretation to s rather than treating it solely as a mathematical parameter. Horwitz and Rabin (1976) noted that Feynman's path integral formulation of the parametrized wave equation is suggestive of diffraction-type experiments. They discussed double-slit experiments as possible tests of the theory. Their tests do not appear to be within physically accessible limits. Technological improvements could change this situation.

Arshansky, Horwitz, and Lavie (1983) argued that physically measurable currents require integrating over the evolution parameter, which is essentially the same idea as Feynman's requirement that s be mathematically removed from the formalism by integration. This integration in Arshansky et al.'s treatment is called "concatenation." A particle is viewed as a concatenation of events along a world line. Events are viewed as the basic dynamical entity, whereas particles are associated with the entire world line. By focusing on events, the Horwitz-Piron approach can

522 Fanchi

be thought of as a theory of events rather than a theory of particles. The concatenation concept was used by Horwitz (1984) to describe radiative interactions of a two-body system.

The relationship between coordinate time and the evolution parameter within the context of relativistic dynamics was discussed by Horwitz, Arshansky, and Elitzur (1988). Their view is essentially the same as Fanchi's (1986, 1987), although Horwitz et al. view the evolution parameter as an unobservable quantity. Rather than integrate over s a la Feynm~tn and Horwitz et al., Fanchi assumes s is physically observable and seeks experimental tests to verify or refute the assumption (e.g., Fanchi, 1986, 1990).

Horwitz, Piron, and Reuse (1975) later extended the spin-0 formalism to include spin-l/2 particles. In this paper, they observed that the formalism of Horwitz and Piron (1973) was invariant with respect to a group of transformations that is equivalent to the group proposed by Moses (1969), Johnson (1969), and Aghassi et al. (1970). Horwitz et al. (1975) did not include the s translation group in their discussion. They noted that the irreducibility condition in Horwitz and Piron's work (1973) had to be abandoned so that a superselection rule for a spin-l/2 particle could be postulated. The superselection rule is a 4-vector n ° that indexes a family of Hilbert spaces H n. The spin-l/2 particle is represented by a two-component eigenfunction that satisfies the field equation

i -~s = K(1 /2) ~ (III.G.11 )

The operator K(1/2 ) is a 2 × 2 matrix that includes spin effects through four 2 x 2 matrices (W,) associated with the timelike unit four-vector n °. Letting L(n ) denote a boost, i.e.,

L(n )~ n~o = n ° (III.G.12) with

the definition of W~ is

where

n~= (1, 0, 0, 0) (III.G.13)

o - o v (III.G.14) W n = L(n)v Wno

W~0 = (0, ½6p) (III.G.15)

and {6p} are the Pauli matrices. The scalar product on the Hilbert space C 2 × L2(R 4, d4x) o f two-component wave functions is

2

(~U, q~) -- I gt+q~ d4x = Z f g t * ~ i d 4 x (III.G.16) i = 1


More discussions on the spin-l/2 particle formalism are given by Piron and Reuse (1978) and Horwitz and Arshansky (1982). The latter paper discusses the transformation properties of the spin-l/2 eigenfunction with respect to discrete CPT symmetries.

Reuse (1978) applied the spin-l/2 formalism to the hydrogen atom for a matrix operator KR with the form

1 K R = ~ m ( -- ion, -- e A . ) ( -- iO u -- e A u) -- g~m It° ( - iou - e A " ) Fuv W~

g)2 2 ~ 2 # O F nVF~n p (III.G.17)

where

F~v 1o ~p;o (III.G.18)

The tensor e,,,p~ is the Levi-Civita tensor, and the constants m and #o denote the observable rest mass and the Bohr magneton, respectively. The constants (g~, g2, g3) are dimensionless phenomenological constants. For a concise overview of this work and a comparison with experimental data, see Reuse (1979). Reuse extended the formalism to hydrogen-like and positronium-like systems without spin (1980a) and with spin (1980b).

The formalism of Horwitz, Piron, and Reuse has been applied to the study of relativistic two-body scattering (Horwitz and Lavie, 1982; Arshansky and Horwitz, 1988; Arshansky and Horwitz, 1989c), three-body scattering (Alt and Hannemann, 1986), classical and quantum statistical mechanics (Horwitz, Schieve, and Piron, 1981; Miller and Suhonen, 1982), including a relativistic Boltzmann equation (Horwitz, Shashoua, and Schieve, 1989) and chaos (Schieve and Horwitz, 1991), a model of atomic and molecular systems (Grelland, 1981), and electromagnetic interactions (Horwitz, 1984). The nonrelativistic limits of an interacting two-body problem and Eq. (IILG.9) were demonstrated by Horwitz and Rotbart (1981). Francisco (1986) studied the evolution of parametrized coherent states in the neighborhood of a strong gravitational field. The mass spectrum of a charged particle moving in a constant magnetic field has been calculated by Arensburg and Horwitz (1991).

Horwitz and Lavie's (1982) scattering theory incorporated a wave operator whose existence was proved by Horwitz and Softer (1980). Horwitz and Lavie defined a three-dimensional cross-section similar to that of Cooke (1968) and Cook (1972). This cross-section is shown to reduce to

524 Fanchi

the more familiar two-dimensional spatial cross-section under certain conditions. Horwitz and Lavie compared their scattering theory with standard relativistic quantum field theory as well as with less well-known theories. They studied electromagnetic interactions and derived the lowest-order Rutherford cross-section.

I note for comparison that an alternative parametrized scattering theory was proposed by Droz-Vincent within the context of a multi- parameter formalism (Droz-Vincent, 1979, 1982a, 1987). Multi-parameter formalisms were suggested as early as 1950 by Feynman (1950) for treating N-body systems. Within the context of parametrized theories, multi- parameter formalisms have been reintroduced by several authors including Davidon (1955b), Enatsu (1956), Cooke (1968), Droz-Vincent (1979), Horwitz and Rohrlich (1981 ), and Fanchi and Wilson (1983).

Droz-Vincent (1980) defined operators in a square integrable space L2(R 4N) for an N-body system. He proposed scattering operators that were extensions of operators defined on off-the-mass-shell states (a state which is not an eigenstate of a mass operator such as Hop =p~p~). Droz-Vincent (1982b, 1984) studied the scattering of two directly interacting particles. He found that the relativistic problem reduced under certain conditions to a nonrelativistic problem. The mass shell of the relativistic problem corresponded to the kinetic energy shell of the reduced problem.

Droz-Vincent's 1980 theory prohibited defining a probabilistic interpretation of the wavefunction in terms of the scalar product in LZ(R4N). A few years later Droz-Vincent (1988) accepted Eq. (III.G.10) and a probabilistic interpretation as appropriate concepts for his parametrized theory. He related the evolution parameter to the evolution of noninter- seeting spacelike surfaces.

In an attempt to reformulate electromagnetism, Horwitz, Arshansky, and Elitzur (1988) found it necessary to introduce the idea of a "pre-Maxwell field." Pre-Maxwell fields are parameter dependent. Four-vector potentials are obtained by integrating pre-Maxwell fields with respect to s:

f o o

A ~ = a ~ d s (III.G.19) o o

Horwitz e t aL (1988) argue that the parameter-dependent pre-Maxwell fields may have an objective dynamical reality, but our perceptions are based on integrations over s--the concatenation idea. Building on parametrized field theories of Fanchi (1979), Hostler (1980, 1981), and Kubo (1985), a Lagrangian density for the matter field and pre-Maxwell gauge fields was constructed and studied by Saad, Horwitz, and Arshansky (1989).


Saad et al.'s (1989) Lagrangian density is

l f l •

+ ~* ( - i ~-~s-ea4) ~ } + h.c. + ~ f=~f=~

/.z = O, 1, 2, 3, ~,fl=0,1,2,3,4 (III.G.20)

where {a~} are pre-Maxwell fields. The index 4 refers to the parameter s. The tensor f ~ is defined by

f~p = 0~ap - ~a~ (III.G.21)

leading to the equation

Oef~=2 j~=_ej= (III.G.22)

where the current is given by

i j~= ---[q/*(O~-i~a~)tp-((O~+iea~)tp*)~9] (III.G.23) 2m

Equation (III.G.23) shows how the usual electric charge e is related to the coupling constants 2 and e. Saad et aL (1989) were able to construct Maxwell's theory of electrodynamics as a subset of their formalism. Preliminary considerations for second-quantizing the theory were presented. Green's functions for the field strength tensor f ~ in Eq. (III.G.21) were derived by Land and Horwitz (1991). They showed how to reproduce the propagator approach of Schwinger (1951) and DeWitt (1975), and studied the causal properties of the propagators.

It is interesting to compare Eq. (III.G.20) with the Lagrangian density introduced by Kubo (1985):

1 1 Z/ K= -- ~ Fuv(x, s) FUr(x, s) - ~ [ ~AU(x, s)] 2

| OA~(x, s) ~A~(x, s) - ~- - ¢'(x, s) ~ ( x , s) 2 Os Os

1 I O¢(x,s) Oq'(x,s) 1 +~ ~(x,s) as as tp(x,s) +j~(x,s)A~(x,s) (III.G.24)

526 Fanchi

Unlike Saad et al. (1989), Kubo lets the four-vector potential be a function of s. Mass does not appear explicitly in Kubo's Lagrangian density or in the corresponding field equations

j~(x, s)= ig "~(x, s) y,~b(x, s): (III.G.25)

[TuO~-i ~-~s]tP(x,s)=ig7~O(x,s)A~(x,s) (III.G.26)

O~O" +-~s 2 A~(x, s)= -j~(x, s) (III.G.27)

Equations (III.G.25) and (III.G.27) apply to matter fields, while Eq. (III.G.26) applies to photons. Notice that Eqs. (III.G.25) and (III.G.27) are extensions of Dirac's equation. We have seen this equation used by several researchers (e.g., Corben, 1961; Johnson, 1970; Lopez and Perez, 1981; Ellis, 1981; Herdegen, 1982; and Evans, 1989). Kubo (1985) reproduced the formalism of quantum electrodynamics from Eq. (III.G.24).

Constraint Hamiltonian Dynamics. Rohrlich (1979) recognized Dirac (1949) as the first person to suggest a Hamiltonian approach to the description of a relativistic system. During the first half of the 1960's Currie et al. (1963) and Leutwyler (1965) showed that standard formulations of relativistic Hamiltonian dynamics were unable to support interactions between particles. These "no-interaction" theorems implied that the usual transformation properties of Hamiltonian dynamics (theories based only on 10 Poincar6 generators) would have to be modified or extended if a nontrivial covariant Hamiltonian dynamics was to be developed. Garrod (1968) and Pearle (1968) avoided the "no-interaction" theorems by working with a classical Hamiltonian dynamics based on Eqs. (III.G.4) and (III.G.5). Pearle's work forms a classical basis for Horwitz and Piron's quantum formalism. Another approach that sought to avoid "no-interaction" theorems is called constraint Hamiltonian dynamics (CHD).

Many of the concepts of CHD touch parametrized theories but are not central to them. An introduction to the subject can be found in Sundermeyer (1982) and a collection of papers edited by Llosa (1982). Sundermeyer presented techniques for investigating singular systems and then applied the techniques to specific examples. He brought together information from several different reviews and the research literature. Llosa (1982) compiled a series of articles from a workshop attended by researchers with interest in relativistic action-at-a-distance theories. Among the articles was Ph, Droz-Vincent's review of his multi-time (multi- parameter) formalism and a discussion of second quantization of a system


of interacting particles. The CHD treatment of N-body systems was examined by Rohrlich (1982a) and Todorov (1982). Additional references can be found in Rohrlich (1979), King and Rohrlich (1980), Marnelius (1982), and Iranzo et al. (1983).

CHD was devised as an attempt to express the concepts of special relativity in a Hamiltonian dynamical form while avoiding the "no-interaction" theorems. Rohrlich (1982b) discussed explanations of how CHD avoids the "no-interaction" theorems. The description of an N-body system is presented in an 8N-dimensional phase space consisting of N space- time four-vectors and N energy-momentum four-vectors. Two types of constraints may be applied to this system. First-class constraints commute with all other constraints, while second-class constraints do not commute with at least one other constraint. Classical CHD specifies N first-class constraints that usually serve as a mass-shell condition by relating p~pp of each particle to the interaction of the particle with all other particles. Other constraints are needed to parametrize the motion of a trajectory in phase space. This is the point where CHD connects with parametrized theories.

Horwitz and Rohrlich (1981) presented a quantized version of CHD and derived a classical limit in agreement with classical CHD. They developed a multiple-parameter formalism for an N-body system and constructed a scattering matrix. Further work on the scattering theory was presented by Horwitz and Rohrlich (1982). Horwitz and Lavie (1982) showed that quantum CHD was an approximation of their single-parameter scattering theory.

Horwitz and Rohrlich's (1985) efforts in the area of CHD eventually led to the discovery that relativistic quantum mechanical constraint dynamics suffers from severe limitations. They showed that two-body interactions were restricted to the elastic case in which interacting particle energies must be conserved in the center-of-mass frame.

Droz-Vincent has pointed out (private communication, 1991) that the significance of the work by Horwitz and Rohrlich (1985) on the limitations of CHD depends on the assumption that the same interaction term exists in both mass-shell constraints. This assumption, although widely applied because of its simplicity, is not the most general approach permitted by first class constraint formalism. Thus, efforts to develop CHD have continued. For example, Samuel (1982a, b) did not assume that mass-shell constraints are first class. This gave him greater freedom in choosing interaction potentials. Crater and van Alstine (1987) studied two-body spin-l/2 particles within the constraints discovered by Horwitz and Rohrlich (1985). Sazdjian (1987) showed the formal equivalence between the two-body problem in CHD and the Bethe-Salpeter (1951) equation. Droz-Vincent (1990) used CHD and the strong compatibility condition [Ha, Hb] = 0

528 Fanchi

between the Hamiltonian operators of particles a and b to study the relativistic Zeeman effect. Plyushchay (1990) developed a relativistic model--including first-class constraints--of a massive spinning particle. Much more CHD work exists, but efforts to link CHD with parametrized theories have been discouraged by the findings of Horwitz and Rohrtich (1985).

III.H. Probabilistic Formalisms

Four-Space Formalism. Nonrelativistic quantum mechanics is based on Born's probability density

pB= ~O*(Y, t) ~kB(Y, t)>~0 (III.H.1)

and the associated normalization condition

l i p 8 d3x = 1 (III.H.2)

where R is the spatial region of interest. Many problems of non- paramettized relativistic quantum theories can be traced (Fanchi, 1981a) to retaining an integral over 3-space as in Eq. (III.H.2) and rejecting Eq. (III.H. t ).

The four-space formalism (FSF) takes the approach that Eq. (III.H.1) is correct and the 3-space normalization needs to be changed (Fanchi and Collins, 1978; Collins and Fanchi, 1978; Fanchi and Wilson, 1983). The question of what to use in place of Eq. (III.H.2) is answered by requiting a mathematical equivalence between space and time coordinates. This suggests that the probability density p should be a joint distribution in the space and time coordinates, though p may be conditioned by some invariant parameters. If it is assumed that p is conditioned by a single invariant scalar s, the normalization condition is

foP(X I S) d4x = 1 (III.H.3)

where D is the four-volume of interest. For p to be differentiable and non-negative (Collins, 1977a), its derivative must satisfy Op/Os = 0 if p = 0. This leads to Born's relation

p(xls) = gt*(x, s) 7t(x, s) ~> 0 (III.H.4)

The name "four-space formalism" is based on the observation that p is a joint probability distribution over four space-time coordinates rather than


the original three of nonrelativistic quantum theory. Also note that the integration in D includes contributions from space-time regions outside of the light cone. The light cone constraint enters the theory as a "macro- scopic causality" principle. The light cone is a constraint on the most probable trajectory of a free particle; it does not apply to the allowed region of integration.

If p is differentiable, differentiation of Eq. (III.H.3) with respect to s gives

fD Op(xls) = (III.H.5) d4x 0 ~s

Equation (II1.H.5) is assured by requiring that p satisfy the continuity equation

0p 0 ~s+~--~x~(PV")=0 (III.H.6)

where V" is a four-vector and p vanishes at spatial or temporal infinity. From probability theory, the expectation value of a quantity Q is

(Q ) = fD Qp(x]s) d4x (III.H.7)

Setting Q equal to the space-time four-vector, differentiating Eq. (III.H.7) with respect to s, and using Eq. (III.H.6) yields

d(x ~' ) p V ~ d4x (III.H.8)

ds JD

Thus, d(x")/ds is the expectation value of V ", hence V" is identified as proper four-velocity. Equation (III.H.8) is the FSF basis of the Stueckelberg-Feynman interpretation.

Given the above probabilistic basis, it is possible to derive a probabilistic formalism which has the attributes of quantum mechanics. The derivation has been performed for single spinless particle systems in nonrelativistic (Collins, 1977a, b, 1979; Gilson, 1968) and relativistic (Fanchi and Collins, 1978; Collins and Fanchi, 1978) motion. Attempts to link the formalism to a particular probabilistic model, namely the Markov process, have been made (e.g., Gilson, 1968; Guerra and Ruggiero, 1978), but are not necessary. An FSF for single particles characterized by internal variables such as spin has been developed (Fanchi, 1981b) and extended to many-body systems (Fanchi and Wilson, 1983). Since the fundamental

530 Fanchi

FSF concepts are exhibited by the single spinless particle system, it is sufficient here to consider the equation

where

O*F= F*O (III.H.9)

F = i8 3 + ~m 2 OX" Ox~

e2e3 I ~O ~ 1 + iSm-~-- A~-~XU +-~x~ (A"qJ) (III.H.10)

The four-vector A ~ is related to the proper four-velocity V" by

[ 0¢ + e2 Au] (III.H.11) V" ~m t_el 0X---~

and the scalar function ¢ represents the phase of the decomposition of the eigenfunction:

tp(x, s) = [p(x l s) ] 1/2 exp[i¢(x, s)] (III.H.12)

Equation (III.H.9) follows from Born's relation, the continuity equation, and the Helmholtz decomposition of V ". It asserts that O*F or F*O is real.

Field equations for a physical system are obtained by specifying the c-numbers ~,~, el, e2, ~3 and by recognizing that Eq. (III.H.9) may be linearly decoupled by writing

F = V'O (III.H.13)

where V' must be Hermitian. Field equations suitable for use in either a quantum mechanical formalism or a field theoretic formalism (Fanchi, 1979) must have the form of Eq. (III.H.13). To obtain the field equation for a spinless particle interacting with an electromagnetic field, we identify ~1 and e3 as Planck's constant and make the identifications

e 2 e2= - e , e,~= 1/m, V' = ~mA"A~ + V (III.H.14)

to get

i ~--~ = 2m + V~, ~ = i--Ox, - eA" (III.H.15)


where V is a Hermitian operator representing as yet unspecified interactions. The expectation value of an observable is defined as

( g2 ) -- fD tp*g2t) d4x (III.H.16)

For comparison, Vatsya (1989) derived Eq. (III.H.15) without the potential V by treating particle momentum as a gauge potential. Other derivations within the context of stochastic theories are discussed below.

Equations (III.H.9) through (III.H.15) arise as a result of the procedure used to derive the wave equation (III.H.15). Wave equations satisfying the probabilistic concepts of the FSF--the normalization condition, Born's relation, and the continuity equation--may be derived by other means. Lopez and Perez (1981) showed that the FSF interpretation is consistent with an equation like Eq. (III.G.26) for particles with spin. [Note the sign error in their. Eq. (9).] Lopez and Perez's derivation differs substantially from that presented by Fanchi (1981b). They avoided introducing mass in the wave equation, but their treatment of the electromagnetic interaction included a four-vector current containing a mass parameter. By contrast, Evans (1989, 1990) derived Lopez and Perez's wave equation but abandoned Eq. (IILH.4). His resulting probability density is interpreted as a charge density and is not positive definite.

The parameter s is viewed as a physically measurable quantity (Fanchi, 1986) that may be connected to entropy as an "arrow of time" by way of a relativistic Boltzmann's H theorem (Fanchi, 1987). A parametrized classical cosmology (Fanchi, 1988) was developed as a pre-requisite for applying a result of the parametrized statistical mechanics developed by Horwitz, Schieve, and Piron (198l) to Big Bang models. An attempt to quantize gravity led Pavsic (1984, 1985) to parametrized theories.

Pavsic sought an independent interpretation of the evolution parameter. He argued (1986) that the part of the wavefunction that describes the external world collapses relative to a "stream of consciousness" which he likens to the memory sequence of an automaton. In Pavsic's view (1991a), the evolution parameter marks an observer's subjective experience of "now" as it evolves in spacetime. The observer perceives the world line one event at a time as his experience of "now" evolves. To Pavsic, this localization of the observer's perception corresponds to restricting the wave packet to a finite region of space-time. In addition to philosophy, Pavsic presented a phase space form of an action with a Lagrange multiplier (Pavsic, 1987). He combined the action with a variational principle to derive parametrized equations of motion without the mass-shell constraint (Pavsic, 1991a) in classical mechanics, and both first and second quantized formulations

532 Fanchi

(Pavsic, 1991b). Although initially Pavsic worked only with a Dirac-like equation (Pavsic, 1984, 1985, 1986), he eventually included a Stueckelberg equation (Pavsic, 1991a, b) in his variational formulation.

The FSF is broad enough to encompass specific applications of other parametrized theories. For example, direct links were provided by Fanchi and Wilson (1983) to parametrized treatments of relativistic two-body scattering (Horwitz and Lavie, 1982), relativistic harmonic oscillator models (Feynman, Kislinger, and Ravndal, 1971; Cook, 1972a, b; Kim and Noz, 1986; Arshansky and Horwitz, 1989a), coulomb potential models (Enatsu and Kawaguchi, 1975), and wave equations without explicit mass parameters (Enatsu, 1963; Hostler, 1980).

The FSF has been applied to particle scattering from electrostatic barriers for spin-0 (Fanchi and Collins, 1978; Thaller, 1981) and spin-l/2 (Fanchi, 1981c) particles. This application takes advantage of the Stueckelberg-Feynman interpretation and avoids reflection and transmission coefficients that exceed unity. Thus, the FSF is not plagued by the Klein paradox (Klein, 1929). Damour (1986) applied Stueckelberg's concept of pair creation to study the Klein paradox associated with particle scattering from a strong gravitational field. Some authors, such as Manogue (1988), use the term "superradiance" to describe the situation when the reflected current exceeds the transmitted current. Evans (1991) uses a different interpretation with a parametrized wave equation to study the Klein paradox. His approach results in reflection and transmission coefficients that exceed unity as in the standard theory (e.g., Bjorken and Drell, 1964; Wergeland, 1983). Manogue (1988) argues that superradiance is a physical phenomenon. Each of the approaches is consistent within its set of assumptions. Experimental guidance is needed to determine the correct interpretation.

A parametrized formulation of relativistic quantum field theory (Fanchi, 1979) includes nonparametrized Lagrangian quantum field theory as a special case. It contains a mass conservation law (Fanchi, 1979; Kyprianidis, 1987) that has kinematical implications (Fanchi, 1990) for particles undergoing mass-state transitions. Letting particles interact via parameter-dependent potentials leads to mass-state transitions. Under appropriate kinematical conditions, quantum transitions across the light cone are theoretically possible. These transitions provide a mechanism for tachyon creation and annihilation as a consequence of tardyon interactions. Experimental tests of the theory have been proposed (Fanchi, 1990).

Hostler. Hostler (1980) presented a formulation that adopted most of the basic concepts of Horwitz and Piron (1973) and Fanchi and Collins (1978). In particular, Hostler accepted the commutator equation (III.G.8),


the normalization condition (III.H.3), the probability density (III.H.4), and the continuity equation (III.H.6). Hostler's work differed primarily by removing mass from the classical equations, e.g., Eqs. (III.A.9) and (III.A.t0). Following Feynman's (1950, 1951) discussion of Fock's (1937) work, Hostler preferred wave equations that were independent of mass, such as Eqs. (III.B.6) and (III.B.7). Hostler's formulation was linked to the FSF by Fanchi and Wilson (1983).

Hostler developed quantum field theory using an action principle derivation. He described fermions with an equation like Eq. (III.B.7). Bosons were described by iterating the free particle form of Eq. (III.B.7) to get

1 2 g2 [ ( - ~ 3 ~ ) +~s2] ~FHos=0 (III.H.17)

Hostler (1981) used Eq. (III.H.17) as a model of photons and considered the interaction of fermions with photons. His formalism was extended to include spin-l/2 particles (Hostler, 1988). The wave equation is

H.(l+itr).H+7~ @no~=0, H~=p,-eA~ (III.H.18)

where the eigenfunction is a 2 x 1 Pauli spinor rather than a 4 x 1 Dirac spinor of Fock's (1937) original formulation.

Stochastic Formalism. The next line of development of parametrized theories continues the debate about the interpretation of quantum mechanics. Born's probabilistic interpretation of the wavefunction has not always been accepted, even in the nonrelativistic theory. Alternative interpretations have been sought, ranging from hidden variable theories to specific probabilistic models. One of the probabilistic models is the Markov process. We can understand the essence of the Markov process by comparing it with the simpler but related random walk process.

In the random walk process, a particle undergoes a sequence of displacements. The magnitude and direction of each displacement is independent of all preceding displacements. By contrast, if we let the displacement of the particle depend on the outcome of the immediately preceding displacement, we have a Markov process. A link between Markov processes and quantum theory was made when Nelson (1966, 1985) showed that the Schr6dinger equation could be formally equated to a Markov process. Although I focus on the most common stochastic method, i.e., the Markov process, we should observe that not all stochastic methods are based on Markov processes, e.g., Serva (1988), and Burdet and Perrin (1990) rely on

534 Fanchi

Bernstein processes. Nelson (1985) pointed out that the fundamental idea of stochastic methods is the background field hypothesis, i.e,, a diffusion process is caused by the interaction of a system with a background field.

Several authors (Haba and Lukierski, 1977; Guerra and Ruggiero, 1978; Vigier, 1979; Cufaro-Petroni and Vigier, 1979; and Namsrai, 1981) extended Nelson's 1966 analysis to derive a parametrized Klein-Gordon equation like Eq. (III.G.9). This approach is referred to as the stochastic interpretation of quantum mechanics (SIQM). An alternative treatment based on stochastic concepts is the stochastic quantization method of Parisi and Wu (1981). It is briefly discussed at the end of this section.

The evolution parameter in SIQM is identified as proper time and is independent of space-time coordinates. Although Eq. (III.G.9) was known to SIQM researchers, it was usually reduced to a mass eigenstate (Klein-Gordon) form. Kyprianidis and Sardelis (1984) showed that the Guerra-Ruggiero-Vigier derivation depended on averaging over the stochastic processes in both spacelike and timelike regions and developed an H-theorem. A mass-independent wave equation

1 ~ ~'~S ~ K(X, S) = ~ 8 ~O~'~ K(X, S ) (III.H.19)

was derived by applying a relativistic stochastic variational principle to the average stochastic action of a relativistic Markov process (Cufaro-Petroni et al., 1985; and Guerra and Mana, 1983). Cufaro-Petroni et al. (1985) interpreted the Hamiltonian on the right-hand side of Eq. (III.H.19) as a mass operator conjugate to the mass squared. The wavefunction is related to the invariant probability density pK(X, S) and the stochastic action functional SK(X, s) by the relation

OK(X, S)= ~ exp[ iSK(X, S)] (III.H.20)

Notice that the functional SK is formally equivalent to the phase of Eq. (III.H.12). A stochastic velocity corresponding to Eq. (III.H.11) has been derived by Kyprianidis (1987) for a particle in the presence of an external field. Substituting Eq. (III.H.20) into Eq. (III.H.19) lets us decompose Eq. (III.H.19) into the Hamilton-Jacobi-type equation

8SK + ~ ~? ,S~?"SK + QK=O (III.H.21) 8s

and the continuity equation

O~SK-} - ~ u(p KOUS K) = 0 (III.H.22)


where QK is the relativistic quantum potential

QK = - ½(8~,8"PK + 8~,PKS~'PI¢), Ply--- ½ In P x (III.H.23)

Equations (III.H.19) through (III.H.23) are viewed as equivalent ways of describing a particle undergoing a random relativistic Markov process (Kyprianidis, 1987).

The idea of a Hamiltonian generator of evolution parameter translations was presented by Garuccio et al. (1984) and Dewdney et al. (1985). A Hilbert space construction with the positive-definite norm [Eqs. (III.H.3) and (III.H.4)] became a part of SIQM in 1986 (Dewdney et aL, 1986a). The realization that SIQM was moving in the direction of earlier parametrized theories came when Dewdney et al. (1986b) became aware of the canonical quantization procedure of Horwitz and Rohrlich (1982). A detailed connection between SIQM and parametrized theories was provided by Kyprianidis (1987).

Kyprianidis (1987) drew connections between SIQM and the FSF [especially its field theoretic formulation (Fanchi, 1979; Hostler, 1980)], and Greenberger's theory of indefinite mass. He discussed the gauge invariance of the theory and proposed an extension of the theory by writing a particle velocity of the form

¢, = 8F'SK + eA ~' + a8~'2 (lII.H.24)

The latter term includes the Clebsch parameters 2 and a. Schonberg (1954) introduced the Clebsch parameters in the context of classical Hamilton- Jacobi theory. The Clebsch parameters modify the total momentum without changing the forces acting on a system. They are new fields in Kyprianidis' (1987) extension whose physical significance was not fully explored.

On a slightly tangential note, we remark that candid appraisals of a nonstandard treatment of quantum theory are rare to find in the literature. An example is the brief review of Kyprianidis' 1987 article by Maddox (1987). Maddox complains that Kyprianidis "fails to say what he thinks is wrong with Dirac's equation or, for that matter, with the development of field theory in the past 40 years." This complaint reflects a lack of apprecia- tion of Section II of Kyprianidis' paper in which Kyprianidis presents several difficulties with the existing paradigm. Kyprianidis listed only a few difficulties. One he did not list, for example, is the need to find a tractable solution procedure for bound-state problems. In addition to ignoring Kyprianidis' list of difficulties, Maddox did not properly interpret the concept of space-time probability density, particularly the requirement that the probability density vanish at spatial and temporal infinity [an explicit

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5 3 6 Fanchi

application of this requirement is given by the noncovariant wave packet presented in Fanchi and Collins (1978); also see Horwitz and Rabin (1976)]. The lesson to be learned is that acceptance of a parametrized theory is going to require a paradigm shift with its many attendant problems (Kuhn, 1970).

One of the questions underlying SIQM is the source of space-time fluctuations. What is the "background field"? Schieve and Horwitz (1991) have demonstrated that two relativistic particles interacting via a potential of the form

1 4 1 2 V(p) = abp - gap , a, b > 0 (III.H.25)

can lead to chaotic orbits. The chaotic behavior of particle orbits led to their speculation that space-time may be chaotic. This is reminiscent of Takano's (1961) introduction of a random fluctuation a' in the space-time metric

( s t ) 2 = S 2 - - (a') 2, S 2 = x u x " (III.H.26)

where a' is associated with a probability distribution. Along these same lines, Yasue (1977) introduced Brownian motion into the trajectory of a relativistic particle to derive a Fock equation. By contrast, Nakagomi (1988) presented a theory in which particle fluctuations are generated in its proper frame of reference. Nakagomi used a model based on the Markov process to show that random particle motion may be parametrized by either proper time (1988) or observers's time (1989).

Other authors have suggested that a subquantum aether is the source of stochastic behavior. In this view, the vacuum is Dirac's aether (Dirac, 1951). It induces frictional effects with a friction coefficient ~/. Kaloyerou and Vigier (1989) derived the nonlinear stochastic wave equation

2i ~ s = ( - i0" -- eA~')( - i~ ~ -- eA ~) ~

v ~ O - 7 [ i l n ( ~ h / O * ) l / 2 + B ] ~ k (III.H.27)

where the last term is a result of stochastic quantization (Namsrai, 1986). The scalar potential is related to the 4-vector potential by

A p = 0UB (III.H.28)

Friction is represented in the subquantum aether by modifying the evolution parameter. The constant 7 relates the modified or "reduced" evolution parameter Sred to the usual evolution parameter by the equation

1 - - e - ~ s~d (III.H.29)


Within this framework, Kaloyerou and Vigier (t989) adopted Fanchi and Collins' (1978) interpretation of the wave function and the evolution parameter. Vigier (1991) constructed soliton-like solutions of Eq. (III.H.27) that follow the trajectories associated with the usual solutions of the Schr6dinger and Klein-Gordon equations. Holland, Kyprianidis, and Vigier (1987) have argued that Feynman paths should be interpreted as possible mean paths and not just mathematical abstractions. They conclude that action-at-a-distance as manifested by the quantum potential of the stochastic wave equation [e.g., Eq. (III.H.23)] is propagated instantaneously in the center-of-mass frame of a two-body problem.

We encountered the concept of instantaneous communication before (Enatsu, 1971). No formal relationship has yet been drawn between the treatments presented by Holland et al. and Enatsu. Enatsu's (1963) approach--particularly his use of Euclidean field theory--may provide a link to another parametrization technique.

Parisi and Wu. Parisi and Wu (1981) have introduced a stochasti- cally based formulation of relativistic quantum theory with a scalar parameter. Considerable work has been done in this area over the past decade, and a relatively recent review has been presented by Damgaard and Huffel (1987).

Following Damgaard and Huffel, I denote a field by qt(x, s) and call Se the Euclidean action. It may have the form

(IILH.30)

in n-dimensional Euclidean space. The evolution parameter s is considered a "fictitious time variable" in Parisi and Wu's method. They build on the analogy between Euclidean quantum field theory and classical statistical mechanics, namely that the Euclidean path integral measure

exp( - Se ) / f exp ( - SE) DO (III.H.31)

may be considered the stationary distribution of a stochastic process. The evolution of the field ~b(x, s) in the parameter s must satisfy a stochastic differential equation such as the Fokker-Ptanck equation or the Langevin equation

0 6S E 8s (~(x, s) - cSqb(x, s) ~- q(x, s) (III.H.32)

538 Fanchi

where t/ is a random variable. The right-hand side of Eq. (III.H.32) contains an explicit decomposition into a nonrandom term and a randomly fluctuating term. Calculating the Gaussian average with respect to the random variable yields the correlation function

(~,(x~, s,).-. ~(xk, sk))

=I [~q(X,, SI)'" "~rl(Xk, Sk)] {exp[-- ¼ ~ ~ tl2(X, s )d"x ds] } Dr 1 {exp[ -- ~ ~ tl2(x, s) a~x ds] } Dt 1

(III.H.33)

The connection between stochastic quantization and quantum field theory is now obtained by asserting that equilibrium is reached in the limit as s approaches infinity. In this limit the equal parameter correlation functions become equivalent to the usual Green's functions of quantum field theory; thus

lim ( ~ q ( X 1 , S 1 ) ' " "~)rl(Xk, Sk) ) = ( ~ ( X 1 ) " ' ' ~ ) ( X k ) ) ( I I I . H . 3 4 ) s --. oo

An alternative presentation to the procedure outlined above can be found in Breit, Gupta, and Zaks (1984).

Our interest in Parisi and Wu's approach is limited because of its treatment of the evolution parameter as a "fictitious time." For a review of applications or more information about Parisi and Wu's method, the reader is referred to Horsley and Schoenmaker (1985) or Daamgard and Huffel (1987).

ACKNOWLEDGMENTS

I would like to thank A. O. Barut, R. E. Collins, Ph. Droz-Vincent, H. Enatsu, L. P. Horwitz, L. Hostler, A. Kyprianidis, H. E. Moses, M. Pavsic, P. M. Pearle, F. Rohrlich, D. Salisbury, W. C. Schieve, and J.-P. Vigier for providing references or reviewing the manuscript. I thank C. Fragapane for bringing the paper by Chodos et al. (1985) to my attention. I especially thank K. F. Fanchi for helping perform the literature search.

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