Stochastic quantization and regularization

Breit, J.D.; Gupta, Subhash; Zaks, A.

doi:10.1016/0550-3213(84)90170-6

Cited by 111 publications

(36 citation statements)

References 5 publications

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“…The other possibility is to smear the TJ probability functional [2). In this scheme, the Langevin equation is left alone, while equation (2.2) is replaced by…”

Section: Stochastic Regularizationmentioning

confidence: 99%

“…It is possible to find a necessary condition on the set of functions that can be used as regularizers by studying the loop of the first order· correction [2]. In this case, the loop is decoupled from the rest of the diagram, so the loop can be studied by itself.…”

Section: Stochastic Regularizationmentioning

confidence: 99%

“…In order for the integral to be finite, a necessary condition on the regularization function is that [2] OA(O) = 0. (3.14)…”

Section: Stochastic Regularizationmentioning

confidence: 99%

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Stochastic regularization of scalar electrodynamics

Bern

1985

Nuclear Physics B

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A regularization scheme, first proposed by Breit, Gupta, and Zaks and based upon the Langevin equation of Parisi and Wu, is used to regularize scalar electrodynamics. This scheme is shown to preserve the masslessness of the photon and the tensor structure of the photon vacuum polarization at the one loop level. The scalar wavefunction renormalization, Z 2 , is shown to be equal to the one photon vertex renormalization, Z 1 , to all orders or the stochastically regularized theory.

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“…The other possibility is to smear the TJ probability functional [2). In this scheme, the Langevin equation is left alone, while equation (2.2) is replaced by…”

Section: Stochastic Regularizationmentioning

confidence: 99%

Section: Stochastic Regularizationmentioning

confidence: 99%

See 1 more Smart Citation

Stochastic regularization of scalar electrodynamics

Bern

1985

Nuclear Physics B

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“…However, the stochastic quantization scheme suggests a new regularization procedure, as pointed out by Breit, Gupta, and Zaks. 2 The idea is to replace the driving noise process 77 with a Gaussian process with smoother sample paths. This is accomplished by replacing the 8 correlation function (2) by an approximate 8 function containing a dimensionful parameter A representing a momentum cutoff.…”

Section: )mentioning

confidence: 99%

Nonperturbative bounds on 〈φn〉 in cutoff (λφn
Doering
¹

1985
Phys. Rev. Lett.
10
0
1
0
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Stochastic quantization is used to obtain upper and lower bounds on the vacuum expectation value of <£" in a cutoff \ n scalar theory in d dimensions. These determine the behavior of ( w ) as the cutoff is removed. The results are given by formulas which depend on whether the values of n and d define a perturbatively superrenormalizable, strictly renormalizable, or nonrenormalizable theory.PACS numbers: 02.50. + s, 03.70. + k, 05.40. +j Most calculations in interacting-field theories are performed perturbatively. Infinite renormalizations are required to remove the accompanying divergences and these lead to the distinction between perturbatively renormalizable and nonrenormalizable theories-the latter often being condemned as pathological or nonpredictive. The exact results in this Letter give some insight into some of the conclusions drawn from perturbation theory.We use stochastic quantization to calculate bounds on ( n ) 9 the vacuum expectation value of n 9 in a cutoff k n self-interacting scalar field theory. The bounds characterize the divergence of this quantity as the cutoff is removed and we find qualitatively different behavior if the theory is renormalizable or nonrenormalizable. The result in the nonrenormalizable case disagrees with the predictions of perturbation theory. The calculation is presented in some detail below, as the technique should be useful in the study of fluctuations of many systems described by Langevin equations.The stochastic quantization procedure introduced by Parisi and Wu 1 consists of evaluating the solution to the Langevin (stochastic differential) equation, d(x,t)/dt=-8S[]/8(f>(x,t) + ri(x,t) 9(1)where t is an auxiliary parameter, x£iR d , S is the Euclidean action, and rj(x 9 t) is a Gaussian white-noise process with E[ V (x 9 t)]=0 9 E[ri{x,t)r){x' 9 t')] = 28(t-t' )8 d (x-x'). 2)In the above, E[ ] denotes the average with respect to the noise ensemble. The Schwinger functions are the equaltime equilibrium values of the moments of the random process:(3) r-^oo Their perturbative computation exhibits the usual divergences. However, the stochastic quantization scheme suggests a new regularization procedure, as pointed out by Breit, Gupta, and Zaks. 2 The idea is to replace the driving noise process 77 with a Gaussian process with smoother sample paths. This is accomplished by replacing the 8 correlation function (2) by an approximate 8 function containing a dimensionful parameter A representing a momentum cutoff. For instance, if we use the process rj A with EbliSx 9(4) (9 is the step function), the result is a regularized theory which converges, as the cutoff A->oo, to the original model. Breit, Gupta, and Zaks considered changing the "time" part of the correlation function, and the modification of the "space" part here is reminiscent of the stochastic regularization proposed by Niemi and Wijewardhana. 3 In terms of the perturbation expansions for the Schwinger functions developed in Ref. 1, the effect of this is to introduce cutoff momentum integrals...

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“…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."…”

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confidence: 99%

Review of invariant time formulations of relativistic quantum theories

Fanchi

1993

Found Phys

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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.

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Stochastic quantization and regularization

Cited by 111 publications

References 5 publications

Stochastic regularization of scalar electrodynamics

Stochastic regularization of scalar electrodynamics

Nonperturbative bounds on 〈φn〉 in cutoff (λφn
Doering
¹

1985
Phys. Rev. Lett.
10
0
1
0
View full text Add to dashboard Cite

Review of invariant time formulations of relativistic quantum theories

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Stochastic quantization and regularization

Cited by 111 publications

References 5 publications

Stochastic regularization of scalar electrodynamics

Stochastic regularization of scalar electrodynamics

Nonperturbative bounds on 〈φn〉 in cutoff (λφnDoering1 1985Phys. Rev. Lett.10010View full textAdd to dashboardCite

Review of invariant time formulations of relativistic quantum theories

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About

Nonperturbative bounds on 〈φn〉 in cutoff (λφn
Doering
¹

1985
Phys. Rev. Lett.
10
0
1
0
View full text Add to dashboard Cite