Equilibrium limit of the stochastic quantization in minkowski space

Fukuda, R.; Higurashi, Hitoshi

doi:10.1016/0370-2693(88)91861-8

Cited by 6 publications

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References 9 publications

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“…This paper is organized as follows. In the next section we succinctly review the generating functional method of the Parisi-Wu stochastic quantization after Fukuda and Higurashi 8 ) and derive the stochastic Schwinger-Dyson equation which generalizes the usual Schwinger-Dyson equation for the generating functional of the connected Green's functions and reduces to the latter in the equilibrium limit. We then derive an evolution equation for the 7J-average of a scalar field for a J..¢} theory in § 3, study properties of its solutions for both unbroken and spontaneously broken theories and solve it in the tree approximation numerically on a computer.…”

Section: )-7)mentioning

confidence: 99%

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Spontaneous Breakdown of Symmetry in Stochastic Quantization

Ito¹,

Morita²

1992

Prog. Theor. Phys.

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Section: )-7)mentioning

confidence: 99%

“…In particular, we define from the translational invariance in the coordinate space for n=I *) There exists a compact form of the solution to Eqs. (7a, b), which is used 8 ) to prove the Feynman-Dyson perturbation formula in the equilibrium limit [---ex) provided f.L2>O. (12) and put…”

Section: )-7)mentioning

confidence: 99%

Spontaneous Breakdown of Symmetry in Stochastic Quantization

Ito¹,

Morita²

1992

Prog. Theor. Phys.

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“…(2) In what follows we simply put n=l except in the series expansion with respect to n. For our purpose it is convenient 8 ) to introduce an external field lex) and define the generating functional for equal-r correlation functions by…”

Section: <7j(x R»1)=o <7j(x R)7j(x' R'»1)=2n0mentioning

confidence: 99%

“…(7a, b), which is used 8 ) to prove the Feynman-Dyson perturbation formula in the equilibrium limit [---ex) provided f.L2>O. (12) and put…”

Section: W[j D= (Ex P ( -!D 4 X](x)¢(x R»))1) =mentioning

confidence: 99%

Spontaneous Breakdown of Symmetry in Stochastic Quantization

Ito

Morita

1992

Progress of Theoretical Physics

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207It is shown that the stochastic quantization method of Parisi and Wu determines the vacuum structure of the spontaneously broken theories by the stochastic Schwinger-Dyson equation which, in the equilibrium limit, reduces to the ordinary Schwinger-Dyson equation and is solved in Pi-expansion of the contributing stochastic diagrams. The method explains the nonanalytic behavior, or the two-valuedness of order parameter at vanishing source. It incidentally avoids the problem of complexity and convexity of the effective potential in a totally different manner. The stochastic Schwinger· Dyson equation for the growing order parameter is explicitly solved in some simple cases including a )..if/ theory with both unbroken and spontaneously broken potentials, the Goldstone and the Nambu-Jona-Lasinio models in the lowest order approximation. Ward-Takahashi identities in the spontaneously broken theories are also derived using the Langevin equation without recourse to the path-integral nor the operator formalisms. § 1. IntroductionIn addition to the canonical and path-integral formalisms the stochastic quantization of Parisi and Wu 1 ) provides the third alternative of the second-quantization of field theory. It is based on the purely classical notions using the Langevin equation well-known in the theory of the Brownian motion. Thus field variables are considered to be the random ones subject to the Langevin equation with respect to an extra time coordinate (or the fifth-time) r, reproducing quantum Green's functions in the hypothetical thermodynamic limit r--'>CO. In euclidean formulation the nonperturbative proof of the equivalence of the method to the conventional quantization schemes is well-established.The Parisi-Wu formalism has been applied 2 ) to various problems in field theory including both non-gauge and gauge theories. In this method the Boltzmann factor e-s in the path-integral formula can be derived from the Langevin equation, but one need not work in the path-integral, thereby avoiding the precise definition of the measure. This is interesting because the path-integral bothers us with the problem of the measure. In this connection we recall gauge theories and anomalies. The purpose of this paper is to apply the stochastic quantization to reformulate the spontaneous breakdown of symmetry, which leads to the concept of the growing order parameter 3 ) in the spontaneously broken theories. It turns out that the Parisi-Wu method is capable of describing the spontaneous symmetry breaking without modifying the symmetric lagrangian, but a set of the stochastic diagrams in broken theories is generated by introducing a constant linear source. The result is formulated by an evolution equation for the growing order parameter, which becomes a quasi-linear, partial differential equation in the tree approximation. We solve the quasi-linear, partial differential equation by the method of the characteristic equations and find

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Regularization in the Minkowski stochastic quantization scheme. Scalar model and QED

Morita

Kase

1990

Physics Letters B

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Equilibrium limit of the stochastic quantization in minkowski space

Cited by 6 publications

References 9 publications

Spontaneous Breakdown of Symmetry in Stochastic Quantization

Spontaneous Breakdown of Symmetry in Stochastic Quantization

Spontaneous Breakdown of Symmetry in Stochastic Quantization

Regularization in the Minkowski stochastic quantization scheme. Scalar model and QED

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