scholarly journals THE STOCHASTIC QUANTIZATION METHOD IN PHASE SPACE AND A NEW GAUGE FIXING PROCEDURE

1994 ◽  
Vol 09 (30) ◽  
pp. 2803-2815
Author(s):  
RIUJI MOCHIZUKI
Keyword(s):  
Phase Space ◽  
Path Integral ◽  
Equilibrium Solution ◽  
Planck Equation ◽  
Langevin Equations ◽  
Stochastic Quantization ◽  
Quantization Method ◽  
Canonical Variables ◽  
Gauge Fixing ◽  
Integral Distribution

We study the stochastic quantization of the system with first class constraints in phase space. Though the Langevin equations of the canonical variables are defined without ordinary gauge fixing procedure, gauge fixing conditions are automatically selected and introduced by imposing stochastic consistency conditions upon the first class constraints. Then the equilibrium solution of the Fokker–Planck equation is identical to the corresponding path-integral distribution.

scholarly journals DYNAMICAL ASPECTS OF INEXTENSIBLE CHAINS

2012 ◽  
Vol 26 (02) ◽  
pp. 1250009
Author(s):  
FRANCO FERRARI ◽  
MACIEJ PYRKA
Keyword(s):  
Path Integral ◽  
Degrees Of Freedom ◽  
Planck Equation ◽  
Integral Form ◽  
Langevin Equations ◽  
Thermal Bath ◽  
Fixed Temperature ◽  
Generating Functional ◽  
A Chain ◽  
Non Gaussian

In the present work, a method to impose the inextensibility constraints on the dynamics of a chain fluctuating in a thermal bath at fixed temperature is investigated. The final goal is to construct the probability function of the chain and the generating functional of the correlation functions of the relevant degrees of freedom of the system. First, we study the dynamics of a freely hinged chain composed by massive beads connected together by massless segments of fixed length. It is shown that a system of this kind may be described by a set of Langevin equations in which the noise is characterized by a non-gaussian probability distribution. Starting from these Langevin equations, the generating functional of the freely hinged chain is derived in path integral form. A connection with a stochastic process governed by a Fokker–Planck equation is established. Next, a chain composed by one-dimensional bars with constant mass distribution is considered. A path integral expression of the generating functional for a chain of this type is derived. Finally, it is verified that in the limit in which the chain becomes continuous, both generating functionals of the freely hinged chain and of the freely jointed bar chain converge to the same result as expected.

Path integral for (2+1)-dimensional Shape Dynamics

2015 ◽  
Vol 30 (12) ◽  
pp. 1550057
Author(s):  
A. González ◽  
H. Ocampo
Keyword(s):  
General Relativity ◽  
Phase Space ◽  
Path Integral ◽  
York Time ◽  
Path Integral Quantization ◽  
Shape Dynamics ◽  
Gauge Fixing ◽  
Dimensional Shape

We studied the path integral quantization for the Shape Dynamics formulation of General Relativity in the 2+1 torus universe. We show that the Shape Dynamics path integral on the reduced phase space is equivalent with the previous results obtained for the ADM 2+1 gravity and we found that the Shape Dynamics Hamiltonian allows us to establish a straightforward relation between reduced systems in the (τ, V)-form and the τ-form through the York time gauge fixing.

GAUGE FIXING, UNITARITY AND PHASE SPACE PATH INTEGRALS

1992 ◽  
Vol 07 (21) ◽  
pp. 5245-5279 ◽  
Author(s):  
MARTIN LAVELLE ◽  
DAVID MCMULLAN
Keyword(s):  
Phase Space ◽  
Path Integral ◽  
Wide Class ◽  
Gauge Theories ◽  
Path Integrals ◽  
Landau Gauge ◽  
Coulomb Gauge ◽  
Restricted Version ◽  
Feynman Gauge ◽  
Gauge Fixing

We analyse the extent to which path integral techniques can be used to directly prove the unitarity of gauge theories. After reviewing the limitations of the most widely used approaches, we concentrate upon the method which is commonly regarded as solving the problem, i.e. that of Fradkin and Vilkovisky. We show through explicit counterexamples that their main theorem is incorrect. A proof is presented for a restricted version of their theorem. From this restricted theorem we are able to rederive Faddeev’s unitary phase space results for a wide class of canonical gauges (which includes the Coulomb gauge) and for the Feynman gauge. However, we show that there are serious problems with the extensions of this argument to the Landau gauge and hence the full Lorentz class. We conclude that there does not yet exist any satisfactory path integral discussion of the covariant gauges.

Parisi–Wu stochastic quantization for relativistic particles in a weak gravitational plane wave

2009 ◽  
Vol 87 (4) ◽  
pp. 399-405 ◽  
Author(s):  
S. Haouat ◽  
N. Chine ◽  
L. Chetouani
Keyword(s):  
Plane Wave ◽  
Thermal Equilibrium ◽  
Wave Functions ◽  
Langevin Equations ◽  
Stochastic Quantization ◽  
Quantization Method ◽  
Equilibrium Limit ◽  
Dirac Particles ◽  
Klein Gordon ◽  
Relativistic Particles

The problem of relativistic particles moving in the background of a weak gravitational plane wave is solved by the use of Parisi–Wu stochastic quantization method. After having solved the corresponding Langevin equations, we have calculated exactly the correlation product of two fields known as a Green's function at the thermal-equilibrium limit for the fictitious time. By addition of an infinitesimal imaginary part of the mass, the existence of the limit of the correlation function at the equilibrium is assured. Analytical and exact expressions of the wave functions are obtained both for Klein–Gordon (KG) and Dirac particles.

scholarly journals Finite BRST–anti-BRST transformations in generalized Hamiltonian formalism

2014 ◽  
Vol 29 (27) ◽  
pp. 1450159 ◽  
Author(s):  
Pavel Yu. Moshin ◽  
Alexander A. Reshetnyak
Keyword(s):  
Dynamical Systems ◽  
Path Integral ◽  
Hamiltonian Path ◽  
Hamiltonian Formalism ◽  
Lagrangian Formalism ◽  
Yang Mills ◽  
Gauge Dependence ◽  
Field Dependent ◽  
Gauge Fixing ◽  
Constrained Dynamical Systems

We introduce the notion of finite BRST–anti-BRST transformations for constrained dynamical systems in the generalized Hamiltonian formalism, both global and field-dependent, with a doublet λa, a = 1, 2, of anticommuting Grassmann parameters and find explicit Jacobians corresponding to these changes of variables in the path integral. It turns out that the finite transformations are quadratic in their parameters. Exactly as in the case of finite field-dependent BRST–anti-BRST transformations for the Yang–Mills vacuum functional in the Lagrangian formalism examined in our previous paper [arXiv:1405.0790 [hep-th]], special field-dependent BRST–anti-BRST transformations with functionally-dependent parameters λa= ∫ dt(saΛ), generated by a finite even-valued function Λ(t) and by the anticommuting generators saof BRST–anti-BRST transformations, amount to a precise change of the gauge-fixing function for arbitrary constrained dynamical systems. This proves the independence of the vacuum functional under such transformations. We derive a new form of the Ward identities, depending on the parameters λaand study the problem of gauge dependence. We present the form of transformation parameters which generates a change of the gauge in the Hamiltonian path integral, evaluate it explicitly for connecting two arbitrary Rξ-like gauges in the Yang–Mills theory and establish, after integration over momenta, a coincidence with the Lagrangian path integral [arXiv:1405.0790 [hep-th]], which justifies the unitarity of the S-matrix in the Lagrangian approach.

TRANSITION STATE THEORY IN EXTENDED PHASE SPACE

1993 ◽  
Vol 07 (19) ◽  
pp. 1263-1268
Author(s):  
H. DEKKER ◽  
A. MAASSEN VAN DEN BRINK
Keyword(s):  
Phase Space ◽  
Transition State Theory ◽  
Unstable Mode ◽  
Planck Equation ◽  
State Theory ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Theoretical Concepts ◽  
Rate Processes ◽  
Mode Energy

Turnover theory (of the escape Γ) à la Grabert will be based solely on Kramers' Fokker–Planck equation for activated rate processes. No recourse to a microscope model or Langevin dynamics will be made. Apart from the unstable mode energy E, the analysis requires new theoretical concepts such as a constrained Gaussian transformation (CGT) and dynamically extended phase space (EPS).

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