Helmuth Huffel

Stochastic quantization in Minkowski space

Abstract We propose a generalization of the euclidean stochastic quantization scheme of Parisi and Wu that is applicable to fields in Minkowski space. A perturbative proof of the equivalence of the new method to ordinary quantization is given for the self-interacting scalar field. It is argued furthermore non-perturbatively that the method generally implies the Schwinger-Dyson equations.

Wilson loop exponentiation and temporal gauges

Abstract Using generalisations of the Mandelstam-Leibbrandt prescription for a Yang-Mills theory in temporal gauge, we perturbatively calculate the Wilson loop. We demonstrate that, unlike the principal-value prescription, these prescriptions give a result that agrees with the Feynman gauge.

Generalized Stochastic Quantization of Yang–Mills Theory

Abstract We perform the stochastic quantization of Yang–Mills theory in configuration space and derive the Faddeev–Popov path integral density. Based on a generalization of the stochastic gauge fixing scheme and its geometrical interpretation this result is obtained as the exact equilibrium solution of the associated Fokker–Planck equation. Included in our discussion is the precise range of validity of our approach.

Nonperturbative Stochastic Quantization of the Helix Model

Abstract The helix model describes the minimal coupling of an abelian gauge field with three bosonic matter fields in 0+1 dimensions; it is a model without a global Gribov obstruction. We perform the stochastic quantization in configuration space and prove nonperturbatively equivalence with the path integral formalism. Major points of our approach are the geometrical understanding of separations into gauge independent and gauge dependent degrees of freedom as well as a generalization of the stochastic gauge fixing procedure which allows us to extract the equilibrium Fokker–Planck probability distribution of the model.

Global path integral quantization of Yang–Mills theory

Based on a generalization of the stochastic quantization scheme recently a modified Faddeev-Popov path integral density for the quantization of Yang-Mills theory was derived, the modification consisting in the presence of specific finite contributions of the pure gauge degrees of freedom. Due to the Gribov problem the gauge fixing can be defined only locally and the whole space of gauge potentials has to be partitioned into patches. We propose a global path integral density for the Yang-Mills theory by summing over all patches, which can be proven to be manifestly independent of the specific local choices of patches and gauge fixing conditions, respectively. In addition to the formulation on the whole space of gauge potentials we discuss the corresponding global path integral on the gauge orbit space relating it to the original Parisi-Wu stochastic quantization scheme and to a proposal of Stora, respectively. q 2000 Elsevier Science B.V. All rights reserved.

Nonlinear Brownian motion and Higgs mechanism

An extension of the stochastic quantization scheme is proposed by adding nonlinear terms to the field equations. Our modification is motivated by the recently established theory of active Brownian motion. We discuss a way of promoting this theory to the case of infinite degrees of freedom. Equilibrium distributions can be calculated exactly and are interpreted as path integral densities of quantum field theories. By applying our procedure to scalar QED, the symmetry breaking potential of the Higgs mechanism arises as the equilibrium solution.

Swarms with canonical active Brownian motion.

We present a swarm model of Brownian particles with harmonic interactions, where the individuals undergo canonical active Brownian motion, i.e., each Brownian particle can convert internal energy to mechanical energy of motion. We assume the existence of a single global internal energy of the system. Numerical simulations show amorphous swarming behavior as well as static configurations. Analytic understanding of the system is provided by studying stability properties of equilibria.

QED revisited: proving equivalence between path integral and stochastic quantization

We perform the stochastic quantization of scalar QED based on a generalization of the stochastic gauge fixing scheme and its geometric interpretation. It is shown that the stochastic quantization scheme exactly agrees with the usual path integral formulation.

Quantizing Yang–Mills theory on a two-point space

We perform the Batalin-Vilkovisky quantization of Yang–Mills theory on a 2-point space, discussing the formulation of Connes–Lott as well as Connes’ real spectral triple approach. Despite of the model’s apparent simplicity the gauge structure reveals infinite reducibility and the gauge fixing is afflicted with the Gribov problem. 1We perform the Batalin–Vilkovisky quantization of Yang–Mills theory on a two-point space, discussing the formulation of Connes–Lott as well as Connes’ real spectral triple approach. Despite the model’s apparent simplicity the gauge structure reveals infinite reducibility and the gauge fixing is afflicted with the Gribov problem.

Generalized stochastic gauge fixing

Abstract We propose a generalization of the stochastic gauge fixing procedure for the stochastic quantization of gauge theories where not only the drift term of the stochastic process is changed but also the Wiener process itself. All gauge invariant expectation values remain unchanged. As an explicit example we study the case of an abelian gauge field coupled with three bosonic matter fields in 0 + 1 dimensions. We nonperturbatively prove equivalence with the path integral formalism.