Gerald Kelnhofer

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.

Functional integration and gauge ambiguities in generalized abelian gauge theories

Abstract We consider the covariant quantization of generalized abelian gauge theories on a closed and compact n -dimensional manifold whose space of gauge invariant fields is the abelian group of Cheeger–Simons differential characters. The space of gauge fields is shown to be a non-trivial bundle over the orbits of the subgroup of smooth Cheeger–Simons differential characters. Furthermore each orbit itself has the structure of a bundle over a multi-dimensional torus. As a consequence there is a topological obstruction to the existence of a global gauge fixing condition. A functional integral measure is proposed on the space of gauge fields which takes this problem into account and provides a regularization of the gauge degrees of freedom. For the generalized p -form Maxwell theory closed expressions for all physical observables are obtained. The Green’s functions are shown to be affected by the non-trivial bundle structure. Finally the vacuum expectation values of circle-valued homomorphisms, including the Wilson operator for singular p -cycles of the manifold, are computed and selection rules are derived.

Abelian gauge theories on compact manifolds and the Gribov ambiguity

We study the quantization of Abelian gauge theories of principal torus bundles over compact manifolds with and without boundary. It is shown that these gauge theories suffer from a Gribov ambiguity originating in the nontriviality of the bundle of connections whose geometrical structure will be analyzed in detail. Motivated by the stochastic quantization approach, we propose a modified functional integral measure on the space of connections that takes the Gribov problem into account. This functional integral measure is used to calculate the partition function, Green’s functions, and the field strength correlating functions in any dimension by using the fact that the space of inequivalent connections itself admits the structure of a bundle over a finite dimensional torus. Green’s functions are shown to be affected by the nontrivial topology, giving rise to nonvanishing vacuum expectation values for the gauge fields.

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.

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.

Field space entanglement entropy, zero modes and Lifshitz models

Abstract The field space entanglement entropy of a quantum field theory is obtained by integrating out a subset of its fields. We study an interacting quantum field theory consisting of massless scalar fields on a closed compact manifold M . To this model we associate its Lifshitz dual model. The ground states of both models are invariant under constant shifts. We interpret this invariance as gauge symmetry and subject the models to proper gauge fixing. By applying the heat kernel regularization one can show that the field space entanglement entropies of the massless scalar field model and of its Lifshitz dual are agreeing.

Remarks on the thermodynamics and the vacuum energy of a quantum Maxwell gas on compact and closed manifolds

Abstract The quantum Maxwell theory at finite temperature at equilibrium is studied on compact and closed manifolds in both the functional integral and Hamiltonian formalism. The aim is to shed some light onto the interrelation between the topology of the spatial background and the thermodynamic properties of the system. The quantization is not unique and gives rise to inequivalent quantum theories which are classified by θ-vacua. Based on explicit parametrizations of the gauge orbit space in the functional integral approach and of the physical phase space in the canonical quantization scheme, the Gribov problem is resolved and the equivalence of both quantization schemes is elucidated. Using zeta-function regularization the free energy is determined and the effect of the topology of the spatial manifold on the vacuum energy and on the thermal gauge field excitations is clarified. The general results are then applied to a quantum Maxwell gas on an n-dimensional torus providing explicit formulae for the main thermodynamic functions in the low- and high-temperature regimes, respectively.

Quantizing Yang-Mills theory: From Parisi-Wu stochastic quantization to a global path integral

Abstract Based on a generalization of the stochastic quantization scheme we recently proposed a generalized, globally defined Faddeev-Popov path integral density for the quantization of Yang-Mills theory. In this talk first our approach on the whole space of gauge potentials is shortly reviewed; in the following we discuss the corresponding global path integral on the gauge orbit space relating it to the original Parisi-Wu stochastic quantization scheme.