Towards a nonperturbative path integral in gauge theories

We propose a modification of the Faddeev–Popov procedure to construct a path integral representation for the transition amplitude and the partition function for gauge theories whose orbit space has a non-Euclidean geometry. Our approach is based on the Kato–Trotter product formula modified appropriately to incorporate the gauge invariance condition, and thereby equivalence to the Dirac operator formalism is guaranteed by construction. The modified path integral provides a solution to the Gribov obstruction as well as to the operator ordering problem when the orbit space has curvature. A few explicit examples are given to illustrate new features of the formalism developed. The method is applied to the Kogut–Susskind lattice gauge theory to develop a nonperturbative functional integral for a quantum Yang–Mills theory. Feynman's conjecture about a relation between the mass gap and the orbit space geometry in gluodynamics is discussed in the framework of the modified path integral.     

Quantum Superconformal Symmetry in N = 2 Harmonic Superspace

In gauge theories, not all rigid symmetries of the classical action can be maintained manifestly in the quantization procedure, even in the absence of anomalies. If this occurs for an anomaly-free symmetry, the effective action is invariant under a transformation that differs from its classical counterpart by quantum corrections. In this note, we set up a harmonic superspace formalism for computing quantum deformations of superconformal symmetry in the N = 4 supersymmetric Yang–Mills theory.     

Path integral quantization of the Poisson‐Sigma model

We apply the antifield quantization method of Batalin and Vilkovisky to the calculation of the path integral for the Poisson‐Sigma model in a general gauge. For a linear Poisson structure the model reduces to a nonabelian gauge theory, and we obtain the formula for the partition function of two‐dimensional Yang‐Mills theory for closed oriented two‐dimensional manifolds.     

Linear bosonic quantum field theories arising from causal variational principles

We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.

     

Minimal Coupling in Koopman-von Neumann Theory

Classical mechanics (CM), like quantum mechanics (QM), can have an operatorial formulation. This was pioneered by Koopman and von Neumann (KvN) in the 1930s. They basically formalized, via the introduction of a classical Hilbert space, earlier work of Liouville who had shown that the classical time evolution can take place via an operator, nowadays known as the Liouville operator. In this paper we study how to perform the coupling of a point particle to a gauge field in the KvN version of CM. So we basically implement at the classical operatorial level the analog of the minimal coupling of QM. We show that, differently than in QM, not only the momenta but also other variables have to be coupled to the gauge field. We also analyze in detail how the gauge invariance manifests itself in the Hilbert space of KvN and indicate the differences with QM. As an application of the KvN method we study the Landau problem proving that there are many more degeneracies at the classical operatorial level than at the quantum one. As a second example we go through the Aharonov-Bohm phenomenon showing that, at the quantum level, this phenomenon manifests its effects on the spectrum of the quantum Hamiltonian while at the classical level there is no effect whatsoever on the spectrum of the Liouville operator.     

Mathematical quantum Yang–Mills theory revisited

A mathematically rigorous relativistic quantum Yang–Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is nonperturbative, without cut-offs, and agrees with the causality and stability principles. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous papers Dynin ([11] and [12]). The principal result is established anew: due to the quartic self-interaction term in the Yang–Mills Lagrangian along with the semisimplicity of the gauge group, the quantum Yang–Mills energy spectrum has a positive mass gap. Furthermore, the quantum Yang–Mills Hamiltonian has a countable orthogonal eigenbasis in a Fock space, so that the quantum Yang–Mills spectrum is point and countable. In addition, a fine structure of the spectrum is elucidated.     

Contextuality in Classical Physics and Its Impact on the Foundations of Quantum Mechanics

It is shown that the hallmark quantum phenomenon of contextuality is present in classical statistical mechanics (CSM). It is first shown that the occurrence of contextuality is equivalent to there being observables that can differentiate between pure and mixed states. CSM is formulated in the formalism of quantum mechanics (FQM), a formulation commonly known as the Koopman–von Neumann formulation (KvN). In KvN, one can then show that such a differentiation between mixed and pure states is possible. As contextuality is a probabilistic phenomenon and as it is exhibited in both classical physics and ordinary quantum mechanics (OQM), it is concluded that the foundational issues regarding quantum mechanics are really issues regarding the foundations of probability.     

On nonlinear gauge theories

In this note, we study non-linear gauge theories for principal bundles, where the structure group is replaced by a Lie groupoid. We follow the approach of Moerdijk–Mr?un and establish its relation with the existing physics literature. In particular, we derive a new formula for the gauge transformation which closely resembles and generalizes the classical formulas found in Yang Mills gauge theories.     

Generalised BRST symmetry and gaugeon formalism for perturbative quantum gravity: Novel observation

In this paper the novel features of Yokoyama gaugeon formalism are stressed out for the theory of perturbative quantum gravity in the Einstein curved spacetime. The quantum gauge transformations for the theory of perturbative gravity are demonstrated in the framework of gaugeon formalism. These quantum gauge transformations lead to renormalised gauge parameter. Further, we analyse the BRST symmetric gaugeon formalism which embeds more acceptable Kugo–Ojima subsidiary condition. Further, the BRST symmetry is made finite and field-dependent. Remarkably, the Jacobian of path integral under finite and field-dependent BRST symmetry amounts to the exact gaugeon action in the effective theory of perturbative quantum gravity.     

No‐go result for supersymmetric gauge theories in the causal approach

We consider the massless supersymmetric vector multiplet in a purely quantum framework and propose a power counting formula. Then we prove that the interaction Lagrangian for a massless supersymmetric non‐Abelian gauge theory (SUSY‐QCD) is uniquely determined by some natural assumptions, as in the case of Yang‐Mills models. The result can be easily generalized to the case when massive multiplets are present, but one finds out that the massive and the massless Bosons must be decoupled, in contradiction with the standard model. Going to the second order of perturbation theory produces an anomaly which cannot be eliminated. We make a thorough analysis of the model working only with the component fields.     

Of Ghosts, Gauge Volumes, and Gauss's Law

The relationship between the canonical operator and the path integral formulation of quantum electrodynamics is analyzed with a particular focus on the implementation of gauge constraints in the two approaches. The removal of gauge volumes in the path integral is shown to match with the presence of zero-norm ghost states associated with gauge transformations in the canonical operator approach. The path integrals for QED in both the Feynman and the temporal gauges are examined and several ways of implementing the gauge constraint integrations are demonstrated. The upshot is to show that both the Feynman and the temporal gauge path integrals are equivalent to the Coulomb gauge path integral, matching the results developed by Kurt Haller using the canonical formalism. In addition, the Faddeev–Popov form for the Feynman gauge and temporal gauge Lagrangian path integrals are derived from the Hamiltonian form of the path integral.     

Quantum Field Formalism for the Electromagnetic Interaction of Composite Particles in a Nonrelativistic Gauge Model

A classical nonrelativistic U(1) × U(1) gauge field model for the electromagnetic interaction of composite particles is proposed and the quantum formalism is constructed. This gauge model containing a Chern–Simons U(1) field and the electromagnetic U(1) field can be coupled to both a bosonic or a fermionic matter field. We explicitly consider the second case, a composite fermion system in the presence of an electromagnetic field, and we carry out the canonical quantization by the Dirac method. The path integral approach is developed and the Feynman rules are established. A simplified model is considered. As an alternative path integral method, the BRST formalism for this gauge model is also treated.     

The Influence of Quantum Field Fluctuations on Chaotic Dynamics of Yang–Mills System

Abstract

On example of the model field system we demonstrate that quantum fluctuations of non-abelian gauge fields leading to radiative corrections to Higgs potential and spontaneous symmetry breaking can generate order region in phase space of inherently chaotic classical field system. We demonstrate on the example of another model field system that quantum fluctuations do not influence on the chaotic dynamics of non-abelian Yang–Mills fields if the ratio of bare coupling constants of Yang–Mills and Higgs fields is larger then some critical value. This critical value is estimated.     

Quantum General Invariance and Loop Gravity

A quantum physical projector is proposed for generally covariant theories which are derivable from a Lagrangian. The projector is the quantum analogue of the integral over the generators of finite one-parameter subgroups of the gauge symmetry transformations which are connected to the identity. Gauge variables are retained in this formalism, thus permitting the construction of spacetime area and volume operators in a tentative spacetime loop formulation of quantum general relativity.     

Functional Analysis for Gauge Fields on the Front-Form and the Light-Cone Gauge

Constrained systems in quantum field theories call for a careful study of diverse classes of constraints and consistency checks over their temporal evolution. Here we study the functional structure of the free electromagnetic and pure Yang–Mills fields on the front-form coordinates with the null-plane gauge condition. It is seen that in this framework, we can deal with strictu sensu physical fields.     

Relativistic quantum mechanics of supersymmetric particles

The quantum mechanics of an electron in an external field is developed by Hamiltonian path integral methods. The electron is described classically by an action which is invariant under gauge supersymmetry transformations as well as worldline reparametrizations. The simpler case of a spinless particle is first reviewed and it is pointed out that a strictly canonical approach does not exist. This follows formally from the gauge invariance properties of the action and physically it corresponds to the fact that particles can travel backwards as well as forward in coordinate time. However, appropriate application of path integral techniques yields directly the proper time representation of the Feynman propagator. Next we extend the formalism to systems described by anticommuting variables. This problem presents some difficulty when the dimension of the phase space is odd, because the holomorphic representation does not exist. It is shown, however, that the usual connection between the evolution operator and the path integral still holds provided one indludes in the action the boundary term that makes the classical variational principle well defined. The path integral for the relativistic spinning particle is then evaluated and it is shown to lead directly to a representation for the Feynman propagator in terms of two proper times, one commuting, the other anticommuting, which appear in a symmetric manner. This representation is used to derive scattering amplitudes in an external field. In this step the anticommuting proper time is integrated away and the analysis is carried in terms of one (commuting) proper time only, just as in the spinless case. Finally, some properties of the quantum mechanics of the ghost particles that appear in the path integral for constrained systems are developed in an appendix.     

Dislocation theory as a 3‐dimensional translation gauge theory

We consider the static elastoplastic theory of dislocations in an elastoplastic material. We use a Yang‐Mills type Lagrangian (the teleparallel equivalent of Hilbert‐Einstein Lagrangian) and some Lagrangians with anisotropic constitutive laws. The translational part of the generalized affine connection is utilized to describe the theory of elastoplasticity in the framework of a translation gauge theory. We obtain a system of Yang‐Mills field equations which express the balance of force and moment.     

Batalin-Vilkovisky field-antifield quantisation of fluctuations around classical field configurations

The Lagrangian field-antifield formalism of Batalin and Vilkovisky (BV) is used to investigate the application of the collective coordinate method to soliton quantisation. In field theories with soliton solutions, the Gaussian fluctuation operator has zero modes due to the breakdown of global symmetries of the Lagrangian in the soliton solutions. It is shown how Noether identities and local symmetries of the Lagrangian arise when collective coordinates are introduced in order to avoid divergences related to these zero modes. This transformation to collective and fluctuation degrees of freedom is interpreted as a canonical transformation in the symplectic field-antifield space which induces a time-local gauge symmetry. Separating the corresponding Lagrangian path integral of the BV scheme in lowest order into harmonic quantum fluctuations and a free motion of the collective coordinate with the classical mass of the soliton, we show how the BV approach clarifies the relation between zero modes, collective coordinates, gauge invariance and the center-of-mass motion of classical solutions in quantum fields. Finally, we apply the procedure to the reduced nonlinear O(3) σ-model.     

Frölicher-smooth geometries,quantum jet bundles and BRST symmetry

We attempt a clarification of geometric aspects of quantum field theory by using the notion of smoothness introduced by Frölicher and exploited by several authors in the study of functional bundles. A discussion of momentum and position representations in curved spacetime, in terms of generalized semi-densities, leads to a definition of quantum configuration bundle which is suitable for a treatment of that kind. A consistent approach to Lagrangian field theories, vertical infinitesimal symmetries and related currents is then developed, and applied to a formulation of BRST symmetry in a gauge theory of the Yang–Mills type.     

On the role of coherent states in quantum foundations

Coherent states, and the Hilbert space representations they generate, provide ideal tools to discuss classical/quantum relationships. In this paper we analyze three separate classical/quantum problems using coherent states, and show that useful connections arise among them. The topics discussed are: (1) a truly natural formulation of phase space path integrals; (2) how this analysis implies that the usual classical formalism is “simply a subset” of the quantum formalism, and thus demonstrates a universal coexistence of both the classical and quantum formalisms; and (3) how these two insights lead to a complete analytic solution of a formerly insoluble family of nonlinear quantum field theory models.