Functional integration and gauge ambiguities in generalized abelian gauge theories

We consider the covariant quantization of generalized abelian gauge theories on a closed and compact n$n$-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$p$-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$p$-cycles of the manifold, are computed and selection rules are derived.     

A rigorous construction of the chiral anomaly on a spin manifold

The quantum field for Dirac fermion is rigorously formulated on a compact spin manifold by using the functional integral defined as the continuum limit of a lattice approximation with a new action. Within this framework, the chiral anomaly for a fermion interacting with gauge as well as gravitational fields is calculated with mathematical rigor.     

BRST-BV quantization of gauge theories with global symmetries

We consider the general gauge theory with a closed irreducible gauge algebra possessing the non-anomalous global (super)symmetry in the case when the gauge fixing procedure violates the global invariance of classical action. The theory is quantized in the framework of BRST-BV approach in the form of functional integral over all fields of the configuration space. It is shown that the global symmetry transformations are deformed in the process of quantization and the full quantum action is invariant under such deformed global transformations in the configuration space. The deformed global transformations are calculated in an explicit form in the one-loop approximation.     

Functional integral of QED in terms of gauge-invariant quantities

We reduce the functional integral of quantum electrodynamics to an integral containing only local gauge invariant quantities. The set of (bosonic) invariants contains bilinear combinations of the spinor field and a real-valued covector field.     

Canonical quantization and the spectral action; a nice example

We study the canonical quantization of the theory given by Chamseddine–Connes spectral action on a particular finite spectral triple with algebra _M2(C)⊕C$M_2\left(\mathbb{C}\right)\oplus \mathbb{C}$. We define a quantization of the natural distance associated with this noncommutative space and show that the quantum distance operator has a discrete spectrum. We also show that it would be the same for any other geometric quantity. Finally we propose a physical Hilbert space for the quantum theory. This spectral triple had been previously considered by Rovelli as a toy model, but with a different action which was not gauge invariant. The results are similar in the two cases, but the gauge invariance of the spectral action manifests itself by the presence of a non-trivial degeneracy structure for our distance operator.     

Simplicial gauge theory and quantum gauge theory simulation

We propose a general formulation of simplicial lattice gauge theory inspired by the finite element method. Numerical tests of convergence towards continuum results are performed for several SU(2) gauge fields. Additionally, we perform simplicial Monte Carlo quantum gauge field simulations involving measurements of the action as well as differently sized Wilson loops as functions of β.     

A Gauge Model for Quantum Mechanics on a Stratified Space

In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified Kähler space, and we make explicit the requisite singular holomorphic quantization procedure on this space. On the quantum level, this procedure yields a costratified Hilbert space, that is, a Hilbert space together with a system which consists of the subspaces associated with the strata of the reduced phase space and of the corresponding orthoprojectors. The costratified Hilbert space structure reflects the stratification of the reduced phase space. For the special case where the structure group is SU(2), we discuss the tunneling probabilities between the strata, determine the energy eigenstates and study the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata in the strong and weak coupling approximations.     

Quantum corrections to ϕ 4 model solutions and applications to Heisenberg chain dynamics

The Heisenberg spin chain is considered in ? ⁴ model approximation. Quantum corrections to classical solutions of the one-dimensional ? ⁴ model within the correspondent physics are evaluated with account of rest d-1 dimensions of a d-dimensional theory. A quantization of the model is considered in terms of spacetime functional integral. The generalized zeta-function formalism is used to renormalize and evaluate the functional integral and quantum corrections to energy in a quasiclassical approximation. The results are applied to appropriate conditions of the spin chain model and its dynamics, for which elementary solutions, energy and the quantum corrections are calculated.     

Quantum dynamics of a massless relativistic string (II)

The massless relativistic free string is studied in the gauge x₀ = τ. It is found that the classical solutions include transverse and longitudinal vibrations. The problem is treated both in the Lagrangian and Hamiltonian formalism. Different ways of quantizing the system are investigated. The path integral quantization leads to a Poincaré invariant quantum theory in any number of dimensions.     

An Alternative to the Gauge Theoretic Setting

The standard formulation of quantum gauge theories results from the Lagrangian (functional integral) quantization of classical gauge theories. A more intrinsic quantum theoretical access in the spirit of Wigner’s representation theory shows that there is a fundamental clash between the pointlike localization of zero mass (vector, tensor) potentials and the Hilbert space (positivity, unitarity) structure of QT. The quantization approach has no other way than to stay with pointlike localization and sacrifice the Hilbert space whereas the approach built on the intrinsic quantum concept of modular localization keeps the Hilbert space and trades the conflict creating pointlike generation with the tightest consistent localization: semiinfinite spacelike string localization. Whereas these potentials in the presence of interactions stay quite close to associated pointlike field strengths, the interacting matter fields to which they are coupled bear the brunt of the nonlocal aspect in that they are string-generated in a way which cannot be undone by any differentiation.     

Variations on a theme of Schwinger and Weyl

The theme of doing quantum mechanics on all Abelian groups goes back to Schwinger and Weyl. This theme was studied earlier from the point of view of approximating quantum systems in infinite-dimensional spaces by those associated to finite Abelian groups. This Letter links this theme to deformation quantization, and explores the set of noncommutative associative algebra structures on the Schwartz-Weil algebra of any locally compact separable Abelian group. If the group is a vector space of even dimension over a non-Archimedean local fieldK, there exists a family of noncommutative (Moyal) structures parametrized by the local field and containing membersarbitrarily close to the classical one, although the classical algebra is rigid in the sense of deformation theory. The-products are defined by Fourier integral operators. The problem of constructing sucharithmetic Moyal structures on the algebra of Schwartz-Bruhat functions on manifolds that are locally likeK ²ⁿ is raised.In memory of Julian Schwinger     

Self-dual Yang-Mills fields in d=7,8, octonions and ward equations

The Yang-Mills theories in d=7 and d=8 with the arbitrary gauge group G are considered. Generalized self-duality-type relations for gauge fields are reduced to systems of nonlinear differential equations on functions of one variable (Ward equations). Ward equations may be reduced to equations which follow from Yang-Baxter equations. This permits us to obtain new classes of explicit solutions of the Yang-Mills equations in d=7 and d=8.     

Monopoles and topological field theory

We present a topological quantum field theory for magnetic monopoles in an SU(N) Yang-Mills-Higgs model. This field theory is obtained by gauge fixing the topological action defining the monopole charge. This work extends to the three-dimensional case the quantization of invariant polynomials in four dimensions. We choose the Bogomolny self-duality equations as gauge conditions for the magnetic monopole topological field theory. In this way the geometrical equation discussed e.g. in Atiyah and Hitchin's work are recovered as ghost equations of motion. We give the cocycles of the corresponding topological symmetry. In the N→∞ limit interesting phenomena occur. The functional integration is forced to cover only the moduli space and the role of the ghosts stemming from the gauge fixing process is to provide a smooth semiclassical approximation.     

Realization of the Schwinger term in the Gauss law and the possibility of correct quantization of a theory with anomalies

The explicit regularization of the Weyl fermion current leading to the anomalous commutation relation of the Gauss law constraint is established. This requires the modification of the canonical quantization of the corresponding gauge model. The modified form of functional integral quantization for this model is proposed.     

BFV quantization on hermitian symmetric spaces

Gauge-invariant BFV approach to geometric quantization is applied to the case of hermitian symmetric spaces G/H. In particular, gauge invariant quantization on the Lobachevski plane and sphere is carried out. Due to the presence of symmetry, master equations for the first-class constraints, quantum observables and physical quantum states are exactly solvable. BFV-BRST operator defines a flat G-connection in the Fock bundle over G/H. Physical quantum states are covariantly constant sections with respect to this connection and are shown to coincide with the generalized coherent states for the group G. Vacuum expectation values of the quantum observables commuting with the quantum first-class constraints reduce to the covariant symbols of Berezin. The gauge-invariant approach to quantization on symplectic manifolds synthesizes geometric, deformation and Berezin quantization approaches.     

Resummation of Mass Terms in Perturbative Massless Quantum Field Theory

The neutral massless scalar quantum field Φ in four-dimensional space-time is considered, which is subject to a simple bilinear self-interaction. Is is well-known from renormalization theory that adding a term of the form to the Lagrangean has the formal effect of shifting the particle mass from the original zero value to m after resummation of all two-leg insertions in the Feynman graphs appearing in the perturbative expansion of the S-matrix. However, this resummation is accompanied by some subtleties if done in a proper mathematical manner. Although the model seems to be almost trivial, is shows many interesting features which are useful for the understanding of the convergence behavior of perturbation theory in general. Some important facts in connection with the basic principles of quantum field theory and distribution theory are highlighted, and a remark is made on possible generalizations of the distribution spaces used in local quantum field theory. A short discussion how one can view the spontaneous breakdown of gauge symmetry in massive gauge theories within a massless framework is presented.      

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.     

The QCD Hamiltonian and nonlinear quantum mechanics

We study the canonical quantization of SU(N) gauge theory in linear, noncovariant gauges. The canonical formalism is first discussed for the classical theory, with special attention to the features involving nonlinearity and the gauge degrees of freedom. The transition to the quantum theory is then performed for an arbitrary linear gauge, using the covariant quantization rules of nonlinear quantum mechanics. When the quantum Hamiltonian is written in the Weyl-ordered form appropriate for the application of the usual Dyson-Wick perturbative techniques, additional ordering terms appear with respects to the classical Hamiltonian. We discuss the relation of our results to those of previous authors, and the relevance of the ordering terms in field theory.     

SU(3) Local Gauge Field Theory as Effective Dynamics of Composite Gluons

The effective dynamics of quarks is described by a nonperturbatively regularized NJL model equation with canonical quantization and probability interpretation. The quantum theory of this model is formulated in functional space and the gluons are considered as relativistic bound states of colored quark-antiquark pairs. Their wave functions are calculated as eigenstates of hardcore equations, and their effective dynamics is derived by weak mapping in functional space. This leads to the phenomenological SU(3) gauge invariant gluon equations in functional formulation, i.e., the local gauge symmetry is a dynamical effect resulting from the dynamics of the quark model.     

Generalized link-invariants on 3-manifolds ∑h × [0, 1] from Chern-Simons gauge and gravity theories

We show that the recently found generalized Jones and Homfly polynomials for links in _h × [0, 1], where _h is a closed oriented Riemann surface, may be also obtained by the canonical quantization of a Chern-Simons non-Abelian gauge theory on _h × [0, 1]. As a particular case, one may consider the 2+1-dimensional Euclidean quantum gravity with a positive cosmological constant.