Scattering on the hyperbolic plane in the Aharonov-Bohm gauge field

The scattering of a quantum charged particle on the hyperbolic plane in the Abelian Aharonov-Bohm field is considered. S-matrix, scattering amplitude and its flat space limit are constructed in a closed form by the Sommerfeld integral approach.     

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.     

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

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.     

Star-product approach to quantum field theory: The free scalar field

The star-quantization of the free scalar field is developed by introducing an integral representation of the normal star-product. A formal connection between the Feynman path integral in the holomorphic representation and the star-exponential is established for the interacting scalar fields.     

Elliptic Wess-Zumino-Witten model from elliptic Chern-Simons theory

This Letter continues the program aimed at analysing of the scalar product of states in the Chern-Simons theory. It treats the elliptic case with group SU₂. The formal scalar product is expressed as a multiple finite-dimensional integral which, if convergent for every state, provides the space of states with a Hilbert space structure. The convergence is checked for states with a single Wilson line where the integral expressions encode the Bethe Ansatz solutions of the Lamé equation. In relation to the Wess-Zumino-Witten conformal field theory, the scalar product renders unitary the Knizhnik-Zamolodchikov-Bernard connection and gives a pairing between conformal blocks used to obtain the genus-one correlation functions.     

On a Selberg–Schur Integral

A generalization of Selberg’s beta integral involving Schur polynomials associated with partitions with entries not greater than 2 is explicitly computed. The complex version of this integral is given after proving a general statement concerning the complex extensions of Selberg–Schur integrals. All these results have interesting applications in both mathematics and physics, particularly in conformal field theory, because the conformal blocks for the Wess–Zumino–Novikov–Witten model with underlying affine structure can be obtained by analytical continuation of these integrals.      

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     

Functional integral in supersymmetric quantum mechanics

The solution of the square root of the Schrödinger equation for supersymmetric quantum mechanics is expressed in the form of series. The formula may be considered as a functional integral of the chronological exponent of the superpseudodifferential operator symbol over the superspace.     

The star exponential and path integrals on compact groups II: The weyl character formula

This is a continuation of the work begun by Cadavid and Nakashima inLett. Math. Phys. 23, 111–115 (1991). An expression for the Weyl Character Formula is obtained in terms of the star-product path integral; and the relationship between the star-product path integral and the path integral developed on coadjoint orbits is established.     

Functional integral and the Feynman-Kac formula in superspace

We consider the Cauchy problem for linear pseudodifferential equations in superspace. The solution is constructed in the form of series. It may be regarded as a definition of a chronological exponent of a pseudodifferential operator symbol and interpreted as a functional integral in superspace.     

The star exponential and path integrals on compact groups

Using the star exponential, a path integral is constructed for arbitrary compact Lie groups. This is a globalization of the results of Alcalde, Cadavid, and Nakashima, in which a path integral was constructed for SU(2) in terms of local coordinates. Also the phase factor encountered by Alcalde, Cadavid, and Nakashima is shown to be zero.     

Functional integral via functional equation

Discretization ofp-adic Grassmann-valued -model leads to a hierarchical model with the Hamtilonian given by a nontrivial functional integral over the Grassmann variables. Using renormalization group arguments, we reduce the calculation of this integral to a functional equation. The problem of the convergence of the perturbation expansion of this integral, realized as a small-divisors problem, is investigated.     

The star exponential and SU(2) path integrals

An expression is found for the Feynman path integral on SU(2) in terms of the star-exponential.     

Scattering theory for a model of an atom in a quantized field

We compute the scattering operator for a simplified model of an atom interacting a with a quantized field. The field is restricted to the vacuum and one-particle sectors, and the atom has only two states. We also solve the inverse scattering problem for the same model. The methods used rely on the particular form of the interaction, which is chosen to mimic the interaction between radiation and matter.This work was partially supported by NSF Grant DMS-8922941.     

Stochastic quantisation of a gauge field in the spatial axial gauge

The stochastic quantisation method of Nelson can be applied to quantise an Abelian gauge field in the spatial axial gauge.     

Spinor electrodynamics in terms of gauge-invariant quantities

We give a formulation of classical spinor electrodynamics in terms of gauge-invariant quantities. The set of invariants consists of bilinear combinations of spinor fields (currents), a real-valued covector field, and a complex scalar field of modulus one. The presented result is a first step towards formulating quantum electrodynamics in terms of gauge-invariant fields.     

Induced classical gravity

We review the induced-gravity approach according to which the Einstein gravity is a long-wavelength effect induced by underlying fundamental quantum fields due to the dynamical-scale symmetry breaking. It is shown that no ambiguities arise in the definition of the induced Newton and cosmological constants if one works with the path integral for fundamental fields in the low-scale region. The main accent is on a specification of the path integral which enables us to utilize the unitarity condition and thereby avoid ambiguities. Induced Einstein equations appear from the symmetry condition that the path integral of fundamental fields for a slowly varying metric is invariant under the local GL(4, R)-transformations of a tetrad, which contain the local Euclidean Lorentz, O(4)-rotations as a subgroup. The relationship to induced quantum gravity is briefly outlined.     

Krichever-Novikov formulation of topological conformal field theory

The Krichever-Novikov (KN) global operator formalism is applied to construct a topological conformal field theory on a compact Riemann surface from an N=2 super-conformal field theory. The topological version of the KN algebra is derived and the BRST charge is shown to be genus-dependent in this formulation. This leads to an interesting cohomology structure for the physical subspace of the Hilbert space.     

Non-relativistic Limit of a Dirac Polaron in Relativistic Quantum Electrodynamics

A quantum system of a Dirac particle interacting with the quantum radiation field is considered in the case where no external potentials exist. Then the total momentum of the system is conserved and the total Hamiltonian is unitarily equivalent to the direct integral of a family of self-adjoint operators acting in the Hilbert space , where is the Hilbert space of the quantum radiation field. The fiber operator is called the Hamiltonian of the Dirac polaron with total momentum . The main result of this paper is concerned with the non-relativistic (scaling) limit of . It is proven that the non-relativistic limit of yields a self-adjoint extension of a Hamiltonian of a polaron with spin 1/2 in non-relativistic quantum electrodynamics.     

Anomalous quantization of the free Dirac field in three-dimensional spacetime

The possibility for Bose quantization of the free spinor field in three-dimensional spacetime is demonstrated, compared with Fermi quantization, and discussed.This work is supported in part by the Bulgarian National Foundation for Scientific Research (Ph-20-1991).