Feynman-Schwinger technique in field theories

In these lectures we introduce the Feynman-Schwinger representation method for solying nonperturbative problems in field theory. As an introduction we first give a brief overview of integral equations and path integral methods for solving nonperturbative problems. Then we discuss the Feynman-Schwinger (FSR) representation method with applications to scalar interactions. The FSR approach is a continuum path integral integral approach in terms of covariant trajectories of particles. Using the exact results provided by the FSR approach we test the reliability of commonly used approximations for nonperturbative summation of interactions for few body systems.     

Green's electron function in a quantized plane wave field

It is shown that the Green's function of an electron that interacts with a quantized plane wave can be expressed in terms of the corresponding Green's function of a scalar particle. By using the known expression for the Green's function of a scalar particle, an integral representation is found with respect to the intrinsic time for the Green's electron function in a quantized plane wave of arbitrary form.     

Feynman diagrams to three loops in three-dimensional field theory

The two-point integrals contributing to the self-energy of a particle in a three-dimensional quantum field theory are calculated to two-loop order in perturbation theory as well as the vacuum ones contributing to the effective potential to three-loop order. For almost every integral an expression in terms of elementary and dilogarithm functions is obtained. For two integrals, the master integral and the Mercedes integral, a one-dimensional integral representation is obtained with an integrand consisting only of elementary functions. The results are applied to a scalar λφ⁴ theory.     

Relativistic quantum field theory with fractional spin and statistics

We construct a relativistic quantum field theory in 2 + 1 dimensions whose Fock states provide a multivalued representation of the Poincaré group. We add a topological term to the action of a scalar field theory and we show that this endows the path integral of the theory with an operator-valued cocycle which modifies the transformation properties of physical states. We demonstrate that one-particle states carry (in general) fractional spin. We determine the spin of many-particle states and we prove a generalized spin-statistics relation. We propose an equation of motion for on-shell states which generalizes naturally the Dirac equation.     

On the geometrical representation of the path integral reduction Jacobian: The case of dependent variables in a description of reduced motion

The geometrical representation of the path integral reduction Jacobian obtained in the problem of the path integral quantization of a scalar particle motion on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group has been found for the case when the local reduced motion is described by means of dependent coordinates. The result is based on the scalar curvature formula for the original manifold which is viewed as a total space of the principal fiber bundle.     

An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs. This integral representation is a suitable starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry. Research supported by NSERC grant # RGPIN 105490-2004. Research supported in part by the NSF, Grant No. DMS-010-3998. Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

Simple one-dimensional integral representations for two-loop self-energies: the master diagram

The scalar two-loop self-energy master diagram is studied in the case of arbitrary masses. Analytical results in terms of Lauricella- and Appell-functions are presented for the imaginary part. By using the dispersion relation a one-dimensional integral representation is derived. This representation uses only elementary functions and is thus well suited for a numerical calculation of the master diagram.     

On the geometrical representation of the Jacobian in the path integral reduction problem

By using the formula for the scalar curvature of the manifold with the Kaluza-Klein metric we obtain the geometrical representation of the Jacobian resulted from the path integral reduction problem in Wiener path integrals for a scalar particle on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group.     

Supersymmetrization of scalar field theories

We develop a formalism for constructing the vacuum functional and supersymmetrizing a scalar field theory with the help of its ground state representation. The field theory problem is first transformed into a quantum mechanical one for which the ground state representation is well defined. The theory is then supersymmetrized by “taking the square root” of the hamiltonian. Standard approximation techniques are used to construct the vacuum functional with which spontaneous supersymmetry breaking can be analyzed.     

Asymptotic expansion of Feynman amplitudes

Employing the technique of Mellin transforms to scalar convergent Feynman amplitude in the Schwinger integral representation, we determine its asymptotic expansion for large Euclidean momenta. The determination of the coefficients of the expansion is effected via the use of generalized Taylor operators.     

Uniform acceleration and the quantum field theory vacuum,I

Free massless scalar and Dirac fields are quantized in two-dimensional Minkowski space in a representation in which the energy operator for a uniformly accelerating observer is diagonalized. The usual vacuum, and many other pure states, appear to be thermally excited in this representation. The crucial role ot PCT symmetry, as regards the general validity of this conclusion, is illustrated.     

New representation of two-loop propagator and vertex functions

We present a new method of calculating scalar propagator and vertex functions in the two-loop approximation, for arbitrary masses of particles. It is based on a double integral representation, suitable for numerical evaluation. Real and imaginary parts of the diagrams are calculated separately, so that there is no need to use complex arithmetics in the numerical program.     

Solutions of the Bethe-Salpeter Equation in Minkowski Space: A Comparative Study

The Bethe–Salpeter equation for a two-scalar, S-wave bound system, interacting through a massive scalar, is investigated within the ladder approximation. By assuming a Nakanishi integral representation of the Bethe–Salpeter amplitude, one can deduce new integral equations that can be solved and quantitatively studied, overcoming the analytic difficulties of the Minkowski space. Finally, it is shown that the Light-front distributions of the valence state, directly obtained from the Bethe–Salpeter amplitude, open an effective window for studying the two-body dynamics.     

Zero momentum calculations for finite temperature field theory

Problems with calculations of Feynaman diagrams with no external leg carrying non-zero momentum for a finite temperature scalar field theory using a real time path integral approach are considered. The necessary extra rules for all such calculations are derived, covering all the usual methods of calculating the effective potential, and several problems with such calculations from the path integral viewpoint are resolved.     

Finite quantum field theory in noncommutative geometry

We describe a self-interacting scalar field on a truncated sphere and perform the quantization using the functional (path) integral approach. The theory possesses full symmetry with respect to the isometries of the sphere. We explicitly show that the model is finite and that UV regularization automatically takes place.     

An exact representation of the space-time characteristic functional of turbulent Navier-Stokes flows with prescribed random initial states and driving forces

By adopting a formal operator viewpoint, the space-time characteristic functional associated with Navier-Stokes turbulence is expressed in terms of a linear operator acting on the space of functionals. Obtained by a simple similarity transformation of the local translation operator generated by the nonlinear terms in the Navier-Stokes equation, this operator is unitary with respect to the formal scalar product of functionals. The equivalence of this operator representation to the functional integral representation of Rosen is shown and, for Gaussian initial velocity and external force fields, some consequences of this representation are presented.     

Integral Representation for the Eigenfunctions of a Quantum Periodic Toda Chain

An integral representation for the eigenfunctions of a quantum periodic Toda chain is constructed for the N-particle case. The multiple integral is calculated using the Cauchy residue formula. This gives the representation which reproduces the particular results obtained by Gutzwiller for the N=2,3 and 4-particle chain. Our method of solving the problem combines the ideas of Gutzwiller and the R-matrix approach of Sklyanin with the classical results in the theory of Whittaker functions. In particular, we calculate Sklyanin's invariant scalar product from the Plancherel formula for the Whittaker functions derived by Semenov-Tian-Shansky thus obtaining a natural interpretation of the Sklyanin measure in terms of the Harish-Chandra function.     

Vacuum polarization of a scalar field on the nonunimodular lie groups

A method for calculating the vacuum average energy-momentum tensors for a scalar field on the nonunimodular Lie groups is developed. To solve this problem, a method of generalized harmonic analysis on the Lie groups is used based on the method of orbits of coadjoint representation.     

Path integral quantization of field theories with second-class constraints

Faddeev's Hamiltonian path integral method for singular Lagrangians is generalized to the case when second-class constraints appear in the theory. The general formalism is then applied to several problems: quantization of the massive Yang-Mills field theory, light-cone quantization of the self-interacting scalar field theory, and quantization of a local field theory of magnetic monopoles.