Relativistic Feynman-type integrals

Takashi Ichinose

Jan Tarski

M. Asorey, Alberto Ibort, Giuseppe Marmo

The path integral approach to quantum mechanics provides a method of quantization of dynamical systems directly from the Lagrange formalism. In field theory the method presents some advantages over Hamiltonian quantization. The Lagrange formalism preserves relativistic covariance which makes the Feynman method very convenient to achieve the renormalization of field theories both in perturbative and non-perturbative approaches. However, when the systems are confined in bounded domains we shall show that the path integral approach does not describe the most general type of boundary conditions. Highly non-local boundary conditions cannot be described by Feynman's approach. We analyse in this note the origin of this problem in quantum mechanics and its implications for field theory.

Vladimir GIvancevic, Tijana TIvancevic

Georg Junker, A. Inomata

The resolvent of supersymmetric Dirac Hamiltonian is studied in detail. Due to supersymmetry the squared Dirac Hamiltonian becomes block-diagonal whose elements are in essence non-relativistic Schrödinger-type Hamiltonians. This enables us to find a Feynman-type path-integral representation of the resulting Green’s functions. In addition, we are also able to express the spectral properties of the supersymmetric Dirac Hamiltonian in terms of those of the non-relativistic Schrödinger Hamiltonians. The methods are explicitly applied to the free Dirac Hamiltonian, the so-called Dirac oscillator and a generalization of it. The general approach is applicable to systems with good and broken supersymmetry.