Path integrals in configuration space in weakly relativistic many-body theory

Henrique Gomes

R Rosenfelder

Luttinger and Lu(1980) have derived a variational method for representing the Yukawa interaction of bosons and fermions in non-relativistic field theories by a general potential between fermions and fictitious particles ('shadows'). In the present work, this is considerably improved by applying Jensen's inequality only once and representing the kinetic term in the path-integral average as a quantum-mechanical N-body problem using the replica technique. Various weak- and strong-coupling approximations as well as variational bounds for this term are discussed.

Gian Fabrizio De Angelis, Maurizio Serva

Ian H. Redmount, Wai-Mo Suen

The simple physics of a free particle reveals important features of the path-integral formulation of relativistic quantum theories. The exact quantum-mechanical propagator is calculated here for a particle described by the simple relativistic action proportional to its proper time. This propagator is nonvanishing outside the light cone, implying that spacelike trajectories must be included in the path integral. The propagator matches the WKB approximation to the corresponding configuration-space path integral far from the light cone; outside the light cone that approximation consists of the contribution from a single spacelike geodesic. This propagator also has the unusual property that its short-time limit does not coincide with the WKB approximation, making the construction of a concrete skeletonized version of the path integral more complicated than in nonrelativistic theory.

A. J. K�lnay

Dah-Wei Chiou

Starting from the canonical formalism of relativistic (timeless) quantum mechanics, the formulation of a timeless path integral is rigorously derived. The transition amplitude is reformulated as the sum, or functional integral, over all possible paths in the constraint surface specified by the (relativistic) Hamiltonian constraint, and each path contributes with a phase identical to the classical action divided by ℏ. The timeless path integral manifests the timeless feature as it is completely independent of the parametrization for paths. For the special case that the Hamiltonian constraint is a quadratic polynomial in momenta, the transition amplitude admits the timeless Feynman's path integral over the (relativistic) configuration space. Meanwhile, the difference between relativistic quantum mechanics and conventional nonrelativistic (with time) quantum mechanics is elaborated on in light of the timeless path integral.