Path Integral Formulation of Relativistic Quantum Mechanics

Gian Fabrizio De Angelis, Maurizio Serva

V. Pažma

Gian Fabrizio De Angelis, Maurizio Serva

David Bailin, A. Love

Feynman showed that quantum mechanics could be formulated in terms of path integrals. This chapter discusses this approach to quantum mechanics in some detail since it provides the key to the path integral formulation of quantum field theory. It considers a system described by a single generalised coordinate with a conjugate momentum. The chapter shows that analogous objects in non-relativistic quantum mechanics are ground-state expectation values of products of operators.

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.

Leon A. Takhtajan