Timeless path integral for relativistic quantum mechanics

Abstract

Starting from the canonical formalism of relativistic (timeless) quantum mechanics, the formulation of 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 $\hslash$. 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 timeless path integral.

41 pages; more references and comments added; version to appear in CQG

References in corpus (7)

• LQG vertex with finite Immirzi parameter
• Experimental realization of Wheeler's delayed-choice GedankenExperiment
• The loop-quantum-gravity vertex-amplitude
• Casting Loop Quantum Cosmology in the Spin Foam Paradigm
• Loop Quantum Cosmology and Spin Foams
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Cited by in corpus (4)

• Loop Quantum Gravity
• Timeless configuration space and the emergence of classical behavior
• Delayed-choice quantum erasers and the Einstein-Podolsky-Rosen paradox
• Transition probability spaces in loop quantum gravity