Path Integrals for (Complex) Classical and Quantum Mechanics

TL;DR

This paper explores how extending classical phase space into complex domains reveals behaviors akin to quantum phenomena, such as tunneling and uncertainty, and discusses implications for understanding quantization.

Contribution

It compares complex classical mechanics with quantum formalisms, highlighting the role of $ar{}$ in bridging classical and quantum descriptions.

Findings

Complex phase space allows quantum-like tunneling without instantons.

Classical trajectories with extra degrees of freedom mimic quantum uncertainties.

Differences in trajectory usage complicate direct comparisons between formalisms.

Abstract

An analysis of classical mechanics in a complex extension of phase space shows that a particle in such a space can behave in a way redolant of quantum mechanics; additional degrees of freedom permit 'tunnelling' without recourse to instantons and lead to time/energy uncertainty. In practice, 'classical' particle trajectories with additional degrees of freedom have arisen in several different formulations of quantum mechanics. In this talk we compare the extended phase space of the closed time-path formalism with that of complex classical mechanics, to suggest that $\hbar$ has a role in our understanding of the latter. However, differences in the way that trajectories are used make a deeper comparison problematical. We conclude with some thoughts on quantisation as dimensional reduction.