Understanding the Path from Classical to Quantum Mechanics

G. Aravind, Ayush Goyal, Narasimhaiengar Mukunda +1 more

Jean Zinn‐Justin

Path integrals are mathematical objects that can be considered as generalizations to an infinite number of variables, represented by paths, of usual integrals. They share the algebraic properties of usual integrals, but have new properties from the viewpoint of analysis. They are powerful tools for the study of quantum mechanics, since they emphasize very explicitly the correspondence between classical and quantum mechanics. Physical quantities are expressed as averages over all possible paths but, in the semi-classical limit, the leading contributions come from paths close to classical paths. Thus, path integrals lead to an intuitive understanding of physical quantities in the semi-classical limit, as well as simple calculations of such quantities. This observation can be illustrated with scattering processes, spectral properties, or barrier penetration effects. Even though the formulation of quantum mechanics based on path integrals seems mathematically more complicated than the usual formulation based on partial differential equations, the path integral formulation is well adapted to systems with many degrees of freedom, where a formalism of Schrödinger type is much less useful. It allows simple construction of a many-body theory both for bosons and fermions.

R. J. Rivers

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 redolent of quantum mechanics; additional dimensions permit ‘tunnelling’ without recourse to instantons and time/energy uncertainties exist. In practice, ‘classical’ particle trajectories with additional degrees of freedom arise 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 ℏ 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.

MASSIMO RUDAN

MASSIMO RUDAN

Jianbin Liu, Yu Zhou, Hui Chen +4 more

Compared to classical optical coherence theory based on Maxwell's electromagnetic theory and Glauber's quantum optical coherence theory based on matrix mechanics formulation of quantum mechanics, quantum optical coherence theory based on Feynman's path integral formulation of quantum mechanics provides a novel tool to study optical coherence. It has the advantage of understanding the connection between mathematical calculations and physical interpretations better. Quantum optical coherence theory based on Feynman's path integral is introduced and reviewed in this paper. Based on the results of transient first-order interference of two independent light beams, it is predicted that the classical model for electric field of thermal light introduced by classical optical textbooks may not be accurate. The physics of two-photon bunching of thermal light and Hong-Ou-Mandel dip of entangled photon pairs is the same, which can be interpreted by constructive and destructive two-photon interference, respectively. Quantum optical coherence theory based on Feynman's path integral is helpful to understand the coherence properties of light, which may eventually lead us to the answer of the question: what is a photon?