Using the recently developed groupoidal description of Schwinger's picture of Quantum Mechanics, a new approach to Dirac's fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call q-Lagrangian, can be described in terms of a new function on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold M, the quadratic expansion of will reproduce the standard Lagrangians on TM used to describe the classical dynamics of particles.
A quantum route to the classical Lagrangian formalism / Ciaglia, F. M.; Di Cosmo, F.; Ibort, A.; Marmo, G.; Schiavone, L.; Zampini, A.. - In: MODERN PHYSICS LETTERS A. - ISSN 0217-7323. - 36:15(2021), p. 2150091. [10.1142/S0217732321500917]
A quantum route to the classical Lagrangian formalism
Ciaglia F. M.;Di Cosmo F.;Marmo G.;Schiavone L.;Zampini A.
2021
Abstract
Using the recently developed groupoidal description of Schwinger's picture of Quantum Mechanics, a new approach to Dirac's fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call q-Lagrangian, can be described in terms of a new function on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold M, the quadratic expansion of will reproduce the standard Lagrangians on TM used to describe the classical dynamics of particles.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.