TY - JOUR

T1 - Lévy-Leblond equation and Eisenhart–Duval lift in Koopman–von Neumann mechanics

AU - Parida, Bikram Keshari

AU - Sen, Abhijit

AU - Dhasmana, Shailesh

AU - Silagadze, Zurab K.

PY - 2023

Y1 - 2023

N2 - The Koopman–von Neumann (KvN) mechanics is an approach that was formulated long ago to answer the question regarding the existence of a Hilbert space representation of classical mechanics. KvN mechanics is a non-relativistic theory, and it is not clear how spin can be included in it, since spin is widely regarded as a relativistic property. Cabrera et al., in Eur. Phys. J. Spec. Top. 227, 2195 (2019) argued that the Spohn equation [Spohn, Ann. Phys. 282, 420 (2000)] is the correct classical framework for the Koopman–von Neumann theory corresponding to the Dirac equation. However, after Lévy-Leblond’s seminal work on this topic, it became clear that spin naturally arises also from the Galilean invariant wave equations, without any need of relativistic considerations. Inspired by this, we propose another possibility of including spin in the KvN formalism: the Lévy-Leblond equation in the Koopman–von Neumann formalism can be obtained as a null reduction of the massless Dirac equation in the Eisenhart–Duval lift metric. To illustrate the idea, we implement it for a one-dimensional classical system without magnetic interactions.

AB - The Koopman–von Neumann (KvN) mechanics is an approach that was formulated long ago to answer the question regarding the existence of a Hilbert space representation of classical mechanics. KvN mechanics is a non-relativistic theory, and it is not clear how spin can be included in it, since spin is widely regarded as a relativistic property. Cabrera et al., in Eur. Phys. J. Spec. Top. 227, 2195 (2019) argued that the Spohn equation [Spohn, Ann. Phys. 282, 420 (2000)] is the correct classical framework for the Koopman–von Neumann theory corresponding to the Dirac equation. However, after Lévy-Leblond’s seminal work on this topic, it became clear that spin naturally arises also from the Galilean invariant wave equations, without any need of relativistic considerations. Inspired by this, we propose another possibility of including spin in the KvN formalism: the Lévy-Leblond equation in the Koopman–von Neumann formalism can be obtained as a null reduction of the massless Dirac equation in the Eisenhart–Duval lift metric. To illustrate the idea, we implement it for a one-dimensional classical system without magnetic interactions.

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85177056137&origin=inward&txGid=33a073414fc7c8ced5ccb25676adb259

UR - https://www.mendeley.com/catalogue/30f0f070-a515-308b-b059-d6fad36b0c73/

U2 - 10.1142/s0217732323501493

DO - 10.1142/s0217732323501493

M3 - Article

JO - Modern Physics Letters A

JF - Modern Physics Letters A

SN - 0217-7323

M1 - 2350149

ER -