TY - JOUR

T1 - On deformations of classical mechanics due to Planck-scale physics

AU - Chashchina, Olga I.

AU - Sen, Abhijit

AU - Silagadze, Zurab K.

PY - 2020/7/1

Y1 - 2020/7/1

N2 - Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman-von Neumann's Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.

AB - Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman-von Neumann's Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.

KW - classical-quantum dynamics

KW - generalized commutation relations

KW - Koopman-von Neumann theory

KW - NONCOMMUTATIVE GEOMETRY

KW - GENERALIZED UNCERTAINTY PRINCIPLE

KW - LENGTH

KW - GRAVITATIONAL-FIELD

KW - GRAVITY

KW - LIENARD EQUATION

KW - SYSTEMS

KW - QUANTUM-MECHANICS

KW - QUANTIZATION

KW - GEODESIC DEVIATION

UR - http://www.scopus.com/inward/record.url?scp=85090570331&partnerID=8YFLogxK

U2 - 10.1142/S0218271820500704

DO - 10.1142/S0218271820500704

M3 - Article

AN - SCOPUS:85090570331

VL - 29

JO - International Journal of Modern Physics D

JF - International Journal of Modern Physics D

SN - 0218-2718

IS - 10

M1 - 2050070

ER -