Research output: Contribution to journal › Article › peer-review
On deformations of classical mechanics due to Planck-scale physics. / Chashchina, Olga I.; Sen, Abhijit; Silagadze, Zurab K.
In: International Journal of Modern Physics D, Vol. 29, No. 10, 2050070, 01.07.2020.Research output: Contribution to journal › Article › peer-review
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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 -
ID: 25678388