On deformations of classical mechanics due to Planck-scale physics

Abstract

<jats:p> 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. </jats:p>

Free fall in KvN mechanics and Einstein's principle of equivalence

“…and thus KvN phase factors can potentially also lead to some physical effects, but the expected magnitude of such effects is vanishingly small [87].…”

“…As we see, the presence of the arbitrary function f (x, p, t) in (D.24) does not affect the observed probability density ρ and, thus, it can be safely ignored. In fact, this gauge freedom in the choice of the Hamiltonian (D.24) (the Liouville operator) is related to the invariance of the KvN probability density function under the phase transformation of the corresponding wave function [87] and reflects the fact that the phase of the wave function is irrelevant in classical theory.…”

“…In this representation x and X operators are diagonal, while Hilbert space approach to classical mechanics was pioneered by Koopman [13] and von Neumann [14]. A detailed up-to-date pedagogical review is given in [90], where the interested reader can also find relevant references regarding contemporary research related to KvN mechanics (in this regard, see also [87]).…”

Position in Models of Quantum Mechanics with a Minimal Length

“…Several different approaches to GUP have been proposed in the past. In fact, GUP can be cast as a modification of the uncertainty relation between position and momentum, with no necessary modification of the representations of the corresponding operators [9][10][11]; alternatively, modifications of Poisson brackets have been considered in the classical sector, inducing modified classical dynamics [12][13][14][15]; finally, a modified commutation relation between position and momentum is studied, related with a modified uncertainty relation via Schrödinger-Robertson relation [16][17][18][19]. In this work, we will consider this last approach.…”

Towards quantum Mechanics on the curved cotangent bundle

“…Therefore, understanding the maximal wavelength as the radius of the cosmological horizon, it combines quantum gravitational as well as horizon-induced effects. Despite a number of open issues such as the inverse soccer ball problem [28], the classical limit [77,78], friction with the second law of thermodynamics [79], combination with gauge symmetries [80] or, depending on the relativistic completion, deformation [81] or straight breaking [82] of Lorentz-invariance, this endeavour has lead to manifold phenomenological applications [83][84][85][86][87][88][89][90][91][92][93][94][95][96][97].…”