On the modification of the commutation relations in generalized mechanics

Joseph V. Lepore

Hadrien Chevalier, Hyukjoon Kwon, Kiran E. Khosla +2 more

Many theories of quantum gravity can be understood as imposing a minimum length scale the signatures of which can potentially be seen in precise table top experiments. In this work, we inspect the capacity for correlated many-body systems to probe non-classicalities of spacetime through modifications of the commutation relations. We find an analytic derivation of the dynamics for a single mode light field interacting with a single mechanical oscillator and with coupled oscillators to first order corrections to the commutation relations. Our solution is valid for any coupling function as we work out the full Magnus expansion. We numerically show that it is possible to have superquadratic scaling of a nonstandard phase term, arising from the modification to the commutation relations, with coupled mechanical oscillators.

Richard Bourret

Modified commutation relations which have been used in a classical stochastic context are shown to be similar, in a simple quantum mechanical example, to the modification of the potential. In quantum field theory their effect is to yield a nonlinear field equation for a free scalar boson. It is concluded that such modified commutation relations are incompatible with quantum mechanics.

Yoshiaki Hashimoto

H. Calisto, Constantino A. Utreras-Díaz

In this article we review some results obtained from a generalization of quantum mechanics obtained from modification of the canonical commutation relation $[q,p]={\rm i}\hbar$. We present some new results concerning relativistic generalizations of previous works, and we calculate the energy spectrum of some simple quantum systems, using the position and momentum operators of this new formalism.

Calvin R. Putnam