On Time-Space Noncommutativity for Transition Processes and Noncommutative Symmetries

Abstract

We explore the consequences of time-space noncommutativity in the quantum mechanics of atoms and molecules, focusing on the Moyal plane with just time-space noncommutativity ($[\hat{x}_\mu ,\hat{x}_\nu]=i\theta_{\mu\nu}$, \theta_{0i}\neqq 0, $\theta_{ij}=0$). Space rotations and parity are not automorphisms of this algebra and are not symmetries of quantum physics. Still, when there are spectral degeneracies of a time-independent Hamiltonian on a commutative space-time which are due to symmetries, they persist when \theta_{0i}\neqq 0; they do not depend at all on $\theta_{0i}$. They give no clue about rotation and parity violation when \theta_{0i}\neqq 0. The persistence of degeneracies for \theta_{0i}\neqq 0 can be understood in terms of invariance under deformed noncommutative ``rotations'' and ``parity''. They are not spatial rotations and reflection. We explain such deformed symmetries. We emphasize the significance of time-dependent perturbations (for example, due to time-dependent electromagnetic fields) to observe noncommutativity. The formalism for treating transition processes is illustrated by the example of nonrelativistic hydrogen atom interacting with quantized electromagnetic field. In the tree approximation, the $2s\to 1s +\gamma$ transition for hydrogen is zero in the commutative case. As an example, we show that it is zero in the same approximation for $\theta_{0i}\ne 0$. The importance of the deformed rotational symmetry is commented upon further using the decay $Z^0 \to 2\gamma$ as an example.Comment: 13 pages, revised version, references adde