On Time-Space Noncommutativity for Transition Processes and Noncommutative Symmetries

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 ([x̂μ, x̂ν ] = iθμν , θ0i ≡/ 0, θij = 0). Space rotations and parity are not automorphisms of this algebra and are not symmetries of quantum physics. Still, the spectral degeneracies of a time-independent Hamiltonian due to symmetries do not depend at all on θ0i. They give no clue about rotation and parity violation when θ0i ≡/ 0. The persistence of degeneracies for θ0i ≡/ 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. As an illustration, we calculate the rate for the process 2s → 1s + γ in hydrogen as a function of θ0i. (It is zero for θ0i = 0 by parity conservation). The importance of the deformed rotational symmetry is commented upon further using the decay Z0 → 2γ as an example. e-mail: [email protected] e-mail: [email protected]