Phase–number uncertainty from Weyl commutation relations

Abstract

We derive suitable uncertainty relations for characteristics functions of phase and number variables obtained from the Weyl form of commutation relations. This is applied to finite-dimensional spinlike systems, which is the case when describing the phase difference between two field modes, as well as to the phase and number of a single-mode field. Some contradictions between the product and sums of characteristic functions are noted.

Uncertainty relations for characteristic functions

“…Moreover, since Theorem 1 involves operators of the form e iλÔ it becomes valuable when the operatorÔ does not exist itself (consequences of so called Stone-von Neumann theorem). A particularly interesting example of the number-phase uncertainty [58][59][60] (phase operators are not well defined) has just been described [61] along the lines of Theorem 1. Here we would like to briefly touch upon the broad theory of Loop Quantum Gravity [73], in which the so called Ashtekar connection [62] A i a (x) [74] plays the role of the canonical "position" variable in a field-theoretical sense [75].…”