Oscillatory phase - Knowlpedia

An oscillatory phase is a real-valued function $\Phi(t,x)$ appearing in a factor $e^{i\kappa\Phi(t,x)}$ , with a fixed nonzero frequency parameter $\kappa$ . Smooth phases permit differentiation by the chain rule. The exponential depends on $\kappa\Phi$ modulo $2\pi$ .

Locally defined phases

On a periodic domain the exponential can be globally defined even when a real-valued phase exists only in local charts. Two local phases give the same exponential on an overlap if their difference lies in $(2\pi/\kappa)\mathbb Z$ . For continuous lifts on a connected overlap this integer difference is constant.

Evaluating a field along an auxiliary map is a separate composition operation. Averaging in independent auxiliary variables before this composition need not equal averaging the composed physical field.