Non-Locality and Ellipticity in a Gauge-Invariant Quantization

TL;DR

This paper extends the quantum theory of particles with boundary conditions to the Maxwell field, introducing gauge-invariant boundary conditions via pseudo-differential operators, and analyzes the resulting elliptic theory and asymptotic expansions.

Contribution

It generalizes boundary condition schemes to gauge fields, incorporating pseudo-differential operators to maintain gauge invariance and analyzing the elliptic properties of the resulting operators.

Findings

Established gauge-invariant boundary conditions using pseudo-differential operators.

Derived conditions for ellipticity and asymptotic expansion of the symbol.

Analyzed ghost operators of different orders within this framework.

Abstract

The quantum theory of a free particle in two dimensions with non-local boundary conditions on a circle is known to lead to surface and bulk states. Such a scheme is here generalized to the quantized Maxwell field, subject to mixed boundary conditions. If the Robin sector is modified by the addition of a pseudo-differential boundary operator, gauge-invariant boundary conditions are obtained at the price of dealing with gauge-field and ghost operators which become pseudo-differential. A good elliptic theory is then obtained if the kernel occurring in the boundary operator obeys certain summability conditions, and it leads to a peculiar form of the asymptotic expansion of the symbol. The cases of ghost operator of negative and positive order are studied within this framework.