Nonlocality and ellipticity in a gauge-invariant quantization

The quantum theory of a free particle in two dimensions with nonlocal bound ary conditions on a circle is known to lead to surface and bulk states. Suc h a scheme is here generalized to the quantized Maxwell field, subject to m ixed boundary conditions. If the Robin sector is modified by the addition o f a pseudo-differential boundary operator, gauge-invariant boundary conditi ons are obtained at the price of dealing with gauge-field and ghost operato rs which become pseudo-differential. A good elliptic theory is then obtaine d if the kernel occurring in the boundary operator obeys certain summabilit y conditions, and it leads to a peculiar form of the asymptotic expansion o f the symbol. The cases of ghost operator of negative and positive order ar e studied within this framework.