  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.
