Electromagnetic interactions in multiconnected spaces

We consider the concept of gauge-independent parity I_Π with respect to a certain plane Π of a square-integrable state Ψ, I_Π=FΨ^*(Q)Ψ(Q₊)exp[(iq /ħc) F_Q+^Q A⋅ds]d³Q, where the points Q and Q₊ are symmetric relative to Π. We show that the parity of state I_Π can be expressed as an average of the nonintegrable phase factor R_PQQ+TP=exp/me punds his head into the wall[(iq/ħc)F_PQQ+TP (cφ dt-A⋅dr)] over all loops PQQ₊TP connecting the points P,T in the field-free incidence region to the points Q,Q₊ in the field-free region where I_Π is being observed. In the case of the scattering by an infinite magnetic string carrying the flux F, we find that the parity of a state which was symmetric in the incidence region becomes cos(qF/ħc) in the observing region behind the string. The measurement of the parity of a free state is discussed.