The symmetry of reciprocity is reviewed in the context of relativistic quantum mechanics with the specific aim of relating to C, P and T invariances. From this investigation global time reversal is found to be a sufficient condition for reciprocity to hold in scattering from a vector potential. The present proof is free from assumptions of small-angle scattering and from restrictions on z-dependent terms in the scattering equation, and by avoiding S-matrix theory is thought to be accessible to undergraduate teaching.
The symmetry of reciprocity is reviewed in the context of relativistic quantum mechanics with the specific aim of relating to C, P and T invariances. From this investigation global time reversal is found to be a sufficient condition for reciprocity to hold in scattering from a vector potential. The present proof is free from assumptions of small-angle scattering and from restrictions on z-dependent terms in the scattering equation, and by avoiding S-matrix theory is thought to be accessible to undergraduate teaching.
Reciprocity in wave propagation usually refers to the symmetry of the Green's function under the interchange of the source and the observer coordinates, but this condition is not gauge invariant in quantum mechanics, a problem that is particularly significant in the presence of a vector potential. Several possible alternative criteria are given and analyzed with reference to different examples with nonzero magnetic fields and/or vector potentials, including the case of a multiply connected spatial domain. It is shown that the appropriate reciprocity criterion allows for specific phase factors separable into functions of the source and observer coordinates and that this condition is robust with respect to the addition of any scalar potential. In the Aharonov-Bohm effect, reciprocity beyond monoenergetic experiments holds only because of subsidiary conditions satisfied in actual experiments: the test charge is in units of e and the flux is produced by a condensate of particles with charge 2e.
We derive general relationships of the S matrix of fields containing evanescent components. Our formalism covers time-independent quantum scattering as well as scattering of classical scalar waves. We show that reciprocity, energy (or probability) conservation, and time-reversal symmetry in the presence of evanescent waves lead to relationships that extend the well-known relations previously derived in asymptotic scattering. On this basis, we discuss the link between reciprocity and time-reversal symmetry. We also address the experimental feasibility of time reversal of a field containing evanescent components.
Abstract Avoiding the assumption that relativistic scattering be describable by interacting fields we find in the Schrödinger picture relativistic scattering closely analogue to the non-relativistic case. On the space of scattering states the invariant mass operator M′ M ′ of the interacting time evolution has to be unitarily equivalent to the invariant mass M=P2 M = P 2 where P, acting on many-particle states, is the sum of the one-particle four-momenta. For an observer at rest P0 P 0 generates the free time evolution. Poincaré symmetry requires the interacting Hamiltonian H′ H ′ to Lorentz transform as 0-component of a four-vector and to commute with the four-velocity U=P/M U = P / M but not with P, else there is no scattering. Even though H′ H ′ does not commute with P, the scattering matrix does. The four-velocity U generates translations of states as they are seen by shifted observers. Superpositions of nearly mass degenerate particles such as a Klong K long are seen by an inversely shifted observer as a shifted Klong K long with an unchanged relative phase. In contrast, the four-momentum P generates oscillated superpositions e.g. a shifted Kshort K short with a changed relative phase. The probability of scattering of massive particles is shown to be approximately proportional to the spacetime overlap of their position wave functions. This is basic to macroscopic locality and justifies to represent the machinery of actual scattering experiments by the vacuum. In suitable variables the relativistic Hamiltonian of many-particle states is not the sum of a Hamiltonian for the motion of the center and a commuting Hamiltonian for the relative motion but factorizes as their product. They act on different variables of the wave functions.
Born's reciprocity theory can be combined with a recently proposed framework for quantum spacetime by requiring that the free test particle propagators obey the Born-Lande equation in addition to the Klein-Gordon equation. If, furthermore, the coordinate transition amplitudes in between various standards are required to be eigenfunctions of Born's metric operator, then a mass formula results which predicts linear dependence on spin of the squared rest mass of elementary particles. This procedure also leads to a guage and reciprocally invariant formulation of the relativistic canonical commutation relations.
The connection between an observable invariance property of the scattering matrix and the interaction, V, generating the scattering, is studied for a particlar case: rotational invariance in nonrelativistic "potential" scattering. It is shown that a determination that the scattering cross section depends only on the angle between the incident and outgoing beams by no means implies that V is invariant under rotation. This is true even if coherent incident beams are used to probe the target. Some aspects of these results and their relevance for the construction of physical theories are briefly discussed.