Little-used Mathematical Structures in Quantum Mechanics I. Galilei Invariance and the welcher Weg Problem

Abstract

Results of the welcher Weg experiment of Duerr, Nonn and Rempe are explained by using ray representations of the Galilei group. The key idea is that the state of the incoming atom be regarded as belonging to an irreducible unitary ray representation of this group. If this is the case, interaction with an interferometer with a which-way detector must split this state into the direct sum of two states belonging to representations with different internal energies. (While the zero of internal energy is arbitrary, the difference between two internal energies is well-defined and is invariant under unitary transformations.) The state of the outgoing atom will then be a superposition of two mutually orthogonal states, so that there will be no interference. Neither complementarity nor entanglement plays a role in this explanation. Furthermore, in atom interferometry it is not enough for a quantum eraser to erase the internal energy difference; to restore interference, two copies of a representation have to be collapsed into one. In a direct sum of copies of the same representation, copies of the same state will still be orthogonal. These assertions may be testable, and two new atom interferometry experiments are suggested. One of them is an `own-goal' experiment which may decisively refute the explanation offered here, and restore the aura of mystery that this paper tries to dispel.