Interference in Time: a Comment

I comment on the interpretation of a recent experiment showing quantum interference in time. It is pointed out that the standard nonrelativistic quantum theory, used by the authors in their analysis, cannot account for the results found, and therefore that this experiment has fundamental importance beyond the technical advances it represents. The recent experiment of Lindner, et al, clearly shows the effect of interference in time for a particle wave function. The results are discussed in that paper in terms of a very precise solution of the time-dependent nonrelativistic Schrödinger equation. In this Comment, I wish to point out that the nonrelativistic Schrödinger theory cannot predict interference phenomena in time, and therefore the very striking results of this beautiful experiment have a fundamental importance which goes beyond the technical advances which it represents. It is well known, in fact, that the nonrelativistic quantum theory cannot predict interference in time. For example, Ludwig has pointed out that the time variable cannot be a quantum observable, a prerequisite for the development of interference phenomena, since there is no imprimitivity system (i.e., no operator exists that does not commute with t in the nonrelativistic theory) involving this variable; Dirac has similarly argued that if t were an operator, then the resulting t, E commutation relation would imply that the energy of the system is unbounded below, confirming the view that the time cannot be an observable in the nonrelativistic quantum theory (note, however, that in a relativistic theory, negative energies correspond to antiparticle states, and are not excluded). Moreover, as the axiomatic treatment of Piron (see also, Jauch) shows, the Hilbert space of the quantum theory is constructed of a set of wave functions satisfying a normalization condition, e.g., for a single particle, ∫ |ψt(x)| dx ≤ ∞, for each value of the parameter t. There is a distinct Hilbert space for each value of the parameter t. As pointed out by Wick, Wightmann and Wigner, a Hilbert space decomposes into independent and incoherent sectors if there is no observable that connects these sectors; hence, if there were a larger Hilbert space containing a representation for t, the absence of any observable that connects different values of t in nonrelativistic physics would decompose the Hilbert space into a (continuous) direct sum of superselection sectors. Therefore, no superposition of vectors for different values of t is admissible in the framework of the standard nonrelativistic quantum theory. This would exclude an interpretation of the experiment, involving the linear superposition of * e-mail:[email protected].