Interference in Time Lawrence

I discuss 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, does not have the property of coherence in time, and hence cannot account for the results found. Therefore, this experiment has fundamental importance beyond the technical advances it represents. Some theoretical structures which consider the time as an observable, and thus could, in principle, have the required coherence in time, are discussed briefly, and the application of Floquet theory and the manifestly covariant quantum theory of Stueckelberg are treated in some detail. In particular, the latter is shown to account for the results in a simple and consistent way. The recent experiment of Lindner, et al 1 , clearly shows the effect of quantum interference in time for the wave function of a particle. The results are discussed in that paper in terms of a very precise solution of the time-dependent nonrelativistic Schrödinger equation. In this note, I wish to point out that the nonrelativistic Schrödinger theory cannot be used to 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 they represent. They imply, in fact, that the time variable t must be adjoined to the set of standard quantum variables so that the standard ket |x, t > for the representation of the quantum state (in Dirac's terminology 2) can be constructed. It is this structure for the wave function ψ(x, t) ≡< x, t|ψ), where x and t are the spectra of self-adjoint operators, that provides the possibility of coherence in t, and therefore, interference phenomena. If the quantum theory is to remain symplectic in form, the variable E must also be adjoined. The standard nonrelativistic quantum theory cannot be used to predict interference in time. For example, Ludwig 3 has pointed out that the time variable cannot be a quantum observable, since there is no imprimitivity system (i.e., no operator exists that does not commute with t in the nonrelativistic theory) involving this variable. Note that the Hamiltonian of the standard theory evolves quantum states in time, but does not act as a shift operator since it commutes with t.Dirac 2 has argued that if t were an operator, then the resulting t, E commutation relation would imply that the …