The Time of an Event in Quantum Mechanics: A History and Prospectus
Thomas Pashby
University of Pittsburgh
www.hps.pitt.edu/profile/pashby.php
In this paper I trace how the expectations of three progenitors of quantum mechanics (Heisenberg, Dirac and Jordan) regarding the role of time in the theory failed to be borne out in the eventual Hilbert space formalism, and present later developments which illustrate how those expectations could have been met. I claim there were three ways in which these expectations were confounded:
- It was expected that time and energy would be conjugate variables (Dirac, Heinsenberg, Jordan)
- It was expected that the formalism would provide probabilities for the occurrence of particular events in time, i.e., so-called "quantum jumps" (Heisenberg, Jordan)
- It was expected that the time-dependent Schrödinger equation should apply to functions of time and space (Dirac)
The replacement of the Dirac-Jordan transformation theory by von Neumann's spectral theory of operators on Hilbert space meant that time must feature as a parameter (as was soon realized by Pauli in 1933) and so the wavefunctions obeying the time-dependent Schrödinger equation would be functions of space alone, with time evolution determined by a group of unitary operators indexed by t. This new orthodoxy was inhospitable to the representation of the time of occurrence of an event, and could not supply a time-energy uncertainty relation in the form that Heisenberg had supposed. This shift in the theory away from the notion of the time of occurrence led to the introduction of von Neumann's projection postulate and the notorious measurement problem.
The expectation that quantum mechanics would allow a time representation and a meaningful time-energy uncertainty relation regarding the time of a “quantum jump” had given the measurement problem a quite different complexion. In Heisenberg’s Uncertainty paper – whose subject was the “intuitive content of quantum theory” – we find repeated reference to the idea that the time at which a jump occurs could be given an empirical meaning, with the uncertainty with which that time was determined being related to the uncertainty in energy. On the other hand, the von Neumann description of the measurement process was designed to be insensitive to the time at which a property was actualized, so long as collapse occurred before observation of the results of an experiment.
While Pauli’s theorem and related results demonstrate that there can be no self-adjoint operator suitable to represent the time of occurrence of an event, recent work has sought to introduce an operationally meaningful notion through the use of generalized observables, known at POVMs. These time POVMs provide the means to represent experiments which take place over extended periods of time, and are necessary to describe even surprisingly basic experimental setups such as electron diffraction recorded by a screen. The usual description of measurement is as an instantaneous process, which is obviously not the case here, and so the interpretation of these operators within the standard formalism of quantum mechanics is problematic.
Observing that the usual self-adjoint operators provide conditional probabilities in the sense that they supply the probability for observing a particular type of event given that the time is t, I suggest that the conditional probabilities supplied by a time POVM answer a different kind of experimental question: given that an event is observed of this type, what is the probability that it occurs at t? The normalization of such POVMs is thus non-trivial, but large steps towards the resolution of this problem were taken in recent work of Brunetti and Fredenhagen (2002). Tellingly, the systematic description of these conditional probabilities found in Brunetti, Fredenhagen and Hoge (2010) requires for its formulation the application of the Schrödinger equation to functions of time and space as a constraint, just as it (the time-dependent Schrödinger equation) was originally envisioned by Dirac.