Time in Quantum Mechanics

Jan Hilgevoord's article, "Time in quantum mechanics," attempts to clarify problems with interpretations of time in quantum mechanics as well as other sciences. The main problem seen by most is that time itself is treated differently from all other physical variables. While space coordinates like x, y, and z are denoted by operators, the time coordinate t is simply a parameter, an ordinary number. This is seen as a problem for quantum mechanics because other fields such as relativity treat time differently, relating it to spatial coordinates through the Lorentz transformation.

The author argues that this problem isn't real, and is due in part to improper terminology. For the position of a particle, one commonly uses x. However, this should only be used when referring to a fixed coordinate in space. It is important to make the distinction between a fixed point in space (x) and the position of a particle in space (q).

Because time in its current quantum mechanical interpretation isn't a physical dynamical variable like the position q of a particle, the author proposes a physical dynamical system that would closely resemble time in the same way that particle position q closely resembles the spatial coordinate x. In this case, the author claims clocks as the dynamical answer to this problem. This "clock variable" that changes with time could be something as simple as an angle measurement. The author then explores implications of this on quantum mechanics.

I found this article very interesting, especially since I had never really considered there to be a difference between particle position q and fixed spatial coordinate x. However, it became apparent that they were not the same, and that quantum mechanics had seemingly been handling time differently (as a parameter) from other variables such as x, y, and z (as operators). Although much of the mathematical derivations was over my head, and I didn't quite understand some of the suppositions, it was still interesting to observe the implications and results of this theoretical investigation.