quantum mechanics - Energy is actually the momentum in the direction of time?

By comparatively examining the operators

a student concludes that `Energy is actually the momentum in the direction of time.' Is this student right? Could he be wrong?

Answer

The student is right in that energy is the analog of momentum in the "time direction" but I wouldn't go so far as to call it "momentum in the direction of time".

It's analogous in two ways that I can think of off the top of my head:

1. It is the time-component of the 4-dimensional energy-momentum vector in special relativity.


2. Noether's theorem relates a symmetry in the laws of physics with respect to a coordinate to a conserved quantity. Momentum is conserved because the laws of physics are invariant with respect to translations in space, and Energy is conserved because the laws of physics are unchanging in time.

I wouldn't call it "momentum in the direction of time" because that phrase, at least to me, implies that energy is more momentum-like than it really is.