Dirac's c-number Time-energy uncertainty Relation

Its boundary condition can be stated in terms of the spatial coordinates in a given Lorentz frame. Then how about other frames where the spatial and time-like coordinates are mixed? In order to deal with this problem, we need the time-separation variable. Does this variable exist? We are quite familiar with the Bohr radius. It measures the spatial separation between the constituent particles. Then, is there a time-like separation? According to Einstein, the answer is YES. This is the starting point of our plan to construct a harmonic oscillator wave function which can be Lorentz transformed.

Let us tackle this problem with the following three figures. The first figure is a space-time diagram of the harmonic oscillator wave function. The second figure describes Lorentz boosts in the light-cone coordinate system. The third figure combines quantum mechanics with special relativity by combining the first two figures.

The present form of quantum mechanics. Let us consider a hadron consisting of two quarks bound together by a harmonic oscillator potential. Then, Heisenberg's uncertainty principle applies to position-momentum coordinates, and there are excited states along the space-like directions. As for the time-like (time-separation between the quarks), there are no excitations, but there still exist a time-energy uncertainty without excitation. According to Dirac, it is a c-number time-energy uncertainty relation.