Is there any good source talking about the quantization of time?

Question

I am doing a little search about the quantization of time, but I didn't find anything explaining it in a conceptual or in a philosophical way? Is there anyone who can help?

Answer 1

There is Pauli's argument: if the time operator existed, it would have a continuous spectrum. However, the time operator, obeying the canonical commutation relation, would also be the generator of the "energy translations". This means that the Hamiltonian operator would also have a "continuous spectrum", in contradiction with the fact that the energy of any stable physical system must be bounded below.

Example by DIY:

let assume that :$\;t=i\frac{\partial}{\partial E}\;\;\,,x=-i\frac{\partial}{\partial p}\;\;$ and the metric:$\;s^{2}=c^{2}t^{2}-x^{2}\;\;,$ by analogy with (E,p)

we obtain from the metric the equation $\left (\frac{\partial^{2}}{\partial E^{2}}-\frac{\partial^{2}}{c^{2}\partial p^{2}}+(\tau/\hbar)^{2}\right)\psi=0 \;\;\;,$ with $s=c\tau$

it has the same form as the Klein-Gordon equation

''This implies that the Hamiltonian operator would also have a "continuous spectrum", in contradiction with the fact that the energy of any stable physical system must be bounded below.''