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

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?

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.''

• Hi The Tiler. Welcome to Phys.SE. This post seems to be more an answer to this question. Commented Jan 26, 2022 at 21:11
• Yes sir, I found some references like: https: researchgate.net/publication/… arxiv.org/abs/1705.09212 Commented Jan 26, 2022 at 21:28