Time as a Hermitian operator in quantum mechanics

In non-relativistic QM, on one hand we have the following relations:

$$\langle x | P | \psi \rangle ~=~ -i \hbar \frac{\partial}{\partial x} \psi(x),$$

$$\langle p | X | \psi \rangle ~=~ i \hbar \frac{\partial}{\partial p} \psi(p).$$

On the other hand, despite the similarities, the relations cannot be directly applied to energy and time:

$$\langle t | H | \psi \rangle ~=~ i \hbar \frac{\partial}{\partial t} \psi(t),$$

$$\langle E | T | \psi \rangle ~=~ -i \hbar \frac{\partial}{\partial E} \psi(E).$$

Just wondering, how can one mathematically prove that the "classical time" (which means no QFT or relativity involved), unlike its close relative "position", is not a Hermitian operator?

I ask your pardon if you feel the question clumsy or scattered. But to be honest, if I can clearly point out where the core issue of the problem is, I may have already answered it by myself :/

Answer

Time is not a variable in Quantum Mechanics (QM), it's a parameter — much in the same way as it is in Classical (Newtonian) Mechanics.

So, if you have a Hamiltonian, e.g., for the harmonic oscillator, you have $\omega$ as a parameter, as well as the masses of the particle(s) involved, say $m$, and you also have time — even though it's not something that shows up explicitly in the Hamiltonian (remember explicit time dependency from Classical Mechanics: Poisson Brackets, Canonical Transformations, etc — in fact, you could get your answer straight from these kinds of arguments).

In this sense, just like you don't have a 'transformation pair' between $m$ and $\omega$, you also don't have one between time and Energy.

What do you say to convince yourself that $\omega \neq -i\, \partial_m$? Why can't you use this same argument to justify $E \neq -i\, \partial_t$? ;-)

I think Roger Penrose makes a nice illustration of how this whole framework works in his book The Road to Reality: A Complete Guide to the Laws of the Universe: check chapter 17.