quantum mechanics - Is there an actual proof for the energy-time Uncertainty Principle?

As I understand, the energy-time uncertainty principle can't be derived from the generalized uncertainty relation. This is because time is a dynamical variable and not an observable in the same sense momentum is.
Every undergraduate QM book I have encountered has given a very rough "proof" of the time-energy uncertainty relation, but not something that is rigorous, or something even remotely close to being rigorous.
So, is there an actual proof for it? If so, could someone please provide me a link to it or even provide me with a proof? Keep in mind that I am not looking a proof using quantum mechanical principles, as comments below pointed out.

EDIT: All the proofs I have found take the generalized uncertainty relation and say "let $Δτ=σ_q/|dq/dt|$", cf. e.g. this Phys.SE post. But this for me does not suffice as a rigorous proof. People give that Δτ a precise meaning, but the relation is proven just by defining Δτ, so I am just looking for a proof(if there is any) that shows that meaning through mathematics. If no better proof exists, so be it. Then I will be happy with just the proof through which we define that quantity. By defining it in this way, there is room for interpretation, and this shows from the multiple meaning that researchers have given to that quantity (all concerning time of course).

Answer

The main problem is, as you say, that time is no operator in quantum mechanics. Hence there is no expectation value and no variance, which implies that you need to state what $\Delta t$ is supposed to mean, before you can write something like $\Delta E \Delta t\geq \hbar$ or similar.

Once you define what you mean by $\Delta t$, relations that look similar to uncertainty relations can be derived with all mathematical rigour you want. The definition of $\Delta t$ must of course come from physics.

Mostly of course, people see $\Delta t$ not as an uncertainty but as some sort of duration (see for instance the famous natural line widths, for which I'm sure there exist rigorous derivations). For example, you can ask the following questions:

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  Given a signal of temporal length $t$ (it takes $t$ from "no signal" to "signal has completely arrived"), what is the variance of energy/momentum? This can be mapped to the usual uncertainty principle, because the temporal length is just a spread in position space. It is also related to the so-called Hardy uncertainty principle, which is just the Fourier uncertainty principle in disguise and completely rigorous.

  
  


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  If you do an energy measurement, can you relate the duration of the measurement and the energy uncertainty of the measurement? This is highly problematic (see e.g. the review here: The time-energy uncertainty relation. Choosing a model of measurement, you can probably derive rigorous bounds, but I don't think a rigorous bound will actually be helpful, because no measurement model probably captures all of what is possible in experiments.

  
  


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  You can ask the same question about preparation time and energy uncertainty (see the review).

  
  


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  You can ask: given a state $|\psi\rangle$, how long does it take for a state to evolve into an orthogonal state? It turns out that there is an uncertainty relation between energy (given from the Hamiltonian of the time evolution) and the duration - this is the Mandelstamm-Tamm relation referred to in the other question. This relation can be made rigorous (this paper here might give such a rigorous derivation, but I cannot access it).

  
  
  


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  other ideas (also see the review)...

  
  

In other words: You first need to tell me what $\Delta t$ is to mean. Then you have to tell me what $\Delta E$ is supposed to mean (one could argue that this is clear in quantum mechanics). Only then can you meaningfully ask the question of a derivation of an energy-time uncertainty relation. The generalised uncertainty principle does just that, it tells you that the $\Delta$ quantities are variances of operators so you have a well-defined question. The books you are reading seem to only offer physical heuristics of what $\Delta t$ and $\Delta E$ mean in special circumstances - hence a mathematically rigorous derivation is impossible. That's not in itself a problem, though, because heuristics can be very powerful.

I'm all in favour of asking for rigorous proofs where the underlying question can be posed in a rigorous manner, but I doubt that's the case here for a universally valid uncertainty relation, because I doubt that a universally valid definition of $\Delta t$ can be given.