I Why do ##t## and ##-i\hbar\partial_t## not satisfy the definition of a linear map/operator in Hilbert space?

Dr_Nate
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It is common to say that ##t## and ##-i\hbar\partial_t## are not operators in quantum mechanics. But an unambiguous mathematical justification seems lacking.
It is common to say that ##t## and ##-i\hbar\partial_t## are not operators in quantum mechanics. But I haven't seen a satisfying justification.

As an example of the precision of our discourse, someone has said that ##-i\hbar\partial_t## satisfies the definition of Hermicity, but it is not an operator in quantum mechanics. That seems wrong to me because Hermicity requires the above expression to be a linear map/operator.

Alternatively, some say that time is a parameter and not a variable in Hilbert space, so it can't be an operator. However, when I look at the definition of a linear map, I don't see the words \emph{parameter} or \emph{variable} used, so there seems to be a gap in the justification.

Interestingly, no one directly answered this post in that same thread linked above.

Applying ##t## or ##-i\hbar\partial_t## to kets, I don't see a case where additivity and scalar multiplication are not preserved. I don't see how they violate the requirements of a linear mapping back to Hilbert space. Note, though, that I'm an experimentalist, so I can't tell you the difference between a vector space and a field in these definitions (wikipedia link).

Question: How does the application of ##t## and ##-i\hbar\partial_t## to a ket not satisfy the mathematical definition of a linear map/operator in Hilbert space as used in the mathematical formalism of quantum mechanics?
 
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The Hilbert space in QM is the space of functions of ##x##, not a space of functions of ##x## and ##t##. The time derivative operator is therefore not an operator in this Hilbert space. The functions in the Hilbert space must have a finite norm, if they were functions of both ##x## and ##t## then the computation of norm would involve integration over both ##x## and ##t##, but the integral over ##t## would not converge because the wave functions satisfying Schrodinger equation do not vanish at ##t\to\pm\infty##. Of course, the wave functions do depend on both ##x## and ##t##, but only the dependence on ##x## is associated with a Hilbert space structure. The dependence on ##t## is also associated with some space of functions, but this space is not a Hilbert space, because the integration over ##t## does not lead to a finite norm.
 
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PeterDonis said:
Yes, there was an answer given, in post #20 of that thread--which was before the one you linked to here (post #30).
The question I am asking is pretty much about straight math. That answer invokes observables, so isn't what I am looking.
 
Demystifier said:
The Hilbert space in QM is the space of functions of ##x##, not a space of functions of ##x## and ##t##. The time derivative operator is therefore not an operator in this Hilbert space. The functions in the Hilbert space must have a finite norm, if they were functions of both ##x## and ##t## then the computation of norm would involve integration over both ##x## and ##t##, but the integral over ##t## would not converge because the wave functions satisfying Schrodinger equation do not vanish at ##t\to\pm\infty##. Of course, the wave functions do depend on both ##x## and ##t##, but only the dependence on ##x## is associated with a Hilbert space structure. The dependence on ##t## is also associated with some space of functions, but this space is not a Hilbert space, because the integration over ##t## does not lead to a finite norm.
Thank you. This is the answer I am looking for. If I recall correctly something like this was said in the linked thread, but now it is clear to me.

I have read many lists of postulates and have never noticed before in those lists that references to Hilbert space were at fixed time. Now, that I know, I can see it's clear on Wikipedia. I'm curious to see how often it's clearly stated in other more formal places.
 
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Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
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