A. Neumaier

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It seems that it is possible, in some way, to recover a Hilbert space from the path integral formulation (and in non-relativistic QM, this Hilbert space is the standard Hilbert space):
The problem is that you need to assume the positivity of the quantum measure. This cannot be proved for the functional integrals used in QFT - else they would produce finite results without the need for regularization.

Even in quantum mechanics, proving positivity requires somewhere a Hilbert space argument....
 

A. Neumaier

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Zeidler bases everything in the QA "magic formula" (basically, the definition of the path integral) and the LSZ "magic formula"
and he recovers only (and only an asymptotic series for) the asymptotic S-matrix, no finite time dynamics.
 
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and he recovers only (and only an asymptotic series for) the asymptotic S-matrix, no finite time dynamics.
Sure, he does not get 100% out of his formulation. Otherwise, it would be massive.

But even getting less than 100%, I believe it is quite interesting that experimental numbers (S-matrix) can be obtained from the classical Lagrangian, without needing any kind of functional analysis, only probability theory.

Of course, everybody has his own taste, but for my taste, this is an interesting statement.
 

A. Neumaier

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I believe it is quite interesting that experimental numbers (S-matrix) can be obtained from the classical Lagrangian, without needing any kind of functional analysis, only probability theory.
He gets out the numbers. But to get out their meaning as probabilities for scattering results, he needs the standard Hilbert space framework! Indeed, Zeidler starts with that.....
 
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He gets out the numbers. But to get out their meaning as probabilities for scattering results, he needs the standard Hilbert space framework! Indeed, Zeidler starts with that.....
Sure, I am not arguing Hilbert spaces are irrelevant, on the contrary. I just want to raise this issue, in a post where the rules of QM are discussed.

It is unfortunate Zeidler only published 3 out of the 5 volumes of QFT. I would have really liked to read the 4th one, at least.
 

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