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This is somewhat philosophical than physics.

In gauge theories, it is true (more like the first principle) that \begin{equation} \text{ physical observable } \Rightarrow \text{gauge invariant} \end{equation}

However, I wonder if the reverse direction also holds. That is:

\begin{equation} \text{ Any gauge-invariant quantity corresponds to some physical observable? } \end{equation}

I vaguely remember that "Local Quantum Physics" book by Rudolf Haag somehow touches on this issue, but cannot recall correctly.

Could anyone please provide any information?

Qmechanic
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Keith
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    What is your definition of "physical observable"? (For other subtleties on notions of observables, see https://physics.stackexchange.com/q/373357/50583) – ACuriousMind Jan 07 '24 at 23:53
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    Depends on your definition of a physical observable. However, at minimum, it has to be a Hermitian operator. Consequently, a gauge-invariant, non-Hermitian operator is not observable. – Prahar Jan 08 '24 at 03:29

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I wouldn't say that this question is philosophical in the way that philosophy is understood. However, it is how physicists use the term philosophy in their vocabulary. It often means a more strategic question in their field.

If anything, I would say that your question is mathematical because it is often true that mathematicians try to build formal equivalences whereas physicists are often content to assert them or expect them until ev9dence turns up to the contrary.

I expect that the question when phrased correctly will have a positive answer. I would say that a gauge physical invariant can be identified with a gauge invariant property of a field theory.

First, I say gauge physical invariant because these invariants are arising from gauge theories. But philosophically speaking, there is no reason to suspect that the physical observables of quantum gravity, will be from a gauge theory. So we ought to identify the provenance of our observables despite the fact that all our observables come from gauge theories at present.

Second, there is a formal theory of classical gauge theories where the above statement can be made precise. I don't have a ref for that. A formal theory of quantum gauge theories is still a work in progress.

Mozibur Ullah
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