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When we work in path integral formalism in some field theory (or even in QM), we usually look on the action, find its symmetries and than treat them as a gauge symmetry - we sum over all possible fields up to this symmetry.

Now let's say that we happand to have a system which have some physically symmetry - for simplicity let's look on free particle propagating from on point on the X axis to another so that we have clear symmetry for rotation around the x axis. The action will also be symmetric but this is not gauge symmetry so we still want to make the path integral over different paths

So my question is how do we know what is gauge symmetry and what is physical symmetry of we start from the action (for example how do we know that the full $weyl$ X $diff$ of the polyakov action is indeed gauge)?

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    $\begingroup$ I'm not sure what exactly the question is asking - it seems you want to know what the difference between a gauge symmetry and a "normal" symmetry is? What about the definitions you have seenwas not satisfactory? For questions on the nature of gauge symmetries see: physics.stackexchange.com/q/266992/50583, physics.stackexchange.com/q/126978/50583 and their linked questions $\endgroup$
    – ACuriousMind ♦
    Commented Jan 6, 2023 at 22:36
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    $\begingroup$ I think the question is indeed justified because some texts just say that gauge symmetries are unphysical, or redundancies, without extra qualification, while in fact there are indeed extra implicit assumptions to make this statement correct. The judge of whether a gauge transformation is a true physical symmetry or not is whether it has a trivial or non-trivial charge. In flat space QFT for example, gauge transformations that do not die off at infinity are physical symmetries and have important consequences like soft theorems and memory effects. $\endgroup$
    – Gold
    Commented Jan 7, 2023 at 2:40
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    $\begingroup$ For more details on this see physics.stackexchange.com/q/719053 $\endgroup$
    – Gold
    Commented Jan 7, 2023 at 2:40
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    $\begingroup$ Is non-triviality of charge the only indication for true physical symmetry? As a counter-example: in YM theory, I can have conserved current which is not gauge invariant, so not a physical observable $\endgroup$
    – paul230_x
    Commented Jan 7, 2023 at 8:31

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