A Non commutator of symmetries giving rise to a gauge symmetry

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In discussions about symmetries of a Lagrangian, the non-commutation of transformations A and B can lead to new symmetries, as indicated by the condition that (BA)^{-1}AB does not return the original field. This situation suggests the existence of an associated conserved charge or current, as per Noether's theorem. The relationship between generators in the Lie algebra, such as the commutation relations in so(3), illustrates how combinations of symmetries can yield new ones. However, whether a new symmetry arises can often be inferred from the structure constants of the Lie algebra without direct computation. The conversation emphasizes the need for clarity in distinguishing between original and derived symmetries and their corresponding Noether charges.
binbagsss
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If there are two symmetries of a Lagrangian, perhaps they are transformations, A and B, and they don't commute ##[A, B] \neq 0##. Let this act on some field, then if ##(BA) ^{-1}AB## does not return the original field, i.e. if ##(BA) ^{-1}AB \neq \mathbb{1}##, then this gives a rise to a new symmetry so it will have an associated charge/ current by Noethers theorem. I feel like this is a common occurrence but I can't find any references on considering this as a symmetry/computing an associated conserved charge/current. Is there certain terminology/framework I need to be searching for? Thanks
 
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binbagsss said:
If there are two symmetries of a Lagrangian, perhaps they are transformations, A and B, and they don't commute ##[A, B] \neq 0##. Let this act on some field, then if ##(BA) ^{-1}AB## does not return the original field, i.e. if ##(BA) ^{-1}AB \neq \mathbb{1}##, then this gives a rise to a new symmetry ...
Not necessarily so. E.g. the commutator between translations and rotations gives a new translation.

In the algebra ##so(3)## one has

##[L_i, L_j] = i \epsilon_{ijk} L_k##

so

##[L_1, L_2] = i L_3##

i.e. the commutator ##L_1## and ##L_2## generate the third symmetry generated by ##L_3##. But those three generators span the whole algebra, so we only need to look at the corresponding conserved charges of those generators.
 
haushofer said:
Not necessarily so. E.g. the commutator between translations and rotations gives a new translation.

In the algebra ##so(3)## one has

##[L_i, L_j] = i \epsilon_{ijk} L_k##

so

##[L_1, L_2] = i L_3##

i.e. the commutator ##L_1## and ##L_2## generate the third symmetry generated by ##L_3##. But those three generators span the whole algebra, so we only need to look at the corresponding conserved charges of those generators.
ok thanks, I will check the translation and rotation one you say. Is there a way to know whether there will be a 'new' symmetry or not from (BA)^{-1} AB from the Lie algebra of the commutators without having to verify and compute each time?
 
I guess those are given by the structure constants of the Lie-algebra.
 
"Not necessarily so. E.g. the commutator between translations and rotations gives a new translation."

So here you are saying that this can be written in terms of the known translations and so the charge/current associated with the new translation would be able to be written in terms of the known charges/currents associated with the original translations?
 
binbagsss said:
"Not necessarily so. E.g. the commutator between translations and rotations gives a new translation."

So here you are saying that this can be written in terms of the known translations and so the charge/current associated with the new translation would be able to be written in terms of the known charges/currents associated with the original translations?
I don't understand what you mean by "original" and "new" translations. There are only 4 of them (or D, in D spacetime dimensions if you're a string- or sugra-theorist). To every generator you can assign a Noether charge, and these Noether charges obey the same algebra as the generators.
 
haushofer said:
I don't understand what you mean by "original" and "new" translations. There are only 4 of them (or D, in D spacetime dimensions if you're a string- or sugra-theorist). To every generator you can assign a Noether charge, and these Noether charges obey the same algebra as the generators.
"so. E.g. the commutator between translations and rotations gives a new translation." is what I meant by it. Obviously it's a simple case but I am asking for the purposes of thinking about other cases. obviusly I am aware its not a *new* translation.
 
Ah yes, sorry, that can be confusing. But I still don't understand your precise question. Maybe you can provide some more context.
 
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