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In a field theory with signature (1,d), conserved charges are obtained by integrating the time component of a conserved current over a spatial region.

What are the corresponding equations and interpretation in a field theory with (0,D) signature?

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How are you defining the notion of "conservation" without reference to that time dimension? (I mean, this is very close to the title of your question, but I guess I'm saying, "things don't have intrinsic definitions--you first have to decide what you're trying to say and then we can define some ideas to help you think through those invariants." So what sort of field theory are you looking at and what do you want to do with it?) – CR Drost Mar 4 at 2:12
I am looking at Conformal Field theories with a Euclidean signature. Given a killing vector, corresponding to one of the generators of the conformal group, is there a notion of a conserved charge, associated to this killing vector? – Srivatsan Balakrishnan Mar 4 at 2:36
Conserved with respect to what? – CR Drost Mar 4 at 2:46
Essentially a duplicate of – Qmechanic Mar 4 at 6:19

2 Answers 2

Not all conserved charges are obtained by integrating the time component of some conserved current. For example, momentum and angular momentum are conserved charges and are obtained by integrating a spatial component of a conserved current.

So the equations and interpretation for conserved charges in a Euclidean theory are the same as in the Lorentzian--only instead of having a conserved energy the system would thenhave a conserved momentum associated with translations in the Euclidean time direction.

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The mathematical concept that I was searching for in this question is the following:

I will not elaborate more, but except to say that, the Hodge dual allows you to define a conserved current corresponding to any choice of the "time axis". (Sorry for the vagueness).

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