I have a very easy, and naive, question: given a field $\mathbf{A}$ on some vector space $V$, we can calculate how the flux or circulation of this field behaves. For example, we have Gauss's laws for the electric field or the magnetic field. We know that the mathematical proof of such a statement involves ingredients from vector calculus: the topology of the space is important, the nature of the field is important (i.e. singularities and so on). However, in a lot of physics texts, one has a certain notion of charge, or source, of the field, and proving theorems such as Gauss's law, sheds light on the nature of such charges or sources (i.e. if there are monopoles and so on).

My question is: is there a mathematical definition of charges for a certain field?

If there is such a mathematical definition for a classical theory, can it be generalized to General Relativity and QFT?

  • $\begingroup$ A problem is, that if you have point-charges the theory is not mathematically consistent. (like Energy of the field in infinite) $\endgroup$ – lalala Feb 11 at 11:26

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