It is often said that the existence of a single monopole would force electric charge to be quantized, due to Dirac's argument. However, one can write down theories like QED that, independently of the existence of monopoles, have charge occurring strictly in integer multiples of some fundamental value. What is an explicit example of a theory containing of matter coupled to a $U(1)$ gauge field for which charge is not integer-quantized?
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2The answer to Is a magnetic monopole really necessary for charge quantization? says that any $U(1)$ theory has quantised charge so the answer to your question is None. – John Rennie Nov 20 '23 at 05:33
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@JohnRennie Ah I see. So the ambiguity is really whether the gauge group is U(1) or R and detecting the monopole proves to you that your group is U(1). – Panopticon Nov 20 '23 at 06:15