I just started reading PCT, Spin and Statistics, and All That. Can someone explain why we use operator valued distributions to describe fields? I read somewhere that it would take infinite energy to measure an observable at a single point. Why don't we instead use functions from some collection of subsets (i.e. the open sets) of space to the operators to emulate the fact that measurements usually occur over some area? In other words, what is the physical meaning of the test functions used to define the operator valued distributions? Are some of these functions non-physical, possibly too narrow in width that they'd violate some uncertainty principle?
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