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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Wightmans fields are supposed to generate the algebra of observables, which in turn generates the Hilbert space when applied to a cyclic vacuum vector.
Wightman treats fields as distributions to avoid annoying problems like the following: If the free scalar field $\hat{\phi}$ were a function, the observable $\hat{\phi}(x,t)$ would also act on the vacuum as a creation operator, creating a particle with wavefunction equal to the delta-function $\delta_x$ supported at $x$. But QM's probability interpretation only makes sense when such wavefunctions are integrable.
It's also necesary to treat fields (local generators of the algebra of observables) as operator-valued distributions because it allows you to get non-trivial singularities in the OPE coefficients, which are required for even free scalar fields in dimension greater than 1. (The quantum mechanics of a finite number of coordinate variables is a lucky exception; all distributions in this case can be approximated as integration against a continuous function $f$ satisfying $|f(x)f(y)| \leq C |x-y|^{1/2}$.)
There is also the fact that to measure the value of a local observable, we have to couple something to it. Such couplings disturb the field values near where we make the measurement, in accordance with the 2-point function.
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?
Test functions generalize and mollify this idea. (A subset is, after, equivalent to a test function which has the value 1 on the subset and 0 elsewhere.) They also allow you the flexibility you need to talk about gauge theories, where you can only set up experiments involving conserved currents.
