Wightman quantum field - Interpretation I have a question regarding the interpretation of the Wightman quantum field in mathematical quantum field theory. 
A quantum field $\phi$ is a operator-valued distribution. This means that $\phi$ is a linear function
$$\phi:\mathcal{S}(\mathbb{R}^{n})\to L(D,\mathcal{H}),$$
where $S(\mathbb{R}^{4})$ denotes the Schwartz space, $\mathcal{H}$ denotes a Hilbert space and $D$ denotes a dense subset of $\mathcal{H}$, such that $\forall\Psi_{1}, \Psi_{2}\in D$
$$\langle\Psi_{1}\mid\phi(\cdot)\Psi_{2}\rangle:\mathcal{S}(\mathbb{R}^{n})\to\mathbb{C}$$ 
is a tempered distribution. From the distribution theory we know that for regular distributions $T(f)$ there exists a function $T(x)$ such that
$$T(f)=\int\mathcal{d}^{4}x\, T(x)f(x).$$
If a distribution is irregular, such a function does not exist, like for the delta distribution. Nevertheless, the notation $T(x)$ is often used in physics, as for the delta-distribution $\delta(x)$ and also for the quantum field $\phi(x)$.
Now to my question: 
If we write $\phi(x)$, the intepretation is quite clear: the value of the quantum field at the space-time point $x$, but if we write $\phi(f)$ as a distribution, how can the function $f$ be interpreted? Does it have a physical meaning, or is it just a relic of the mathematical description?
Thank you!
 A: I like to think of it this way: We can only measure something with finite spatial resolution and for a finite time. So any experiment only measures an average over a small spacetime region.  This is basically
$$
\phi(f)=\int \phi(x)f(x) d^4x
$$
for some compactly supported   smooth function $f$. This basically is what the distribution definition is doing.  For technical reasons (we like Fourier transforms) people prefer Schwartz class to compactly supported test functions, but I doubt that it makes much difference to the physics.
A: The smearing (test-) function is not a relic of the mathematical description, but a key ingredient of the theory. To quote from the fathers Wightman and Streater (PCT, Statistics, and all That):

It was recognized early in the analysis of field measurements for 
  the electromagnetic field in quantum electrodynamics that, in their 
  dependence on a space-time point, the components of fields are in general more singular than ordinary functions. This suggests that 
  only smeared fields be required to yield well-defined operators. For 
  example, in the case of the electric field, $\mathcal{E}(x,t)$ is not a well-defined 
  operator, while $\int dx ~ dt ~ f(x) \mathcal{E}(x,t) = \mathcal{E}(f) $ is.

Another quote comes from BLT (Introduction to Aximatic Field Theory, 1975):

We define a quantum (or quantized) field as an operator- 
  valued tensor distribution. Such a definition corresponds 
  better to the real physical situation than the more familiar 
  notion of a field as a quantity defined at each point of space- 
  time. Indeed, in experiments the field strength is always 
  measured not at a mathematical point $x$ but in some region of 
  space and in a finite interval of time. Such a measurement is 
  naturally described by the expectation value of the field as a distribution applied to a test function with support in the 
  given space-time region. 

It is also worth noting that classical fields are also distributions. 
A: These smearing functions are closely related to the single particle wave functions.  
Fix a decomposition of spacetime into space times time.  Fourier transform the space coordinate.  In the special case $f(t,p) = \delta_0(t) \psi(p)$, which can be reached by taking a family of gaussian approximations, the operator $\phi^\dagger(f)$ creates a particle with momentum-space wavefunction $\psi(p)$ at time $t=0$.
