# 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!

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.
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.
$$\phi(f)=\int \phi(x)f(x) d^4x$$ 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.