# 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!

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.

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.

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 $$t=0$$.