Ron and Luboš's Answers, +1. FWIW, however, I have found it worthwhile to take QFT to be a stochastic signal processing formalism in the presence of Lorentz invariant (quantum) noise.

The devil is in the details, and I cannot claim to be able to say much, or even anything, about interacting quantum fields, but it is possible to construct random fields that are empirically equivalent, in a specific sense, to the quantized complex Klein-Gordon field (EPL 87 (2009) 31002, http://arxiv.org/abs/0905.1263v2) and to the quantized electromagnetic field (http://arxiv.org/abs/0908.2439v2, completely rewritten a few weeks ago). Needless to say, the fact that a random field satisfies the trivial commutation relation $[\hat\chi(x),\hat\chi(y)]=0$ instead of the nontrivial commutation relation $[\hat\phi(x),\hat\phi(y)]=\mathrm{i}\!\Delta(x-y)$ plays out in numerous ways, and this is not for anyone who wants to stay in the mainstream.

I take it as a significant ingredient that we treat quantum fields as (linear) functionals from a Schwartz space $\mathcal{S}$ of window functions into a $\star$-algebra $\mathcal{A}$ of operators, $\hat\phi:\mathcal{S}\rightarrow\mathcal{A};f\mapsto\hat\phi_f$, instead of dealing with the operator-valued distribution $\hat\phi(x)$ directly, even though we may construct $\hat\phi_f$ directly from $\hat\phi(x)$, by "smearing", $\hat\phi_f=\int\hat\phi(x)f(x)\mathrm{d}^4x$. In terms of these operators, the algebraic structure of the quantized free Klein-Gordon field algebra is completely given by the commutator $[\hat\phi_f,\hat\phi_g]=(f^*,g)-(g^*,f)$, where $(f,g)=\int f^*(x)\mathrm{i}\!\Delta_+(x-y)g(y)\mathrm{d}^4x\mathrm{d}^4y$ is a Hermitian (positive semi-definite) inner product.
A window function formalism is part of the Wightman axiom approach to QFT.
Note that what are called "window functions" in signal processing are generally called "test functions" in QFT. The signal processing community works with Fourier and other transforms in a way that has close parallels with quantum theory and implicitly or explicitly uses Hilbert spaces.

In the vacuum state of the quantized free Klein-Gordon field, we can compute a Gaussian probability density using the operator $\hat\phi_f$,
$$\rho_f(\lambda)=\left<0\right|\delta(\hat\phi_f-\lambda)\left|0\right>=\frac{e^{-\frac{\lambda^2}{2(f,f)}}}{\sqrt{2\pi(f,f)}},$$
which depends on the inner product $(f,f)$ [take the Fourier transform of the Dirac delta, use Baker-Campbell-Hausdorff, then take the inverse Fourier transform]. The reason it's good to work with smeared operators $\hat\phi_f$ instead of with operator-valued distributions $\hat\phi(x)$ is that the inner product $(f,f)$ is only defined when $f$ is square-integrable, which a delta function at a point is not.
We can compute a probability density using an operator $\hat\phi_f$ in *all* states, but, of course, we can only compute a *joint* probability density such as
$$\rho_{f,g}(\lambda,\mu)=\left<0\right|\delta(\hat\phi_f-\lambda)\delta(\hat\phi_g-\mu)\left|0\right>$$ if $\hat\phi_f$ commutes with $\hat\phi_g$; in other words, whenever, but in general only when, the window functions $f$ and $g$ have space-like separated supports. At space-like separation, the formalism is perfectly set up to generate probability densities, which is why QFT is like stochastic signal processing, but at time-like separation, the nontrivial commutation relations prevent the construction of probability densities.

I take it to be significant that the scale of the quantum field commutator, the imaginary component of the inner product $(f,g)$, is the same as the scale of the fluctuations in the probability density $\rho_f(\lambda)$, determined by the diagonal component, $(f,f)$. It's possible, indeed, to construct a quantum field state in which the two scales are different (Phys. Lett. A 338, 8-12(2005), http://arxiv.org/abs/quant-ph/0411156v2), so the equivalence could be thought as surprising as the equivalence of gravitational and inertial mass.

Needless to say, there many people who are working away at QFT. The approach I've outlined is only one, with one person working on it, in contrast to string theory, supersymmetry, noncommutative space-time geometry, etc., all of which have had multiple Physicist-decades or centuries of effort poured into them. It's probably better to follow the money. Also, please note that I have hacked this out in an hour, which I have done because rehearsal is always good. Did anyone read all of this?

This post imported from StackExchange Physics at 2015-04-11 10:30 (UTC), posted by SE-user Peter Morgan