What is the dynamical $\phi^4_3$ model in quantum field theory and what does it mean intuitively? I'm a math student studying the SPDE: $u_t = \Delta u - \kappa u^3 +\dot{W}(t)$, where $\dot{W}(t)$ is space-time white noise. In many papers I'm reading(here for example), I see references to a model in quantum field theory called $\Phi^4_3$ and it is somehow related to this spde.  I don't have a physics background so I'm confused about what it is and why it's important to physicists. And what does this model have to do with SPDEs? Are there any introductory sources I can skim, just so I can get a 1-2 sentence idea on the connection?
 A: The theory $\varphi^4_3$ describes a quantum scalar field in dimension $2+1$ (two spatial and one time dimension). Its $3+1$-dimensional counterpart is physically relevant as the foremost example of a simple but realistic scalar theory (such type of quartic self-interaction should be the one of the Higgs boson if I recall correctly). The lower dimensional version has the advantage to be "solvable", i.e. it is possible to define rigorously the associated quantum theory, albeit it is physically less relevant.
The classical Hamilton-Jacobi equation for the $\varphi^4_3$ model is the Klein-Gordon equation the OP writes, without the white noise part. Without entering into too many gory details, these equations have to be quantized in order to define the quantum field theory. The stochastic equation comes into play when one considers a special type of quantization called stochastic quantization. This quantization, originally proposed by Parisi and Wu in the 80s, has been then developed in several directions (see e.g. this mononography).
Recent rigorous results on quantization that use techniques from SPDEs and stochastic quantization are, among others, the ones given by Hairer, that the OP cites, and Gubinelli-Hofmanova.
