What does stochastic quantization have to do with quantization?

Question

Let me define the way I understand stochastic quantization. Stochastic quantization to me is a way of constructing infinite dimensional measures with a given "density".

Suppose we are given an action functional $S=\int L(q,v)dx$ on some infinite dimensional space. Stochastic quantization is interested in constructing the measure with "density" $$\mu(dx)=\frac{1}{Z}\exp(-S(x))D^\infty x,$$ where $D^\infty x$ is the non existent infinite dimensional Lebesgue measure and $Z$ is a normalization constant.

Stochastic quantization makes this rigorous by considering the stochastic gradient descent equation

$$\partial_t u= -\left(\frac{\delta L}{\delta q}-\nabla \cdot \frac{\delta L}{\delta v}\right)+\xi,$$

where $\xi$ is space-time white noise. If this SPDE has an invariant measure $\mu$, then $\mu$ is defined to be the measure we hoped to construct. The reason is the analogy for the invariant measure of stochastic ODEs. This all makes sense to me.

However, in what sense is this "quantization"? To me, quantization means discretization. What is being discretized in stochastic quantization?

Answer 1

This is a problem of semantics. While "quantize" can indeed mean "make discrete", it can also mean "make quantum", as in "quantizing the Maxwell theory gives QED" or "quantizing non-relativistic mechanics gives quantum mechanics". It is in the latter sense that stochastic quantization makes a classical theory into a quantum theory.

Answer 2

In stochastic quantization, a quantum field theory in $d$ dimensions is represented as the equilibrium limit of a classical stochastic process in $d+1$ dimensions, where the extra dimension is a fictitious "stochastic time". There is no notion of quantization of phase space in stochastic quantization, so perhaps a better terminology would be "stochastic quantum mechanics" or "stochastics quantum field theory".