Parisi-Wu stochastic quantization provides the proper nonperturbative particle formalism, although this is not at all what it was originally proposed for.

The main issue with considering the Feynman expansion in a particle interpretion is the fact that the particles go *back and forth* in physical time, but their interactions are when they are at the same point in space-time. So you can't really say that "this particle got formed, then travelled some time, then met another particle" and so on, because the internal proper times are integrated over, so the other particle's proper time parameter has no order relation with the first particle's proper time parameter. The interactions are not ordered causally in terms of particle proper time, each propagator has a completely separate proper time which is not put together into a global notion of one universal proper time for all the particles.

This gives the Feynman diagram picture an a-causal character, the integrals over the internal proper time aren't "this particle emitted that particle, and then later it was absorbed", because the notion of "later" is not there, the proper times don't fit together coherently to make a global proper time which is ticking along.

This interpretive problem is overcome (trivially!) in stochastic quantization. Recall that stochastic quantization involves a fictitious internal stochastic time, and one is only interested in the real space-time properties, so one considers states which are stationary in the stochastic time. I'll call the stochastic time $\tau$, and the stochastic quantization evolution equation is derived from varying the action:

$$\partial_\tau \phi = - {\delta S \over \delta \phi} + \eta $$

which gives, say for scalar $\phi^4$ theory, the equation

$$\partial_\tau \phi = \nabla^2 \phi - m^2 \phi - {\lambda \over 6} \phi^3 + \eta$$

The kinetic stochastic equation by construction makes a stationary Boltzmann distribution $e^{-S}$, so that it reproduces the original field theory in steady state.

The stochastic equation has a perturbation expansion also, which involves a purely classical (tree level) relation between $\phi$ and $\eta$, and then, if you are interested in stationary correlation functions, you contract the $\eta$s using their trivial correlations:

$$\langle \eta(x,\tau)\eta(x',\tau')\rangle = \delta(x-x')\delta(\tau-\tau')$$

This has the effect of sewing together the tree diagrams into loops, and in the case that one is only calculating expectations which are integrated over all $\tau$, you sew together exactly the Feynman diagrams of the original field theory.

But surprise! You've sewn the original Feynman diagrams together from a completely causal proper-time picture! The particle proper time is nothing other than the stochastic time, and the Parisi-Wu evolution is completely causal in proper time.

The particle propagation in stochastic time is only forward in time--- the propagator for $\tau$ is like a nonrelativistic propagator, not like a relativistic propagator. There is no backward in $\tau$ propagation for anything. At any given perturbative order in the coupling, you produce the same field theory diagram from several different chopped up stochastic versions of the diagram into two trees (this is explained in several places, including Parisi-Wu and reviews), but each tree only has causal in time particle interactions, the splitting of particles is according to the derivative of the original interaction, and the sewing together is once only at the joints.

What happened here is that the Schwinger parameter is promoted to a universal time, particles only interact at the same instant of stochastic time, but the physical space-time correlation functions are found by making an average over all stochastic times in steady state, and this removes the center of the $tau$ integrals, and turns tree-chopped but causal integrals into acausal but still particle based integrals.

Stochastic quantization is completely nonperturbative. The particle diffusion is due to the $\nabla^2$ term, the mass term is interpreted directly as a rate of production or removal of particles, and the cubic term as a 1-3 splitting of the particles causally in $\tau$.

Further, one can in principle analytically continue $\tau$ to a time variable to produce a nonrelativistic quantum mechanics (nonrelativistic in $\tau$) whose ground state is the field theory vacuum, and separately or together continue one of the x variables to be a physical time variable, so that one produces a nonperturbative nonrelativistic particle formalism.