Stochastic partial differential equations in Euclidean quantum field theories

Stochastic partial differential equations (SPDEs) play a significant role in the formulation and analysis of Euclidean quantum field theories (EQFTs). By introducing stochastic processes into PDEs, these equations allow for the modeling of quantum fields in a probabilistic framework, facilitating the study of fluctuations and non-deterministic behaviors inherent in quantum systems. Below, we explore how SPDEs are used in the context of EQFTs.

Euclidean Formulation: In EQFTs, time is treated as an imaginary variable ( $\tau = it$ ) through a Wick rotation, transforming the Minkowski space signature to a Euclidean one. This reformulation makes path integrals more mathematically tractable because it converts oscillatory integrals into well-defined integrals that are easier to handle: $Z = \int \mathcal{D}\phi \, e^{-S_E[\phi]},$ where $S_E[\phi]$ is the Euclidean action.
Fields as Random Variables: In EQFTs, fields can be treated as stochastic variables, leading to a natural connection with SPDEs, where the quantum fluctuations of these fields are represented by stochastic terms in the equations.

Stochastic Quantization: A key method that bridges SPDEs and EQFTs is stochastic quantization. Proposed by Parisi and Wu, this approach quantizes a field theory by introducing a fictitious "stochastic time" and defining a Langevin-type SPDE for the field $\phi(x, \tau)$ : $\frac{\partial \phi(x, \tau)}{\partial \tau} = -\frac{\delta S_E[\phi]}{\delta \phi(x)} + \eta(x, \tau),$ where $\eta(x, \tau)$ is a Gaussian white noise term with properties: $\langle \eta(x, \tau) \rangle = 0, \quad \langle \eta(x, \tau) \eta(x', \tau') \rangle = 2\delta(x - x')\delta(\tau - \tau').$
Equilibrium Distribution: The SPDE evolves the field $\phi$ over the fictitious time $\tau$ towards an equilibrium distribution that coincides with the Euclidean path integral measure: $P[\phi] \propto e^{-S_E[\phi]}.$ This approach allows for a probabilistic interpretation where the field configuration $\phi$ is sampled according to the path integral measure.

Langevin Equation: The SPDE above is analogous to a Langevin equation in statistical physics, describing the time evolution of $\phi$ under the influence of deterministic and stochastic forces.
Fokker-Planck Equation: The corresponding Fokker-Planck equation describes how the probability distribution $P[\phi, \tau]$ evolves in the space of field configurations: $\frac{\partial P[\phi, \tau]}{\partial \tau} = \int dx \, \frac{\delta}{\delta \phi(x)} \left(\frac{\delta S_E[\phi]}{\delta \phi(x)} P[\phi, \tau] + \frac{\delta P[\phi, \tau]}{\delta \phi(x)}\right).$ This equation confirms that, at large fictitious times $\tau \to \infty$ , the system reaches the equilibrium distribution $e^{-S_E[\phi]}$ .

Lattice Field Theory: SPDEs are particularly useful in lattice simulations where fields are defined on discrete grids, enabling the use of numerical stochastic quantization to sample field configurations.
Non-Perturbative Analysis: SPDEs provide insights into non-perturbative phenomena such as instantons, solitons, and confinement, which are difficult to analyze with purely perturbative methods.
Renormalization: SPDEs facilitate the study of renormalization in EQFTs by allowing the stochastic evolution of fields to be analyzed under different scales.

Gaussian White Noise: The most common type of noise used in SPDEs is Gaussian white noise, as it represents uncorrelated fluctuations that simplify mathematical treatments.
Colored Noise: In some advanced field theories, more complex noise structures (e.g., colored noise) are used to model correlated fluctuations that can influence the dynamics in more realistic ways.

$\phi^4$ Theory: For a scalar field theory with a potential $V(\phi) = \frac{\lambda}{4!} \phi^4$ , the SPDE takes the form: $\frac{\partial \phi(x, \tau)}{\partial \tau} = \nabla^2 \phi(x, \tau) - m^2 \phi(x, \tau) - \frac{\lambda}{3!} \phi^3(x, \tau) + \eta(x, \tau),$ where $m$ is the mass of the scalar field and $\lambda$ is the coupling constant.
Gauge Theories: For non-Abelian gauge theories, SPDEs can be extended to model the stochastic quantization of gauge fields with constraints to maintain gauge invariance.