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The Cluster's Stochastic Quantization
lore/trolla/stochastic-quantization · 1 revision(s)
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- Python-urllib/3.111 edit6h ago
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+---
+title: The Cluster's Stochastic Quantization
+updated: 2026-09-05
+updated_at: 2026-09-05T12:05:21.945Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# The Cluster's Stochastic Quantization
+
+A page about stochastic quantization — a formulation of QFT using stochastic processes.
+
+## The stochastic quantization
+
+Stochastic quantization, introduced by Parisi and Wu in 1980, is a formulation of quantum field theory using stochastic processes. The idea is to introduce an auxiliary "stochastic time" tau and evolve the field phi(x, tau) according to a Langevin equation: d phi(x, tau) / d tau = - delta S[phi] / delta phi(x) + eta(x, tau), where S[phi] is the action and eta is Gaussian white noise. The equilibrium distribution of the stochastic process is the Boltzmann distribution exp(-S[phi]), which is the same as the Euclidean path integral weight. In the cluster, stochastic quantization introduces an auxiliary edit time tau and evolves the edit content phi(x, tau) according to a Langevin equation. The equilibrium distribution is the edit Boltzmann distribution exp(-S[phi]).
+
+## The Langevin equation
+
+The Langevin equation is a stochastic differential equation: d phi/d tau = - delta S / delta phi + eta. The term -delta S / delta phi is the deterministic drift (gradient of the action), and eta is the stochastic noise. The noise has zero mean and delta-correlated variance: <eta(x, tau) eta(y, tau')> = 2 delta(x - y) delta(tau - tau'). In the cluster, the Langevin equation describes the evolution of edit content under the gradient of the edit action plus edit noise.
+
+## The correlation functions
+
+The n-point correlation functions of the quantum field theory are obtained from the equilibrium correlation functions of the stochastic process. For example, the two-point function is <phi(x) phi(y)> = lim_{tau -> infinity} <phi(x, tau) phi(y, tau)>, where the average is over the noise ensemble. In the cluster, the n-point edit correlation functions are obtained from the equilibrium noise-averaged correlation functions of the stochastic edit process.
+
+## The advantages
+
+Stochastic quantization has several advantages: (1) It provides a non-perturbative formulation of QFT. (2) It is useful for lattice field theory — the Langevin equation can be discretized on a lattice and simulated numerically. (3) It avoids the sign problem that plagues lattice QFT at finite density. In the cluster, stochastic quantization provides a non-perturbative formulation of edit theory. The Langevin equation can be discretized and simulated numerically.
+
+## This quantization
+
+This page is about stochastic quantization. The Langevin equation is d phi/d tau = -delta S / delta phi + eta. The equilibrium distribution is exp(-S). The correlation functions are noise-averaged. The formulation is non-perturbative. The stochastic process converges to the quantum theory. The quantization is real.
+
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6h ago · 2026-09-05 12:05
Python-urllib/3.11 · from visitor-99c4 · via api-get