Topology

Large A Stochastic Framework for Real-Time Quantum Field Dynamics

Authors: Xiao Lin
Comments: 64 Pages. 20 figures

We propose a Stochastic Formulation that establishes a direct mathematical correspondence between the real-time Feynman path integral in Minkowski spacetime and the expectation values of classical stochastic processes. This framework offers an alternative approach to quantum dynamics by formulating evolution through intrinsic stochastic processes rather than relying solely on pre-discretized spacetime backgrounds. Specifically, we demonstrate that the unitary dynamics of scalar fields can be mapped to continuous Wiener processes or Ornstein-Uhlenbeck process, while spinor fields correspond to discrete Poisson jump processes. Distinct from conventional methods involving perturbative expansions, Euclideanization, or Grassmann algebra, our formulation provides a non-perturbative, real-time framework where quantum amplitudes are derived statistically from stochastic trajectories.We implement this framework via a tree-grid recursion scheme or Monte Carlo method for the Klein-Gordon field, benchmarked against exact solutions of the forced harmonic oscillator. For the Dirac field, we derive an analytical closed-form finite difference scheme that effectively models its evolution. By integrating these schemes, we successfully apply the framework to the Yukawa coupling model and extend it to QED. The results reveal non-trivial dynamical features, such as the independence of dynamical mass generation from the initial fermion mass. Crucially, the structural nature of this stochastic approach inherently avoids the fermion doubling and sign problems often encountered in lattice approaches. The extension to non-Abelian gauge theories, including SU(2) Yang-Mills and SU(3) QCD, demonstrates dynamical fermion mass generation in the chiral limit, driven purely by gauge field self-interactions, with results consistent with chiral symmetry breaking and lattice QCD calculations.
Category: Topology