Stochastic quantization with discrete fictitious time (2501.14260v2)

Abstract: We present a new approach to stochastic quantization `a la Parisi-Wu with a discrete fictitious time. The noise average is modified by weights, which results in the equivalence in the large time limit to the correlation function of the corresponding quantum field theory {\it without taking any continuum limit of fictitious time}. We test our method in a zero-dimensional toy model both perturbatively and numerically.

• The paper presents a stochastic quantization method using discrete fictitious time and weighted noise averages to reproduce QFT correlation functions.
• The method is validated in a zero-dimensional toy model, confirming it matches QFT results perturbatively and numerically without the continuum limit.
• Reducing computational burden by avoiding the continuum limit makes this method promising for large-scale QFT simulations like lattice QCD.

Stochastic Quantization with Discrete Fictitious Time

The paper presented in this paper explores an innovative method for stochastic quantization in quantum field theory, emphasizing applications to a discrete fictitious time framework. This approach modifies the traditional Parisi-Wu stochastic quantization by introducing weights to the noise average, aiming to establish equivalence with quantum field theory (QFT) correlation functions without requiring a continuum limit in the temporal dimension.

Core Concepts and Methodology

The traditional Parisi-Wu stochastic quantization adds an extra fictitious time dimension to a $d$-dimensional field theory, employing Langevin dynamics integrated with Gaussian noise. Long-term behavior of such a system reproduces correlation functions of the associated QFT. The challenge lies in taking the continuum limit of the fictitious time for practical computations, often demanding significant resources.

In contrast, the method proposed in this paper embraces discrete fictitious time processes, maintaining the desired equivalence to QFT correlations. It achieves this by weighting noise averages with specific functions determined by two key discretized drift terms ($W_n$ and $\overline{W}_n$), each satisfying precise technical criteria, including convergence in the continuum limit and maintaining a stable determinant condition for computational viability.

Theoretical Implications and Supersymmetry

An integral aspect of this paper revolves around utilizing supersymmetry to bridge the conceptual gap introduced by discrete fictitious time. Supersymmetry ensures that the stochastic dynamics described by Supersymmetric Quantum Mechanics (SQM) can robustly yield results consistent across time discretizations. Despite the inherent symmetry-breaking that occurs at the boundaries in discrete systems, the work successfully compensates for this deficit using a carefully constructed weight function and maintains theoretical equilibrium.

Numerical and Perturbative Validation

The authors validate their approach through a zero-dimensional toy model traditionally used to probe theoretical consistency in stochastic schemes. Both perturbative and numerical analyses confirm that weighted averages can match expected results from traditional QFT calculations without requiring the discretized time to approach an infinitesimal continuum limit.

Particularly notable is the method's performance in strong coupling scenarios, where traditional stochastic simulations can falter. The use of different drift force formulations demonstrates flexibility and affirms the broad applicability of the approach across various scalar field scenarios.

Practical and Future Implications

Beyond its immediate, practical advantage in reducing the convergence burden for stochastic simulations in QFT, the proposed method stimulates future developments in large-scale lattice QCD computations and similar simulations where managing computational resources is paramount.

In future explorations, translation of these theoretical models to higher-dimensional QFTs and quantum mechanics should yield critical insights. They might also help confront current computational bottlenecks in simulations demanding tricky limit-taking processes. Extending this stochastic framework to non-linear and complex systems could further enrich simulation techniques available in theoretical physics.

Overall, this paper contributes an indispensable methodological tool to the landscape of stochastic quantization, promising a more resource-efficient path for simulations of quantum fields while continuing to uphold the rigorous theoretical and numerical integrity expected by the field.