# Schwinger-Keldysh on the lattice: a faster algorithm and its application to field theory

## TL;DR

This paper introduces a faster Monte Carlo algorithm for real-time field theory simulations that reduces computational costs and improves proposal efficiency, demonstrated on a 1+1 dimensional scalar ^4 theory.

## Contribution

A novel algorithm that avoids explicit Jacobian calculations, significantly reducing computational complexity in real-time lattice field theory simulations.

## Key findings

- Reduced update cost from O(N^3) to less than O(N^2)
- Improved Monte Carlo proposal efficiency
- Successfully applied to scalar ^4 theory with various couplings

## Abstract

A new algorithm is developed allowing the Monte Carlo study of a 1 + 1 dimensional theory in real time. The main algorithmic development is to avoid the explicit calculation of the Jacobian matrix and its determinant in the update process. This improvement has a wide applicability and reduces the cost of the update in thimble-inspired calculations from O(N^3) to less than O(N^2). As an additional feature, the algorithm leads to improved Monte Carlo proposals. We exemplify the use of the algorithm to the real time dynamics of a scalar {\phi}^4 theory with weak and strong couplings.