Variational method in relativistic quantum field theory without cutoff

TL;DR

This paper introduces a relativistic variational method using continuous matrix product states that avoids UV cutoff in 1+1 dimensions, enabling efficient and accurate solutions to quantum field theory problems.

Contribution

It presents a novel relativistic variational approach that satisfies extensivity, computability, and UV insensitivity simultaneously without cutoff in 1+1D QFT.

Findings

Error decreases faster than any power law with more parameters

Method remains computationally efficient with polynomial cost

Successfully applied to self-interacting scalar field without UV cutoff

Abstract

The variational method is a powerful approach to solve many-body quantum problems non perturbatively. However, in the context of relativistic quantum field theory (QFT), it needs to meet 3 seemingly incompatible requirements outlined by Feynman: extensivity, computability, and lack of UV sensitivity. In practice, variational methods break one of the 3, which translates into the need to have an IR or UV cutoff. In this letter, I introduce a relativistic modification of continuous matrix product states that satisfies the 3 requirements jointly in 1+1 dimensions. I apply it to the self-interacting scalar field, without UV cutoff and directly in the thermodynamic limit. Numerical evidence suggests the error decreases faster than any power law in the number of parameters, while the cost remains only polynomial.