Variational method in relativistic quantum field theory without cutoff

The variational method is a powerful approach to solve many-body quantum problems nonperturbatively. However, in the context of relativistic quantum field theory, it needs to meet three seemingly incompatible requirements outlined by Feynman: extensivity, computability, and lack of UV sensitivity. In practice, variational methods break one of the three, 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 three 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.

Automated summary

Variational method is a powerful approach to solve many-body quantum problems nonperturbatively, but faces challenges in relativistic quantum field theory due to the need to meet three seemingly incompatible requirements. In the case of numerical evidence, the error decreases faster than any power law in the number of parameters, while the cost remains only polynomial.