Abstract
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.
Highlights
Quantum field theory (QFT) lies at the root of fundamental physics and is the most fundamental approach we so far have to understand microscopic phenomena
The results for the ground state energy density of φ42 are shown in Fig. 1 and compared with the renormalized Hamiltonian truncation (RHT) computations of [5]
For g 1⁄4 1 and g 1⁄4 2, I pushed the computations to D 1⁄4 32, to get a near exact point of comparison to evaluate the error at lower bond dimensions
Summary
Quantum field theory (QFT) lies at the root of fundamental physics and is the most fundamental approach we so far have to understand microscopic phenomena. A vexing problem of theoretical physics is that QFTs are rarely ever solvable. We seem to know the rules of particle physics, at least to a good precision, but hardly know what they give in general. There were essentially two approaches to deal with QFT approximately: perturbation theory [1] and lattice Monte Carlo methods [2,3]. The first provides results without cutoff in momenta, valid “all the way down” for a true QFT, but accurate only for small coupling. The second works at strong coupling, but introduces a short (UV) and long (IR) distance cutoff
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