Number-theory renormalization of vacuum energy

TL;DR

This paper explores a number-theoretic approach to renormalizing vacuum energy in lattice quantum field theory, revealing conditions under which the vacuum energy vanishes based on the Hamiltonian's form.

Contribution

It introduces a novel number-theoretic method for analyzing vacuum energy in lattice QFT, linking it to sums of squares in residue class rings.

Findings

Vacuum energy is zero for Hamiltonians depending on squared momentum in certain lattice models.

The calculation is connected to a number theory problem involving sums of squares modulo N.

The approach provides a new perspective on vacuum energy renormalization in lattice QFT.

Abstract

For QFT on a lattice of dimension d>=3, the vacuum energy (both bosonic and fermionic) is zero if the Hamiltonian is a function of the square of the momentum, and the calculation of the vacuum energy is performed in the ring of residue classes modulo N. This fact is related to a problem from number theory about the number of ways to represent a number as a sum of $d$ squares in the ring of residue classes modulo N.