# An Alternative Lattice Field Theory Formulation Inspired by Lattice Supersymmetry

## TL;DR

This paper introduces a novel lattice field theory formulation inspired by lattice supersymmetry, addressing fermion doubling and derivative operator issues, and proposing a non-local star product to maintain key properties.

## Contribution

It presents a general lattice formulation that resolves fermion doubling and derivative operator problems, with a one-to-one correspondence to continuum theories, and introduces a non-local star product for field multiplication.

## Key findings

- Fermion doublers can be identified or removed while preserving chirality.
- The lattice derivative is a periodic function ensuring Leibnitz rule compliance.
- The formulation achieves equivalence between lattice and continuum theories.

## Abstract

We propose an unconventional formulation of lattice field theories which is quite general, although originally motivated by the quest of exact lattice supersymmetry. Two long standing problems have a solution in this context: 1) Each degree of freedom on the lattice corresponds to $2^d$ degrees of freedom in the continuum, but all these doublers have (in the case of fermions) the same chirality and can be either identified, thus removing the degeneracy, or, in some theories with extended supersymmetry, identified with different members of the same supermultiplet. 2) The derivative operator, defined on the lattice as a suitable periodic function of the lattice momentum, is an addittive and conserved quantity, thus assuring that the Leibnitz rule is satisfied. This implies that the product of two fields on the lattice is replaced by a non-local "star product" which is however in general non-associative. Associativity of the "star product" poses strong restrictions on the form of the lattice derivative operator (which becomes the inverse gudermannian function of the lattice momentum) and has the consequence that the degrees of freedom of the lattice theory and of the continuum theory are in one-to-one correspondence, so that the two theories are eventually equivalent. Regularization of the ultraviolet divergences on the lattice is not associated to the lattice spacing, which does not act as a regulator, but may be obtained by a one parameter deformation of the lattice derivative, thus preserving the lattice structure even in the limit of infinite momentum cutoff. However this regularization breaks gauge invariance and a gauge invariant regularization within the lattice formulation is still lacking.