$\mathbb{1}$-Loop Theory

TL;DR

This paper introduces the $$-Loop formalism for lattice gauge theory that maintains Poincaré symmetry, reformulating classical equations as identity loops, and derives a lattice gravity theory recovering Einstein's equations.

Contribution

It presents a novel $$-Loop formalism for lattice gauge theories that preserves Poincaré symmetry and leads to a new lattice gravity model consistent with Einstein's equations.

Findings

Defines the $$-Loop as a generalization of Wilson loops.

Develops a lattice Poincaré gauge theory of gravity.

Recovers Einstein's vacuum equations in the continuum limit.

Abstract

A new formalism for lattice gauge theory is developed that preserves Poincar\'e symmetry in a discrete universe. We define the $\mathbb{1}$-loop, a generalization of the Wilson loop that reformulates classical differential equations of motion as identity-valued multiplicative loops of Lie group elements of the form ${[g_1\cdots g_n]=\mathbb{1}}$. A lattice Poincar\'e gauge theory of gravity is thus derived that employs a novel matter field construction and recovers Einstein's vacuum equations in the appropriate limit.