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# [WSS20] Full Discretization of Local Gauge Invariance

Posted 9 months ago

Motivation: Integer Precision One of the greatest achievements of modern physics is its powerful use of symmetry. At the human scale, symmetries abound: by simple experimentation as children we find that the laws of our natural world are unaffected by translations, rotations, and the passage of time. These are global symmetries, and they correspond to invariant quantities, the familiar conservation laws which determine so many natural phenomena. As we study physics at higher energy scales, revealing the particle foundations of nature, these symmetries largely persist -- but we also begin to observe new, stricter symmetries. These symmetries require that physics be invariant under transformations of local “gauge” degrees of freedom, which occur independently at each point in spacetime. Three of the fundamental forces of nature (the exception, as always, being gravity) have been modelled with incredible precision by gauge theories, where the local degrees of freedom are represented by certain well-known Lie groups. Despite our limited understanding of the properties which a unified quantum theory of gravity should fulfill, many physicists have come to strongly suspect that the continuous nature of spacetime is an approximation which breaks down at the Planck scale. If this hypothesis holds, a quantum theory of gravity must describe dynamics with some discretized model, whose continuum limit reproduces both the Standard Model of particle physics and General Relativity. The aim of this project is to study how local gauge symmetries may be represented by undirected graphs, as motivated by the hypergraph-replacement models for fundamental physics proposed in [1]. We use the language of fibre bundles to build an intuition for the characteristics of gauge theories which must be present in our models, and demonstrate familiar physical features of the resultant bundle graphs for U(1) electromagnetism on 3
Background: Fibre Bundles The usual intuition for defining a fibre bundle is a topological space E BxF π:EB BxF [0,1]⊂ 1 S Δx Δe G B G E C:E[ G B G E G E EE
Demonstration: Discrete Hopf U(1)-Bundle As an illustrative example, we turn to the work of Manton [3], who studied the problem of discrete gauge theories in the topologically non-trivial example of a Hopf bundle. Manton makes use of a 24-vertex regular 4D polytope as an approximation to the manifold 3 S E= 3 S 2 S U(1) Any 2D or 3D projection of this polytope is necessarily limited in scope, and blurs the symmetry of the object. Fortunately, from graph connectedness alone, it is possible to understand this symmetry: all vertices have identical local linkedness and global positioning, as should be expected from rotational invariance. A specific labelling scheme for these vertices is chosen in [3] which aids in understanding their connectedness. Embedding coordinates in 4 2 3 S {{0,1},{0,},{0,-1},{0,-},{1,0},{,0},{-1,0},{-,0},{1+,1+},{1+,1-},{1+,-1-},{1+,-1+},{1-,1+},{1-,1-},{1-,-1-},{1-,-1+},{-1-,1+},{-1-,1-},{-1-,-1-},{-1-,-1+},{-1+,1+},{-1+,1-},{-1+,-1-},{-1+,-1+}} Out[]= The edge connectivity under this labelling scheme was entered by hand to confirm various minor results in [3], and confirmed to produce a graph isomorphic to the 24-polytope native to Mathematica. With this labelling, an immediate pattern is visible -- for every vertex V={a,b} V={a,b} Z=a· -1 b {0,∞,1,,-1,-} {0,∞,1,,-1,-} We choose to add edges to the graph which visually identify the group orbits -- it is important to note that vertices are not otherwise connected within each fibre, as no information necessary for calculation is gained. The true edges between fibres are colored in white. Several key features are identifiable by eye, once we establish the procedure of selecting a gauge. {∞,1,,∞} {,-1,-} 12 2
Construction: Are We Braiding? (or are we dancer) With this example in mind, we proceed to constructing fibre bundles on a more general space: a lattice in n
Building U(1)-Bundles on Lattices In this example we consider a d=3 lattice graph with size 5 in each dimension. We will demonstrate how to build a discrete U(1)-bundle on this lattice base space, using the cyclic group C 6 As explained in [4], the key quantities in a lattice gauge theory are the parallel transport operators which indicate how the gauge group action is communicated between neighboring lattice sites. These operators take values from G (as opposed to the continue case where they live in the Lie algebra of G). By assigning an element of G to each edge of our base graph, we therefore totally define the gauge field configuration. By Cayley’s Theorem, any finite group may be represented as a permutation group over its set of elements. Ideally we can identify a minimal permutation representation of G -- in the case of cyclic group C n |