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9
Graham Van Goffrier
[WSS20] Full Discretization of Local Gauge Invariance
Graham Van Goffrier, University College London
Posted
9 months ago
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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
. While gauge theories built upon discrete networks are well-studied, and form the basis of lattice gauge theory, there has been significantly less investigation of entirely discrete models, where the gauge group itself is not permitted to be continuous. In a sense we are returning to a world of integers, where the only meaningful quantities are counts of network connections, with well-founded hope that the complexities of our physical world will be fully describable.
Background: Fibre Bundles
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4
are reduced to
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3
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.
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
which will serve as the base space of our gauge theory.
Building U(1)-Bundles on Lattices
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.
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.
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