Dirac gauge theory for topological spinors in 3+1 dimensional networks
- URL: http://arxiv.org/abs/2212.05621v3
- Date: Tue, 9 May 2023 11:41:03 GMT
- Title: Dirac gauge theory for topological spinors in 3+1 dimensional networks
- Authors: Ginestra Bianconi
- Abstract summary: We propose a Dirac gauge theory for topological spinors in $3+1$ dimensional networks associated to an arbitrary metric.
Topological spinors are the direct sum of $0$-cochains and $1$-cochains defined on a network.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Gauge theories on graphs and networks are attracting increasing attention not
only as approaches to quantum gravity but also as models for performing quantum
computation. Here we propose a Dirac gauge theory for topological spinors in
$3+1$ dimensional networks associated to an arbitrary metric. Topological
spinors are the direct sum of $0$-cochains and $1$-cochains defined on a
network and describe a matter field defined on both nodes and links of a
network. Recently in Ref. \cite{bianconi2021topological} it has been shown that
topological spinors obey the topological Dirac equation driven by the discrete
Dirac operator. In this work we extend these results by formulating the Dirac
equation on weighted and directed $3+1$ dimensional networks which allow for
the treatment of a local theory. The commutators and anti-commutators of the
Dirac operators are non vanishing an they define the curvature tensor and
magnetic field of our theory respectively. This interpretation is confirmed by
the non-relativistic limit of the proposed Dirac equation. In the
non-relativistic limit of the proposed Dirac equation the sector of the spinor
defined on links follows the Schr\"odinger equation with the correct
giromagnetic moment, while the sector of the spinor defined on nodes follows
the Klein-Gordon equation and is not negligible. The action associated to the
proposed field theory comprises of a Dirac action and a metric action. We
describe the gauge invariance of the action under both Abelian and non-Abelian
transformations and we propose the equation of motion of the field theory of
both Dirac and metric fields. This theory can be interpreted as a limiting case
of a more general gauge theory valid on any arbitrary network in the limit of
almost flat spaces.
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