Dirac gauge theory for topological spinors in \(3+1\) dimensional networks. (English) Zbl 1525.83012
Summary: 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 [G. Bianconi, J. Phys. Complex. 2, No. 3, Article ID 035022, 23 p. (2021; doi:10.1088/2632-072X/ac19be)] 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 Schrödinger 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.
MSC:
| 83C45 | Quantization of the gravitational field |
| 68Q12 | Quantum algorithms and complexity in the theory of computing |
| 81P68 | Quantum computation |
| 90C35 | Programming involving graphs or networks |
| 39A12 | Discrete version of topics in analysis |
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
| 35J93 | Quasilinear elliptic equations with mean curvature operator |
| 78A30 | Electro- and magnetostatics |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
| 83C10 | Equations of motion in general relativity and gravitational theory |
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