Abstract
We show that standard identities and theorems for lattice models with $U(1)$ symmetry get re-expressed discretely in the tensorial formulation of these models. We explain the geometrical analogy between the continuous lattice equations of motion and the discrete selection rules of the tensors. We construct a gauge-invariant transfer matrix in arbitrary dimensions. We show the equivalence with its gauge-fixed version in a maximal temporal gauge and explain how a discrete Gauss's law is always enforced. We propose a noise-robust way to implement Gauss's law in arbitrary dimensions. We reformulate Noether's theorem for global, local, continuous or discrete Abelian symmetries: for each given symmetry, there is one corresponding tensor redundancy. We discuss semi-classical approximations for classical solutions with periodic boundary conditions in two solvable cases. We show the correspondence of their weak coupling limit with the tensor formulation after Poisson summation. We briefly discuss connections with other approaches and implications for quantum computing.
Highlights
In this article we show that the basic features of continuous Abelian symmetries in the conventional formulation of field theory have discrete counterparts in Tensor field theory (TFT)
The matter fields can be decoupled by setting βl: 1⁄4 0. As they do not appear in the action, their integration yields a factor 1 and we are left with the pure gauge (PG) Uð1Þ lattice model with partition function
We have shown that some standard theorems and identities associated with the Uð1Þ symmetry that can be derived in the conventional formulation of field theory have a discrete counterpart in TFT
Summary
Tensor field theory (TFT) is a recently developed approach of models studied in the context of lattice gauge theory [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. Functions over compact groups can be expanded in terms of discrete sums of characters for Abelian groups [27] or more generally of group representations [28] This property was exploited to calculate strong coupling expansions [29] and introduce dual variables [30,31,32] for the type of lattice models mentioned above. Noether’s theorem for Abelian symmetries can be reexpressed in the tensor reformulation context as follows: for each symmetry, there is a corresponding tensor redundancy This applies with a remarkable generality to local, global, continuous or discrete Abelian symmetries. The practical consequences of the results for coarse graining, the continuum limit and quantum computations, are briefly discussed in the conclusions
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