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We show that standard identities and theorems for lattice models with $U(1)$symmetry get re-expressed discretely in the tensorial formulation of thesemodels. We explain the geometrical analogy between the continuous latticeequations of motion and the discrete selection rules of the tensors. Weconstruct a gauge-invariant transfer matrix in arbitrary dimensions. We showthe equivalence with its gauge-fixed version in a maximal temporal gauge andexplain how a discrete Gauss's law is always enforced. We propose anoise-robust way to implement Gauss's law in arbitrary dimensions. Wereformulate Noether's theorem for global, local, continuous or discrete Abeliansymmetries: for each given symmetry, there is one corresponding tensorredundancy. We discuss semi-classical approximations for classical solutionswith periodic boundary conditions in two solvable cases. We show thecorrespondence of their weak coupling limit with the tensor formulation afterPoisson summation. We briefly discuss connections with other approaches andimplications for quantum computing.

Discrete aspects of continuous symmetries in the tensorial formulation of Abelian gauge theories