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In these lecture notes, we review some recent works on Hamiltonian latticegauge theories, that involve, in particular, tensor network methods. Theresults reviewed here are tailored together in a slightly different way fromthe one used in the contexts where they were first introduced, by looking atthe Gauss law from two different points of view: for the gauge field it is adifferential equation, while from the matter point of view, on the other hand,it is a simple, explicit algebraic equation. We will review and discuss whatthese two points of view allow and do not allow us to do, in terms of unitarilygauging a pure-matter theory and eliminating the matter from a gauge theory,and relate that to the construction of PEPS (Projected Entangled Pair States)for lattice gauge theories.

Gauss Law, Minimal Coupling and Fermionic PEPS for Lattice Gauge Theories