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This is a question is a follow-up to the answer by @tparker about what in simple terms is gauge invariance. I want to know in detail the subtleties of the definitions for gauge theory (#3) and lattice gauge theory (#4).

See the quote below from the above page:

  • Definition 3: A Lagrangian is sometimes said to posses a "gauge symmetry" if there exists some operation that depends on an arbitrary continuous function on spacetime that leaves it invariant, even if the degrees of freedom being changed are physically measurable.

  • Definition 4: For a "lattice gauge theory" defined on local lattice Hamiltonians, there exists an operator supported on each lattice site that commutes with the Hamiltonian. In some cases, this operator corresponds to a physically measurable quantity.

The cases of Definitions 3 and 4 are a bit conceptually subtle so I won't go into them here - I can address them in a follow-up question if anyone's interested.

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  • $\begingroup$ I'm moving this discussion to chat to keep the question comment thread about the physics. $\endgroup$
    – Emilio Pisanty
    Commented Jul 10, 2016 at 20:03
  • $\begingroup$ It helps to think about both the smooth and lattice cases together as just giving topological spaces. The lattice might be be just the vertices as a 0-skeleton or the bonds can be kept to give the 1-skeleton. The continuous space has higher cells as well. $\endgroup$
    – AHusain
    Commented Jul 10, 2016 at 23:19
  • $\begingroup$ @user122066 see additional comments on the chat room. $\endgroup$
    – Emilio Pisanty
    Commented Jul 11, 2016 at 11:31
  • $\begingroup$ I've written follow-up answers regarding whether there's any sense in which the gauge degrees of freedom can be physically measurable in the Hamiltonian case and the Lagrangian case. $\endgroup$
    – tparker
    Commented Jul 14, 2016 at 4:42

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