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For positive definite infinite dimensional Hilbert spaces, there is the standard Cauchy norm topology. What if this state space has an indefinite norm or a positive semidefinite one, as in gauge theories or Faddeev-Popov ghosts? Which infinite sums are valid, and which aren't?

Similarly, for the algebra of operators, which norm topology do we choose? Not the W*-one? The C* one?

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The topology is imposed only on the physical Hilbert space, which has a positive definite metric.

If you need a topology outside, you are free to choose any that suits your purposes, but there is no canonical one.

As there currently is no mathematically rigorous version of interacting quantum gauge fields, the question of which infinite sums, limits, etc., are valid can currently not be answered.

There is significant rigorous work by Strocchi on a C^*-algebraic framework for gaunge fields in an indefinite metric setting (probably in his book ''Selected Topics on the General Properties of Quantum Field Theory'', though I don't have it available to check). But it hasn't lead so far to substantial results in the interacting case.

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