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Taking Euclidean space as an analytic continuation of Minkowski
spacetime, topological sectors in QCD follow from the principle of
finite action (or fluctuations about finite action configurations).
Thus, the limit of spacetime volume must be taken before summing over
integer topological sectors, leaving the correlation functions
parity-even and theta immaterial. It is now of interest to see what
happens in finite spacetime volumes. As time is compactified, there is
no meaningful analytic continuation of the system to real time but of
course, there is the interpretation in terms of the trace of the
canonical density operator (which is the basis of lattice simulations).
To compute the trace, the theory must be canonically quantized which
requires a normalizable inner product. The latter requires gauge fixing.
In particular, redundant configuration related by large gauge
configurations must not be summed over. The only path to consistently
constrain the wave functionals then leads to a system where parity and
the Hamiltonian commute.
arXiv:2403.00747 [hep-th]