Wednesday, October 12, 2022
Vacuum correlation functions are usually derived from Euclidean path integrals in the limit of infinite time. This way, even without placing constraints on the boundary conditions, there is a projection on the vacuum states. Moreover, in gauge theories for the strong interactions, only configurations with integer topological winding number then contribute to the path integral. In contrast, placing fixed boundary conditions at finite times will not lead to vacuum correlations. Thus pursuing a calculation where the infinite volume limit is taken before the sum over integer winding numbers, one finds correlation functions that do not indicate strong CP violation, in contrast with the usual procedure where limits are taken the other way around.
The complementary picture, relies on theta-vacua as states that have physical meaning without going to infinite Euclidean time. However, the standard expressions for these states are not normalizable in the proper sense. The vacuum state rendered normalizable (and then are elements of a Hilbert space as required by the postulates of QM) when fixing the gauge so as not to integrate over configurations that are redundant under large gauge transformations. With the gauge fixed inner product, the Heisenberg equations of motion are shown only to be consistent for Hamiltonians and states from which the CP odd parts can be simultaneously removed via a unitary transformation.