Global constraints of Gauss' law and the solution to the strong CP problem

Abstract

It is shown that the global constraints of Gauss' law ensure that the vacuum angle must be quantized in gauge theories in the presence of magnetic monopoles. Our quantization rule for the vacuum angle is derived as θ = 2 πN/ n ( n ≠ 0) with integer n being the relevant topological charge [1] of the magnetic monopoles and N is an integer which is not fixed by this method. This provides interesting new understandings and ideas for the strong CP problem and its new solution with the magnetic monopoles as originally proposed recently by the author. Therefore, we conclude [1] again that the strong CP problem can be automatically solved in the presence of magnetic monopoles, the fact that the strong CP-violation can be only so small or vanishing may be a signal for the existence of magnetic monopoles, and the universe is open. If there exists actually the axion field a( x), the value of the physical axion field a phy( x) will not be quantized. However, the expectation value of the CP violating density ϵ μνγσ F μν a F γσ a will be quantized. For the special values of topological number n, it can be vanishing or very small even without using the Peccei-Quinn dynamical adjusting.