Theory of Complex Particle without Extra Dimensions

Complex particle is a kind of bilocal particle having unexpected symmetry, which was proposed by the authour. In the present paper, we show that critical dimension of the complex particle in Minkowski spacetime is $D=4$, while $D=2,4$ or $6$ are permitted in Euclid spacetime. The origin of the restriction to the dimension is the existence of tertiary constraint in the canonical theory, quantization of which leads to an eigenvalue equation having single-valued and bounded solutions only in particular dimension of spacetime. The derivation is based on a detailed analysis of Laplace-Beltrami operator on $S^{1,D-2}$ or $S^{D-1}$.