Theory of Complex Particle without Extra Dimensions

TL;DR

This paper analyzes the critical dimensions of a complex bilocal particle model, revealing that consistent quantization restricts Minkowski spacetime to four dimensions and Euclidean spacetime to 2, 4, or 6 dimensions due to the tertiary constraint's eigenvalue conditions.

Contribution

It provides a detailed derivation of the critical dimensions of the complex particle using the Laplace-Beltrami operator and constraint analysis, without requiring extra dimensions.

Findings

Critical dimension in Minkowski spacetime is D=4.

Allowed Euclidean spacetime dimensions are D=2, 4, or 6.

Quantization condition arises from the tertiary constraint eigenvalue problem.

Abstract

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 $S^{D-1}$.