Theory of Complex Particle without Extra Dimensions

TL;DR

The paper investigates a complex particle model without extra dimensions and demonstrates that the Minkowski-space theory is consistent only in $D=4$, while the Euclidean version permits $D=2,4,6$, determined by a tertiary constraint and its associated Laplace-Beltrami eigenproblem on $S^{D-1}$ and $S^{1,D-2}$. By combining a Lagrangian with complex coordinates, canonical quantization, BRST structure, and an explicit LB spectral analysis, the authors derive the necessary spectrum and show the 2nd-order secondary constraint fixes the Minkowski dimension. The main contribution is the identification of a four-dimensional critical dimension in Minkowski space for a toy model without extra dimensions, with potential for a field-theoretic extension via a Chern-Simons-like action. This work provides a novel perspective on dimensionful consistency conditions arising from gauge structure and spectral constraints, offering a possible route toward realistic 4D physics without invoking extra dimensions.

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 $S^{D-1}$.