Why odd-space and odd-time dimensions in even-dimesional spaces?

Details Why odd-space and odd-time dimensions in even-dimesional spaces? Why odd-space and odd-time dimensions in even-dimesional spaces?

2000-05-31

arxiv.org/abs/hep-ph/0005327v3

Phys.Lett. B486 (2000) 314-321

Physics

High Energy Physics

High Energy Physics - Phenomenology

(only small corrections made)

Scientific paper

10.1016/S0370-2693(00)00775-9

We are answering the question why 4-dimensional space has the metric 1+3 by making a general argument from a certain type of equations of motion linear in momentum for any spin (except spin zero) in any even dimension d. All known free equations for non-zero spin for massless fields belong to this type of equations. Requiring Hermiticity(This is a generalization of an earlier work which shows that without assuming the Lorentz invariance -which in the present work is assumed- the Weyl equation follows using Hermiticity.) of the equations of motion operator as well as irreducibility with respect to the Lorentz group representation, we prove that only metrics with the signature corresponding to q time + (d - q) space dimensions with q being odd exist. Correspondingly, in four dimensional space, Nature could only make the realization of 1+3 dimensional space.

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