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

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 lin ear in momentum for any spin (except spin zero) in any even dimension d. Al l known free equations of motion for non-zero spin for massless fields belo ng to this type of equations. Requiring Hermiticity(1) of the equations of motion operator as well as irreducibility with respect to the Lorentz group representation, we Drove that only metrics with the signature correspondin g to q time + (d - q) space dimensions with q being odd exist, Correspondin gly, in four dimensional space, Nature could only make the realization of 1 + 3-dimensional space. (C) 2000 Elsevier Science B.V. All rights reserved.