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

Abstract 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 of motion for non-zero spin for massless fields belong 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 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.