Generalizing Cross Products and Maxwell's Equations to Universal Extra Dimensions

Abstract

In the past five years, there has been considerable research on the collider phenomenology of TeV-scale extra compact dimensions which are universal - i.e. accessible to all the Standard Model fields. We consider in detail a fundamental conceptual issue in the higher-dimensional theory: Since the cross product is only physically meaningful in three spatial dimensions, how do traditional vector formulas involving the cross product or curl operation generalize in the case of universal extra dimensions? For example, is angular momentum physically meaningful, and, if so, how is it related to the position and momentum vectors? We find that if angular velocity, torque, angular momentum, and magnetic field, for example, are regarded as anti-symmetric second-rank tensors instead of pseudovectors, we can construct tensor forms of the standard cross product and curl equations that do generalize to higher dimensions. Although Maxwell's equations have been generalized to extra dimensions in terms of differential forms, we show that our simple technique casts the generalized Maxwell's equations in a form very similar to the standard vector calculus. We also provide direct integrals for the fields in a practical tensor calculus form. We explore the electric and magnetic fields in detail to show that there are no fundamental conceptual problems hidden in the theory - e.g. how the Lorentz force can be meaningfully expressed in terms of the velocity and magnetic field when the traditional cross product and right-hand rule do not apply. Finally, we explore the effects that compactification has on the effective form of Maxwell's equations, which could be useful for direct tests of Coulomb's law.