# Lorentz-covariant four-vector formalism for two-measure theory Academic Article

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### abstract

• In the conventional two-measure theory, the scalar density function $\ensuremath{\Phi}$ is taken to be $\ensuremath{\Phi}\ensuremath{\equiv}{ϵ}^{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}\ensuremath{\sigma}}{ϵ}_{abcd}({\ensuremath{\partial}}_{\ensuremath{\mu}}{\ensuremath{\varphi}}^{a})({\ensuremath{\partial}}_{\ensuremath{\nu}}{\ensuremath{\varphi}}^{b})({\ensuremath{\partial}}_{\ensuremath{\rho}}{\ensuremath{\varphi}}^{c})({\ensuremath{\partial}}_{\ensuremath{\sigma}}{\ensuremath{\varphi}}^{d})$, where the indices $a,b,c,d=1$, 2, 3, 4 are internal-space indices. It is more natural to replace the four scalars ${\ensuremath{\varphi}}^{a}$ by a Lorentz-covariant four-vector ${\ensuremath{\varphi}}^{m}$ with a local Lorentz index $m=(0)$, (1), (2), (3). We entertain this possibility, and show that the newly proposed Lagrangian respects not only Lorentz covariance, but also global-scale invariance. The crucial equation ${\ensuremath{\partial}}_{\ensuremath{\mu}}L=0$ in the conventional two-measure theory also arises in our new formulation, as the ${\ensuremath{\varphi}}^{m}$-field equation.

### publication date

• January 7, 2013