On the square of first order differential operators of Dirac type and the Einstein–Hilbert action

The aim of this paper is to present a new global formula for the Lichnerowicz decomposition of the square of a general Dirac type first order differential operator. Concerning gauge theories, this formula permits re-writing the Yang–Mills action linearly in the curvature so that it becomes similar to the Einstein–Hilbert action. In fact, it is shown that the two action functionals can be expressed in identical geometric forms. This holds true even more also for Yang–Mills–Yukawa-like gauge theories. From this one infers that, in particular, the full bosonic action of the (minimal) Standard Model can be expressed geometrically in complete accordance with Einstein’s theory of gravity.