A gravitational gauge field theory based on Stephenson–Kilmister–Yang gravitation with scalar and spinor fields as gravitating matter sources

Abstract

A gravitational gauge theory with a spin–affine connection (Lorentz connection) as a rotational gauge potential (fundamental dynamical variable) is suggested for reformulating the theory of Stephenson–Kilmister–Yang gravity, in which the Einstein field equation of gravity is a first-integral solution of a spin-connection gravitational gauge field equation. A heavy intermediate field \(\phi \) that accompanies a matter field \(\varphi \) is introduced in order to remove the conventional dimensionful gravitational coupling. Such a \(\varphi \) – \(\phi \) coupling can lead to dimensionless gravitational coupling (i.e., the gravitational constant is dimensionless) in the present gravitational gauge field theory. A low-energy effective Lagrangian density for the matter field can be obtained by integrating out the accompanying heavy field in generating functional of path integral formalism, and therefore, a dimensionful gravitational coupling coefficient (Einstein gravitational constant) emerges. Such a dimensionless coupling of gravity, where the dimensionful coupling is emergent at low energies, is considered for scalar and spinor fields, which serve as gravitating matter fields (gravitational source). Though there are higher derivatives (e.g., third- and fourth-order partial derivatives) of the scalar and spinor fields in the low-energy effective Lagrangian densities, the ordinary equations of motion of the scalar and spinor fields can also be emergent from the present gravitational gauge theory. Therefore, the Einstein gravity can be recovered from the present gravitational gauge theory. In addition to the gravitational Lagrangian of the spacetime-rotational gauge potential (i.e., spin–affine connection), the Lagrangian of a spacetime-translational gauge potential (i.e., vierbein) is also constructed. Thus, the present dimensionless gravitational gauge coupling preserves local rotational and translational gauge symmetries. Since the spin-connection gravitational gauge field equation is a third-order differential equation of metric (the Einstein field equation of gravity is a first-integral solution), it could provide a new route to the vacuum energy cosmological constant problem.

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