Loop Quantum Vector-Tensor Gravity and Its Spherically Symmetric Model

TL;DR

The paper extends loop quantum gravity methods to a vector-tensor theory of gravity by performing a Hamiltonian analysis and recasting the geometric sector in an $SU(2)$ connection framework. It constructs a complete quantum kinematical setup, promotes the scalar constraint to a well-defined operator on a vertex Hilbert space, and demonstrates a deparametrization procedure in the spherically symmetric sector using the vector field as a clock, yielding a time-dependent physical Hamiltonian. The reduced phase space is quantized to yield physical Hilbert spaces $\mathcal{H}_{phy,\pm}$ with corresponding Hamiltonians $\hat{H}_{phy,\pm}$, and the full physical space is the direct sum $\mathcal{H}_{phy}=\mathcal{H}_{phy,+} \oplus \mathcal{H}_{phy,-}$; the construction is underpinned by an anomaly-free constraint algebra and a well-defined dual action on vertex states. The work lays a solid foundation for quantum dynamics of vector-tensor gravity and paves the way for applications to quantum cosmology and black hole physics, while leaving open important questions about the kernel of the scalar constraint and operator self-adjointness.

Abstract

The Hamiltoinian analysis of the vector-tensor theory of gravity is performed. The resulting geometrical dynamics is reformulated into the connection dynamics, with the real SU(2)-connection serving as one of the configuration variables. This formulation allows us to extend the loop quantization scheme of general relativity to the vector-tensor theory, thereby rigorously constructing its quantum kinematical framework. The scalar constraint is promoted to a well-defined operator in the vertex Hilbert space, to represent quantum dynamics. Moreover, the spherically symmetric model of the vector-tensor theory is obtained by the symmetric reduction. Following the general deparametrization strategy for theories with diffeomorphism invariance, the spherically symmetric model can be fully deparametrized in terms of the degrees of freedom of the vector field. The corresponding reduced phase space quantization is carried out. The physical Hamiltonian generating relative evolution is promoted to a well-defined operator on the physical Hilbert space.