Quantum Geometry of the Light Cone: Fock representation and Spectrum of Radiated Power

Abstract

Starting from the symplectic potential for the $\gamma$-Palatini--Holst action on a null hypersurface, we identify an auxiliary conformal field theory (CFT), which carries a representation of the constraint algebra of general relativity on a null surface. The radiative data, which is encoded into the shear of each null generator, is mapped into an $SU(1,1)$ current algebra on each light ray. We study the resulting quantum theory for both bosonic and fermionic representations. In the fermionic representation, the central charge on each null ray is positive, for bosons it is negative. A negative central charge implies a non-unitary CFT, which has negative norm states. In the model, there is a natural $SU(1,1)$ Casimir. For the bosonic representations, the $SU(1,1)$ Casimir can have either sign. For the fermionic representations, the $SU(1,1)$ Casimir is always greater or equal to zero. To exclude negative norm states, we restrict ourselves to the fermionic case. To understand the physical implications of this restriction, we express the $SU(1,1)$ Casimir in terms of the geometric data. In this way, the positivity bound on the $SU(1,1)$ Casimir translates into an upper bound for the shear of each null generator. In the model, this bound must be satisfied for all three-dimensional null hypersurfaces. This in turn suggests to apply it to an entire null foliation in an asymptotically flat spacetime. In this way, we obtain a bound on the radiated power of gravitational waves in the model.