Abstract
We perform a covariant (Lagrangian) quantization of perturbative gravity in the background of a Schwarzschild black hole. The key tool is a decomposition of the field into spherical harmonics. We fix Regge-Wheeler gauge for modes with angular momentum quantum number $l \geq 2$, while for low multipole modes with $l$ $=$ $0$ or $1$ -- for which Regge-Wheeler gauge is inapplicable -- we propose a set of gauge fixing conditions which are 2D background covariant and perturbatively well-defined. We find that the corresponding Faddeev-Popov ghosts are non-propagating for the $l\geq2$ modes, but are in general nontrivial for the low multipole modes with $l = 0,1$. However, in Schwarzschild coordinates, all time derivatives acting on the ghosts drop from the action and the low multipole ghosts have instantaneous propagators. Up to possible subtleties related to quantizing gravity in a space with a horizon, Faddeev's theorem suggests the possibility of an underlying canonical (Hamiltonian) quantization with a manifestly ghost-free Hilbert space.
Highlights
We would like to perform a perturbative quantization of gravity in the background of a Schwarzschild black hole, in a setting originally studied by Regge and Wheeler [1], and later studied by Zerilli [2] and Martel and Poisson [3], which was recently discussed in the context of quantization in Refs. [4,5]
A key obstacle to a straightforward perturbative quantization of gravity is the presence of a gauge symmetry—diffeomorphism invariance—which leaves the naive Feynman path integral ill defined
A procedure for defining the covariant (Lagrangian) path integral for quantum field theories with gauge symmetries was proposed in Refs. [6,7] and involves breaking the gauge symmetry using a set of gauge-fixing conditions and introducing compensating Faddeev-Popov (FP) ghost fields, whose action is computed according to the rules introduced in Refs. [6,7]
Summary
We would like to perform a perturbative quantization of gravity in the background of a Schwarzschild black hole, in a setting originally studied by Regge and Wheeler [1], and later studied by Zerilli [2] and Martel and Poisson [3], which was recently discussed in the context of quantization in Refs. [4,5]. [18] the choice of the gauge-fixing condition involved in addition to the standard de Donder term a set of extra terms nonlinear in hμν (for example, terms like hνλ∇λhμν), with arbitrary, numerical or depending on scalars of the theory, coefficients The result, in this case of the two-loop UV divergence in gravity, is independent of the seven additional parameters defining the generalized gauge-fixing condition. The functional determinant resulting from the transformation from C to Cmay contribute to the local measure of integration some field-dependent divergent term proportional to δDð0Þ, where D is the dimension of the spacetime on which x is a coordinate (in our case, we will have D 1⁄4 2 since our fields will become effectively twodimensional after decomposition into spherical harmonics) Terms of this nature can be neglected in a regularized theory or can be shown to cancel, as explained in Ref. We leave the study and resolution of such subtleties for future work
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
