Abstract
A transformation is introduced in momentum representation to keep a covariant description at every stage of a loop computation in gravity. The procedure treats on equal footing local internal and space-time symmetries althought the complete transformation is known for the former [1] whereas in gravity we solve for the first few orders in an expansion. As an explicit application the one loop UV divergences of Hilbert-Einstein gravity with a cosmological constant and spin 0, 1/2 and 1 matter are computed with functional methods and in a field-covariant formalism.
Highlights
This letter adds to the techniques for loop computations by introducing a covariant momentum representation which treats on equal footing local internal and space-time symmetries
What we mean by this can be sketched for local space-time symmetries as follows: the naive transformation to momentum representation ∇ → iq + Γ (∇ → iq + A) does not display gauge covariance when one integrates over ddq leaving Γμνρ (Aμ) behind; this is addressed in the covariant derivative expansion (CDE) with a transformation that trades the dependence on connection Γ for curvature
Functional methods have been applied to particle physics over the decades and the recent literature contains complete and accessible descriptions [20, 21] to which we refer the reader for the detailed formulation; here rather we shall start from a number of results in the literature whose combination is required to tackle gravity
Summary
Functional methods have been applied to particle physics over the decades and the recent literature contains complete and accessible descriptions [20, 21] to which we refer the reader for the detailed formulation; here rather we shall start from a number of results in the literature whose combination is required to tackle gravity. Where parenthesis around indixes denotes symmetrization V(αWβ) = VαWβ + VβWα and with the opposite placing of indices as usual yet this convention follows from our component field gμν This somewhat unfamiliar language might be more accessible if we note that the graviton propagator or the inverse of the two point action contains the inverse of the metric G, G−αβ1,ρσ = gα(σgρ)β − gαβgρσ. Otherwise this treatment for a covariant result is not new in gravity and is related to what is at times termed a Vilkovisky’s action [27].
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