A remark on quantum gravity☆
Introduction
A renormalizable theory poses a computational problem for a theoretical physicist: even if only a finite number of amplitudes need renormalization, the quantum equations of motion—the Dyson–Schwinger equations (DSE)—ensure that these amplitudes must be calculated as iterated integrals based on a skeleton expansion for the Green functions. There is an infinite series of skeletons, of growing computational complexity, and thus a formidable challenge at hand. Order is brought to this situation by the fact that the skeletons can be organized in terms of the underlying Hochschild cohomology of the Hopf algebra of a renormalizable theory, the computational challenge remains though in the analytic determination of the skeletons and their Mellin transforms [1], [2], [3]. This approach, combining the analysis of the renormalization group provided in [4] with the analysis of the mathematical structure of DSE provided in [5], [6], [1], has led to new methods in solving DSE beyond perturbation theory [5], [2], [3].
A nice fact is that internal symmetries can be systematically understood in terms of this Hochschild cohomology: Slavnov–Taylor identities are equivalent to the demand that multiplicative renormalization is compatible with the cohomology structure [7], leading to the identification of Hopf ideals generated by these very Ward and Slavnov–Taylor identities [8].
For a non-renormalizable theory the situation is worse: the computational challenge for the theorist is repeated infinitely as there is now an infinite number of amplitudes demanding renormalization, each of them still based on an infinite number of possible skeleton iterations.
But the interplay with Hochschild cohomology leads to surprising new insights into this situation, which in this first paper we discuss at an elementary level for the situation of pure gravity.
Section snippets
The structure of Dyson–Schwinger equations
To compare the situation for a renormalizable QFT with the situation for an unrenormalizable one, we consider QED in four and six dimensions of spacetime.
Gravity
We consider pure gravity understood as a theory based on a graviton propagator and n-graviton couplings as vertices. A fuller discussion incorporating ghosts and matter fields is referred to future work.
An immediate observation concerns powercounting in such a theory. If we work with Feynman rules as given in [11], we see that each n-graviton vertex is a quadric in momenta attached to the vertex. This has an immediate consequence. Corollary 2 Let ∣Γ∣ = k. Then ω(Γ) = −2(∣Γ∣ + 1). Proof A 1PI one-loop graph has as many
Remarks and conclusions
We finish this short paper with a few remarks concerning the structure of theories with a powercounting as above such that propagators and vertices cancel in their contributions to the superficial degree of divergence. We call theories with such a powercounting leg-renormalizable, to contrast them from the ordinary (loop)-renormalizable theories.
Acknowledgments
It is a pleasure to thank David Broadhurst, John Gracey and Karen Yeats for discussions.
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Talk given at “NCG-Conference in honor of Alain Connes”, Paris, 29/03/07–06/04/07, dedicated to Alain in friendship and gratitude. Work supported in parts by Grant NSF-DMS/0603781. Author supported by CNRS.