# Can Horava gravity be formulated in LQG framework?

1. Jul 11, 2010

### marcus

My impression was that much of the 2009 interest in Horava gravity had waned. A number of problems with it were brought out at the Perimeter Institute conference on it that was held in November 2009. Ted Jacobson did the final summary session of the conference and gave an informative wrap-up.

However, whether or not Horava gravity is a good idea, wouldn't it be interesting if it could be reformulated using the tools of canonical LQG? Chopin Soo and collaborators, have made an attempt along these lines.

This just appeared on arxiv:

http://arxiv.org/abs/1007.1563
New formulation of Horava-Lifgarbagez quantum gravity as a master constraint theory
Chopin Soo, Jinsong Yang, Hoi-Lai Yu
4 pages
(Submitted on 9 Jul 2010)
"Horava-Lifgarbagez theory of quantum gravity attempts to preserve unitarity by relinquishing space-time covariance, and improve renormalizability by including higher order (spatial) derivatives. For theories without full space-time covariance, departures of the constraint algebra from the Dirac algebra are to be expected. In the non-projectable version of Horava-Lifgarbagez gravity, the commutator of two local Hamiltonian constraints leads to severely restrictive secondary constraints and perplexing 'troubles'. On the other hand, the projectable version has an integrated Hamiltonian constraint and consistent constraint algebra. But an extra graviton mode which can be problematic is then allowed, whereas in Einstein's theory the spurious mode is eliminated precisely by the local Hamiltonian constraint. A new formulation of Horava-Lifgarbagez gravity, naturally realized as a representation of the master constraint algebra studied by loop quantum gravity researchers, is presented in this work. This reformulation yields a consistent canonical theory with 1st class constraints. It captures the essence of Horava-Lifgarbagez gravity in retaining only spatial diffeomorphisms (instead of full space-time covariance) as the physically relevant non-trivial gauge symmetry; at the same time the local Hamiltonian constraint which is needed to remove the spurious mode is equivalently enforced by the master constraint."

"Master constraint" is a version of the Hamiltonian constraint in a form of canonical LQG developed by Thiemann. It seems from Soo-Yang-Yu's work that it might be possible to formulate Horava gravity using LQG methods.
http://www.cqis.ncku.edu.tw/eindex.php?name=english/detail/ecs.htm

I remember some connection of Chopin Soo with Ashtekar and Smolin. He was postdoc in Astekar's group at Penn State, and i think I remember he co-authored a paper with Smolin, a few years back. Yes, here it is:
http://arxiv.org/abs/gr-qc/9405015
The Chern-Simons Invariant as the Natural Time Variable for Classical and Quantum Cosmology
Lee Smolin, Chopin Soo
32 pages, CGPG-94/4-1
(Submitted on 6 May 1994)
We propose that the Chern-Simons invariant of the Ashtekar-Sen connection is the natural internal time coordinate for classical and quantum cosmology. The reasons for this are a number of interesting properties of this functional, which we describe here. 1)It is a function on the gauge and diffeomorphism invariant configuration space, whose gradient is orthogonal to the two physical degrees of freedom, in the metric defined by the Ashtekar formulation of general relativity. 2)The imaginary part of the Chern-Simons form reduces in the limit of small cosmological constant, Lambda, and solutions close to DeSitter spacetime, to the York extrinsic time coordinate. 3)Small matter-field excitations of the Chern-Simons state satisfy, by virtue of the quantum constraints, a functional Schroedinger equation in which the matter fields evolve on a DeSitter background in the Chern-Simons time. We then n propose this is the natural vacuum state of the theory for $$\Lambda \neq 0$$. 4)This time coordinate is periodic on the configuration space of Euclideanized spacetimes, due to the large gauge transformations, which means that physical expectation values for all states in non-perturbative quantum gravity will satisfy the KMS condition, and may then be interpreted as thermal states. 5)Forms for the physical hamiltonians and inner product which support the proposal are suggested, and a new action principle for general relativity, as a geodesic principle on the connection superspace, is found.

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