Abstract
We formulate the path integral for Jackiw-Teitelboim gravity in the second order formalism working directly with the metric and the dilaton. We consider the theory both in Anti-de Sitter(AdS) and de Sitter space(dS) and analyze the path integral for the disk topology and the “double trumpet” topology with two boundaries. We also consider its behavior in the presence of conformal matter. In the dS case the path integral evaluates the wavefunction of the universe which arises in the no-boundary proposal. In the asymptotic AdS or dS limit without matter we get agreement with the first order formalism. More generally, away from this limit, the path integral is more complicated due to the presence of modes from the gravity- dilaton sector and also matter sector with short wavelengths along the boundary that are smaller than the AdS or dS scales. In the double trumpet case, for both AdS and dS, we find that bosonic matter gives rise to a diverging contribution in the moduli space integral rendering the path integral ill-defined. The divergence occurs when the size of the wormhole neck vanishes and is related to the Casimir effect. For fermions this divergence can be avoided by imposing suitable boundary conditions. In this case, in dS space the resulting path integral gives a finite contribution for two disconnected universes to be produced by quantum tunneling.
Highlights
In the double trumpet case, for both Anti-de Sitter (AdS) and de Sitter (dS), we find that bosonic matter gives rise to a diverging contribution in the moduli space integral rendering the path integral ill-defined
For the disk topology we find in the asymptotic AdS or dS limit, obtained by taking the dilaton and length of the boundary to diverge while keeping their ratio fixed, that the results of the second order path integral quantization agree with those obtained from the first order formalism
Once matter is added we find that its quantum effects give rise to a contribution in the path integral which diverges when the neck goes to zero size
Summary
Jackiw-Teitelboim (JT) gravity is a theory of two-dimensional gravity which has received considerable attention recently [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110]. For the disk topology we find in the asymptotic AdS or dS limit, obtained by taking the dilaton and length of the boundary to diverge while keeping their ratio fixed, that the results of the second order path integral quantization agree with those obtained from the first order formalism. It is worth mentioning that simple dimensional counting shows that the quantum effects of matter only arise away from the asymptotic limit when one is working at finite boundary length, and including them in a systematic manner along with the quantum effects from the gravity-dilaton sector is quite non-trivial Such an analysis would need to be carried out to go beyond the semi-classical limit which has been analyzed in considerable detail recently where the number of matter fields N → ∞, and the gravity-dilaton is treated as being classical.
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