One-loop quantum gravity in the Einstein universe

Abstract

Abstract:We study quantum gravity with the Einstein-Hilbert action including the cosmological constant on the Euclidean Einstein universe S 1 × S 3 . We compute exactly the spectra and the heat kernels of the relevant operators on S 3 and use these results to compute the heat trace of the graviton and ghost operators and the exact one-loop effective action on S 1 × S 3 . We show that the system is unstable in the infrared limit due to the presence of the negative modes of the graviton and the ghost operators. We study the thermal properties of the model with the temperature T = (2πa 1 ) −1 determined by the radius a 1 of the circle S 1 . We show that the heat capacity C v is well defined and behaves like ∼ T 3 in the high temperature limit and has a singularity of the type ∼ (T − T c ) −1 , indicating a second-order phase transition, with the critical temperature T c determined by the cosmological constant Λ and the radius a of the sphere S 3 . We also discuss some peculiar properties of the model such as the negative heat capacity as well as possible physical applications.

On the Weyl anomaly of 4D conformal higher spins: a holographic approach

“…This clearly generalizes the spin two for which one actually has the part involving the Riemann tensor 2R (c d) a b (see e.g. [70]) that gives the only Weyl-tensor contribution 2 W…”

“…This yields for unconstrained spin two fields spin two: tr V {Ω 2 2 ) ∼ − (D + 2) W 2 (73) in accordance, for example, with [70] (eqn.3.54).…”

“…We first write in terms of gpd's to proceed again by contracting to extract the overall dependence on s and D until we reach now the s = 1 term for which we also already know the answer (see e.g. [70], eqn.3.12) spin one:…”

Virial coefficients expressed by heat kernel coefficients

“…As for Fermi gases, the treatment is exactly the same, and the results only have some minor differences: Besides the grand potential (1) changes to ln Ξ = n ln 1 + ze −βεn and the Bose-Einstein integrals in eqs. (14) and (16) change to the Fermi-Dirac integrals, the main difference is that all of the even-order virial coefficients b 2 , b 4 , · · · will have an extra negative sign while the odd-order ones b 1 , b 3 , · · · are the same as the Bose case. Therefore, the virial expansion for Fermi gases can be obtained from the above Bose results directly.…”

“…Since the first several heat kernel coefficients can be analytically expressed in terms of geometric invariants of the manifold [8,9,10], the asymptotic expansion of heat kernel, i.e., the heat kernel expansion, becomes a very important tool [8,11,12]. The heat kernel expansion has been applied in many fields of physics, such as quantum field theory [13,14,15], quantum gravity [16,17], and string theory [18]. In quantum statistical mechanics, the sum over the spectrum is a key component to calculate the grand partition function, so the heat kernel approach also plays an important role [11,19,20].…”