Quantum gravity at a Lifshitz point

QUANTUM GRAVITY AT A LIFSHITZ POINT PHYSICAL REVIEW D 79, 084008 (2009)H ij ¼ ~H ij þ 1 @ i @ j2 ij@ 2 H: (53)D. At the free-field fixed pointThe action (38) of the z ¼ 3 theory with detailed balancecontains three dimensionless coupling constants: , , andw. However, only one of them, w, controls the strength ofinteractions. The noninteracting limit corresponds to sendingw ! 0, while keeping and the ratio ¼ w(47)This choice of variables diagonalizes the equations ofmotion in our gauge. Since the kinetic term is universal,its analysis in the z ¼ 3 theory is identical to that presentedfor z ¼ 2 in Sec. 4.5 of [3]. In our gauge and in terms of thenew variables, the kinetic term takes the formfixed. This limit yields a two-parameter family of free-fieldfixed points, parametrized by and .In preparation for the study of the full interacting theory,it is useful to first investigate the properties of this family offree-field fixed points. The linearization of the z ¼ 3 theoryis performed in exactly the same way as in [3] for theanalogous case of the z ¼ 2 gravity and we will thereforebe relatively brief, referring the reader to [3] for furtherdetails.We expand the theory in small fluctuations h ij , n, and n iaround the flat background,g ij ij þ wh ij ; N 1 þ wn; N i wn i :(48)The reference background is a solution of the equations ofmotion of the z ¼ 3 theory (38). Keeping only quadraticterms in the action, n drops out from the theory. A naturalgauge choice isn i ¼ 0: (49)This fixes most of the Diff F ðMÞ gauge symmetry, leavingtime-independent spatial diffeomorphisms DiffðÞ unfixed.The residual DiffðÞ gauge symmetry can be convenientlyfixed by setting@ i h ij @ j h ¼ 0; (50)where h h ii . Imposing this condition at some fixed timeslice t ¼ t 0 effectively fixes the residual DiffðÞ invariance.The Gauss constraint@ i_ h ij@ j_ h ¼ 0 (51)(which follows from varying the original action with respectto n i ) then ensures that (50) stays valid at all times.In order to diagonalize the linearized equations of motionand read off the dispersion relation of the propagatingmodes implied by our gauge choice (49) and (50), it isconvenient to first redefine the variables by introducingH ij h ij ij h; (52)the gauge condition (50) implies that H ij is transverse. Wethen decompose the transverse tensor H ij into its transversetraceless part ~H ij and its trace H,S K 1 Z dtd 32 2 x ~H _ij_~H ij þ 1 H:2ð1 3Þ _ 2 (54)It would appear that the dependence of the kinetic term ofH on can be absorbed into a rescaling of H, but wechoose not to do so, because it would obscure the geometricorigin of H in the full nonlinear theory.On the other hand, the potential term of the z ¼ 3 theoryreduces toS V 28Zdtd 3 x ~H ij ð@ 2 Þ 3 ~H ij : (55)Because of the conformal properties of the Cotton tensor,the potential term in the Gaussian approximation dependsonly on ~H ij and not on H.As pointed out in [3], the kinetic term (54) indicates thattwo values of play a special role. At ¼ 1=3, the theorybecomes compatible with the local anisotropic Weyl invariancediscussed in Sec. II C 3 above. At that value of ,the scalar mode H is a gauge artifact. The kinetic term forH also appears singular at ¼ 1. As explained in [3], thishappens because at this special value of , the linearizedtheory exhibits an extra gauge invariance, which can beused to eliminate physical excitations of H as well.The transverse traceless tensor ~H ij contains two propagatingphysical polarizations. These gravitons satisfy anonrelativistic gapless dispersion relation,! 2 ¼ 44 ðk2 Þ 3 : (56)For values of outside of the two special values 1 and1=3, the scalar mode H will represent a physical degree offreedom, with its linearized equation of motion givensimply by €H ¼ 0. When the theory is deformed by relevantoperators, the equation of motion for H will contain termswith spatial derivatives up to fourth order, which is notenough to yield a propagator with good ultraviolet properties.It appears that, in order to make the theory powercountingrenormalizable at generic values of not equal to1or1=3, either the scalar mode would have to be eliminatedby an extra gauge symmetry, or super-renormalizableterms which give short-distance spatial dynamics to thescalar mode need to be added to the potential. We willbriefly return to this point in Sec. III B.084008-9