Generalized uncertainty principle, quantum gravity and Hořava-Lifshitz gravity
aa r X i v : . [ h e p - t h ] S e p Generalized uncertainty principle, quantum gravity andHoˇrava-Lifshitz gravity
Yun Soo Myung Institute of Basic Science and School of Computer Aided ScienceInje University, Gimhae 621-749, Korea
AbstractWe investigate a close connection between generalized uncertainty principle (GUP) anddeformed Hoˇrava-Lifshitz (HL) gravity. The GUP commutation relations correspond to theUV-quantum theory, while the canonical commutation relations represent the IR-quantumtheory. Inspired by this UV/IR quantum mechanics, we obtain the GUP-corrected gravitonpropagator by introducing UV-momentum p i = p i (1 + βp ) and compare this with tensorpropagators in the HL gravity. Two are the same up to p -order. e-mail address: [email protected] Introduction
Recently Hoˇrava has proposed a renormalizable theory of gravity at a Lifshitz point [1],which may be regarded as a UV complete candidate for general relativity. At short dis-tances the theory of z = 3 Hoˇrava-Lifshitz (HL) gravity describes interacting nonrelativisticgravitons and is supposed to be power counting renormalizable in (1+3) dimensions. Re-cently, the HL gravity theory has been intensively investigated in [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]. The equations ofmotion were derived for z = 3 HL gravity [29, 30], and its black hole solution was firstfound in asymptotically anti-de Sitter spacetimes [30] and black hole in asymptotically flatspacetimes [31].It seems that the GUP-corrected Schwarzschild black hole is closely related to blackholes in the deformed Hoˇrava-Lifshitz gravity [32, 33]. Also, the GUP provides naturally aUV cutoff to the local quantum field theory as quantum gravity effects [34, 35].On the other hand, one of main ingredients for studying quantum gravity is the GUP,which has been argued from various approaches to quantum gravity and black hole physics [36].Certain effects of quantum gravity are universal and thus, influence almost any system witha well-defined Hamiltonian [37]. The GUP satisfies the modified Heisenberg algebra [38][ x i , p j ] = i ¯ h (cid:16) δ ij + βp δ ij + 2 βp i p j (cid:17) , [ x i , x j ] = [ p i , p j ] = 0 (1)where p i is considered as the momentum at high energies and thus, it can be interpreted tobe the UV-commutation relations. Here p = p i p i . In this case, the minimal length whichfollows from these relations is given by δx min = ¯ h p β. (2)On the other hand, introducing IR-canonical variable p i with x i = x i through the replace-ment p i = p i (cid:16) βp (cid:17) , (3)these variables satisfy canonical commutation relations[ x i , p j ] = i ¯ hδ ij , [ x i , x j ] = [ p i , p j ] = 0 . (4)Here p i is considered as the momentum at low energies with p = p i p i . It is easy toshow that Eq. (1) is satisfied to linear-order β when using Eq. (4). Hence, the replace-ment (3) could be used as an important low-energy window to investigate quantum gravityphenomenology up to linear-order β . 2t was known for deformed HL gravity that the UV-propagator for tensor modes t ij take a complicated form Eq. (32), including up to p -term from the Cotton bilinear term C ij C ij . We have explored a connection between the GUP commutator and the deformedHL gravity [39]. Explicitly, we have replaced a relativistic cutoff function K ( p Λ ) by anon-relativistic density function D D ( β~p ) to derive GUP-corrected graviton propagators.These were compared to (32). It was pointed out that two are qualitatively similar , butthe p -term arisen from the crossed term of Cotton and Ricci tensors did not appear inthe GUP-corrected propagators. Also, it was unclear why the D = 2 GUP-corrected tensorpropagator (not the D = 3 GUP-corrected propagator) is similar to the UV-propagatorderived from the z = 3 HL gravity.In this work, we investigate a close connection between GUP and deformed HL grav-ity. At high energies, we assume that the UV-propagator takes the conventional form G UV ( ̟, p ) in Eq. (34), whereas at low energies, the IR-propagator takes the conventionalform G IR ( ̟, p ) in Eq. (35). It is very important to understand how the UV-propagator isrelated to the IR-propagator in the non-relativistic gravity theory. We find a GUP-correctedgraviton propagator by applying (3) to G UV ( ̟, p ) and compare it with the UV-tensor prop-agator (32) in the HL gravity. Two are the same up to p -order, although the p -term arisenfrom a crossed term of Cotton tensor and Ricci tensor is still missed in the GUP-correctedgraviton propagator. This indicates that a power-counting renormalizable theory of the HLgravity is closely related to the GUP. z = 3 HL gravity
Introducing the ADM formalism where the metric is parameterized ds ADM = − N dt + g ij (cid:16) dx i − N i dt (cid:17)(cid:16) dx j − N j dt (cid:17) , (5)the Einstein-Hilbert action can be expressed as S EH = 116 πG Z d x √ gN h K ij K ij − K + R − i , (6)where G is Newton’s constant and extrinsic curvature K ij takes the form K ij = 12 N (cid:16) ˙ g ij − ∇ i N j − ∇ j N i (cid:17) . (7)Here, a dot denotes a derivative with respect to t . An action of the non-relativistic renor-malizable gravitational theory is given by [1] S HL = Z dtd x h L K + L V i , (8)3here the kinetic terms are given by L K = 2 κ √ gN K ij G ijkl K kl = 2 κ √ gN (cid:16) K ij K ij − λK (cid:17) , (9)with the DeWitt metric G ijkl = 12 (cid:16) g ik g jl − g il g jk (cid:17) − λg ij g kl (10)and its inverse metric G ijkl = 12 (cid:16) g ik g jl − g il g jk (cid:17) − λ λ − g ij g kl . (11)The potential terms is determined by the detailed balance condition as L V = − κ √ gN E ij G ijkl E kl = √ gN ( κ µ − λ ) (cid:16) − λ R + Λ W R − W (cid:17) − κ η C ij − µη R ij ! C ij − µη R ij ! ) . (12)Here the E tensor is defined by E ij = 1 η C ij − µ (cid:16) R ij − R g ij + Λ W g ij (cid:17) (13)with the Cotton tensor C ij C ij = ǫ ikℓ √ g ∇ k (cid:18) R jℓ − Rδ jℓ (cid:19) . (14)Explicitly, E ij could be derived from the Euclidean topologically massive gravity E ij = 1 √ g δW T MG δg ij (15)with W T MG = 1 η Z d xǫ ikl (cid:16) Γ mil ∂ j Γ lkm + 23 Γ nil Γ ljm Γ mkn (cid:17) − µ Z d x √ g ( R − W ) , (16)where ǫ ikl is a tensor density with ǫ = 1.In the IR limit, comparing L with Eq.(6) of general relativity, the speed of light,Newton’s constant and the cosmological constant are given by c = κ µ s Λ W − λ , G = κ π c , Λ cc = Λ W . (17)The equations of motion were derived in [29] and [30]. We would like to mention that theIR vacuum of this theory is anti-de Sitter (AdS ) spacetimes. Hence, it is interesting to4ake a limit of the theory, which may lead to a Minkowski vacuum in the IR sector. To thisend, one may deform the theory by introducing “ µ R ” ( ˜ L V = L V + √ gN µ R ) and then,take the Λ W → c = κ µ , G = κ π c , λ = 1 . (18)The deformed HL gravity has an important parameter [31] ω = 8 µ (3 λ − κ , (19)which takes the form for λ = 1 ω = 16 µ κ . (20)Actually, ω plays the role of a charge in the Kehagias-Sfetsos (KS) black hole with λ = 1and K ij = C ij = 0 [32] derived from the Lagrangian˜ L λ =1 V = √ gN µ (cid:16) R + 34 ω R − ω R ij R ij (cid:17) . (21)and a spherically symmetric metric ansatz. Furthermore, it was shown that the entropy ofKS black hole could be explained from the entropy of GUP-corrected Schwarzschild blackhole when making a connection of β → ω [33]. A meaningful prediction of various theories of quantum gravity (string theory) and blackholes is the presence of a minimum measurable length or a maximum observable momen-tum. This has provided the generalized uncertainty principle which modifies commutationrelations shown by Eq. (1). A universal quantum gravity correction to the Hamiltonian isgiven by H UV = p m + V ( x i ) = p m + V ( x i ) + βm p + β m p (22) ≡ H IR + H (23)with H IR = p m + V ( x i ) , H = βm p + β m p . (24)5e note that Eq. (23) may be used for a perturbation study with p = − i ¯ hd/dx i . Wesee that any system with a well-defined quantum (or even classical) Hamiltonian H IR , isperturbed by H near the Planck scale. In this sense, the quantum gravity effects are insome sense universal. Some examples were performed in [37, 40, 41, 42]. It turned out thatthe corrections could be interpreted in two ways when considering linear-order perturbation H = βm p : either that for β = β l / h with β ∼
1, they are exceedingly small, beyondthe reach of current experiments or that they predict upper bounds on the quantum gravityparameter β ≤ for the Lamb shift. z = 3 HL gravity
The field equation for tensor modes propagating on the Minkowski spacetimes is givenby [24] ¨ t ij − µ κ △ t ij + µ κ △ t ij − µκ η ǫ ilm ∂ l △ t j m − κ η △ t ij = T ij (25)with external source T ij and the Laplacian △ = ∂ i → − p . We could not obtain thecovariant propagator because of the presence of ǫ -term. Assuming a massless gravitonpropagation along the x -direction with p i = (0 , , p ), then the t ij can be expressed interms of polarization components as [28] t ij = t + t × t × − t +
00 0 0 . (26)Using this parametrization, we find two coupled equations for different polarizations¨ t + − µ κ △ t + + κ µ △ t + + κ µ η ∂ △ t × − κ η △ t + = T + , (27)¨ t × − µ κ △ t × + κ µ △ t × − κ µ η ∂ △ t + − κ η △ t × = T × . (28)In order to find two independent components, we introduce the left-right base defined by t L/R = 1 √ (cid:16) t + ± it × (cid:17) (29)where t L ( t R ) represent the left (right)-handed modes. After Fourier-transformation, we findtwo decoupled equations − ̟ t L + c p t L + κ µ
16 ( p ) t L − κ µ η p ( p ) t L + κ η ( p ) t L = T L , (30) − ̟ t R + c p t R + κ µ
16 ( p ) t R + κ µ η p ( p ) t R + κ η ( p ) t R = T R . (31)6e have UV-tensor propagators with ω = 16 µ /κ t L/R = − T L/R ̟ − c (cid:16) p + ω p ∓ η µω p p + η κ ω p (cid:17) . (32)We note that the left-handed mode is not allowed because it may give rise to ghost ( − c η µω p p ),while the right-handed mode is allowed because there is no ghost ( c η µω p p ). At this stage,we mention that p (= √ p i p i ) is a magnitude of momentum p i but not a time component ̟ . Finally, we find UV-propagators in the four dimensional frame with p µ = ( ̟, , , p ) as t L/R = − T L/R ̟ − c (cid:16) p + ω p ∓ η µω p + η κ ω p (cid:17) . (33) It is known for deformed HL gravity that the UV-propagator for tensor modes t ij take acomplicated form shown in Eq. (32), including up to p -term from the Cotton bilinear term C ij C ij .At high energies, we assume that the UV-propagator takes the conventional form G UV ( ̟, p ) = 1 ̟ − c p , (34)whereas at low energies, the IR-propagator takes the conventional form G IR ( ̟, p ) = 1 ̟ − c p . (35)Considering (3), the UV-propagator (34) takes the form G UV ( ̟, p ) = 1 ̟ − c (cid:16) p + 2 βp + β p (cid:17) . (36)The GUP-corrected tensor propagator is determined by t GUPij = − G UV ( ̟, p ) T ij = − T ij ̟ − c (cid:16) p + 2 βp + β p (cid:17) , (37)where scaling dimensions are given by [ β ] = − , [ ̟ ] = 3 , and [ c ] = 2 for the z = 3 HLgravity. This is exactly the same form as the UV-tensor propagator (32) up to p whenusing the replacement of β → /ω which was derived for entropy of the Kehagias-Sfetsosblack hole without the Cotton tensor ( C ij = 0) [33]. However, considering terms beyond p ( p and p ), we could not make a definite connection between two propagators eventhough highest space derivative of sixth order are found in both propagators. Explicitly,the p -term is absent for the GUP-corrected propagator and coefficients in the front of p are different. Two coefficients are the same for η = 128 /κ .7 Discussions
We have explored a close connection between generalized uncertainty principle (GUP) anddeformed Hoˇrava-Lifshitz (HL) gravity. It was proposed that the GUP commutation rela-tions describe the UV-quantum theory, while the canonical commutation relations representthe IR-quantum theory. Inspired by this UV/IR quantum mechanics, we obtain the GUP-corrected graviton propagator by introducing UV-momentum of p i = p i (1 + βp ) with p i the IR momentum. We compare this with tensor propagators in the HL gravity. Two arethe same up to p -order, but the p -term arisen from the crossed term of Cotton and Riccitensors did not appear in the GUP-corrected propagators.Importantly, we confirm that the deformed HL gravity with ω parameter contains effectsof quantum gravity implied by the GUP with the linear-order of β when using a relation of β = 1 /ω . This means that the deformed z = 2 HL gravity without Cotten tensor could bewell described by the GUP [2]. This Lagrangian is given by˜ L z =2 = √ gN " κ (cid:16) K ij K ij − λK (cid:17) + µ (cid:16) R + 12 ω λ − λ − R − ω R ij R ij (cid:17) . (38)The tensor propagator is derived from the above Lagrangian on the Minkowski backgroundwhere Ricci-square term R does not contribute to the bilinear term of t ij t ij . Hence, it iseasily shown that ω p -term in the tensor propagator (32) comes from R ij R ij -term. On theother hand, the modified Heisenberg commutation relation (1) is satisfied to linear-order β when calculating the GUP-corrected propagator (37). Therefore, it is valid that thedeformed z = 2 HL gravity without Cotten tensor is well explained by the GUP.However, it needs a further study in order to make a clear connection between z = 3 HLgravity and the GUP with second-order of β ( β ) because the former contains the Cottontensor C ij and the replacement (3) is obscure. Acknowledgement
This work was supported by Basic Science Research Program through the National ScienceFoundation (KRF) of Korea funded by the Ministry of Education, Science and Technology(2009-0086861).
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