Generalized uncertainty principle, quantum gravity and Ho\v{r}ava-Lifshitz gravity

We investigate a close connection between generalized uncertainty principle (GUP) and deformed Ho\v{r}ava-Lifshitz (HL) gravity. The GUP commutation relations correspond to the UV-quantum theory, while the canonical commutation relations represent the IR-quantum theory. Inspired by this UV/IR quantum mechanics, we obtain the GUP-corrected graviton propagator by introducing UV-momentum $p_i=p_{0i}(1+\beta p_{0}^2)$ and compare this with tensor propagators in the HL gravity. Two are the same up to $p_0^4$-order.

It seems that the GUP-corrected Schwarzschild black hole is closely related to black holes in the deformed Hořava-Lifshitz gravity [32,33]. Also, the GUP provides naturally a UV 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 with a well-defined Hamiltonian [37]. The GUP satisfies the modified Heisenberg algebra [38] where p i is considered as the momentum at high energies and thus, it can be interpreted to be the UV-commutation relations. Here p 2 = p i p i . In this case, the minimal length which follows from these relations is given by On the other hand, introducing IR-canonical variable p 0i with x i = x 0i through the replace- these variables satisfy canonical commutation relations Here p 0i is considered as the momentum at low energies with p 2 0 = p 0i p 0i . It is easy to show that Eq. (1) is satisfied to linear-order β when using Eq. (4). Hence, the replacement (3) could be used as an important low-energy window to investigate quantum gravity phenomenology up to linear-order β.
It 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 6 0 -term from the Cotton bilinear term C ij C ij . We have explored a connection between the GUP commutator and the deformed HL gravity [39]. Explicitly, we have replaced a relativistic cutoff function K( p 2 Λ 2 ) by a non-relativistic density function D D (β p 2 ) to derive GUP-corrected graviton propagators.
These were compared to (32). It was pointed out that two are qualitatively similar, but the p 5 -term arisen from the crossed term of Cotton and Ricci tensors did not appear in the GUP-corrected propagators. Also, it was unclear why the D = 2 GUP-corrected tensor propagator (not the D = 3 GUP-corrected propagator) is similar to the UV-propagator derived from the z = 3 HL gravity.
In this work, we investigate a close connection between GUP and deformed HL gravity. At high energies, we assume that the UV-propagator takes the conventional form G UV (̟, p 2 ) in Eq. (34), whereas at low energies, the IR-propagator takes the conventional form G IR (̟, p 2 0 ) in Eq. (35). It is very important to understand how the UV-propagator is related to the IR-propagator in the non-relativistic gravity theory. We find a GUP-corrected graviton propagator by applying (3) to G UV (̟, p 2 ) and compare it with the UV-tensor propagator (32) in the HL gravity. Two are the same up to p 4 0 -order, although the p 5 0 -term arisen from a crossed term of Cotton tensor and Ricci tensor is still missed in the GUP-corrected graviton propagator. This indicates that a power-counting renormalizable theory of the HL gravity is closely related to the GUP.

z = 3 HL gravity
Introducing the ADM formalism where the metric is parameterized the Einstein-Hilbert action can be expressed as where G is Newton's constant and extrinsic curvature K ij takes the form Here, a dot denotes a derivative with respect to t. An action of the non-relativistic renormalizable gravitational theory is given by [1] where the kinetic terms are given by with the DeWitt metric and its inverse metric The potential terms is determined by the detailed balance condition as Here the E tensor is defined by with the Cotton tensor C ij Explicitly, E ij could be derived from the Euclidean topologically massive gravity with where ǫ ikl is a tensor density with ǫ 123 = 1.
In the IR limit, comparing L 0 with Eq.(6) of general relativity, the speed of light, Newton's constant and the cosmological constant are given by The equations of motion were derived in [29] and [30]. We would like to mention that the IR vacuum of this theory is anti-de Sitter (AdS 4 ) spacetimes. Hence, it is interesting to take a limit of the theory, which may lead to a Minkowski vacuum in the IR sector. To this end, one may deform the theory by introducing "µ 4 R" (L V = L V + √ gN µ 4 R) and then, take the Λ W → 0 limit [31]. We call this the deformed HL gravity without detailed balance condition. This does not alter the UV properties of the theory, while it changes the IR properties. That is, there exists a Minkowski vacuum, instead of an AdS vacuum. In the IR limit, the speed of light and Newton's constant are given by The deformed HL gravity has an important parameter [31] which takes the form for λ = 1 Actually, 1 2ω plays the role of a charge in the Kehagias-Sfetsos (KS) black hole with λ = 1 and K ij = C ij = 0 [32] derived from the Lagrangiañ and a spherically symmetric metric ansatz. Furthermore, it was shown that the entropy of KS black hole could be explained from the entropy of GUP-corrected Schwarzschild black hole when making a connection of β → 1 ω [33].

GUP-quantum mechanics
A meaningful prediction of various theories of quantum gravity (string theory) and black holes is the presence of a minimum measurable length or a maximum observable momentum. This has provided the generalized uncertainty principle which modifies commutation relations shown by Eq. (1). A universal quantum gravity correction to the Hamiltonian is given by with We note that Eq. (23) may be used for a perturbation study with p 0 = −ihd/dx 0i . We see that any system with a well-defined quantum (or even classical) Hamiltonian H IR , is perturbed by H 1 near the Planck scale. In this sense, the quantum gravity effects are in some sense universal. Some examples were performed in [37,40,41,42]. It turned out that the corrections could be interpreted in two ways when considering linear-order perturbation H 1 = β m p 4 0 : either that for β = β 0 l 2 Pl /2h 2 with β 0 ∼ 1, they are exceedingly small, beyond the reach of current experiments or that they predict upper bounds on the quantum gravity parameter β 0 ≤ 10 34 for the Lamb shift.

Tensor modes for deformed z = 3 HL gravity
The field equation for tensor modes propagating on the Minkowski spacetimes is given by [24] with external source T ij and the Laplacian △ = ∂ 2 i → −p 2 0 . We could not obtain the covariant propagator because of the presence of ǫ-term. Assuming a massless graviton propagation along the x 3 -direction with p 0i = (0, 0, p 3 ), then the t ij can be expressed in terms of polarization components as [28] Using this parametrization, we find two coupled equations for different polarizations In order to find two independent components, we introduce the left-right base defined by where t L (t R ) represent the left (right)-handed modes. After Fourier-transformation, we find two decoupled equations We have UV-tensor propagators with ω = 16µ 2 /κ 2 We note that the left-handed mode is not allowed because it may give rise to ghost (− 8c 2 η 2 µω p 3 p 4 0 ), while the right-handed mode is allowed because there is no ghost ( 8c 2 η 2 µω p 3 p 4 0 ). At this stage, we mention that p 0 (= √ p 0i p 0i ) is a magnitude of momentum p 0i but not a time component

GUP-corrected propagator
It is known for deformed HL gravity that the UV-propagator for tensor modes t ij take a complicated form shown in Eq. (32), including up to p 6 0 -term from the Cotton bilinear term C ij C ij .
At high energies, we assume that the UV-propagator takes the conventional form whereas at low energies, the IR-propagator takes the conventional form Considering (3), the UV-propagator (34) takes the form The GUP-corrected tensor propagator is determined by where scaling dimensions are given by [β] = −2, [̟] = 3, and [c] = 2 for the z = 3 HL gravity. This is exactly the same form as the UV-tensor propagator (32) up to p 4 0 when using the replacement of β → 1/ω which was derived for entropy of the Kehagias-Sfetsos black hole without the Cotton tensor (C ij = 0) [33]. However, considering terms beyond p 4 0 (p 5 0 and p 6 0 ), we could not make a definite connection between two propagators even though highest space derivative of sixth order are found in both propagators. Explicitly, the p 5 0 -term is absent for the GUP-corrected propagator and coefficients in the front of p 6 0 are different. Two coefficients are the same for η 4 = 128/κ 2 .

Discussions
We have explored a close connection between generalized uncertainty principle (GUP) and deformed Hořava-Lifshitz (HL) gravity. It was proposed that the GUP commutation relations describe the UV-quantum theory, while the canonical commutation relations represent the IR-quantum theory. Inspired by this UV/IR quantum mechanics, we obtain the GUPcorrected graviton propagator by introducing UV-momentum of p i = p 0i (1 + βp 2 0 ) with p 0i the IR momentum. We compare this with tensor propagators in the HL gravity. Two are the same up to p 4 0 -order, but the p 5 0 -term arisen from the crossed term of Cotton and Ricci tensors did not appear in the GUP-corrected propagators.
Importantly, we confirm that the deformed HL gravity with ω parameter contains effects of 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 be well described by the GUP [2]. This Lagrangian is given bỹ The tensor propagator is derived from the above Lagrangian on the Minkowski background where Ricci-square term R 2 does not contribute to the bilinear term of t ij t ij . Hence, it is easily shown that 2 ω p 4 0 -term in the tensor propagator (32) comes from R ij R ij -term. On the other 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 the deformed 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 HL gravity and the GUP with second-order of β (β 2 ) because the former contains the Cotton tensor C ij and the replacement (3) is obscure.