Quantum Gravity at a Lifshitz Point (0901.3775v2)

Abstract: We present a candidate quantum field theory of gravity with dynamical critical exponent equal to z=3 in the UV. (As in condensed matter systems, z measures the degree of anisotropy between space and time.) This theory, which at short distances describes interacting nonrelativistic gravitons, is power-counting renormalizable in 3+1 dimensions. When restricted to satisfy the condition of detailed balance, this theory is intimately related to topologically massive gravity in three dimensions, and the geometry of the Cotton tensor. At long distances, this theory flows naturally to the relativistic value z=1, and could therefore serve as a possible candidate for a UV completion of Einstein's general relativity or an infrared modification thereof. The effective speed of light, the Newton constant and the cosmological constant all emerge from relevant deformations of the deeply nonrelativistic z=3 theory at short distances.

• The paper introduces an innovative quantum gravity theory that achieves power-counting renormalizability in 3+1 dimensions using anisotropic scaling with z=3.
• It employs a framework based on foliation-preserving diffeomorphisms and detailed balance, incorporating higher-order spatial derivatives to overcome issues in classical gravity.
• The theory illustrates a transition from nonrelativistic UV behavior to emergent relativistic IR physics, offering a potential route to a UV completion of Einstein’s gravity.

Quantum Gravity at a Lifshitz Point: A Paradigm for Anisotropic Scaling

Petr Hořava's work, titled "Quantum Gravity at a Lifshitz Point," presents an innovative approach to constructing a quantum field theory of gravity characterized by the dynamical critical exponent $z = 3$. This paper offers a structured framework where space and time exhibit distinct scaling properties, drawing inspiration from theoretical progress in condensed matter physics, specifically quantum critical phenomena. The model proposes a theory that is power-counting renormalizable in 3+1 dimensions, diverging substantially from conventional relativistic frameworks by inherently incorporating nonrelativistic scaling at high energies.

Theoretical Framework and Methodology

The principal challenge addressed by Hořava is the lack of perturbative renormalizability in Einstein's gravity, attributed to the negative mass dimension of the gravitational coupling. Traditional higher-derivative approaches introduce non-unitary ghost states. The proposed solution eschews the requirement of Lorentz invariance at high energies, exploiting an anisotropic scaling between space and time characterized by $z > 1$.

In this framework, the spatial metric $g_{ij}$, along with the lapse function $N$ and shift vector $N_i$, navigates a manifold structured by a foliation-preserving diffeomorphism symmetry, diverging from full spacetime diffeomorphisms. The kinetic term is universally constructed, while the potential term $S_V$ maintains symmetry under this restricted group, including higher-order spatial derivative terms to secure renormalizability.

A significant innovation is the imposition of the detailed balance condition, simplifying the spectrum of interaction terms by tying their form to a variational principle associated with $W$, a classical action defined in fewer dimensions. This reduces the plethora of potential terms, drawing parallels to methodologies in nonequilibrium statistical mechanics. The intricate geometry of the Cotton tensor, a third-order derivative construct, is fundamental to this theory, acting as a potential candidate to satisfy detailed balance.

UV Completion and Infrared Behavior

The theory is distinguished by its nonrelativistic behavior in the ultraviolet (UV) region, with $z = 3$ playing a pivotal role in establishing renormalizability in 3+1 dimensions. Short-distance dynamics are dominated by terms of the highest derivative order, while infrared (IR) physics emerge naturally with relativistic scaling. This transition empowers the theory as a potential UV completion of General Relativity (GR) or its modifications. Notably, the emergent speed of light, Newton’s constant, and cosmological constant arise from relevant deformations, aligning the long-wavelength limit with Einstein’s relativity.

Potential Applications and Future Directions

Hořava's theory defines a prototype for quantum gravity models potentially extendable to other dimensions and critical exponents. For instance, exchanging parameters allows exploration of a 4+1 dimensional universe with $z = 4$, or engaging with super-renormalizable constructs in lower dimensions. The concept also naturally draws connections to the bulk/boundary dualities reminiscent of AdS/CFT, providing potential insights into nonrelativistic holographic correspondences.

This methodological shift could revolutionize how quantum gravity engages with cosmological phenomena, offering novel interpretations of fundamental quandaries like the information paradox and the role of holography in quantum gravity. Moreover, the emergence of relativity from nonrelativistic constructs propels new inquiries into the nature of spacetime and the early universe, potentially revising understandings of event horizons and cosmological gravitational dynamics.

Conclusion

Ultimately, Hořava's exploration into "Quantum Gravity at a Lifshitz Point" lays functional groundwork for continuing investigation into gravity with anisotropic scale symmetries. While intricate in its construction, it promises a fertile domain for theoretical developments that might reconcile quantum mechanics with gravity, offering a distinctive perspective on emergent spacetime symmetries. The progress in theoretical physics, as exemplified by this work, forges new pathways to understand our universe's most profound mysteries.