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Quantization of Emergent Gravity Preprint typeset in JHEP style - HYPER VERSION arXiv:1312.0580 Quantization of Emergent Gravity Hyun Seok Yang Center for Quantum Spacetime, Sogang University, Seoul 121-741, Korea E-mail: [email protected] Abstract: Emergent gravity is based on a novel form of the equivalence principle known as the Darboux theorem or the Moser lemma in symplectic geometry stating that the elec- tromagnetic force can always be eliminated by a local coordinate transformation as far as spacetime admits a symplectic structure, in other words, a microscopic spacetime becomes noncommutative (NC). If gravity emerges from U(1) gauge theory on NC spacetime, this picture of emergent gravity suggests a completely new quantization scheme where quan- tum gravity is defined by quantizing spacetime itself, leading to a dynamical NC spacetime. Therefore the quantization of emergent gravity is radically different from the conventional approach trying to quantize a phase space of metric fields. This approach for quantum gravity allows a background independent formulation where spacetime as well as matter fields is equally emergent from a universal vacuum of quantum gravity. Keywords: Models of Quantum Gravity, Gauge-Gravity Correspondence, arXiv:1312.0580v4 [hep-th] 24 Dec 2014 Non-Commutative Geometry. Contents 1. Introduction 1 2. U(1) gauge theory on Poisson manifold 8 3. Deformation quantization 16 4. Noncommutative gauge theory 24 5. Emergent gravity from U(1) gauge fields 32 5.1 Interlude for local life 32 5.2 Equivalence principle and Riemann normal coordinates 40 5.3 Fedosov manifolds and global deformation quantization 42 5.4 Generalization to Poisson manifolds 55 5.5 Towards a global geometry 59 6. Noncommutative geometry and quantum gravity 68 6.1 Quantum geometry and matrix models 68 6.2 Emergent time 74 6.3 Matrix representation of Poisson manifolds 79 6.4 Noncommutative field theory representation of AdS/CFT correspondence 83 7. Discussion 88 A. Darboux coordinates and NC gauge fields 91 B. Modular vector fields and Poisson homology 93 C. Jet bundles 98 1. Introduction This paper grew out of the author’s endeavor, after fruitful interactions with his colleagues, to clarify certain aspects of the physics of emergent gravity [1, 2, 3] proposed by the author himself a few years ago. The most notable feedbacks (including some confusions and fallacies) may be classified into three categories: (I) Delusion on noncommutative (NC) spacetime, (II) Prejudice on quantization, (III) Globalization of emergent geometry. – 1 – Let us first defend an authentic picture from the above actions (I) and (II). But we have to confess our opinion still does not win a good consensus. (III) will be one of main issues addressed in this paper. We start with the discussion why we need to change our mundane picture about gravity and spacetime if emergent gravity picture makes sense. According to the general theory of relativity, gravity is the dynamics of spacetime geometry where spacetime is realized as a (pseudo-)Riemannian manifold and the gravitational field is represented by a Riemannian metric [4, 5]. Therefore the dynamical field in gravity is a Riemannian metric over spacetime and the fluctuations of metric necessarily turn a (flat) background spacetime into a dynamical structure. Since gravity is associated to spacetime curvature, the topology of spacetime enters general relativity through the fundamental assumption that spacetime is organized into a (pseudo-)Riemannian manifold. A main lesson of general relativity is that spacetime itself is a dynamical entity. The existence of gravity introduces a new physical constant, G. The existence of the gravitational constant G, together with another physical constants c and ~ originated from the special relativity and quantum mechanics, implies that spacetime at a certain scale G~ −33 known as the Planck length LP = 3 = 1.6 10 cm, is no longer commuting, instead c × spacetime coordinates obey the commutationq relation [yµ,yν]= iθµν . (1.1) Note that the NC spacetime (1.1) is a close pedigree of quantum mechanics which is the i i formulation of mechanics on NC phase space [x ,pj] = i~δj. Once Richard Feynman said (The Character of Physical Law, 1967) “I think it is safe to say that no one understands quantum mechanics”. So we should expect that the NC spacetime similarly brings about a radical change of physics. Indeed a NC spacetime is much more radical and mysterious than we thought. However we understood the NC spacetime too easily. The delusion (I) on NC spacetime is largely rooted to our naive interpretation that the NC spacetime (1.1) is an extra structure (induced by B fields) defined on a preexisting spacetime. This naive picture inevitably brings about the interpretation that the NC spacetime (1.1) necessarily breaks the Lorentz symmetry. See the introduction in [6] for the criticism of this viewpoint. One of the reasons why the NC spacetime (1.1) is so difficult to understand is that the concept of space is doomed and the classical space should be replaced by a state in a complex vector space . Since the NC spacetime (1.1), denoted by R2n, is equivalent to H θ the Heisenberg algebra of n-dimensional harmonic oscillator, the Hilbert space in this H case is the Fock space (see eq. (6.2)). Since the Hilbert space is a complex linear vector H space, the superposition of two states must be allowed which necessarily brings about the interference of states as in quantum mechanics. We may easily be puzzled by a gedanken experiment that mimics the two slit experiment or Einstein-Podolsky-Rosen experiment in quantum mechanics. Furthermore any object defined on the NC spacetime R2n becomes O θ an operator acting on the Hilbert space . Thus we can represent the object in the Fock H O space , i.e., End( ). Since the Fock space has a countable basis, the representation H O∈ H of End( ) is given by an N N matrix where N = dim( ) [7, 8, 9]. In the case at O ∈ H × H – 2 – hand, N . This is the point we completely lose the concept of space. And this is the →∞ pith of quantum gravity to define the final destination of spacetime. To our best knowledge, quantum mechanics is the more fundamental description of nature. Hence quantization, understood as the passage from classical physics to quantum physics, is not a physical phenomenon. The world is already quantum and quantization is only our poor attempt to find the quantum theoretical description starting with the classical description which we understand better. The same philosophy should be applied to a NC spacetime. In other words, spacetime at a microscopic scale, e.g. LP , is intrinsically NC, so spacetime at this scale should be replaced by a more fundamental quantum algebra such as the algebra of N N matrices denoted by N . Therefore the usual classical spacetime × A does not exist a priori. Rather it must be derived from the quantum algebra like classical mechanics is derived from quantum mechanics. But the reverse is not feasible. In our case the quantum algebra is given by the algebra N of N N matrices. Since N = End( ), A × A ∼ H it is isomorphic to an operator algebra θ acting on the Hilbert space . The algebra A H θ is NC and generated by the Heisenberg algebra (1.1). The spacetime structure derived A from the NC ⋆-algebra θ is dubbed emergent spacetime. But this emergent spacetime A is dynamical, so gravity will also be emergent via the dynamical NC spacetime because gravity is the dynamics of spacetime geometry. It is called emergent gravity. But it turns out [1, 2, 3] that the dynamical NC spacetime is defined as a deformation of the NC spacetime (1.1) (see eq. (6.1)) and the deformation is related to U(1) gauge fields on the NC spacetime (1.1). This picture will be clarified in section 4. In this emergent gravity picture, any spacetime structure is not assumed a priori but defined by the theory. That is, the theory of emergent gravity must be background independent. Hence it is necessary to define a configuration in the algebra θ, for instance, A like eq. (1.1), to generate any kind of spacetime structure, even for flat spacetime. A beautiful picture of emergent gravity is that the flat spacetime is emergent from the Moyal- Heisenberg algebra (1.1). See section 6.2 for the verification. Many surprising results immediately come from this dynamical origin of flat spacetime [2, 3], which is absent in general relativity. As a result, the global Lorentz symmtry, being an isometry of flat spacetime, is emergent too. If true, the NC spacetime (1.1) does not break the Lorentz symmetry. Rather it is emergent from the NC spacetime (1.1). This is the picture how we correct the delusion (I). The dynamical system is described by a Poisson manifold (M,θ) where M is a differ- entiable manifold whose local coordinates are denoted by yµ µ = 1, , d = dim(M) and · · · the Poisson structure 1 ∂ ∂ θ = θµν Γ(Λ2T M) (1.2) 2 ∂yµ ∂yν ∈ ^ is a (not necessarily nondegenerate) bivector field on M. Quantum mechanics is defined by quantizing the Poisson manifold (M,θ) where M is the phase space of particles with local µ i coordinates y = (x ,pi). Let us call it ~-quantization. Similarly the NC spacetime (1.1) is defined by quantizing the Poisson manifold (M,θ) where M is a spacetime manifold, e.g., M = R2n and yµ’s are local spacetime coordinates. Let us call it θ-quantization. The first order deviation of the quantum or NC multiplication from the classical one is given by the – 3 – Poisson bracket of classical observables.
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