aa r X i v : . [ h e p - t h ] F e b Towards A Background Independent QuantumGravity
Hyun Seok Yang
Institute for the Early Universe, Ewha Womans University, Seoul 120-750, KoreaE-mail: [email protected]
Abstract.
We recapitulate the scheme of emergent gravity to highlight how a backgroundindependent quantum gravity can be defined by quantizing spacetime itself.
1. Introduction
According to general relativity, gravity is the dynamics of spacetime geometry where spacetimeis a (pseudo-)Riemannian manifold and the gravitational field is represented by a Riemannianmetric [1]. The gravitational field equations are given by the Einstein equations defined by R MN − g MN R = 8 πGc T MN . (1)A beauty of the equation (1) may be phrased in a poetic diction (John A. Wheeler) that mattertells spacetime how to curve, and spacetime tells matter how to move. However the notoriousdifficulty to elevate the intimate cooperation between spacetime geometry and matters to aquantum world has insinuated doubt into us for the harmonious conspiracy. Therefore we willtake a closer look at (1) aiming to reveal an inmost conflict in (1) behind the superficial harmony.The first observation [2] is that the gravitation described by (1) presupposes a physicallyinviable vacuum. If one consider a flat spacetime whose metric is given by g MN = η MN , theleft-hand side of (1) identically vanishes and so the right-hand side must cultivate a completelyempty space with T MN = 0. But the concept of empty space in Einstein gravity is in acutecontrast to the concept of vacuum in quantum field theory (QFT) where the vacuum is notempty but full of quantum fluctuations. As a result, QFT claims that such an empty spaceof nothing is inviable in Nature and, instead, the vacuum is extremely heavy whose weight ismaximally of Planck mass, i.e., ρ vac ∼ M P . Thus, if there is no way to completely suppressquantum fluctuations, Einstein gravity probably either presupposes a physically inviable spaceor incorrectly identifies the genetic origin of flat spacetime.The second observation [3] is that flat spacetime in general relativity behaves like an elasticbody with tension although the flat spactime itself is the geometry of special relativity. We haveseen that slides for colloquium introducing general relativity to the public or undergraduatestudents often contain the image like Figure 1 below. The Figure 1 illustrates how a massivebody changes the geometry of spacetime around the mass point. In general relativity, this(curved) geometry is described by the gravitational field g MN ( x ) = η MN + h MN ( x ) andinterpreted as gravity. When the massive body moves to another place, the original pointhere the body was placed will recover a (nearly) flat geometry like a rubber band. Thatis, the (flat) spacetime behaves like a metrical elasticity which opposes the curving of space. Figure 1.
Two-dimensional analogyof spacetime distortion. (Image fromWikipedia: Spacetime)But this picture rather exhibits a puzzling nature offlat spacetime because the flat spacetime should be acompletely empty space without any kind of energy aswe remarked above. How is it possible for an emptyspace of nothing to behave like an elastic body withtension ? Moreover we know that the gravitational forceis extremely weak which implies that the space stronglywithstands the curving and so the tension of spacetimewould be extremely big, maybe, of the Planck energy.The third observation [4] is that the gravitationalfield g MN ( x ) = η MN + h MN ( x ) has a vacuumexpectation value (vev), i.e., h g MN ( x ) i vac = η MN likethe Higgs field φ ( x ) = v + h ( x ). These two fields also describe particles (either spin-2 gravitonor spin-0 Higgs) as usual quantum fields in Standard Model. However these two particles arevery exotic because all other fields, denoted as Ψ, in Standard Model have a zero vev; h Ψ i vac = 0.We are reasonably understanding why the Higgs field has the nonzero vev which triggers theelectroweak symmetry breaking. In effect, Standard Model is defined in a nontrivial vacuumwith the Higgs condensate v = h φ i vac whose dynamical scale is around v ∼
246 GeV. Thereforeany particle interacting with the Higgs field feels a resistance in vacuum and acquires a mass.What about the flat spacetime η MN = h g MN ( x ) i vac ? Is it also originated from some kind ofvacuum condensate ? If so, what is the dynamical scale of the condensate ? Note that thegravitation is characterized by its own intrinsic scale given by the Newton constant G = L P where L P = M − P ∼ − cm is the Planck length and classical gravity corresponds to L P → M (cid:16) ∂∂s (cid:17) = g MN ( x ) ∂∂x M ⊗ ∂∂x N . (2)It is well-known that the metric (2) in the tetrad formalism can be defined by the tensor productof two vector fields E A = E MA ( x ) ∂∂x M ∈ Γ( T M ) as follows (cid:16) ∂∂s (cid:17) = η AB E A ⊗ E B . (3)Mathematically, a vector field X on a smooth manifold M is a derivation of the algebra C ∞ ( M ).Here the vector fields E A ∈ Γ( T M ) are the smooth sections of tangent bundle
T M → M which are dual to the vector space E A = E AM ( x ) dx M ∈ Γ( T ∗ M ), i.e., h E A , E B i = δ AB . Theexpression (3) glimpses the avatar of gravity that a spin-two graviton might arise as a compositeof two spin-one vector fields . In other words, the tensor product (3) can be abstracted bythe relation (1 ⊗ S = 2 ⊕
0. Note that any field Ψ for fundamental particles in StandardModel cannot be written as the tensor product of other two fields, so to say, Ψ = Ψ ⊗ Ψ .Only composite particles (or bound states) such as mesons can be represented in such a way.Therefore graviton represented by the tensor product (3) is certainly different from fundamentalparticles in Standard Model.he final observation [7, 8] is that there is an acute mismatch of symmetry between gravityand matters because gravity is the only interaction sensitive to a shift of the Lagrangian by anadditive constant. To be precise, if one shift a matter Lagrangian L M by a constant Λ, that is, L M → L ′ M = L M − , (4)it results in the shift of the energy-momentum tensor by T MN → T MN − Λ g MN in the Einsteinequation (1) although the equations of motion for matters are invariant under the shift (4).Definitely the Λ-term in (4) will appear as the cosmological constant in Einstein gravity and itaffects the spacetime structure. For instance, a flat spacetime is no longer a solution of (1). Evenworse is that this clash of symmetry brings about the stability problem of spacetime. As weremarked in the first observation, the vacuum in QFT is a stormy sea of quantum fluctuationswhich accommodates the vacuum energy of the order of M P . Fortunately the vacuum energydue to the quantum fluctuations, regardless of how large it is, does not make any trouble toQFT thanks to the symmetry (4). However the general covariance requires that gravity couplesuniversally to all kinds of energy. Therefore the vacuum energy ρ vac ∼ M P will induce a highlycurved spacetime whose curvature scale R would be ∼ M P according to (1). If so, the QFTframework in the background of quantum fluctuations must be broken down due to a large back-reaction of background spacetime. But we know that it is not the case. QFT is well-defined asever in the presence of the vacuum energy because the background spacetime still remains flat,as we empirically know. What is wrong with this argument ?After consolidating all the suspicions inferred above, we throw a doubt on the genesis that flatspacetime is free gratis, i.e., costs no energy. All the above reasonings imply that the negligenceabout the dynamical origin of flat spacetime defining a local inertial frame in general relativitymight be a core root of the incompatibility inherent in (1). It should be remarked that thegenesis about spacetime cannot be addressed within the context of general relativity becauseflat spacetime is a geometry of special relativity rather than general relativity and so it is assumedto be a priori given without reference to its dynamical origin. All in all, it is tempted to inferthat flat spacetime may be not free gratis but a result of Planck energy condensation in vacuum[9, 10]. Surprisingly, if that inference is true, it appears as the H´oly Gr´ail to cure several notoriousproblems in theoretical physics; for example, to resolve the cosmological constant problem, tounderstand the nature of dark energy and to explain why gravity is so weak compared to otherforces. After all, the target is to formulate a background independent theory to correctlyexplain the dynamical origin of flat spacetime. Note that Einstein gravity is not completelybackground independent since it assumes the prior existence of a spacetime manifold. But itturns out [11, 12, 5] that the emergent gravity from noncommutative (NC) geometry preciselyrealizes the desired property, as will be surveyed in the next sections.
2. Einstein gravity from electromagnetism on symplectic space
Now we will show that the vierbeins in (3) and so the Riemannian metrics arise fromelectromagnetic fields living in a space (
M, B ) supporting a symplectic structure B [13, 14,15, 16]. See [17, 18, 19, 5, 20] for recent reviews of this subject. The symplectic structure B isa nondegenerate, closed 2-form, i.e. dB = 0 [21]. Therefore the symplectic structure B definesa bundle isomorphism B : T M → T ∗ M by X A = ι X B where ι X is an interior productwith respect to a vector field X ∈ Γ( T M ). One can invert this map to obtain the inverse map θ ≡ B − : T ∗ M → T M defined by α X = θ ( α ) such that X ( β ) = θ ( α, β ) for α, β ∈ Γ( T ∗ M ). Here we refer to a background independent theory where any spacetime structure is not a priori assumed butdefined by the theory. From now on, we will work in Euclidean space though we still use the term “spacetime”. After illuminatinghow a space is emergent from U (1) gauge fields, we will speculatively touch the issue of emergent time. he bivector θ ∈ Γ(Λ T M ) is called a Poisson structure of M which defines a bilinear operationon C ∞ ( M ), the so-called Poisson bracket, defined by { f, g } θ = θ ( df, dg ) (5)for f, g ∈ C ∞ ( M ). Then the real vector space C ∞ ( M ), together with the Poissonbracket {− , −} θ , forms an infinite-dimensional Lie algebra, called a Poisson algebra P =( C ∞ ( M ) , {− , −} θ ). First note that the orthonormal tangent vectors E A = E MA ( x ) ∂ M ∈ Γ( T M )satisfy the Lie algebra [ E A , E B ] = − f ABC E C . (6)In general, the composition [ X, Y ], the Lie bracket of X and Y , on Γ( T M ), together withthe real vector space structure of Γ(
T M ), forms a Lie algebra V = (Γ( T M ) , [ − , − ]). There isa natural Lie algebra homomorphism between the Lie algebra V = (Γ( T M ) , [ − , − ]) and thePoisson algebra P = ( C ∞ ( M ) , {− , −} θ ) defined by [21] C ∞ ( M ) → Γ( T M ) : f X f (7)such that X f ( g ) = θ ( df, dg ) = { f, g } θ (8)for f, g ∈ C ∞ ( M ). It is easy to prove the Lie algebra homomorphism X { f,g } θ = [ X f , X g ] (9)using the Jacobi identity of the Poisson algebra P .Let us take M = R and a constant symplectic structure B = B MN dx M ∧ dx N , for simplicity.A remarkable point is that the electromagnetism on a symplectic manifold ( R , B ) is completelydescribed by the Poisson algebra P = ( C ∞ ( M ) , {− , −} θ ) [22, 23, 12]. For example, the actionis given by S = 14 g Y M Z d x { C A , C B } θ (10)where C A ( x ) = B AB x B + A A ( x ) ∈ C ∞ ( M ) , A = 1 , · · · , x A ,i.e. { x A , x B } θ = θ AB , and { C A ( x ) , C B ( x ) } θ = − B AB + ∂ A A B − ∂ B A A + { A A , A B } θ ≡ − B AB + F AB ( x ) ∈ C ∞ ( M ) . (12)It is clear [5] that the equations of motion as well as the Bianchi identity can be representedonly with the Poisson bracket {− , −} θ : { C B ( x ) , { C A ( x ) , C B ( x ) } θ } θ = 0 , (13) { C A ( x ) , { C B ( x ) , C C ( x ) } θ } θ + cyclic = 0 , (14)where { C A , { C B , C C } θ } θ = ∂ A F BC + { A A , F BC } θ ≡ D A F BC ∈ C ∞ ( M ) . (15)A peculiar thing for the action (10) is that the field strength F AB in (12) is nonlinear due tothe Poisson bracket term although it is the curvature tensor of U (1) gauge fields. Thus one canconsider a nontrivial solution of the following self-duality equation F AB = ± ε ABCD F CD . (16)n fact, after the canonical Dirac quantization (40) of the Poisson algebra P =( C ∞ ( M ) , {− , −} θ ), the solution of the self-duality equation (16) is known as noncommutative U (1) instantons [24, 25, 26]. When applying the Lie algebra homomorphism (9) to (12), we getthe identity X F AB = [ V A , V B ] (17)where the vector fields V A ≡ X C A ∈ Γ( T M ) are obtained by the map (8) from the set of thecovariant coordinates C A ( x ) in (11). As a result, the self-duality equation (16) is mapped to theself-duality equation of the vector fields V A [22, 23]:[ V A , V B ] = ± ε ABCD [ V C , V D ] . (18)Note that the vector fields V A = V MA ∂ M are divergence free, i.e., ∂ M V MA = 0 by the definition (8)and so preserves a volume form ν because L V A ν = ( ∇ · V A ) ν = 0 where L V A is a Lie derivativewith respect to the vector field V A . Furthermore it can be shown [12] that V A can be related tothe vierbeins E A by V A = λE A with λ ∈ C ∞ ( M ) to be determined.If the volume form ν is given by ν ≡ λ − ν g = λ − E ∧ · · · ∧ E (19)or, in other words, λ = ν ( V , · · · , V ), one can easily check that the triple of K¨ahler forms for ahyper-K¨ahler manifold M is given by [5] J a + = 12 η aAB ι A ι B ν, J ˙ a − = − η ˙ aAB ι A ι B ν, (20)where ι A is the interior product with respect to V A and η aAB and η ˙ aAB are self-dual and anti-self-dual ’t Hooft symbols [27]. One can prove that gravitational instantons satisfying the self-dualityequation R MNAB = ± ε ABCD R MNCD (21)are hyper-K¨ahler manifolds, i.e., dJ a + = 0 or dJ ˙ a − = 0 and vice versa. It is straightforward toprove [5] that the hyper-K¨ahler conditions dJ a + = 0 or dJ ˙ a − = 0 are precisely equivalent to (18)which can easily be seen by applying to (20) the formula [21] d ( ι X ι Y α ) = ι [ X,Y ] α + ι Y L X α − ι X L Y α + ι X ι Y dα (22)for vector fields X, Y and a p -form α .In retrospect, the self-dual Lie algebra (18) was derived from the self-duality equation (16) of U (1) gauge fields defined on the symplectic manifold ( R , B ). As a consequence, U (1) instantonson the symplectic manifold ( R , B ) are gravitational instantons [28, 29, 22, 23] ! We wantto emphasize that the emergence of Riemannian metrics from symplectic U (1) gauge fieldsis an inevitable consequence of the Lie algebra homomorphism between the Poisson algebra P = ( C ∞ ( M ) , {− , −} θ ) and the Lie algebra V = (Γ( T M ) , [ − , − ]) if the underlying action of U (1) gauge fields is given by the form (10). Moreover, the equivalence between U (1) instantonsin the action (10) and gravitational instantons turns out to be a particular case of more generalduality between the U (1) gauge theory on a symplectic manifold ( M, B ) and Einstein gravity[12, 30], as will be sketched below.First of all, we draw general results derived from the Lie algebra isomorphism betweenthe Poisson algebra P = ( C ∞ ( M ) , {− , −} θ ) and the Lie algebra V = (Γ( T M ) , [ − , − ]). Since V A = λE A ∈ Γ( T M ) where λ = det V MA , the Riemannian metric (3) is given by (cid:16) ∂∂s (cid:17) = δ AB E A ⊗ E B = λ − δ AB V A ⊗ V B (23)r ds = δ AB E A ⊗ E B = λ δ AB V A ⊗ V B (24)where V A = λ − E A ∈ Γ( T ∗ M ) is a dual basis of V A ∈ Γ( T M ). Note that the smoothfunctions C A ( x ) ∈ C ∞ ( M ) ( A = 1 , · · · ,
4) in (11) are linearly independent and thus the vectorfields V A ∈ Γ( T M ) defined by (7) are also linearly independent. Accordingly the vector fields V A ( A = 1 , · · · ,
4) span a full four-dimensional space. In effect, the metric (24) is completelydetermined by the set (11) of U (1) gauge fields and it describes a general Riemannian manifold.So far we did not impose the equations of motion (13) and the Jacobi identity (14) on themetric (23). Eventually we have to impose them because the set (11) of U (1) gauge fields obey(13) and (14). In order to do that, let us apply the Lie algebra homomorphism (9) again to (15)to yield X D A F BC = [ V A , [ V B , V C ]] ∈ Γ( T M ) . (25)It is then straightforward to get the following correspondence [12] D B F AB = 0 ⇔ [ V B , [ V A , V B ]] = 0 , (26) D A F BC + cyclic = 0 ⇔ [ V A , [ V B , V C ]] + cyclic = 0 . (27)Now a critical question is whether the equations of motion (26) for gauge fields together with theJacobi (or Bianchi) identity (27) can be written as the Einstein equations for the metric (23).A quick notice is that (26) and (27) will end in some equations related to Riemann curvaturetensors because they differentiate the metric (23) twice.To see what they are, recall that, in terms of covariant derivative, the torsion T and thecurvature R can be expressed as follows [1] T ( X, Y ) = ∇ X Y − ∇ Y X − [ X, Y ] , (28) R ( X, Y ) Z = [ ∇ X , ∇ Y ] Z − ∇ [ X,Y ] Z, (29)where X, Y and Z are vector fields on M . Because T and R are multilinear differential operators,we get the following relations [31] T ( V A , V B ) = λ T ( E A , E B ) , (30) R ( V A , V B ) V C = λ R ( E A , E B ) E C . (31)After imposing the torsion free condition T ( E A , E B ) = 0, it is straightforward, using (30) and(31), to derive the identity below R ( E A , E B ) E C + cyclic = λ − (cid:16) [ V A , [ V B , V C ]] + cyclic (cid:17) . (32)Therefore we immediately see [12] that the Bianchi identity (27) for U (1) gauge fields isequivalent to the first Bianchi identity for Riemann curvature tensors, i.e., D A F BC + cyclic = 0 ⇔ R ( E A , E B ) E C + cyclic = 0 . (33)The mission for the equations of motion (26) is more involved. An underlying idea is tocarefully separate the right-hand side of (26) into a part related to the Ricci tensor R AB and aremaining part. Basically, we are expecting the following form of the Einstein equations D B F AB = 0 ⇔ R AB = 8 πG (cid:16) T AB − δ AB T (cid:17) . (34)fter a straightforward but tedious calculation [12], we get a remarkably simple but crypticresult R AB = − λ h g (+) aD g ( − )˙ bD (cid:16) η aAC η ˙ bBC + η aBC η ˙ bAC (cid:17) − g (+) aC g ( − )˙ bD (cid:16) η aAC η ˙ bBD + η aBC η ˙ bAD (cid:17)i . (35)To get the result (35), we have defined the structure equation of vector fields V A ∈ Γ( T M )[ V A , V B ] = − g ABC V C (36)and the canonical decomposition g ABC = g (+) aC η aAB + g ( − ) ˙ aC η ˙ aAB . (37)A notable point is that the right-hand side of (35) consists of purely interaction terms betweenself-dual and anti-self-dual parts in (37) which is the feature withheld by matter fields only[32, 33]. Incidentally, the self-duality equation (18) can be understood as g ( − ) ˙ aC = 0 (self-dual)or g (+) aC = 0 (anti-self-dual) in terms of (37) and so R AB = 0 in (35), i.e., (18) describes aRicci-flat manifold. Of course, this is consistent with the fact that a gravitational instanton isa Ricci-flat, K¨ahler manifold. Nevertheless a unique property of (35) is to contain a nontrivialtrace contribution, i.e., a nonzero Ricci scalar, due to the second part which is not existent inEinstein gravity as was recently shown in [33]. By comparing (35) with (34), a surprising contentof the energy-momentum tensor was found in [12], which will be discussed in section 4.We come to the conclusion that general relativity or gravity can emerge from theelectromagnetism supported on a symplectic spacetime ( M, B ), which is an interacting theorydefined by the action (10). How is it possible to realize the equivalence principle or generalcovariance, the most important property in the theory of gravity (general relativity), fromthe U (1) gauge theory on a symplectic or Poisson manifold ? It turns out [11, 12] that thePoisson structure (5) of spacetime admits a novel form of the equivalence principle even forthe electromagnetic force, known as the Darboux theorem or the Moser lemma in symplecticgeometry, and consequently the electromagnetism on a symplectic spacetime can be realized asa geometrical property of spacetime. In the end, the symplectization of spacetime geometrywould be a novel and authentic way to quantize gravity [5, 20].
3. Noncommutative geometry and quantum gravity
We have observed that Einstein gravity can be emergent from electromagnetism as long asspacetime admits a symplectic structure and its underlying theory is completely defined by thePoisson algebra P = ( C ∞ ( M ) , {− , −} θ ). For instance, an underlying dynamical system forgravity is described by the action (10) which leads to the equations of motion (13). Note thatthe Jacobi identity (14) is an important property for P to be a Poisson algebra and to form aLie algebra. One can understand the Lie algebra morphism (7) as the adjoint map defined byad f : g
7→ { f, g } θ (38)and thence the action of any element on the algebra is a derivation, i.e.,ad f ( g · h ) = (ad f g ) · h + g · ad f h (39)for f, g, h ∈ C ∞ ( M ). The Jacobi identity of the Poisson algebra P is then equivalent to theidentity (9) between the operators of the adjoint representation. This identity implies that themap (38) sending each element to its adjoint action is a Lie algebra homomorphism of the originalalgebra P into the Lie algebra V = (Γ( T M ) , [ − , − ]) of its derivations. This is a mathematicalasis to explain how gravity is emergent from the electromagnetism on a symplectic manifold( M, B ).Using the isomorphism between the Lie algebras P and V , we showed that a standard(commutative) dynamical system for gravity can be described in terms of vector fields in V . Recall that vector fields on a usual (commutative) space are derivations of the algebra C ∞ ( M ) of smooth functions on this space. And vector fields, being a global concept, has itsnoncommutative (NC) generalization, called a derivation of NC algebra. Now we will show howthe derivation of NC algebra can be obtained by canonically (`a la Dirac) quantizing the Poissonalgebra P = ( C ∞ ( M ) , {− , −} θ ).The Dirac quantization of the Poisson algebra P = ( C ∞ ( M ) , {− , −} θ ) consists of a complexHilbert space H and of a quantization map Q to attach to functions f ∈ C ∞ ( M ) on M operators b f ∈ A θ acting on H [34, 35]. The map Q : C ∞ ( M ) → A θ by f
7→ Q ( f ) ≡ b f should be C -linearand an algebra homomorphism: f · g d f ⋆ g = b f · b g (40)and f ⋆ g ≡ Q − (cid:16) Q ( f ) · Q ( g ) (cid:17) (41)for f, g ∈ C ∞ ( M ) and b f , b g ∈ A θ . The Poisson structure (5) controls the failure of commutativity[ b f , b g ] ∼ i { f, g } θ + O ( θ ) . (42)For example, the coordinate generators of A θ are noncommuting with the Heisenberg algebrarelation [ x A , x B ] = iθ AB (43)where we omit the hat for the coordinate generators for a notational convenience. From thedeformation quantization point of view, the NC algebra of operators in A θ is equivalent to thedeformed algebra of functions defined by the Moyal ⋆ -product (41) which is, according to theWeyl-Moyal map [34, 35], given by b f · b g ∼ = ( f ⋆ g )( x ) = exp (cid:16) i θ AB ∂ xA ∂ yB (cid:17) f ( x ) g ( y ) | x = y . (44)According to the quantization map (40), every expressions in P are mapped to correspondingoperators in A θ . For instance, for the symplectic gauge fields in (11), we have the map C A ( x ) ∈ C ∞ ( M ) ⇒ b C A ( x ) = B AB x B + b A A ( x ) ∈ A θ (45)and, for the Poisson bracket in (12), { C A ( x ) , C B ( x ) } θ ⇒ − i [ b C A ( x ) , b C B ( x )] ⋆ = − B AB + ∂ A b A B − ∂ B b A A − i [ b A A , b A B ] ⋆ ≡ − B AB + b F AB ( x ) ∈ A θ (46)where [ b f , b g ] ⋆ = b f ⋆ b g − b g ⋆ b f . The quantized action for NC U (1) gauge fields is then given by b S = − g Y M Z d x [ b C A , b C B ] ⋆ . (47)Similarly, one can lift the adjoint map (7) or (38) to derivations of the NC algebra A θ :ad ⋆ b f : b g
7→ − i [ b f , b g ] ⋆ (48)hat satisfies the Leibniz rule, i.e.,ad ⋆ b f ( b g ⋆ b h ) = (ad ⋆ b f b g ) ⋆ b h + b g ⋆ ad ⋆ b f b h (49)for b f , b g, b h ∈ A θ . And the Jacobi identity of the ⋆ -commutator (48) leads to the conclusion thatthe polydifferential operator on A θ [36], whose set is denoted as Γ θ ( d T M ),ad ⋆ b f ≡ X ⋆ b f = X f + ∞ X n =2 ξ A ··· A n X f ∂ A · · · ∂ A n (50)is again a derivation of A θ satisfying the deformed Lie algebra[ X ⋆ b f , X ⋆ b g ] = X ⋆ [ b f, b g ] ⋆ . (51)It should be noted that the polydifferential operator (50) recovers the usual vector field in thecommutative limit θ →
0. Hence it is obvious that the left-hand side of (51) is a deformation ofthe ordinary Lie bracket of vector fields. See [36] for an explicit formula up to second order.It is easy to “quantize” the Lie algebra homomorphism (25) using the above relation (51) X ⋆ b D A b F BC = [ V ⋆A , [ V ⋆B , V ⋆C ]] ∈ Γ θ ( d T M ) (52)where V ⋆A ≡ X ⋆ b C A ∈ Γ θ ( d T M ) are generalized vector fields defined by (50). Accordingly we havea NC generalization of the correspondence (26) given by [12, 5] b D B b F AB = 0 ⇔ [ V ⋆B , [ V ⋆A , V ⋆B ]] = 0 , (53) b D A b F BC + cyclic = 0 ⇔ [ V ⋆A , [ V ⋆B , V ⋆C ]] + cyclic = 0 . (54)Since the leading order in (50) recovers the usual vector fields, the Einstein equations (34) willappear as the leading order of NC gauge fields described by (53) and (54) and higher orderterms will generate derivative corrections of the Einstein gravity. Although there is no concreteverification so far, it was conjectured in [11] that the resulting emergent gravity from NC gaugefields will be based on the NC geometry defined by b T ( X ⋆ , Y ⋆ ) = b ∇ X ⋆ Y ⋆ − b ∇ Y ⋆ X ⋆ − [ X ⋆ , Y ⋆ ] , (55) b R ( X ⋆ , Y ⋆ ) Z ⋆ = [ b ∇ X ⋆ , b ∇ Y ⋆ ] Z ⋆ − b ∇ [ X ⋆ ,Y ⋆ ] Z ⋆ , (56)where X ⋆ , Y ⋆ , Z ⋆ ∈ Γ θ ( d T M ) and b ∇ X ⋆ is a generalized affine connection on A θ evaluated at thevector field X ⋆ . If so, the NC gravity in [37, 38] would be defined by NC gauge fields.Now we will argue that the NC geometry described by (53) and (54) has to define aquantum gravity at a microscopic scale, e.g., Planck scale L P and provides a clue to realizea background independent formulation of quantum gravity [20]. One can meaningfully speakof a NC dynamics (without reference to local concepts such as that of points or time instant)provided that one describes NC dynamics in terms of derivations of the corresponding algebra.This is precisely the geometric basis underpinning the above construction. In this approach, thecrux for the background independentness is that the NC spacetime defined by the Heisenbergalgebra (43) admits a separable Hilbert space H which is an infinite-dimensional Fock spaceof two-dimensional quantum harmonic oscillators. Therefore any NC fields in A θ , which areoperators acting on H , can be represented in the Fock space H as N × N matrices where = dim H → ∞ [23]. In the end, we have a matrix representation, denoted as A N , for the NC U (1) gauge theory described by (53) and (54) that is given by [39] b D B b F AB = 0 ⇔ [ C B , [ C A , C B ]] = 0 , (57) b D A b F BC + cyclic = 0 ⇔ [ C A , [ C B , C C ]] + cyclic = 0 , (58)where C A = B AB x B + A A ∈ A N is a matrix representation in H of the NC field b C A ( x ) ∈ A θ .Note that the matrix equations of motion (57) can be derived from the 0-dimensional IKKTmatrix model [40, 41] whose action is given by S M = − g Tr[ C A , C B ] . (59)An underlying picture for the emergent gravity [5] will become clear by recasting thearguments so far with the matrix action (59). The action (59) is zero-dimensional and so it doesnot assume any kind of spacetime structure. There are only four N × N Hermitian matrices C A ( A = 1 , · · · ,
4) which are subject to a couple of algebraic relations defined by the right-handsides of (57) and (58). Therefore, in order to create a Universe (or any existence) , first we haveto specify a vacuum of the theory where all fluctuations are supported. For consistency, thevacuum should also satisfy (57) and (58). Since the action (59) allows infinitely many solutionseven with different topologies, it is not unique but there is a natural “primitive” vacuum definedby h C A i vac ≡ b A (0) A = B AB x B (60)where B AB is a constant matrix of rank 4. The vacuum (60) describes a uniform condensate ofNC gauge fields and obeys (57) and (58) if x B ∈ A N satisfy the Heisenberg algebra (43). Wecan also introduce fluctuations over the vacuum (60) which are represented by A A . Becausethere is a Hilbert space H as a representation space of the Heisenberg algebra (43), we canregard the matrices C A ∈ A N as operators in A θ acting on the Hilbert space H . According tothe Weyl-Moyal map (44), these adjoint operators are in turn mapped to NC fields representedby (45). It is worthy of remark that the pith for the duality between geometry and algebraoriginates from the coherent condensation (60) of quantum harmonic oscillators in vacuum,which grants a symplectic structure to the vacuum. It is well-known [17] that the NC algebra A θ generated by (43) admits a nontrivial inner automorphism whose infinitesimal form is calledan inner derivation defined by (48). As a result, the dynamics of fluctuations on the vacuum(43) can always be described by the inner derivations of the algebra A θ , as was verified in (53)and (54). We showed that their commutative limit is nothing but the Einstein gravity or generalrelativity.Now we are ready to disclose the secret nature of spacetime we have posed in the introduction.Because the IKKT matrix model (59) does not assume any prior existence of spacetime fromthe beginning, in other words, (59) is a background independent theory, it is necessary to definea configuration in the algebra A θ , for instance, like (60), to generate any kind of spacetimestructure, even for flat spacetime. So the question is: What is the spacetime emergent from thevacuum (60) which signifies a uniform condensate of NC gauge fields in vacuum ? The definition(48) immediately says that the corresponding vector field V (0) A = δ BA ∂ B for the vacuum gaugefield b A (0) A is precisely that of flat spacetime, i.e., h g AB i vac = δ AB . See (24) for the metric where λ = 1 in this case. Remarkably the vacuum (60) responsible for the flat spacetime is not anempty space unlike general relativity. Instead the flat spacetime is emergent from a uniformcondensation of gauge fields in vacuum [12, 5]. Its surprising consequences will be discussed innext section. . Emergent spacetime and dark energy To recapitulate, it was shown that the symplectic structure of spacetime M leads to anisomorphism between symplectic geometry ( M, B ) and Riemannian geometry (
M, g ) wherethe deformations of symplectic structure B in terms of electromagnetic fields F = dA aretransformed into those of Riemannian metric g . This approach for quantum gravity allowsa background independent formulation which provides a novel and authentic way to quantizegravity [20]. As such, the theory should be formulated in a way that the spacetime geometryarises from a vacuum solution to the field equation of the theory. One should not have tospecify a preferred background spacetime in order to write down the field equations. The theory(59) described by large N matrices is precisely the case. Indeed, the flat spacetime arisesfrom the vacuum solution (60). All other fluctuations over the vacuum are represented by NC U (1) gauge fields with the action (47), whose commutative limit θ → ∂ A → E A = ∂ A + h MA ( x ) ∂ M or δ AB → g AB = δ AB + h AB , according to the map (7). Oneimmediate conclusion is, thus, that a uniform condensation of gauge fields such as the vacuum(60) does not gravitate [9, 12] because it is simply responsible for the generation of flat spacetime.A natural question is then what is the dynamical scale of the vacuum condensate (60). Because gravity emerges from NC U (1) gauge fields described by the action (10) or (47), theparameters, g Y M and θ = ( B ), defining the NC gauge theory should be related to the Newtonconstant G in emergent gravity. A simple dimensional analysis leads to the relation [12] G ¯ h c ∼ g Y M | θ | . (61)Then one can immediately estimate the vacuum energy ρ vac caused by the condensate (60): ρ vac ∼ g Y M | B AB | ∼ g Y M M P ∼ − M P (62)where M P = (8 πG ) − / ∼ GeV is the Planck mass and g Y M ∼ . Finally the emergentgravity reveals a remarkable picture that the condensation of Planck energy in vacuum is actuallythe origin of flat spacetime. That is to say, the huge Planck energy (62) was simply used to makea flat spacetime. Hence we can conclude that the vacuum energy ρ vac ∼ M P does not gravitate,which is a tangible difference from Einstein gravity. It is of prime importance that the emergentgravity should not contain a coupling of cosmological constant like R d x √ g Λ. This importantconclusion may be more strengthened by looking at the definition (7) of emergent metric whichis insensitive to a constant vacuum energy because it requires the identity X F AB − B AB = X F AB ∈ Γ( T M ) (63)for a constant field strength B AB . Consequently, the emergent gravity clearly dismisses thecosmological constant problem [9, 2].As a necessary consequence, the emergent gravity respects the shift symmetry (4). Forexample, under a shift in the B -field, B → B ′ = B + b , with b a constant antisymmetric tensor,the NC field theory (47) defined in the new background θ ′ = ( B + b ) is physically equivalent tothat in the old one θ = ( B ) due to the well-known Seiberg-Witten equivalence [42]. Moreoverthe Hilbert spaces H ( θ ′ ) and H ( θ ) for the Heisenberg algebra (43) are isomorphic to each other. We do not have an understanding yet how time emerges in this context. Therefore it may be rude to talk aboutthe dynamics, energy, etc. But a qualitative nature of the underlying physics can equally be addressed even inEuclidean space. Hence, for some time, we will reluctantly stay in Euclidean space not to get astray due to thenotorious issue of emergent time. lso the vector fields V ′ (0) A and V (0) A determined by B ′ and B backgrounds are equally flat as longas they are constant. This means that the shift symmetry (4) belongs to a global automorphism(a Darboux transformation) of the NC algebra A θ , i.e. A θ ′ ∼ = A θ , which can be interpreted asa global Lorentz transformation [5].If a flat spacetime emerges from the Planck energy condensation (62) in vacuum, we candraw several interesting implications (though necessarily speculative) which seem to resolve allthe puzzles posed in the introduction. First of all, it implies that spacetime will behave like anelastic body with the tension of Planck energy [3]. In other words, gravitational fields generatedby the deformations of the background (60) will be very weak because the spacetime vacuum isvery solid with a stiffness of the Planck mass. Therefore the dynamical origin of flat spacetimeexplains the metrical elasticity opposing the curving of space as depicted in Figure 1 and thestunning weakness of gravitational force [2]. Furthermore the emergent spacetime implies thatthe global Lorentz symmetry, being an isometry of flat spacetime, should be a perfect symmetryup to the Planck scale because the flat spacetime was originated from the condensation of themaximum energy in Nature.The emergence of spacetime by the vacuum condensate (60) probably also has very interestingimplications to cosmology [6]. It is worthwhile to notice that the vacuum algebra (43) describesan extremely coherent condensation because it is the Heisenberg algebra of two-dimensionalquantum harmonic oscillator. As a result, the spacetime vacuum (60) should describe a zero-entropy state in spite of the involvement of Planck energy like as the electrical resistance ofsuperconductors is zero because the Cooper pair condensate moves as a coherent quantummechanical entity. It is very mysterious but it should be the case, because the flat spacetime isa completely empty space from the viewpoint of general relativity and so has no entropy. Thethermodynamic laws then suggest that a global time evolution of universe will have an arrowsince the entropy of universe has to increase anyway since its birth. It was argued in [6] that thecoherent condensate (60) of spacetime “monads” may explain the arrow of time in the cosmicevolution of our universe.One may envisage a spontaneous creation of an exponentially expanding universe. But onecould not say that the de Sitter universe was created out of field energy in a preexisting space.If we intend to understand the Planck energy condensation in vacuum as a dynamical process,the time scale for the condensate will be roughly of the Planck time. Thus it is natural toconsider that the explosive inflation era that lasted roughly 10 − seconds at the beginning ofour univese corresponds to the dynamical process for the instantaneous condensation of vacuumenergy ρ vac ∼ M P to enormously spread out a spacetime. A simple consideration of energyconservation for the condensate leads to the expansion rate12 V I − πGρ vac R = 0 ⇒ V I = H I R (64)where H I = q πGρ vac ∼ M P . This implies that the dynamical process for the vacuumcondensate may explain a cosmic inflation to generate an extremely large spacetime ∼ e L H .Unfortunately, it is not clear how to microscopically describe this dynamical process by usingthe matrix action (59). Nevertheless, it is quite obvious that the cosmic inflation should be a We may emphasize that it is an inevitable consequence if quantum gravity should be formulated in a backgroundindependent way and so the spacetime geometry emerges from a vacuum configuration of some fundamentalingredients in the theory. It is then reasonable to assume that the gravitational constant G = M − P would set anatural dynamical scale for the emergence of gravity and spacetime. Recently there was a very interesting work [43, 44] addressing this issue using the Monte Carlo analysis of thetype IIB matrix model in Lorentzian signature. In this work it was found that three out of nine spatial directionsstart to expand at some critical time after which exactly 3+1 dimensions dynamically become macroscopic. ynamical condensation in vacuum for the generation of spacetime according to our emergentgravity picture [6].Note that the vacuum condensate (60) causes the microscopic spacetime to be NC and sointroduces a spacetime uncertainty relation. Therefore, a further accumulation of energy overthe NC spacetime (43) will be subject to the UV/IR mixing [45]. This spacetime exclusion willprevent a very localized energy from further condensing into the vacuum, which may correspondto the stability of spacetime. This reasoning implies that the condensation of the vacuum energy(62) happened at most only once. By the same reason, the cosmic inflation should take placeonly once and, thereby, eternal inflation and cyclic universe seem to be inconsistent with ourpicture [6].The special relativity unifies space and time into a single entity – spacetime. Hence it will bedesirable to put space and time on an equal footing. If a space is emergent, so should time. Butthe concept of time is more stringent since it is difficult to give up the causality and unitarity.We believe that a naive introduction of NC time, e.g., [ t, x ] = iθ , will be problematic becauseit is impossible to keep the locality in time with the NC time and so to protect the causalityand unitarity. How can we define the emergent time together with the emergent space ? Howis it entangled with the space to unfold into a single entity – spacetime and to take the shapeof Lorentz covariance ? A decisive lesson comes from quantum mechanics.Quantum mechanics is the formulation of mechanics in NC phase space[ x i , p j ] = i ¯ hδ ij . (65)In quantum mechanics, the time evolution of a dynamical system is described by the quantumHamilton’s (Heisenberg) equation i ¯ h ddt b f ( t ) = [ b f ( t ) , b H ] (66)where b f ( t ) , b H ∈ A ¯ h are operators in the set of observables A ¯ h . This equation can be integratedto give a finite time evolution given by b f ( t ) = U ( t ) † b f (0) U ( t ) (67)where U ( t ) = exp( − i b Ht/ ¯ h ) is the time evolution operator. Therefore, as we know well, anintrinsic (particle) time in quantum mechanics is defined as an inner automorphism of NCalgebra A ¯ h . Note that the time evolution (67) is meaningful only if the underlying algebra A ¯ h is NC but it does not require an operator or NC time to describe the history of the particlesystem.A notable point is that any Poisson manifold ( M, θ ) always admits a dynamical Hamiltoniansystem on M where the Poisson structure θ is a bivector in Γ(Λ T M ) and the dynamics of asystem is described by the Hamiltonian vector field X f = θ ( df ) of an underlying Poisson algebra[21]. After a (Dirac or deformation) quantization of the Poisson algebra, one can describe thedynamics of the system in terms of derivations of an underlying NC algebra. For example, in theclassical limit ¯ h →
0, the Heisenberg equation (66) reduces to the classical Hamilton’s equation ddt f ( t ) = X H ( f ( t )) = { f ( t ) , H } ¯ h (68)where X H is a Hamiltonian vector field defined by ι X H ω = dH with the symplectic structure ω = P i dx i ∧ dp i of the particle phase space. In order to get an insight about the emergenttime, it would be worthwhile to realize that the mathematical structure of emergent gravity isasically the same as quantum mechanics. The former is based on the NC space (43) while thelatter is based on the NC phase space (65).Therefore we can also apply the same philosophy to the case of NC spacetime [12]. Wesuggest the concept of time in emergent gravity as the corresponding Hamilton’s equation in theNC space (43) ddt b f ( t ) = − i [ b f ( t ) , b H ] ⋆ (69)where b f ( t ) , b H ∈ A θ . Its finite version will be given by an inner automorphism, similar to (67),of the NC algebra A θ . In the commutative limit θ →
0, (69) reduces to the similar equation as(68) ddt f ( t ) = X H ( f ( t )) = { f ( t ) , H } θ (70)where X H is a Hamiltonian vector field defined by ι X H B = dH . Conversely, the Heisenbergequation (69) is the quantization of the classical Hamilton’s equation (70) following the rule(39). But, if the Hamiltonian b H is time-independent, one can infer by analogy with quantummechanics that the Heisenberg equation (69) describes only an internal time evolution over thespace.Let us consider a dynamical evolution described by the change of a symplectic structure from ω = B to ω t = ω + t ( ω − ω ) for all 0 ≤ t ≤ ω − ω = dA . A remarkable point(due to the Moser lemma [21]) is that there exists a one-parameter family of diffeomorphisms φ : M × R → M such that φ ∗ t ( ω t ) = ω , ≤ t ≤
1. Then the evolution of the symplecticstructure is locally described by the flow φ t starting at φ = identity and generated by timedependent vector fields X t = dφ t dt ◦ φ − t satisfying the equation ι X t ω t + A = 0 . (71)Actually the covariant coordinates X A ( x ) = θ AB C B ( x ) = x A + θ AB A B ( x ) in (11) correspondto the Darboux transformation φ : x A X A obeying φ ∗ ( B + F ) = B [11]. Note thatthe emergence of gravity originates from the global existence of the one-parameter family ofdiffeomorphisms φ t ∈ Diff( M ) describing the local deformation of an initial symplectic structure ω = B due to the electromagnetic force F = dA . Here we observe that the fluctuation ofbackground geometry (determined by ω = B ) due to the deformation of symplectic structuresnecessarily accompanies the time evolution of entire geometry.There are two facts known in symplectic geometry material to the concept of emergent time.The time evolution of a time-dependent system can again be defined by the inner automorphismof an extended phase space whose extended Poisson bivector is given by [21] e θ = θ + ∂∂t ∧ ∂∂H . (72)As usual, the generalized Hamiltonian vector field is defined by e X H = − e θ ( dH ) = θ AB ∂H∂x B ∂∂x A + ∂∂t (73)and the corresponding Hamilton’s equation is given by V ( f ) ≡ e X H ( f ) = { C , f } e θ = ∂f∂t + { C , f } θ (74)where C ≡ − H ∈ C ∞ ( M × R ). Basically we regard C = − H as an another dynamicalvariable and so we have a set of coordinates denoted by e C e A = ( C , C A ) in M × R and, followinghe same philosophy as (8), we have introduced a vector field V = e X H ∈ Γ( T ( M × R )).But note that the original vector fields V A ≡ X C A remain intact because of the relation e V A = e θ ( dC A ) = θ ( dC A ) = V A . Another important point is the theorem of Souriau and Sternberg[46] stating that a nontrivial time evolution in the presence of electromagnetic fields can bedescribed by the Hamilton’s equation with a free Hamiltonian H = H but with a new Poissonstructure Θ = (cid:16) B + F (cid:17) deformed by the electromagnetic force F = dA . In the case of (74), thistheorem means [5] that V ( f ) = ∂f∂t − { H , f } Θ (75)where Θ AB = { X A , X B } θ and H is a Hamiltonian function when F = 0, viz., for flat spacetime.The result (75) reveals a consistent picture with general relativity about time [12]. If Θ AB is constant (homogeneous), e.g. Θ AB = θ AB , a clock will tick everywhere at the same ratebecause (75) is exactly the same as the time evolution on flat spacetime. But, if Θ AB ( x ) is notconstant (inhomogeneous) and so an underlying geometry is curved, the time evolution will notbe uniform and a clock will tick at the different rate at different places. Also it is quite plausiblethat the local Lorentz symmetry would be recovered on a local chart because Θ AB ( x ) on thelocal chart will not be significantly changed and thus the time evolution there is locally the sameas the flat spacetime.The above picture can be more illuminated by evaluating the metric on M × R defined bythe vector fields ( V , V A ) ∈ Γ( T ( M × R )) or their dual one-forms ( V , V A ) ∈ Γ( T ∗ ( M × R )).The resulting (4+1)-dimensional metric is given by ds = λ (cid:16) − dt + V AM V AN ( dx M − A M )( dx N − A N ) (cid:17) (76)where A M = − (cid:16) θ MN ∂C ∂x N (cid:17) dt and λ = ν ( V , V , · · · , V ) with volume form ν = dt ∧ d x . Onecan easily see that the metric (76) reduces to the (4+1)-dimensional Minkowski spacetime afterturning off all fluctuations. Interestingly, the metric (76) appears as an emergent geometry ofmatrix quantum mechanics – the BFSS matrix model [47] – whose action is given by S MQM = 1 g Z dt Tr (cid:16) −
12 ( D C A ) + 14 [ C A , C B ] (cid:17) (77)where D C A = ∂ C A ∂t − i [ A , C A ]. Using the relationship between large N matrix model and NCfield theory under the Moyal vacuum (60) with h A i vac = 0, one can show [23] that the matrixquantum mechanics (77) is equivalent to (4+1)-dimensional NC U (1) gauge theory. From theNC gauge theory representation of the matrix model (77), it is straightforward to reproduce theemergent metric (76) using the vector fields V A ∈ Γ( T M ) determined by NC fields. In this way,we may get some deep insight about the formidable issue of emergent time [5].Now let us return to the result (35) to discuss a content of the energy-momentum tensordefined by its right-hand side. First it will be convenient to decompose the right-hand side (35)into two parts [12]: 8 πGT ( M ) AB = − λ (cid:16) g ACD g BCD − δ AB g CDE g CDE (cid:17) , (78)8 πGT ( L ) AB = 12 λ (cid:16) ρ A ρ B − Ψ A Ψ B − δ AB ( ρ C − Ψ C ) (cid:17) , (79) We do not understand why the time emerges with an opposite sign, i.e., with the Minkowski signature. Thesignature is just our wishful choice. here ρ A ≡ g BAB and Ψ A ≡ − ε ABCD g BCD . Recall that the result (35) was obtainedin Euclidean space. In order to get a corresponding result in (3+1)-dimensional Lorentzianspacetime just like above, we need to start with a three-dimensional NC space. But we cannotcomplete a full three-dimensional NC space with the Moyal algebra (43) since it is possibleonly with even dimensions. Instead, it may be necessary to have a Lie algebra vacuum [30],e.g. [ x A , x B ] = iε ABC x C ( A, B, C = 1 , , x A , x B , x C ] = iε ABC .Unfortunately, the calculation for these cases is much more difficult. Even it is quite demandingto define derivations (i.e., vector fields) for the latter case although the former case is rather well-known from the representation theory of Lie algebra. Therefore we will take a simple-mindedrecipe – the Wick rotation. (We are not happy with this trick.)The Wick rotation will be defined by x = ix . Under this Wick rotation, we get the results δ AB → η AB , ε = 1 → − ε = − A → i Ψ A in Minkowski spacetime. There aresome reasons that the energy-momentum tensor (78) has to be mapped to the one of the usualMaxwell theory in commutative spacetime. Indeed it was argued in [12] that it can be doneby reversing the map (7). But, as we already remarked in section 2, the engrossing part is(79) since it is absent in Einstein gravity and would be a rather unique feature of emergentgravity. Since we are eventually interested in a long-wavelength limit, we will take only thescalar mode in (79) which will be a source of the expansion/contraction of spacetime. This willbe a good approximation since the remaining term corresponds to a quadruple mode which givesrise to the shear distortion of spacetime and can thus be neglected. For the same reason, wecan ignore the Maxwell energy-momentum tensor (78) since it is also a purely quadruple mode.Hence, in the long-wavelength limit where h ρ M ρ N i ≃ g MN ρ P and h Ψ M Ψ N i ≃ g MN Ψ P , theenergy-momentum tensor (79) simply reduces to [12, 5] T ( L ) MN ≃ − R πG g MN (80)where R = λ (cid:16) ρ M ρ N + Ψ M Ψ N (cid:17) g MN is the Ricci-scalar of the metric (24). Note that λ = √− g and ρ M = ∂ M λ . Since Ψ M is in some sense Hodge-dual to ρ M , we expect that their fluctuationswill be of the same order with a characteristic wavelength L H . Therefore, by a simple dimensionalargument, we estimate that R ∼ L H . In the end, the energy-momentum tensor (80) proves tobe T ( L ) MN ∼ − L P L H g MN . (81)We will argue [9, 10] that the weird energy (81) would be originated from “vacuumfluctuations” with the largest possible wavelength L H due to the UV/IR mixing triggered by thenoncommutativity of spacetime [45]. Recall that the vacuum energy ρ vac ∼ M P does not coupleto gravity since it was used to create a flat spacetime in our picture. Thus vacuum fluctuationsover the vacuum (60) will be a leading contribution to the deformation of spacetime curvature.So let us calculate the vacuum fluctuation energy by the action (47) with the largest possiblewavelength L H : ρ = ρ vac + δρ = 14 g Y M (cid:16) B AB − b F AB ( x ) (cid:17) = 14 g Y M B AB (cid:16) θ b F ( x ) (cid:17) In the Lorentzian signature, the sign of the Ricci scalar R depends on whether fluctuations are spacelike ( R >
R <
0) [12, 5]. In consequence the spacelike perturbations act as a repulsive force whereas the timelikeones act as an attractive force. When considering the fact that the fluctuations in (79) are random in nature andwe are living in (3+1) (macroscopic) dimensions, the ratio of the repulsive and attractive components will end in : = 75 : 25. Is it outrageous to conceive that this ratio curiously coincides with the dark composition of ouruniverse ? M P (cid:16) L P L H (cid:17) ∼ M P + 1 L P L H , (82)where it is natural to assess that b F ( x ) ∼ L H for a dimensional reason. Note that the vacuumfluctuation energy δρ ∼ g Y M B AB b F AB ( x ) ∼ L P L H (83)is a total derivative term and so a boundary term on a hypersurface of radius L H . If we assume L H to be the size of cosmic horizon, the vacuum fluctuation energy (83) is in good agreementwith the observed value of current dark energy ρ DE ∼ (10 − eV) [12].Although our conclusion about the dark energy may be too speculative and so a full-fledgedformulation is further required, we believe that the emergent gravity from NC gauge fields hasplenty of rooms to explain the nature of dark energy and our underlying arguments must betrue even in a full-fledged theory.
5. A novel unification in noncommutative spacetime
It has been hoped that a physically viable theory of quantum gravity would unify into a singleconsistent model all fundamental interactions and describe all known observable interactions inthe universe, at both subatomic and astronomical scales. We have argued that the gravitationcan emerge from NC gauge fields and a background independent quantum gravity can be definedby quantizing spacetime itself. The upshot is that if gravity is emergent, then the spacetimeshould be emergent too. If so, every structures supported on the spacetime must also be emergentfor an internal consistency of the theory. Hence it should be natural that matter fields as wellas non-Abelian gauge fields for weak and strong forces have to be emergent together with thespacetime. Thus an urgent question is the following. How to define matter fields as well asnon-Abelian gauge fields describing quarks and leptons in the context of emergent geometry ?In order to figure out an underlying picture for emergent matters [12, 5], it would be usefulto start with the Feynman’s observation about the electrodynamics of charged particles [48]. In1948, Feynman got a beautiful idea how to understand electrodynamics in terms of symplecticgeometry of particle phase space. Briefly speaking, Feynman asks a question what is the mostgeneral form of interactions consistent with particle dynamics defined in the quantum phasespace (65). Surprisingly he ends up with the electromagnetic force. In other words, theelectromagnetic force is only a consistent interaction with a quantum particle satisfying thecommutation relation (65). But the Feynman’s observation raises a curious question. We knowthat, beside the electromagnetic force, there exist other interactions, weak and strong forces, inNature. Thus the problem is how to incorporate the weak and strong forces together into theFeynman’s scheme. Because he started with only a few very natural axioms, there seems to beno room to relax his postulates to include the weak and strong forces except introducing extradimensions. Remarkably it works with extra dimensions !To be more precise, consider a particle dynamics defined on R × F with an internal space F whose coordinates are { x i : i = 1 , , } ∈ R and { Q I : I = 1 , · · · , n − } ∈ F . The dynamics ofthe particle carrying an internal charge in F is defined by a symplectic structure on T ∗ R × F whose commutation relations are given by [49, 50][ x i , x j ] = 0 , m [ x i , ˙ x j ] = i ¯ hδ ij , (84)[ Q I , Q J ] = i ¯ hf IJK Q K , (85)[ x i , Q I ] = 0 . (86) Unfortunately, the significance of boundary terms such as (83) due to the UV/IR mixing was overlooked in[12, 5]. This boundary contribution caused by the UV/IR mixing could be consistent with the holographic natureof dark energy. he internal coordinates Q I satisfy SU ( n ) algebra and so carry their own Poisson structureinherited from the Lie algebra. One more condition, the so-called Wong’s equation [51], isimplemented by ˙ Q I + 12 f IJK ( A Ji Q K ˙ x i + ˙ x i A Ji Q K ) = 0 (87)to ensure that the internal charge Q I is parallel-transported along the trajectory of a particleunder the influence of non-Abelian gauge fields A i ( x, t ) = A Ii ( x, t ) Q I [52]. Actually, the Wong’sequation is the Heisenberg equation (66) for Q I with the Hamiltonian b H = ( m ˙ x i + Q I Q I ), i.e.,˙ Q I = − i ¯ h [ Q I , b H ] . (88)One can easily show it using the fact m ˙ x i = p i − A i ( x, t ) and [ p i , Q I ] = 0.If we repeat the Feynman’s question, we can arrive at the conclusion that the most generalinteraction of a quantum particle on R carrying an internal charge Q I satisfying (87) and thecommutation relations (84)-(86) is a non-Abelian interaction of SU ( n ) gauge fields [49]. Fromour perspective, we thus need extra dimensions with the Poisson structure F satisfying the abovecommutation relations to realize leptons and quarks interacting with SU (2) and SU (3) gaugefields. Wishfully, it will be more desirable to find a mechanism to realize this structure togetherwith the emergence of spacetime geometry.With this motivation, we consider a U ( N ) gauge theory in four dimensions whose action isgiven by S Y M = − G s Z d x Tr (cid:16) F µν F µν + 12 D µ Φ a D µ Φ a −
14 [Φ a , Φ b ] (cid:17) (89)where Φ a ( a = 1 , · · · ,
6) are adjoint scalar fields in U ( N ). For our purpose, we are interestedin a large N limit, in particular, N → ∞ . The action (89) is then exactly the bosonic partof 4-dimensional N = 4 supersymmetric U ( N ) Yang-Mills theory, which is the large N gaugetheory of the AdS/CFT correspondence [53, 54, 55]. Suppose that a vacuum of the theory (89)is given by h Φ a i vac = B ab y b , h A µ i vac = 0 (90)where B ab is a constant matrix of rank 6. And assume that the vacuum expectation values y a ∈ U ( N → ∞ ) satisfy the algebra [ y a , y b ] = iθ ab N × N (91)where θ ab = ( B ) ab . It is then obvious that the vacuum (90) in the N → ∞ limit is definitely asolution of the theory (89) and the vacuum algebra (91) is familiar with the Heisenberg algebraof NC space R θ . Consequently the large- N matrices on R , in the action (89) can be mappedto NC fields in C ∞ ( R , ) ⊗ A θ .Let us consider fluctuations b A M ( X ) = ( b A µ , b A a )( x, y ) , M = 0 , , · · · , N matricesin the action (89) around the vacuum (90) D µ ( x, y ) = ∂ µ − i b A µ ( x, y ) , Φ a ( x, y ) = B ab y b + b A a ( x, y ) , (92)where the fluctuations are assumed to also depend on the vacuum moduli in (90). Therefore letus introduce 10-dimensional coordinates X M = ( x µ , y a ) and 10-dimensional connections definedby D M ( X ) = ∂ M − i b A M ( X ) ≡ ( D µ , D a = − i Φ a )( x, y ) . (93)s a result, the large- N matrices in the action (89) are now represented by their master fieldswhich are higher-dimensional NC U (1) gauge fields in (93) whose field strength is given by b F MN = ∂ M b A N − ∂ N b A M − i [ b A M , b A N ] ⋆ . (94)In the end, the 4-dimensional U ( N ) Yang-Mills theory (89) has been transformed into a 10-dimensional NC U (1) gauge theory and the action (89) can be recast into the simple form[23] b S = − g Y M Z d X (cid:16) b F MN − B MN (cid:17) . (95)To find a gravitational metric dual to the large- N gauge theory (89) or, equivalently, to findan emergent metric determined by the NC gauge theory (95), we can apply the adjoint operation(48) to the 10-dimensional NC gauge fields in (93) after switching the index M → A = 0 , , · · · , b V A [ b f ]( X ) = [ D A , b f ] ⋆ ( x, y )= V MA ( x, y ) ∂ M f ( x, y ) + O ( θ ) (96)for b f ( x, y ) ∈ C ∞ ( R , ) ⊗ A θ . In the commutative limit, the vector fields V A = V MA ∂ M ∈ Γ( T M )on a 10-dimensional Lorentzian manifold M is given by V A = ( ∂ µ + A aµ ∂ a , D ba ∂ b ) (97)or their dual basis V A = V AM dX M ∈ Γ( T ∗ M ) is given by V A = (cid:16) dx µ , V ab ( dy b − A bµ dx µ ) (cid:17) , (98)where V ca D bc = δ ba and A aµ ≡ − θ ab ∂ b A µ ∂y b , D ba ≡ δ ba − θ bc ∂ b A a ∂y c . (99)Hence the 10-dimensional geometry dual to the gauge theory (89) or (95) can easily bedetermined by [23, 12] ds = λ η AB V A ⊗ V B = λ (cid:16) η µν dx µ dx ν + δ ab V ac V bd ( dy c − A c )( dy d − A d ) (cid:17) (100)where A a = A aµ dx µ and the conformal factor is defined by λ = ν ( V , V , · · · , V ) (101)for a 10-dimensional volume form ν = d x ∧ d y or, more generally, ν = d x ∧ ν .It has been known from the AdS/CFT duality [53, 54, 55] that the large- N gauge theory (89)is a nonperturbative formulation of type IIB string theory on AdS × S background. We haveverified above that the 4-dimensional U ( N ) gauge theory (89) gives rise to a 10-dimensionalgravity with the metric (100) [23]. We see that the existence of nontrivial gauge fields A µ ( x )causes the curving of the original flat spacetime R , and so the four-dimensional spacetimealso becomes dynamical together with an entirely emergent 6-dimensional space. Therefore, thelarge- N gauge theory (89) almost provides a background independent description of spacetimegeometry except the original background R , whose existence was a priori assumed at theoutset. We confirm again the important picture [5] that, in order to describe a classical geometryrom a background independent theory, it is necessary to have a nontrivial vacuum defined by a“coherent” condensation of gauge fields, e.g., the vacuum defined by (90).A remarkable aspect of the large- N gauge theory (89) is that it admits a rich variety oftopological objects. So our curiosity is what kind of geometry emerges from such a topologicalobject (according to the map (96)) when the topological solution has been defined by the gaugetheory (89) or (95) and what kind of object is materialized in four-dimensional spacetime fromthe stable solution. We will assert that consolidating some generic features of emergent geometryand the Feynman’s picture about the weak and strong forces leads to a remarkable picture forwhat matter is. In particular, a matter field such as leptons and quarks may simply arise asa stable localized geometry in extra dimensions, which is a topological object in the definingalgebra (NC ⋆ -algebra) of quantum gravity.Consider a stable class of time-independent solutions in the action (89) satisfying theasymptotic boundary condition (90). For such kind of solutions, we may forget about timeand work in the temporal gauge, A = 0. Since the adjoint scalar fields asymptotically approachthe common limit (90) (which does not depend on x i := x ), we can think of R as having thetopology of a three-sphere S = R ∪ {∞} , with the point at infinity included. In particular,the matrices Φ a ( x ) are nondegenerate along S and so Φ a defines a well-defined mapΦ a : S → GL( N, C ) (102)from S to the group of nondegenerate complex N × N matrices. If this map represents anontrivial class in the third homotopy group π (GL( N, C )), the solution (102) will be stableunder small perturbations, and the corresponding nontrivial element of π (GL( N, C )) representsa topological invariant [56]. In the stable regime where N > /
2, the homotopy groups ofGL( N, C ) or U ( N ) define a generalized cohomology theory, known as K-theory K ( X ) [57]. Forexample, for X = R , , this group with compact support is given by [56] K ( R , ) = π (GL( N, C )) = Z . (103)Note that the map (102) is contractible to the group of maps from S to U ( N ).We now come to the connection with K-theory, via the classic Atiyah-Bott-Shapiro (ABS)construction [58] which relates the Grothendieck groups of Clifford modules to the K-theoryof spheres. The ABS isomorphism relates complex and real Clifford algebras to K-theory [59]:Such a relation is somehow expected, given that the periodicity of K-theory is similar to theperiodicity of Clifford algebras [31]. Note that the group K ( X ) also classifies D-branes in type IIsuperstring theory on a manifold X [60, 61, 62, 63]. In particular, the RR-charge of type IIB D-branes is measured by the K-theory class of their transverse space, so that K ( S p ) = π p − ( U ( N ))classifies (9 − p )-branes in type IIB string theory on flat R , spacetime.The ABS construction uses the gamma matrices of the Lorentz group SO (3 ,
1) to constructan explicit generator of the K-theory group (103) [59]. Let S ± be two irreducible spinorrepresentations of Spin (4) and define the gamma matrices Γ µ : S + → S − to satisfy the Diracalgebra { Γ µ , Γ ν } = 2 η µν . Let us also introduce the Dirac operator D : H × S + → H × S − suchthat D = Γ µ p µ + · · · (104)where p µ = ( ω, p ) is a four-momentum and the abbreviation denotes possible higher ordercorrections in higher energies. Here the Dirac operator (104) is regarded as a linear operatoracting on a Hilbert space H as well as the spinor vector space S ± . The Hilbert space H wouldbe possibly much smaller than the primitive Fock space for the Heisenberg algebra (91) becausethe Dirac operator (104) actually acts on collective (coarse grained) modes of the solution (102)[56].n order to construct stable topological objects that take values in the K-theory (103), it isnatural to consider topological solutions made out of Φ a ( x ) ∈ U ( N ) according to the homotopymap (102). As was shown before, the large- N matrices in U ( N → ∞ ) gauge theory can bedescribed by NC U (1) gauge fields with the action (95) in higher dimensions. In particular, theadjoint scalar fields Φ a ( x ) ∈ U ( N ) are mapped to NC U (1) gauge fields in extra dimensions andobey the relation − i [Φ a , Φ b ] = − B ab + b F ab . (105)Therefore, the topological solutions made out of Φ a ( x ) ∈ U ( N ) will be given by NC U (1)instantons in four or six dimensions. For instance, in four-dimensional subspace, one can considerNC U (1) instanton solutions given by [24, 25, 26] b F ab = ± ε abcd b F cd (106)where a, · · · , d = 1 , · · · , U (1)instantons defined by [64] b F ab = ± ε abcdef b F cd J ef , (107) J ab b F ab = 0 , (108)where J ab = κ B ab is a nondegenerate symplectic matrix defined by the vacuum (90). Weshowed in section 2 that the NC U (1) instantons in commutative limit are equivalent togravitational instantons which are hyper-K¨ahler manifolds and also called Calabi-Yau 2-folds[22, 23]. Similarly it can be shown [64] that the 6-dimensional NC Hermitian U (1) instantonssatisfying (107) and (108) can be recast into Calabi-Yau 3-folds using the vector fields definedby (96). If we define a gravitational instanton as a Ricci-flat, K¨ahler manifold, the Calabi-Yau3-fold corresponds to a 6-dimensional gravitational instanton.It is well-known that gravity can be formulated as a gauge theory of Lorentz group. In thisgauge theory formulation, Calabi-Yau n -folds correspond to SU ( n ) Yang-Mills instantons in 2 n -dimensions and the gauge group SU ( n ) appears as the holonomy group of a Calabi-Yau n -fold[27, 65]. Combining the relationship between NC U (1) instantons, SU ( n ) Yang-Mills instantonsand Calabi-Yau n -folds altogether, we get the trinity of instantons [33] depicted in Figure 2.According to the above construction, the topological solution (102) takes a value in the K-theory group (103) and is realized as a stable geometry, i.e., a Calabi-Yau manifold, in extradimensions. And the ABS theorem says that the stable topological solution is represented by theDirac operator (104) on the spacetime R , . In other words, the Calabi-Yau manifold constructedfrom a NC U (1) instanton in extra dimensions will be realized as a four-dimensional fermion χ ( t, x ) whose dispersion relation in low energies is given by the relativistic Dirac equation i Γ µ ∂ µ χ + · · · = 0 (109)as was already suggested by the Dirac operator (104). Although the emergence of 4-dimensionalspinors from large- N matrices or NC gauge fields is just a consequence of the fact that the ABSconstruction uses the Clifford algebra to construct explicit generators of π ( U ( N )), its physicalorigin is mysterious and difficult to understand.Recall that the Weyl spinor in (109) is originated from NC U (1) instantons in extra 2 n -dimensions which are also realized as Calabi-Yau n -folds with SU ( n ) holonomy. Therefore the4-dimensional spinor χ should be charged under the SU ( n ) ⊂ SO (2 n ) gauge group and so cancouple to SU ( n ) gauge fields in four dimensions. If so, our last question is how non-Abeliangauge fields A Iµ ( x ) ∈ SU ( n ) on R , can arise from the U (1) gauge fields on R , × R θ . igure 2. Trinity of instantons. (Image from [33])To recapitulate, the U ( N → ∞ ) gauge theory (89) in the Moyal background (90) has beenmapped to the 10-dimensional NC U (1) gauge theory (95) defined on the space R , × R θ . Thenthe K-theory (103) for any sufficiently large N can be identified with the K-theory K ( A θ ) for theNC ⋆ -algebra A θ . But, if we consider low-energy excitations around the solution (102) whoseK-theory class is given by K ( A θ ) and that would be a sufficiently localized state described bya compact (bounded self-adjoint) operator in A θ , it will not appreciably disturb the ambientgravitational field. This means [12] that we may reduce the problem to a quantum particledynamics on R , × F where F is an internal space describing collective modes of the solution(102). It is natural to identify the coordinate of F with an internal charge carried by the Weylfermion χ in (109). We observed above that the (collective) coordinates of F will take values inthe SU ( n ) Lie algebra such as the isospins or colors and will be denoted by Q I ( I = 1 , · · · , n − y a , y b ] = iθ ab ⇔ [ a i , a † j ] = δ ij (110)where a, b = 1 , · · · , i, j = 1 , ,
3. There is a well-known fact that the n -dimensionalharmonic oscillator can realize an SU ( n ) symmetry and the generators of the SU ( n ) Lie algebraare given by Q I = X i,j a † i T Iij a j (111)where T I are constant n × n matrices satisfying the SU ( n ) Lie algebra [ T I , T J ] = i ¯ hf IJK T K . Itis easy to check that the Schwinger representation (111) satisfies the commutation relations in(84)-(86). As was reasoned above, the SU ( n ) generators in (111) can be regarded as low-energycollective modes (or order parameters) in the vicinity of the solution (102).Since the chiral fermion χ in (109) is charged under the SU ( n ) symmetry whose generatorsare given by (111), it can interact with four-dimensional SU ( n ) gauge fields A Iµ ( x ). Let ρ ( H )be a representation of the Lie algebra (111). We will take an n -dimensional representationin H = L ( C n ) which is much smaller than the original Fock space of (110). Since weare considering a low-energy limit where gravitational back-reactions are ignored, it will bereasonable to take only the lowest modes of NC U (1) gauge fields b A µ ( x, y ) ∈ C ∞ ( R , ) × A θ as low-energy approximation. So we will expand the U (1) gauge fields b A µ ( x, y ) in (92) with the SU ( n ) basis in (111) b A µ ( x, y ) = A µ ( x ) + A Iµ ( x ) Q I + A IJµ ( x ) Q I Q J + · · · (112)where it is assumed that each term in (112) belongs to an irreducible representation of ρ ( H ).Through the expansion (112), we get SU ( n ) gauge fields A Iµ ( x ) as well as ordinary U (1) gaugefields A µ ( x ) as low lying excitations [12].The coarse-grained fermion χ in (109) behaves like a stable relativistic particle in thespacetime R , . Hence, when the particle moves along R , there will be bosonic excitationsarising from changing the position in R of the internal charge F according to the relation m ˙ x i = p i − A i ( x , t ) and the Wong’s equation (88). That is, we can think of the Dirac operator(104) as an operator H × S + → H × S − where H = L ( C n ) and introduce a minimal couplingwith the U (1) and SU ( n ) gauge fields in (112) by the replacement p µ → p µ − eA µ ( x ) − A Iµ ( x ) Q I .Then the Dirac equation (109) becomes i Γ µ (cid:16) ∂ µ − ieA µ − iA Iµ ( x ) Q I (cid:17) χ + · · · = 0 . (113)Here we see that the chiral fermion χ in the homotopy class π ( U ( N )) is in the fundamentalrepresentation of SU ( n ). As a result, the spinor in (113) can be identified with a quark, an SU (3) multiplet of chiral Weyl fermions interacting with gluons A Iµ ( x ) ( I = 1 , · · · ,
8) for n = 3and with a lepton, an SU (2) doublet of chiral Weyl fermions interacting with isospin gauge fields A Iµ ( x ) ( I = 1 , · · · ,
3) for n = 2 [12, 5].We want to point out that the emergent matters from stable geometries in extra dimensionsare consistent with the Calabi-Yau compactification in string theory. In string theory, a Calabi-Yau manifold serves as an internal geometry of string theory with 6 extra dimensions and theirshapes and topology determine a detailed structure of the multiplets for elementary particlesand gauge fields through the compactification, which leads to a low-energy phenomenology infour dimensions. A very similar picture seems to be also realized in the context of emergentgeometry via the ABS theorem and the trinity of instantons illustrated in Figure 2.To conclude, we have observed that the theory (89) allows topologically stable solutions aslong as the homotopy group (103) is nontrivial. Remarkably, a matter field such as leptonsand quarks simply arises from such a stable solution and non-Abelian gauge fields correspondto collective zero-modes of the stable localized solution (102). Although the solution (102)is interpreted as particles and gauge fields ignoring their gravitational effects, we have torecall that it is a stable excitation over the vacuum (90) and so originally a part of spacetimegeometry according to the map (96). Consequently, we get a remarkable picture, if any, thatmatter fields such as leptons and quarks simply arise as a stable localized geometry, which isa topological object in the defining algebra (NC ⋆ -algebra) of quantum gravity. This approachfor quantum gravity thus allows a novel unification where spacetime as well as matter fields isequally emergent from a universal vacuum of quantum gravity [20]. We believe that such anelegant unification of geometry and matters is a unique feature realized only in the backgroundindependent formulation of quantum gravity. Acknowledgments
We are grateful to Jungjai Lee and John J. Oh for helpful discussions, collaborations andencouragements over the years. This research was supported by Basic Science Research Programthrough the National Research Foundation of Korea (NRF) funded by the Ministry of Education,Science and Technology (2011-0010597) and by the RP-Grant 2010 of Ewha Womans University. eferences [1] Misner C W, Thorne K S and Wheeler J A 1973
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