Covariant Non-Commutative Space-Time 4 1 0 2 n Jonathan J. Heckman1∗ and Herman Verlinde 2† a J 8 1Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA ] h 2Department of Physics, Princeton University, Princeton, NJ 08544, USA t - p e h [ 1 v 0 1 Abstract 8 1 . We introduce a covariant non-commutative deformation of 3+1-dimensional conformal 1 field theory. The deformation introduces a short-distance scale (cid:96) , and thus breaks scale 0 p 4 invariance, but preserves all space-time isometries. The non-commutative algebra is defined 1 on space-times with non-zero constant curvature, i.e. dS or AdS . The construction makes : 4 4 v essential use of the representation of CFT tensor operators as polynomials in an auxiliary i X polarization tensor. The polarization tensor takes active part in the non-commutative al- r gebra, which for dS takes the form of so(5,1), while for AdS it assembles into so(4,2). a 4 4 The structure of the non-commutative correlation functions hints that the deformed theory contains gravitational interactions and a Regge-like trajectory of higher spin excitations. January 2014 ∗e-mail: [email protected] †e-mail: [email protected] 1. Introduction One of the basic ways to generalize the classical notion of space-time is non-commutative geometry [1]. An especially attractive feature of non-commutative generalizations of quan- tum field theory is the natural appearance of a minimal resolution length scale. Non- commutativity may thus act as a UV regulator. However, most known implementations of space-time non-commutativity have the substantial drawback that they explicitly violate Lorentz invariance. For many reasons, it would therefore be of special interest to find exam- ples of covariant non-commutative space-times, that preserve all global symmetries of the underlying classical space-time [2–4]. In this paper we propose a covariant non-commutative (CNC) deformation of a general 3+1-D conformal field theory (CFT) defined a homogeneous space-time with constant posi- tive or negative curvature. We focus on the application to 3+1-D de Sitter space-time dS , 4 though the discussion can be easily generalized to 3+1-D anti-de Sitter space-time AdS . 4 3+1-D de Sitter space-time is described by an embedding equation of the form ηABX X = R2 (1) A B whereA,B runfrom0to4,η = (−,+,+,+,+)isthe5DMinkowskimetric,andRspecifies AB the curvature radius. For AdS , we take η = (−,+,+,+,−), and replace R2 → −R2. 4 AB Like any known non-commutative deformation of space-time, we will postulate that the position operators X satisfy a non-trivial commutation relation of the general form A (cid:2)X ,X (cid:3) = i(cid:126)(cid:96)2S . (2) A B AB Here (cid:96) is a short-distance length scale, and S is a (dimensionless) anti-symmetric tensor, AB with , = 0,...,4. If S where some fixed anti-symmetric tensor, equation (2) would break A B AB Lorentz invariance. To obtain a covariant non-commutative deformation, the isometry group of 3+1-D de Sitter space-time would need to act both on the coordinates X and the tensor A S , via the infinitesimal SO(4,1) generators M AB AB [M ,X ] = i(cid:126)(cid:0)η X −η X (cid:1), (3) AB C AC B BC A [M ,S ] = i(cid:126)(cid:0)η S +η S −η S −η S (cid:1). (4) AB CD AC BD BD AC AD BC BC AD However, to justify its non-trivial transformation property, S should represent an active AB degree of freedom similar to the space-time coordinates X . How can this be arranged? A Recent studies of CFT correlation functions of primary tensor operators [5] have made successfuluseofanextensionoftheembeddingformalism[6,7], inwhichthetensoroperators are encoded by polynomials in an auxiliary polarization vector P . Tangent directions to the A 1 de Sitter embedding equation (1) can be folded into anti-symmetric tensors S = X P . AB [A B] Tensor operators O (X) can thus be promoted into functions O(X,S) defined on an A1..Aj extension of space-time TdS . This formalism has the payoff that CFT correlators of tensor 4 operators take the form of correlators of scalar operators O(X,S) on the extended space- time [5]. Space-time isometries act on these correlators by transforming all positions and spin variables simultaneously, as in equation (3)-(4). This use of an extended space-time sheds new light on how to obtain covariant non- commutativity. It is natural to look for possible ways to identify the anti-symmetric tensor S that encodes the spin of tensor CFT operators with the tensor S that appears in the AB AB space-time commutator algebra (2). In this paper we will show that this idea can indeed be utilized to construct a covariant non-commutative deformation of a general 3+1-D CFT. To obtain a self-consistent implementation, the spin variables S also need to acquire AB a non-trivial commution relation. The non-commutative version of S is obtained from its AB commutative cousin via the replacement S → S +M (5) AB AB AB with M the so(4,1) generators (3). The quantized S thus satisfy an so(4,1) algebra. AB AB Combinedwithequation(2), thetotalCNCalgebraextendstoso(5,1), theLiealgebraofthe Lorentz group in 6 dimensions, under which the space-time coordinates and spin variables transform as an anti-symmetric tensor Z , defined via Z = X and Z = (cid:96)S . IJ 5A A AB AB The scale of non-commutativity is set by the short distance length scale (cid:96) = (cid:126)(cid:96). We p also introduce the dimensionless ratio N = R/(cid:96) . (6) p We will assume that (cid:126) is small and that N is very large. The interpretation of the so(5,1) Lie algebra as a covariant non-commutative space-time algebra was first proposed in a short note by C.N. Yang, published in 1947 [3], soon after Snyder’s earlier work [2]. Somewhat surprisingly, possible concrete physical realizations of Yang’s proposal have to our knowledge not been actively investigated since. This paper is organized as follows. In section 2 we summarize the embedding formal- ism for tensor operators in 3+1-D CFT and introduce the SO(5,1) invariant notation. In section 3 define the covariant non-commutative space-time algebra, and write it in terms of local Minkowski coordinates. In sections 4 and 5 we construct the star product and use this to define the covariant non-commutative deformation of the CFT correlation functions. In section 6 we present a spinor formulation of the covariant non-commutative space-time algebra. We end with some concluding comments in section 7. 2 2. Tensor Operators as Polynomials Here we briefly summarize the generalized embedding formalism for CFT correlators of tensor primary operators. A more detailed discussion can be found in [5]. Let X denote the 5D embedding coordinates of R4,1, in which 3+1-D de Sitter space is A defined as the subspace (1). A general spin j primary operator O (X) in a CFT defined A1...Aj on 3+1-D de Sitter space-time (1) is symmetric, traceless and transverse XAO (X) = 0, ηABO (X) = 0. (7) AA2...Aj AA2..B..Aj The transversality constraint ensures that the tensor indices represent tangent vectors to the space-time manifold dS . Any such tensor operator can be represented as a polynomial in 4 an auxilary polarization vector P via A O(X,P) = O (X)PA1···PAj . (8) A1...A(cid:96) As long as the P mutually commute, the polynomial O(X,P) automatically represents a A symmetric tensor. The transversality and tracelessness conditions (7) furthermore imply that the polynomials O(X,P) satisfy the differential equations ∂ X O(X,P) = 0 (transverse) (9) A∂P A ∂2 O(X,P) = 0 (traceless) (10) ∂PA∂P A Traceless tensors thus correspond to harmonic polynomials in the polarization vector P . A The transversality condition (9) implies that O(X,P +αX) = O(X,P) for any α. So- lutions to this condition are characterized by the property that the polarization variable P A only appears in the anti-symmetric combination [5] S = X P −X P . (11) AB A B B A So instead of using the vector P , we can choose to write the transverse tensor operators as A functions O(X,S). The property that S takes the form (11) is enforced by requiring that AB εABCDES X = 0. (12) AB C One readily sees that this condition implies that S must have one leg in the X direction. AB A Note that it automatically follows that εABCDES S = 0. In terms of the variable S , AB CD AB 3 the tracelessness condition (10) takes the form (cid:16) ∂ (cid:17)2 XA O(X,S) = 0. (13) ∂SAB To build in the anomalous scale dependence of CFT correlation functions, it is often convenient to extend the embedding formalism to a 6D space-time R4,2 with (4,2) signature (−,+,+,+,+,−), by adding a time-like coordinate X . The 6D coordinates are then re- 5 stricted to lie on the null cone XAX − X2 = 0, on which the conformal group SO(4,2) A 5 naturally acts [6,7]. Conformal operators are homogenous functions of dimension −∆ O(λX,λS) = λ−∆O(X,S). (14) The projection down to dS amounts to gauge fixing the projective symmetry by setting 4 X = R. Below, we will work in the gauge fixed formulation. This is sufficient for our 5 purpose, since scale invariance will be broken anyhow by the non-commutative deformation. Summary: We can represent primary tensor operators on 3+1-D de Sitter space-time as harmonic functions of the generalized coordinates (X ,S ) subject to the constraints (1) A AB and (12). As a simple example of how the formalism works, the 2-point function of two spin j operators with conformal dimension ∆ is given by [5] (cid:0) (cid:1)j (cid:68) (cid:69) S ·S 1 2 O(X ,S )O(X ,S ) = const. . (15) 2 2 1 1 (cid:0) (cid:1)∆+j R2−X ·X 1 2 For now, the coordinates X and spin variables S seem to stand on rather different footing: A AB the former refer to actual space-time points, while the latter are just a convenient packaging of polarization indices. The two types of variables can however be naturally unified as coordinates of an extended space-time, given by the tangent space TdS to 3+1-D de Sitter. 4 We can think of the extended space-time TdS as analogous to superspace, used for grouping 4 supermultiplets in supersymmetric theories into single superfields, except that now the extra polarization variables are bosonic rather than fermionic. SO(5,1) Symmetric Notation One may assemble the coordinates and spin variables into a single anti-symmetric 6×6 matrix Z = −Z , with I,J = 0,...,5, via IJ JI Z = X , Z = (cid:96)S . (16) 5A A AB AB Here (cid:96) denotes an infinitesimal expansion parameter with the dimension of length. To raise 4 and lower indices, we introduce the flat metric η with signature (−,+,+,+,+,+). The IJ embedding equation (1) and tranversality constraint (12) can be recognized as the (cid:96) → 0 limit of the following SO(5,1) invariant relations 1 Z ZIJ = R2, εIJKLMNZ Z = 0. (17) 2 IJ KL MN Taking (cid:96) → 0 amounts to performing a Inonu-Wigner contraction of SO(5,1), which yields ISO(4,1), the Poincar´e group of the 4+1-D Minkowski space, the embedding space of dS . 4 The 8D extended space-time parametrized by Z then reduces to the tangent space TdS . IJ 4 The above SO(5,1) symmetric notation will prove to be convenient for our purpose, but the symmetry is of course explicitly broken to the SO(4,1) subgroup: interactions and correlationfunctionsofatypicalCFTdonotrespectthefullSO(5,1)orISO(4,1)symmetry. The following two basic statements remain true, however: • Correlation functions of tensor operators in a 3+1-D CFT can be written as correlators (cid:68) (cid:69) O (Z )...O (Z )O (Z ) (18) n n 2 2 1 1 of scalar operators defined on the 8-dimensional extended space-time TdS , parametrized 4 by the coordinates (16), subject to (17). • Any pair of points Z and Z on the extended space-time are related via an SO(5,1) 1 2 rotation Λ (or after performing the IW contraction, an ISO(4,1) transformation) via 12 ZIJ = (Λ )I ZKL(Λ )J . (19) 1 12 K 2 21 L Here (Λ )I (Λ )KJ = ηIJ. So ZIJ transforms in the adjoint representation. 12 K 21 The above two statements are all that we need to proceed with our construction. Note that in order to obtain finite correlators from (18), we need to first extract a power of ((cid:96))j from each spin j tensor operator before taking the Inonu-Wigner limit (cid:96) → 0. Even if SO(5,1) is not a full symmetry, the Z notation sometimes still provides a convenient IJ packagingofhigherspincorrelators. E.g. the2-pointfunctions(15)ofspinj tensoroperators can be be summarized into a single quasi-SO(5,1) invariant formula via (cid:34) (cid:35) (cid:68) (cid:69) 1 O(Z )O(Z ) = const. (20) 2 1 (cid:0) (cid:1)∆ R2 −Z ·Z 1 2 j where the subscript j indicates the projection onto the term proportional to ((cid:96))2j. This projection breaks SO(5,1) to SO(4,1). We will use the formula (20) later on. 5 ForCFTsdefinedonAdS , theZ notationisnaturallyinvariantunderSO(4,2)instead 4 IJ of SO(5,1). Given that SO(4,2) is identical to the conformal group in 3+1 dimensions, it is perhaps tempting to look for a relation between the above 6D notation and the more conventional 6D embedding formalism for 3+1-D CFTs [5–7]. However, it is important to not confuse the two. While both 6D notations and associated symmetry groups include the space-time isometries SO(4,1) (for dS ) or SO(3,2) (for AdS ) as a subgroup, the 4 4 two extended rotation groups are really distinct. Conformal transformations, including the conformal boosts, are true symmetries of the CFT, but do not mix coordinates and spin variables. The 6D rotations (19) that act on the Z coordinates (16), on the other hand, IJ arenotallsymmetries: besidesspace-timeisometries, equation(19)includestransformations that mix the coordinates and spin variables. The latter symmetries are explicitly broken. 3. Covariant Non-Commutative Space-Time We now introduce the covariant non-commutative deformation of the extended space time. The deformation involves two distinct steps. First we postulate that the coordinates and spin variables satisfy non-trivial commutation relations. Secondly, we replace the de Sitter space to the SO(5,1) invariant embedding equations (17). To indicate the transition to quantized space-time, we introduce a dimensionless expansion parameter (cid:126). We also identify a short distance length scale (cid:96) related to the de Sitter radius and the scale (cid:96) via p R = N(cid:96) , (cid:96) = (cid:126)(cid:96). (21) p p We will assume that (cid:126) is small and that N is extremely large, so we will often work to leading order in 1/N. We adopt the SO(5,1) invariant notation (16). The CNC deformed theory is obtained by promoting the generalized coordinates Z to IJ quantum operators that satisfy commutation relations isomorphic to the so(5,1) Lie algebra (cid:0) (cid:1) [Z ,Z ] = i(cid:96) η Z +η Z −η Z −η Z . (22) IJ KL p IK JL JL IK IL JK JK IL In terms of the coordinates X and spin variables S , the commutator algebra reads [3] A AB (cid:2)X ,X (cid:3) = i(cid:126)(cid:96)2S , [S ,X ] = i(cid:126)(cid:0)η X −η X (cid:1), (23) A B AB AB C AC B AB C [S ,S ] = i(cid:126)(cid:0)η S +η S −η S −η S (cid:1). (24) AB CD AC BD BD AC AD BC BC AD In addition to postulating the above operator algebra, we deform the embedding equation of the de Sitter space-time to its SO(5,1) invariant version (17), with (cid:96) small but finite. 6 As with any quantum deformation, it is important to verify the correspondence principle, thatis,thatthereexistsalimitinwhichthequantumdeformedtheoryreducestotheclassical theory. In the above parametrization, we can consider two different classical limits. We can (i) set the length scale (cid:96) = 0, while keeping the non-commutativity parameter (cid:126) finite, or (ii) turn off the non-commutativity by setting (cid:126) = 0, while keeping the length scale (cid:96) finite. It is clear that if we take both limits at the same time, we get back the undeformed CFT. But what happens if we take only one of the two limits? We claim that in both cases, we recover the undeformed CFT by performing a simple similarity transformation. Taking limit (i) produces a scale invariant theory with commutative coordinates X , but A with non-commutative spin variables S . The correlation functions are directly obtained AB from the commutative correlators by performing the replacement (5), with M the vector AB field that implements the SO(4,1) rotations (3)-(4). This redefinition does not amount to a true deformation of the CFT, but rather to a natural extension of the embedding formalism, in which the descendant operators (obtained by acting with the M generators AB on primary operators) are mixed in with tensor primary operators. In other words, the geometric meaning of S has been deformed, rather than the CFT. AB Taking limit (ii) produces a commutative theory, defined on the deformed de Sitter space- time (17). For given value of S , equation (17) amounts to a rescaling of the de Sitter radius AB (cid:18) S SAB(cid:19)1/2 R → λ R with λ = 1− AB . (25) S S 2(cid:126)2N2 Thanks to the conformal invariance of the original CFT, this geometric deformation can be absorbed into a conformal transformation of local operators (c.f. equations (14) and (15)) O(X,S) → λ∆+jO(cid:0)λ X,S(cid:1). (26) S S CFT correlation functions are invariant under the combined transformation (25) and (26). We conclude that taking limit (ii) again produces the commutative CFT. The actual size of the CNC deformation is set by (cid:96) = R/N. The other constants (cid:126) p and the scale (cid:96) are just useful formal parameters. Indeed, from the commutator (23) and Heisenberg, we learn that the coordinates X become fuzzy at the scale A (cid:10)(cid:0)∆XA(cid:1)2(cid:11) >∼ (cid:96)2p(cid:112)C2, with C2 = 21(cid:126)2SABSAB. (27) We can identify C with the second Casimir of the SO(4,1) space-time isometry group. 2 In the next sections, we will translate the CNC algebra of the generalized coordinated (X,S) into an explicit and well-defined deformation of the CFT correlation functions. 7 Flat Space Limit It is instructive to write the CNC deformation in a local Minkowski space-time region, say, where X (cid:39) R and X (cid:28) R for µ = 0,..,3. We thus isolate the X coordinate and use 4 µ 4 SO(3,1) notation. To write the commutation relations we identify S with the SO(4,1) AB symmetry generators M , into 6 Lorentz generators/angular momenta J = M and 4 AB µν µν translations/momentaP = 1M . Thespecialrelations(17)implythatangularmomentum µ R 4µ is purely orbital angular momentum J = P X . We also introduce the notation h = (cid:126)X . µν [µ ν] R 4 Indeed, h will play the role of Planck’s constant (not to be confused with our (cid:126) parameter). The positions and momenta satisfy the following covariant commutation relations i(cid:126) [X ,P ] = ihη , [P ,P ] = J , [X ,X ] = i(cid:126)(cid:96)2J . (28) µ ν µν µ ν R2 µν µ ν µν The first and second relation look like the standard commutation relations for coordinates and momenta in a local patch of de Sitter space. The third relation is the covariant non- commutative deformation. In the above notation, it appears that translation invariance is broken, because the Lorentz generators J act relative to a preferred origin in space-time. µν Translation symmetry is preserved, however, because (i) it acts via X → X +ha , and (ii) µ µ µ the effective Planck constant h is an operator with non-trivial commutation relations [3] i(cid:126) [h,X ] = i(cid:126)(cid:96)2P , [h,P ] = − X , [h,J ] = 0. (29) µ µ µ R2 µ µν The magnitude of h is determined by the Casimir relation (17) (cid:16) 1 (cid:17) 1 h2 = (cid:126)2 1− X Xµ −(cid:96)2P Pµ − J Jµν. (30) R2 µ µ 2N2 µν Since, without loss of generality, we can assume that Xµ (cid:28) R, we see that h (cid:39) (cid:126) as long as the mass squared is small compared to 1/(cid:96)2. We further note that the above algebraic relations enjoy an intriguing T-duality symmetry under the interchange of coordinates and momenta 1X ↔ (cid:96)P combined with the reflection h ↔ −h. R µ µ 4. Star Product We will now construct the star product between functions on the non-commutative ex- tended space-time. In its most basic form, the star product deforms the product of two functions G(Z) and F(Z) evaluated at the same point Z on the extended space-time. Since in our case the commutator algebra between the coordinates Z takes the form of a Lie algebra, this star product is well understood and directly related to the CBH formula [8]. 8 The abstract definition is as follows. We can make a unique mapping between commu- ˆ tative functions F(Z) and symmetrized functions F(Z) of the non-commutative coordinates ˆ Z . (Here we temporarily decorate the operator valued coordinates with a hat.) The prod- IJ ˆ ˆ uctG(Z)F(Z)oftwosymmetrizednon-commutativefunctionsisnotasymmetrizedfunction ˆ in the non-commutative coordinates Z. But it can always be reorderered such that it be- comes symmetrized. The extra terms produced by the re-ordering process are encoded in the ˆ ˆ ˆ star product G(Z)(cid:63)F(Z). This prescription thus identifies (G(cid:63)F)(Z) = (G(Z)F(Z)) . sym This rule associates a unique star product to any Lie algebra. The resulting star product takes the following general form [8] (cid:110) (cid:111) G(Z)(cid:63)F(Z) = exp (cid:126)ZIJD (cid:0)←∂− ,−→∂ (cid:1) G(Z)F(Z) (31) IJ Z Z ←− −→ where ∂ acts on G(Z) and ∂ on F(Z). An explicit, albeit formal, definition of the symbol Z Z D (a,b) in terms of the CBH formula is given in [8] and in the Appendix. The CBH star IJ product(31)hasanobviousgeneralizationtoann-foldstarproductF (Z)(cid:63)...(cid:63)F (Z)(cid:63)F (Z) n 2 1 between n functions of the same variable Z. This n-fold star product is associative. For our application, however, we will need the star product G(Z ) (cid:63) F(Z ) between 2 1 functions evaluated at two different locations on the non-commutative extended space-time. This may seem like a completely new concept, since a natural first guess would be to treat the two locations Z and Z as two independent mutually commuting operators. However, 1 2 this would lead to a trivial star product. In our case, we are helped by the fact that any two points Z and Z on the commutative space can be related to each other via an SO(5,1) 1 2 rotation. This means that we can view the non-commutative points Z and Z as two 1 2 quantum coordinates related via a classical SO(5,1) rotation Λ , as in equation (19). 12 Let us pick some fixed but otherwise arbitrary base point Z. We can then factorize the SO(5,1) rotation Λ that relates Z to Z into a product 12 1 2 Λ = Λ Λ−1, Z = Λ ·Z, Z = Λ ·Z, (32) 12 1 2 1 1 2 2 where Λ and Λ are the SO(5,1) rotations that relate each corresponding point to the base 1 2 point Z. Here Λ ·Z is short-hand for the action of Λ on Z, as defined in equation (19). 1 1 We can now promote the base point Z to a non-commutative coordinate, while leaving the SO(5,1) rotations Λ and Λ as classical quantities, and write the star product 1 2 (cid:0) (cid:1) (cid:0) (cid:1) G(Z )(cid:63)F(Z ) = G Λ ·Z (cid:63)F Λ ·Z . (33) 2 1 2 1 Here the (cid:63)-symbol is defined as in equation (31), with G◦Λ and F ◦Λ treated as functions 2 1 of the base point Z. Note that this definition treats the points Z and Z symmetrically. 1 2 9