CAUSALITY IN NONCOMMUTATIVE FIELD THEORY

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1 CAUSALITY IN NONCOMMUTATIVE FIELD THEORY C. S. ACATRINEI National Institute for Nuclear Phsics and Engineering, P.O.Bo MG-6, RO Bucharest-Magurele, Romania, Received August 30, 008 Operatorial methods offer an insightful alternative to the usual (Wigner- Wel-)Moal approach to noncommutative field theories. In particular one can prove easil that the elementar degrees of freedom are bilocal, and live on a reduced configuration space. Causalit issues become particularl transparent in this framework. 1. INTRODUCTION A field theor on a noncommutative (NC) space refers to the dnamics of a set of fields depending on independent variables which have an operatorial character. For the simplest case of a single scalar field φ one has φ=φ ( ˆμ), [ ˆμ, ˆν] = iθ μν( ˆ). (1) with μ representing the space-time coordinates ( 0 = t, 1,, 3 ). We will consider here the case of a commuting time t and z coordinate (which we will often omit for simplicit), and of noncommuting and directions: [ ˆ, ˆ] = iθ I, () with I the identit operator and θ a constant having the dimension of an area. It is not clear whether Professor ieica would have liked the idea of noncommutative space, given that no eperimental support for it is available. M first ecuse for still discussing it is that the idea originated with ieica s own doctoral supervisor, Werner Heisenberg. Heisenberg contemplated NC space presumabl in analog with his well known noncommutativit of quantum phase space hoping to find a solution to the problem of the infinities in quantum electrodnamics. Through Pauli and Oppenheimer, the idea reached Snder, who wrote the first paper on NC space [1]. The second ecuse is that NC leads to important but controlable modifications in field theor. In particular one manages to keep control over a field theor which is not local. Thus, although Heisenberg s original hope was not confirmed (at least when θ μν in (1) is constant), NC field theor presents an interesting development, a useful Rom. Journ. Phs., Vol. 53, Nos. 9 10, P , Bucharest, 008

2 1008 C. S. Acatrinei theoretical laborator in which ideas can be tested, all in all a welcome variation around the established path of usual quantum field theor. We will briefl stress the conceptual part, and discuss causalit after assessing the bilocal character of the theor. Causalit is an essential feature of an phsical model. In usual relativistic field theor, it is defined via the ordering of precise events taking place in spacetime. For a field φ depending on space-time coordinates μ = ( 0 = t, 1,, 3 ), the (micro)causalit condition reads [ φ ( ),φ(0)] = 0, t 0. (3) It reflects the assumption that two events having space-like separation cannot influence each other. In the NC field theor literature, causalit is discussed in a relativel small number of papers (about one in a hundred), and even there with contradictor results. The reason is the use of the Wel-Moal quantization procedure, in which NC space is mapped to a continuum of same dimensionalit, parametrized b the so-called Wel smbols. The correspondence to phsical space (assumed NC) is at best statistical. Thus, a point in Wel smbol space has no precise correspondent in the phsical (NC) space. On the other hand the product of functions gets deformed to the Moal star product ( ) ( ) ( ) ( ) lim ep i f g f g μν θ μ ν f( ) g( ). (4) In consequence, if one wants to generalize the causalit condition (3) to NC fields, one encounters two ambiguities. 1. It is not clear whether one should take the commutator or the starcommutator of two fields []. We will dispose of this ambiguit triviall through the canonical formulation [4], to be briefl reviewed below.. It is not clear what a space-like interval means when some of the coordinates are noncommuting hence not all can be sharpl measured simultaneousl. Several conditions were used in the literature, for events separated b the quadri-vector ( Δ,Δ t r ) in Wel space: a) the usual light-cone condition, see e.g. [9]: Δt Δ Δ Δz 0, too weak to alwas ensure vanishing of the commutator of two fields. b) the light-wedge condition: Δt Δz 0 [6, 7], which drops completel the noncommuting coordinates; it is too strong, since it operates onl on the commuting part of the space; it cannot be implemented if no commuting coordinate z is available. c) an intermediate condition: Δt Δz θ [8], with the RHS tring to account for the average spreading Δ +Δ. It involves a statistical correction

3 3 Causalit in noncommutative field theor 1009 with respect to the condition (b) above and is still inappropriate [8], as we will also see. We will remove both ambiguities, using the canonical framework of [4, 5]. The first ambiguit is disposed of simpl b using an operatorial formulation, in which the commutator is uniquel defined. The second, more veing, problem is solved since, as it will be shown, it is natural to drop one of the noncommuting coordinates, sa, and then require zero commutator provided Δt Δ Δz 0, but in phsical space, not in Wel space. In consequence, we will show disproving previous claims, that NC theories with commuting time are causal.. BILOCALITY Consider a ( + 1)-dimensional scalar Φ ( t, ˆ, ˆ), defined over a commuting time t and a pair of NC coordinates satisfing (). The operators ˆ and ŷ act on a harmonic oscillator Hilbert space in the usual wa. Out of an infinite number of possibilities, ma be given a discrete basis { n } formed b eigenstates of ˆ + ˆ [5], or a continuous one { }, composed of eigenstates of, sa, ˆ [4]. We will use this last basis. To quantize Φ [4], start with a usual classical commuting field, epanded into normal modes with coefficients a and a *. Upon usual field quantization, a and a * become operators acting on a standard Fock space. To make the underling space noncommutative, let us introduce () and appl the Wel (not ik ( + k Wel-Moal!) quantization procedure to the eponentials e ). The result is dk dk Φ= +. π i( ωt kˆ ˆ) ( ˆ ˆ) ˆ k k iωt k ˆ k k akke a kke ω k which means the following: Φ is a doubl -quantum field operator, acting on a direct product of two Hilbert spaces, Φ:. Φ creates (destros), via aˆ ( a ˆ ), an ecitation represented b an operatorial plane wave kk kk i( t kˆ kˆ e ω k ). One could work with Φ as an operator read to act on both and. It is however simpler to saturate its action on, working with epectation values Φ :. At this point bilocalit appears. For, consider the famil { } of eigenstates of ˆ : equation is (5) ˆ =, ˆ = θ i. A simple but ke

4 1010 C. S. Acatrinei 4 ( ˆ ˆ ( ) ik + ik + k ik + kθ/ e = e δ( k θ ) = e δ( k θ ). (6) This is a bilocal epression, and we alread see that its span along the ais, ( ), is proportional to the momentum along the conjugate direction, i.e. ( ) =θ k. In general, for n pairs of NC directions, one can keep onl one coordinate out of ever pair; commutativit is gained on the reduced space, at the epense of localit. Using Eqs. (5) and (6) one sees that ( ω + k ) + ( ω + k ) i t k i t k ˆ ˆ k, k k, k k, k dk Φ = a e + a e π ω where k = ( ) /θ. Thus, Φ annihilates a linear combination of rods of (arbitrar) momentum k and (fied) length θk, and creates rods of momentum k and length θk. It is not anmore a local operator, in contrast to usual field theor. Failure to recognize that feature eplicitel in the Moal formulation ma lead to erroneous conclusions about causalit (although the interesting paper [3] should be noted). The so-called IR/UV connection is also rather obscure in the Moal representation, but is seen as an intrinsic feature of the fundamental degrees of freedom in the operatorial approach [4]. We turn to a discussion of causalit. (7) 3. CAUSALITY It has been shown in [4] that free NC fields behave eactl like commutative fields living in a lower-dimensional space. In fact a free (1 + 1)-dimensional dipole [resulting from the + 1 NC theor we were discussing] with endpoints situated at and behaves like a commutative (1 + 1) point particle centered at +, but with a modified dispersion relation ( ) ) ω = k +. In conclusion all usual manipulations performed on θ propagators in (1 + 1)-dimensions can be carried over, including those used to demonstrate causalit. This immediatel shows that at the free level NC field theories are causal, contrar to previous claims []. For interacting fields, one epects the dipolar character of the degrees of freedom to manifest, as e.g. in perturbation theor [4]. It is however remarkable that as far as causalit issues are concerned, bilocalit has little influence, and a proof of causalit can be given like for commutative theories. For, consider the vanishing of the commutator to hold,

5 5 Causalit in noncommutative field theor 1011 [ φ ( t, ),φ( t, )] = 0 (8) with 1 =, = being the average positions (centers of mass) of the two dipoles considered. We want (8) to be true when the interval, defined with respect to the average dipole positions, is space-like, 1 1 ( t t ) ( ) 0. (9) Equations (8, 9) are however genericall equivalent to [ φ ( t, ),φ( t, )] = 0,, (10) provided one can appl a boost along to render equal the two times appearing in Eq. (8). This requires the (1 + 1)-dimensional dipole theor to be invariant under boosts in the -direction (a fact completel overlooked in the literature, which claims that the onl invariance left after NC is imposed is a product of O() for the NC part and of the Lorentz group, e.g. SO(1, 1), for the rest). The invariance follows from the form of the classical Lagrangian for dipoles, 1 L = ( φ) ( φ) [( θ Δ ) + m ] φ V( φ ). (11) t Above, V(φ) is the potential for the field, genericall denotes the average position of a dipole, +, whereas Δ denotes its span,. The onl think to prove is the invariance of the third term in the RHS which immediatel follows from the tensorial character of the inverse of θ=θ ~ and the usual Lorentz transformation of Δ. At the quantum level one has no reason to worr about an anomal, since the integration measure in the path integral is invariant. In consequence, Eqs. (8, 9) are tantamount, via a boost, to ih t ih t e [ φ (0, ),φ(0, )] e = 0 (1) which is true at t = 0, since this is b definition the time at which the fields behave like free fields (H denotes the interacting part of the Hamiltonian, including V, in the interaction representation). We stress that the above causalit argument works for a full interacting theor. Adding now the (passive) commutative coordinate z, we conclude that the correct criterion for causalit is [ φ ( t,, z ),φ( t,, z )] = 0, ( t t ) ( ) ( z z ) 0, (13) and that it is satisfied in NC field theor.

6 101 C. S. Acatrinei 6 Acknowledgements. This contribution is respectousl dedicated to the memor of Professor ªerban Þiþeica. Partial financial support from the Romanian Ministr of Education and Research, grant CEEX-05-D11-49, is acknowledged. REFERENCES 1. H. Snder, Phs. Rev. 71, 38 (1947).. O. W. Greenberg, Phs. Rev. D73, (006). 3. N. Seiberg, L. Susskind, N. Toumbas, JHEP 06, 44 (000). 4. C. S. Acatrinei, Phs. Rev. D67, (003). 5. C. S. Acatrinei, J. Mod. Phs. A41, (008). 6. L. Alvarez-Gaume, M. Vazquez-Moto, Nucl. Phs. B668, 93 (003). 7. C.S. Chu, K. Furuta, T. Inami, Int. J. Mod. Phs. A1, 67 (006). 8. A. Haque, S. D. Joglekar, J. Mod. Phs. A41, 1540 (008). 9. M. Chaichian, K. Nishijima, A. Tureanu, Phs. Lett. B586, 146 (003).