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High Energy Commutators in Particle, String and Membrane Theories 5 W. Chagas-Filho 0 Departamento de Fisica, Universidade Federal de Sergipe, Brazil 0 2 February 1, 2008 n a J Abstract 7 We study relativistic particle, string and membrane theories as defin- 1 ing field theories containing gravity in (0+1), (1+1) and (2+1) space- v timedimensions, respectively. Weshowhowanoff shellinvarianceofthe 5 4 masslessparticleactionallowstheconstructionofanextensionofthecon- 0 formalalgebraandinducesatransitiontoanon-commutativespace-time 1 geometry. Thisnon-commutativegeometryisfoundtobepreservedinthe 0 space-timesupersymmetricmasslessparticletheory. Itisthenshownhow 5 thebasicbosoniccommutatorswefoundforthemasslessparticlemayalso 0 be encountered in the tensionless limit of string and membrane theories. h/ Finally we speculate on how the non-locality introduced by these com- t mutators could beused to construct a covariant Newtonian gravitational - p field theory. e h 1 Introduction : v i X Since some time ago there has been some interest in explaining a very small r measured value for the cosmological constant. It is believed that this tinny a value is a reflection of the presence of large amounts of dark matter in the universe [1,2]. Because the measured value is positive, dark matter should be contributingwithpositivegravitationalpotentialenergyasaconsequenceofits interactionwithusualmatter. Thisimpliesthatdarkmattershouldrepelusual matter. We may add to this situation the observation[3], using high resolution measurementsofthe cosmicmicrowavebackgroundradiation,thatthe universe is effectively flat. On theoreticalgrounds there is an interesting result obtained bySiegel[4],basedinscalarparticletheory,thattherearestringycorrectionsto gravitationandthatthesecorrectionsmakethegravitationalforcelessattractive at very short distances. This work is an attempt to sketch a mathematical pictureinwhichalltheabovementionedsubjectswouldoccupymathematically natural and consistent positions. The central mathematical concept in this attempt will be non-commutativity of space-time coordinates 1 Non-commutativity is the central mathematical concept expressing uncertainty in quantum mechanics, where it applies to any pair of conjugate variables. But there are other situations where non-commutativity may manifest itself. For instance, in the presence of a magnetic field even momenta fail to mutually commute. The modern trend is to believe that one might postulate non-commutativity for a number of reasons. One of these is the belief that in quantum theories including gravity, space-time must change its nature at the Planck scale. This is because quantum gravity has an uncertainty principle which prevents one from measuring positions to better accuracies than the Planck length. The momentum and energy required to make such a measure- ment will itself modify the space-time geometry at this scale [5]. This effect could then be modeled by a non-vanishing commutation relation between the space-time coordinates. For most theories, postulating an uncertainty relation between space-time coordinates will lead to a non-local theory and this may conflict with Lorentz invariance. However, there is a remarkable exception to this situation: string theory. String theory is not local in any sense we now understand and indeed it has more than one parameter characterizing this non-locality. One of these is the string length l , the average size of a string. As we will see in this work, s another useful parameter is the string tension T , the string analogue of the particle’s mass. The string tension defines an energy scale in string theory and the vanishing tension limit is expected to describe the string dynamics at the Planck scale [6]. Historically,the firstuse of non-commutativegeometry in string theory was in Witten’s formulation of open string field theory [7]. Witten’s formulation of non-commutative string field theory is based in the introduction of a star product which is defined as the overlap of half of two strings. The central idea in this formulation is to decompose the string coordinates xµ[σ] as the direct sum xµ[σ]=lµ[σ] rµ[σ] , where lµ[σ] and rµ[σ] are respectively left and right ⊕ coordinatesofthestring,definedrelativetothemidpointatσ =π/2.Ifwenow consider the fields of two strings, say A(lµ[σ],rµ[σ]) and B(lµ[σ],rµ[σ]), then 1 1 2 2 Witten’s star product gives A(lµ[σ],rµ[σ])⋆B(lµ[σ],rµ[σ])=C(lµ[σ],rµ[σ]) (1.1) 1 1 2 2 1 2 where the field C(lµ[σ],rµ[σ]) is given by the functional integral 1 2 C(lµ[σ],rµ[σ])= [dz]A(lµ[σ],zµ[σ])B(zµ[σ],rµ[σ]) (1.2) 1 2 Z 1 2 and zµ[σ] = rµ[σ] = lµ[σ] corresponds to the overlap of half of the first string 1 2 with half of the second string. Witten’s formulation of open bosonic string field theory has experienced a rebirth through new physical insights and technical advances. In particular, it was shown by Bars [8] that Witten’s star product, originally defined as a path integral that saws two strings into a third one, is equivalent to the Moyal star product [9], which is the central mathematical concept in the familiar formulation of non-commutative geometry. But this equivalence is rather subtle. As 2 shownin[8],Witten’sstarproductisequivalenttotheusualMoyalstarproduct intherelativisticphasespaceofonlytheevenmodesofoscillationofthestring. ThisisbecausethemappingoftheWitten⋆totheMoyal⋆necessarilyrequires that the Fourier space for the odd positions be named as the even momenta since (x ,p ) arecanonicalunder the Moyalstar-product. These developments, e e and many others that have recently appearedin the literature, led to the belief that non-commutative geometry naturally arises in string field theory. However,becauseitwasdefinedforapplicationsintraditionalquantumme- chanics,the Moyalstar product leaves unchanged the Heisenberg commutation relationsbetweenthecanonicaloperators. Intheusualformulationsoffieldthe- ory on non-commutative spaces, the position coordinates satisfy commutation relations of the form [10,11] [qµ,qν]=iθµν (1.3) whereθµν isaconstantanti-symmetrictensor. Theeffectofsuchamodification isreflectedinthe momentumspaceverticesofthe theoryby factorsofthe form [12] exp iθµνp q (1.4) µ ν { } Thefirstquantizedstringinthe lightconegauge,asaperturbativenon-commu tative theory based in a commutation relation such as (1.3), was studied in [12]. It was found by the authors in [12] that, apart from trivial factors, the structure of the non-commutative theory is identical to the structure of the commutative one. It thus seems that nothing substantially new is introduced instring theory by the use of a Moyalstar-productstructure, orby postulating another star-product structure based on commutators such as (1.3). In this workwe givea smallcontributionto this subject by showinghowwe may use the special-relativistic ortogonality between velocity and acceleration toinducetheappearanceofanewinvarianceofthemasslessrelativisticparticle action. This invariance may then be used to generate transitions to new space- time coordinates which, in the simplest case, satisfy commutator relations of the form [x ,x ] iM∗ (1.5) µ ν ∼ µν where M∗ is an off shell anti-symmetric operator that generates space-time µν rotations. This commutation relation is part of a modified Heisenberg commutator structure which reduces to the usual canonical commutators when the mass shell condition is imposed. Most part of this work is the explicit verifi- cation that the basic commutator structure we found for the massless particle mayalsobeencounteredinstringandmembranetheory. Aspointedoutin[11], anyfinite-dimensionaldeformedPoissonstructurecanbecanonicallyquantized. It may be then interesting to investigate what quantum effects we may find in particle, string and membrane theories if we construct the corresponding non- commutative theories based on commutators such as (1.5) instead of (1.3). Animportantpointtoourconsiderationshereisthatspace-timenon-commutativity introduces an associated non-locality [10]. In non-commutative gauge 3 theories based on the commutator (1.3) this non-locality allows the interpre- tation of the gauge particles as dipoles carrying opposite charges of the corresponding gauge theory [10, 12]. In the context of string theory, it was shown [13] that an open string with both ends on a (D2-D0)-brane system in which a magnetic field is defined looks like an electric dipole if a background electric field is added. The dipole-background interaction gives the flux modifications needed for the Born-Infeld action on the non-commutative torus and results in a Hamiltonian that depends on two integers,n and m, whose ratio m gives the n density of D0-branes distributed on the D2-brane [13]. In this work we study relativistic particle, string and membrane theories as gaugesystemsinwhichthegaugeinvarianceisgeneralcovariance. Weinterpret these relativistic objects as defining field theories containing gravity in (0+1), (1+1) and (2+1) space-time dimensions, respectively. The paper is organized as follows: For the task of clarity, in section two we briefly review the concept of a space-time conformalvector field and the associated conformalalgebra. In section three we discuss particle theory and show how we may induce an off shell invariance of the massless particle action and how this invariance allows the construction of an off shell extension of the conformal algebra and how it permits a transition to non-commutative space-time coordinates. In section four we show that the bosonic commutator structure we found for the massless particle is preserved in the space-time supersymmetric extension of the theory. Section five deals with bosonic strings. We present an alternative string action thatin principle does notrequirea criticaldimensionandthat allowsa smooth transitiontothetensionlesslimitofstringtheory. Weshowhowthisalternative stringactionmay be gauge-fixedto yielda non-commutativetheory atthe uni- tary tension value and to give the extension of the particle commutators in the vanishing tension limit. Section six extends these results to superstring theory and section seven briefly describes the same situation in relativistic membrane theory. We present our conclusions in section eight, where we also speculate on the possibility of using the non-locality introduced by our commutation relations to provide the conceptual basis for a covariant Newtonian gravitational field theory. 2 Conformal Vector Fields Consider the Euclidean flat space-time vector field Rˆ(ǫ)=ǫµ∂ (2.1) µ such that ∂ ǫ +∂ ǫ =δ ∂.ǫ (2.2) µ ν ν µ µν The vector field Rˆ(ǫ) gives rise to the coordinate transformation δxµ =Rˆ(ǫ)xµ =ǫµ (2.3) 4 Thevectorfield(2.1)isknownastheKillingvectorfieldandǫµ isknownasthe Killing vector. One can show that the most general solution for equation (2.2) in a four-dimensional space-time is ǫµ =δxµ =aµ+ωµνx +αxµ+(2xµxν δµνx2)b (2.4) ν ν − The vector field Rˆ(ǫ) for the solution (2.4) can then be written as 1 Rˆ(ǫ)=aµP ωµνM +αD+bµK (2.5) µ µν µ − 2 where P =∂ (2.6) µ µ M =x ∂ x ∂ (2.7) µν µ ν ν µ − D =xµ∂ (2.8) µ K =(2x xν δνx2)∂ (2.9) µ µ − µ ν P generatestranslations,M generatesrotations,D generatesdilatationsand µ µν K generates conformal transformations in space-time. The generators of the µ vector field Rˆ(ǫ) obey the commutator algebra [P ,P ]=0 (2.10a) µ ν [P ,M ]=δ P δ P (2.10b) µ νλ µν λ µλ ν − [M ,M ]=δ M +δ M δ M δ M (2.10c) µν λρ νλ µρ µρ νλ νρ µλ µλ νρ − − [D,D]=0 (2.10d) [D,P ]= P (2.10e) µ µ − [D,M ]=0 (2.10f) µν [D,K ]=K (2.10g) µ µ [P ,K ]=2(δ D M ) (2.10h) µ ν µν µν − [M ,K ]=δ K δ K (2.10i) µν λ νλ µ λµ ν − 5 [K ,K ]=0 (2.10j) µ ν The commutator algebra (2.10) is the conformal space-time algebra in four dimensions. Noticethatthecommutators(2.10a-2.10c)correspondtothePoincaré algebra. The Poincaré algebra is a sub-algebra of the conformal algebra. Let us now see how we can extend the conformal algebra, and how this extended algebra is related to a non-commutative space-time geometry. 3 Relativistic Particles A relativistic particle describes is space-time a one-parametertrajectory xµ(τ). The dynamics of the particle must be independent of the parameter choice. A possible formof the actionis the one proportionalto the arclength traveledby the particle and given by S = m ds= m dτ x˙2 (3.1) − Z − Z − p In this work we take the parameter τ to be the particle’s proper time, m is the particle’s mass and ds2 = δ dxµdxν. A dot denotes derivatives with respect µν − to τ and we use units in which h¯ =c=1. Action(3.1)isobviouslyinadequatetostudythemasslesslimitofthetheory andsowemustfindanalternativeaction. Suchanactioncanbeeasilycomputed by treating the relativistic particle as a constrained system. In the transition to the Hamiltonian formalism action (3.1) gives the canonical momentum m p = x˙ (3.2) µ µ √ x˙2 − and this momentum gives rise to the primary constraint 1 φ= (p2+m2)=0 (3.3) 2 WefollowDirac’s[14]conventionthataconstraintissetequaltozeroonlyafter allcalculationshavebeenperformed. ThecanonicalHamiltoniancorresponding toaction(3.1),H =p.x˙ L,identicallyvanishes. Thisisacharacteristicfeature − of reparametrization-invariantsystems. Dirac’s Hamiltonian for the relativistic particle is then 1 H =H +λφ= λ(p2+m2) (3.4) D 2 where λ(τ) is a Lagrange multiplier. The Lagrangianthat correspondsto (3.4) is L = p.x˙ H D − 1 = p.x˙ λ(p2+m2) (3.5) − 2 6 Solvingthe equationofmotionfor p that followsfrom(3.5)andinsertingthe µ result back in it, we obtain the particle action 1 1 S = dτ( λ−1x˙2 λm2) (3.6) Z 2 − 2 The great advantage of action (3.6) is that it has a smooth transition to the m=0 limit. Action (3.6) is invariant under the Poincarétransformations δxµ =aµ+ωµxν (3.7a) ν δλ=0 (3.7b) Invariance of action (3.6) under transformation (3.7a) implies that we can construct a space-time vector field corresponding to the first two generatorsin the right of equation (2.5). These generators realize the Poincaré algebra (2.10a- 2.10c). Now we make a transition to the massless limit. This limit is described by the action 1 S = dτλ−1x˙2 (3.8) 2Z The massless limit is usually considered to describe the high energy limit of relativistic particle theory. The canonical momentum conjugate to xµ is 1 p = x˙ (3.9) µ µ λ The canonical momentum conjugate to λ identically vanishes and this is a pri- maryconstraint,p =0. ConstructingthecanonicalHamiltonian,andrequiring λ the stability of this constraint, we are led to the mass shell condition 1 φ= p2 =0 (3.10) 2 Let us now study which space-time symmetries are present in this limit. Being the m=0 limit of (3.6), action(3.8) is also invariantunder the Poincarétrans- formation(3.7). Themasslessaction(3.8)howeverhasalargersetofspace-time invariances. It is also invariant under the scale transformation δxµ =αxµ (3.11a) δλ=2αλ (3.11b) where α is a constant, and under the conformal transformation δxµ =(2xµxν δµνx2)b (3.12a) ν − 7 δλ=4λx.b (3.12b) where b is a constant vector. Invariance of action (3.8) under transformations µ (3.7a), (3.11a) and (3.12a) then implies that the full conformal field (2.5) can be defined in the massless sector of the theory. Itisconvenientatthispointtorelaxthemassshellcondition(3.10)because the massless particle will be off mass shell in the presence of interactions [15]. We may now use the fact that we are dealing with a special-relativistic system. Special relativity has the characteristic kinematical feature that the relativistic velocity is always ortogonal to the relativistic acceleration ( see, for instance, [16]). Whilethemathematicsinvolvedatthispointisverysimple,thephysical implications of this ortogonality condition are rather unexplored. In this work wemakeanattempttopartiallyclarifythesephysicalimplicationsinthecontext of massless relativistic particle theory. We start by adopting the point of view that specialrelativity automaticallyimplies the ortogonalityconditionbetween the velocity and the acceleration. Reasoning in this way, we find that we may use this ortogonalityto induce the presence of a new invariance of the massless particleaction. Then,asaconsequenceofthefactthataction(3.8)isaspecial- relativistic action, it is invariant under the transformation xµ x˜µ =exp β(x˙2) xµ (3.13a) → { } λ exp 2β(x˙2) λ (3.13b) → { } where β is an arbitrary function of x˙2. We emphasize that although the ortogonality condition must be used to get the invariance of action (3.8) under transformation(3.13),this conditionisnotanexternalingredientinthe theory. Infact,the ortogonalitybetweenthe relativisticvelocityandthe accelerationis an unavoidable condition here, it is an imposition of special relativity. We are just using this aspect of special relativity to get an invariant action. Now, invariance of the massless action under transformations (3.13) means that infinitesimally we can define the scale transformation δxµ = αβ(x˙2)xµ , where α is the same constant that appears in equations (2.4) and (2.5). These transformations then lead to the existence of a new type of dilatations. These new dilatations manifest themselves in the fact that the vector field D of equation (2.8) can be changed according to D =xµ∂ D∗ =xµ∂ +β(x˙2)xµ∂ =D+βD (3.14) µ µ µ → Infact,becauseallvectorfieldsinequation(2.5)involvepartialderivativeswith respecttoxµ andβ isafunctionofx˙µ only,wecanalsointroducethegenerators P∗ =P +βP (3.15) µ µ µ M∗ =M +βM (3.16) µν µν µν 8 K∗ =K +βK (3.17) µ µ µ and define the new space-time vector field 1 V∗ =aµP∗ ωµνM∗ +αD∗+bµK∗ (3.18) 0 µ − 2 µν µ The generators of this vector field obey the algebra [P∗,P∗]=0 (3.19a) µ ν [P∗,M∗ ]=(δ P∗ δ P∗)+β(δ P∗ δ P∗) (3.19b) µ νλ µν λ − µλ ν µν λ − µλ ν [M∗ ,M∗ ]=(δ M∗ +δ M∗ δ M∗ δ M∗ ) µν λρ νλ µρ µρ νλ− νρ µλ− µλ νρ +β(δ M∗ +δ M∗ δ M∗ δ M∗ ) (3.19c) νλ µρ µρ νλ− νρ µλ− µλ νρ [D∗,D∗]=0 (3.19d) [D∗,P∗]= P∗ βP∗ (3.19e) µ − µ − µ [D∗,M∗ ]=0 (3.19f) µν [D∗,K∗]=K∗+βK∗ (3.19g) µ µ µ [P∗,K∗]=2(δ D∗ M∗ )+2β(δ D∗ M∗ ) (3.19h) µ ν µν − µν µν − µν [M∗ ,K∗]=(δ K∗ δ K∗)+β(δ K∗ δ K∗) (3.19i) µν λ λν µ− λµ ν λν µ− λµ ν [K∗,K∗]=0 (3.19j) µ ν Noticethatthevanishingbracketsoftheconformalalgebra(2.10)arepreserved as vanishing in the above algebra, but the non-vanishing brackets of the conformal algebra now have linear and quadratic contributions from the arbitrary functionβ(x˙2). Algebra(3.19)is anoffshellextensionofthe conformalalgebra (2.10). Now consider the commutator structure induced by transformation(3.13a). We assume the usual commutation relations between the canonical variables, [x ,x ] = [p ,p ] = 0 , [x ,p ] = iδ . Taking β(x˙2) = β(λ2p2) in transfor- µ ν µ ν µ ν µν mation (3.13a) and transforming the p in the same manner as the x , we find µ µ that the new transformed canonical variables (x˜ ,p˜ ) obey the commutators µ µ [p˜ ,p˜ ]=0 (3.20) µ ν 9 [x˜ ,p˜ ]=iδ (1+β)2+(1+β)[x ,β]p (3.21) µ ν µν µ ν [x˜ ,x˜ ]=(1+β) x [β,x ] x [β,x ] (3.22) µ ν µ ν ν µ { − } written in terms of the old canonical variables. These commutators obey the non trivial Jacobi identities (x˜ ,x˜ ,x˜ ) = 0 and (x˜ ,x˜ ,p˜ ) = 0. They also µ ν λ µ ν λ reduce to the usual canonical commutators when β(λ2p2)=0. Itmaybequestionedhere,onthebasisofthedefinition(3.9)oftheclassical canonical momentum, if the quantum momentum should not transform in the inverse way of the xµ under (3.13), as is the case for the classical momemtum. Thisisaninterestingpointinthemathematicaldevelopmentshere. Infact,the identificationofthe correctquantumcanonicalmomentumis a typicalproblem in field theories defined over a non-commutative space-time [24]. An example is non-commutative quantum string field theory, where the momentum space Fouriertransformoftheoddvibrationalpositionmodesbehaveasthequantum canonicalmomentaconjugateto theevenvibrationalpositionmodes [8]. Inthe context of this work, transforming the quantum momenta as the classical ones implies substituting the commutator (3.21) by the commutator [x˜ ,p˜ ]=(1+β) iδ (1 β) [x ,β]p µ ν µν µ ν { − − } ThiscommutatoralsoreducestotheusualHeisenbergcommutatorwhenβ(λ2p2)= 0. However, the Jacobi identity (x˜ ,x˜ ,p˜ ) = 0, based on commutator (3.22) µ ν λ and on the above commutator does not seem to close. If this is the case, the classical and quantum canonical momenta of the massless particle on a non- commutativespace-timearerelatedbyascaletransformationofthetype(3.13). That is, p =exp 2β p . q c { } The simplest example of non-commutative space-time geometry induced by transformation(3.13)is the casewhenβ(λ2p2)=λ2p2. The newpositions then satisfy [x˜ ,x˜ ]= 2iλ2M∗ (3.23) µ ν − µν where M∗ is the extended off shell operator of Lorentz rotations given by µν equation (3.16). The commutator (3.23) satisfies Tr[x˜ ,x˜ ]=0 (3.24) µ ν Z asis the casefora generalnon-commutativealgebra[10]. Fromequation(3.23) we may say that there exists an inertial frame in which the uncertainty introduced by two simultaneous position measurements is an off shell rotation in space-time. This appears to confirm the observation in [25] that the non- commutative space-time is to be interpreted as having an internal angular momentum throughout. Wemaynowconsiderthequestionofifwecanfindamathematicalstructure that wouldbehave like a dipole in the context ofmassless particle theory. Such 10