D-Branes and the Non-Commutative Structure of Quantum Space Time

Corfu Summer Institute on Elementary Particle Physics, 1998

PROCEEDINGS

D-branes and the Non-Commutative Structure of Quantum Space Time

N.E. Mavromatosa∗and R.J. Szabob a Department of Physics - Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, U.K. E-mail: [email protected] b The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark E-mail: [email protected]

Abstract: A worldsheet approach to the study of non-abelian D-particle dynamics is presented based on viewing matrix-valued D-brane coordinate fields as coupling constants of a deformed σ-model which defines a logarithmic conformal field theory. The short-distance structure of spacetime is shown to be naturally captured by the Zamolodchikov metric on the corresponding moduli space which encodes the geometry of the string interactions between D-particles. Spacetime quantization is induced directly by the string genus expansion and leads to new forms of uncertainty relations which imply that general relativity at very short-distance scales is intrinsically described by a non-commutative geometry. The indeterminancies exhibit decoherence effects suggesting the natural incorporation of quantum gravity by short-distance D-particle probes. Some potential experimental tests are briefly described.

1. New Uncertainty Principles in bound ∆x ≥O(`s) on the measurability of dis- String Theory tances in spacetime. This result means that, if one uses only string states as probes of short- A long-standing problem in string theory is to de- distance structure, the conventional ideas of gen- termine the structure of spacetime at very short eral relativity break down at distances smaller distance scales, typically at lengths smaller than than `s. the finite intrinsic length of the strings. One of However, until very recently there has been the first analytical approaches to this problem no systematic derivation of (1.1) based on some was to study the effects of high-energy string set of fundamental principles. The appearence scattering amplitudes on the accuracy with which of new solitonic structures in string theory, which one can measure position and momentum [1]. incorporate defects in spacetime, suggest the pos- This implies the conventional string-modified sibility of using such objects as probes of short- Heisenberg uncertainty principle distance structure. These non-perturbative objects are known as D-branes and can be analyt- ¯h O 2 ∆x>∼ + (`s)∆p+... (1.1) ically described beyond the conventional string ∆p worldsheet approach. In many instances how- The modifications to the usual phase space un- ever, such as the cases that will be studied in the certainty relation in (1.1) come from stringy cor- following, a perturbative string loop-expansion rections which are due to the finite minimum approach is still sufficient. As we will discuss in this paper, such an approach leads to new forms length `s of the string. In fact, minimizing the right-hand side of (1.1) shows that the string of uncertainty relations, in addition to (1.1), length scale gives an absolute minimum lower which are attributed to the recoil of the spacetime defects in the process of the scattering of ∗Conference speaker Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob string matter off the D-brane solitons. In the dimensional quantum gravity wormholes [6], that case of multi-brane systems, this leads to a non the genus expansion of the worldsheet theory will commutative spacetime structure at very short lead to a quantization of the couplings {gI } [7]. distances. The quantum field theory is described by the In this paper we will give an exposition of fixed-genus Euclidean path integral these results, based mostly on the articles [2]– Z I I −LQFT[Φ;{g }] [5]. We will begin with a brief review of soli- ZQFT[{g }]= DΦe , ton structures in string theory, emphasizing the Z I 2 I worldsheet σ-model approach to T -duality and LQFT[Φ; {g }]=L∗[Φ] + d zgOI[Φ] Dirichlet branes. We will then describe how to Σ (1.2) view spacetime coordinates and momenta in this framework as σ-model coupling constants, such where Φ denotes a collection of fields defined on that the genus expansion leads to a canonical a Riemann surface Σ. The action L∗[Φ] defines a phase space quantization. We then specialize conformal field theory and OI [Φ] are a set of local to the case of a system of multi-D-particles and deformation vertex operators. The sum over all show how one passes from Lie algebraic to space- topologies of the two-dimensional quantum field time non commutativity. In this way the quan- theory (1.2) can be evaluated exactly in the di- tum spacetime which follows from the many-body lute wormhole gas approximation (fig. 1) and I D-particle dynamics is induced directly by the it induces statistical fluctuations ∆g of the σ- quantum string theory itself. The important as- model couplings, Z pects of the construction are a logarithmic con- X Z [{gI }] ' Dα P[{α }] formal field theory formalism for the relevant re- QFT I I genera coil operators, an effective target space Z −L { I I } Lagrangian for the σ-model couplings which is DΦe QFT[Φ; g +∆g (α) ] (1.3) described by non-abelian Born-Infeld theory, and the interpretation of the target space time as the where the fields αI are wormhole parameters. For Liouville mode of the underlying two-dimensional a dilute gas, the wormhole probability distribution quantum gravity. We will show how this for- function is given by malism leads to new uncertainty principles in D- 1 IJ P[{αI }]=Nexp − αI G αJ (1.4) brane quantum gravity. We conclude with some 2Γ2 general remarks and outlook on the nature of where Γ is the width of the distribution and GIJ is an time in D-brane quantum gravity, including pos- appropriate metric on the moduli space of coupling sible experimental tests of spacetime non com- constants {gI }. This promotes the couplings gI to mutativity from γ-ray burst observations, neu- quantum operators bgI = gI +∆gI on target space. tral kaon physics, and atom interferometry mea-

surements.

h h

h ~h h~h Quantization of Collective Coordinates: ~ + + + ... The Basic Idea

It is worthwhile to give a quick overview of the Figure 1: Resummation of the genus expansion in formalism which will follow. To use D-branes as the pinched approximation. The solid circles are worldsheet discs (or spheres) and the thin lines are probes of the short distance properties of space- strips attached to them with inﬁnitesimal pinching time, we shall view the collective coordinates and size δ. Each strip corresponds to an insertion of a momenta of D-particles as a set of σ-model cou- bilocal operator on the worldsheet. plings {gI } (this includes the case of multi-D0- brane conﬁgurations which inherently contain non- commutative structures). It follows from the Cole- Let us now specialize to the case of a system of man approach to probabilistic couplings via two- N non-relativistic heavy D-particles. In this case, the

2 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

{ I } { i 0 cd 0 } couplings g are given by the set Yab(X ),Pj (X ) , by imposing Neumann boundary conditions on the where Y are the collective coordinates of the D-particles embedding fields XM , M =0,1,...,9, of the string and P are their collective momenta. The indices worldsheet Σ, which we assume has the topology of a i, j label the directions in spacetime while a, b, c, d = disc. At the circular boundary of Σ the fields are con- M 1,...,N label the U(N) isospin gauge symmetry stant along the normal directions, ∂nX =0,and present in any multi-brane system. The field X0 is are allowed to vary as arbitrary functions XM (s)on the worldsheet temporal embedding coordinate and ∂Σ. Here ∂n is the normal derivative to the boundary ab ab ˜ the momenta are given by Pi = MsUi ,where ∂ΣinΣ.AT-duality transformation X → X,de- M αβ ˜ M 1 fined on the worldsheet by ∂αX = ∂β X ,maps Ms = (1.5) the Neumann boundary conditions into the Dirich- gs`s M let boundary conditions ∂tX˜ =0,orequivalently is the mass of a D-particle, with gs the string coupling M M X˜ |∂Σ =Y ,where∂t is the derivative tangent to constant. According to the above general prescrip- ∂Σ. Now the fields are fixed at the specified values tion, the multi-brane dynamics will induce position- Y M on ∂Σ but can vary in the normal directions. momentum (or phase space), space-space, and space– If the T -duality mapping is applied to 9 − p spatial time indeterminancies as follows. From the associ- directions, then the Dirichlet conditions define a hy- ated wormhole distribution (1.4) there follows a set of persurface in 10-dimensional spacetime. The hyper- i position uncertainties ∆Yab corresponding to the ef- surface is embedded into target space from an effec- 2 fective widths Γ /GIJ, where Γ is proportional to the tive p + 1 dimensional worldvolume in which the em- h i string coupling gs and GIJ = VIVJ is the Zamolod- bedding fields XM are allowed to vary freely. These { } chikov metric [8] with VI the set of local vertex objects are known as Dp-branes. They are solitons operators associated with the D-particle couplings of the open superstring theory and supersymmetry { I } ab g . The momentum uncertainties ∆Pj arise upon guarantees their stability. They have a BPS mass M canonical quantization in the moduli space of σ- given by (1.5) and are characterized as being topo- { I } model couplings g which leads to the quantum logical defects which are fixed in the 9 − p spacetime commutator hh ii directions. Open string excitations can attach them- b i bcd i c d b i − δ selves to the D-brane domain walls. Y ab, Pj = i ¯h M δ j δ a δ b , P ab = i ¯hM i δYab (1.6) In this paper we shall specialize to the case of where the effective Planck constant ¯hM of M is of D0-branes, or D-particles. In this case the set of string embedding fields can be written as XM = order gs. Furthermore, the target space time T in 0 i 0 the “physical” frame depends on the positions and (X ,X ), where X is the worldline time coordinate, momenta of the D-particles and therefore becomes which satisfies Neumann boundary conditions, and i a target space operator Tb upon summation over all X , i =1,...,9, are the coordinates of the D-particle worldsheet topologies. From this we will derive a which obey Dirichlet boundary conditions. The dy- space–time uncertainty relation of the form [9] namics of these excitations can be described by de- i O 2 forming the usual free string σ-model action S0[X] ∆Yab∆T>∼ (gs`s) which will imply, in particu- lar, that extremely heavy D-particles can probe very by a worldsheet boundary vertex operator [10] small distances. I 1 S = S [X] − ds Y (X0) ∂ Xi(s)“; C 0 `2 i n Z s ∂Σ 2. String Solitons and Non-abelian 1 S [X]= d2z∂XM∂X¯ N η (2.1) 0 2`2 MN D-particle Dynamics s Σ

where ηMN is a (critical) flat Minkowski spacetime Short-distance Spacetime Structure metric. The non-relativistic motion of heavy The discovery [10, 11] of new solitonic structures D-particles can be described by the Galilean-boosted i 0 i i 0 i in superstring theory have dramatically changed our configurations Y (X )=Y +U X where U is the understanding of target space structure. These new non-relativistic velocity of the branes. non-perturbative objects are known as Dirichlet-branes The interesting situation arises when one consid- and they can be seen to arise from the implementa- ers a configuration of N D-branes (fig. 2). The mul- tion of T -duality as a canonical transformation in the tiple D-brane assembly leads to a non-commutative path integral [12, 13] for the usual open superstring structure at very short distances in the spacetime. with free endpoints. The latter object is described The situation is actually quite simple. Consider a

3 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

system of N parallel D-branes. When the separa- [14]. The reduced Yang-Mills potential tion between branes is of a sub-Planckian distance X9 scale, the solitons can interact with each other via 1 1 i j 2 V0[Y ]= 4 tr Y ,Y (2.2) the exchange of open strings. It is this property `s 4gs i,j=1 of D-brane dynamics that makes them good probes of the short-distance structure of spacetime and it then governs the dynamics of the branes, where tr implies that spacetime at very small distance scales denotes the trace in the fundamental representation is described by some sort of non-commutative ge- of U(N). There are two limiting regimes of this the- ometry. A tractable limit of this situation is the ory. In the weak-coupling limit gs → 0, the branes case of overlapping branes or infinitesimal separa- are very far apart and do not interact with each tions. More complicated situations can also arise, for other. The potential (2.2) is minimized by those instance when the branes interact via the exchange matrix-valued configurations which satisfy [Y i,Yj]= of other D-branes, in addition to their string interac- 0, ∀i, j. These solutions correspond to states of max- tions. One would then need to employ a formalism imal supersymmetry. In this case the generic U(N) for intersecting D-brane configurations. In the fol- gauge symmetry of the Yang-Mills theory is broken lowing we will concentrate on the simpler situation down to U(1)N , i.e. there is one U(1) gauge field of only open string excitations between the D-branes. on each D-brane. Moreover, the matrix fields can be simultaneously diagonalized by a gauge transfor- i {i } mation to give Y =diagy(a) a=1,...,N , where the i eigenvalues y(a) represent the coordinates of each D- particle. Now consider the opposite case where the

D D branes are almost on top of each other. Since the en- 1 2 ergy of a fundamental string which stretches between 1 | − | two different branes is 2`s ~y(a) ~y(b) , it follows that in this case more massless vector states appear in the spectrum of the U(N) gauge theory. Now the configurations of the Yang-Mills potential (2.2) satisfy [Y i,Yj] =0for6 i=6 jand correspond to states of broken supersymmetry which must be incorporated into the quantum gauge theory. Thus the limit of co- Figure 2: Emergence of the enhanced U(N) gauge inciding branes restores the full U(N) gauge symme- symmetry for bound states of N parallel D-branes try and leads to a Lie algebraic non-commutativity i (planes). An oriented fundamental string (wavy in the spacetime coordinates Yab. The components lines) can start and end either at the same or dif- with a =6 b are to be interpreted as the coordinates of 2 ferent D-brane, giving N massless vector states in the short open string degrees of freedom stretched be- the limit of coinciding branes. These states form a tween the branes, so that the D-brane coordinates are representation of U(N). viewed as adjoint Higgs fields in this picture. These ideas are depicted schematically in fig. 2. In the following we shall be interested in estab- lishing the manner in which this Lie algebraic non- commutativity implies a genuine quantum spacetime Low-energy Effective Field Theories non-commutativity. For this, we consider an alterna- tive description to the Yang-Mills matrix quantum To understand why a non-commutative structure is mechanics which is given by a non-local deforma- implied at very short distance scales, one needs to tion of a free worldsheet σ-model. In the simplest examine the effective field theory description for N case of non-relativistic motion, it follows from the parallel D-branes. In the multi-D-particle case, the BPS mass formula (1.5) that the limit of heavy D- assembly is described by a set of N × N Hermitian particles is equivalent to taking gs 1. This is pre- i matrices Yab, i =1,...,9, in the adjoint representa- cisely the regime in which worldsheet perturbation tion of the unitary group U(N) which are obtained theory can be trusted. The partition function for as the remnant fields from the dimensional reduction the uniform motion of the multi-D0-brane system in of 10-dimensional maximally supersymmetric U(N) the T -dual Neumann picture is defined as the expec- Yang-Mills theory to the worldlines of the D-particles tation value of a path-ordered Wilson loop operator

4 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

along the worldsheet boundary in the free σ-model, Z D −S0[X] ZN [Y ]= DX e × I D 0 M tr P exp igs AM (X (s)) dX (s) ∂Σ (2.3) (a) × where the field A is an N N Hermitian matrix D in the adjoint representation of U(N)withAM = 0 1 i (A,−2Y) the components of the U(N) gauge po- `s tential dimensionally reduced to the worldline of the D-particles. However, the model (2.3) on its own is a bit D too naive and needs to be supplemented with some auxilliary prescriptions. There are problems with D D summing over worldsheet genera in the Dirichlet picture, which could be related to the breaking of the (b) T -duality symmetry from an anomaly in the non- abelian case [15, 16]. Modular logarithmic diver- Figure 3: (a) World-sheet annulus diagram for the gences appear in matter field amplitudes when the leading quantum correction to the propagation of a string state V (wavy lines) in a D-brane background, string propagator L0 is computed with Dirichlet boundary conditions. Consider a string matter field polar- and (b) the pinched annulus configuration which is ization tensor V in the D-brane background at the the dominant divergent contribution to the quantum level of an annular topology (fig. 3). Corresponding recoil. to a string state of conformal weight ∆, the string propagator containsR a modular parameter integra- dq ∆ tion of the form q q . In the pinched annulus where limit, with infinitesimal pinching size δ,thestates 0 0 0 0 0 of zero conformal dimension ∆ = 0 therefore yield C(X )=θ(X ) ,D(X)=X θ(X ) logarithmic divergences of the form log δ. (2.5) and Z These modular divergences are cancelled by +∞ 1 dq 0 adding logarithmic recoil operators [17] to the matrix θ (X0)= eiqX (2.6) 2πi q − i σ-model action (2.3). If one is to use low-energy −∞ probes to observe short-distance spacetime structure, is the regulated step function whose → 0+ limit such as a generalized Heisenberg microscope, then is the usual step function. The operators (2.5) have one needs to consider the scattering of string matter non-vanishing matrix elements between different string off the D-particles. The worldsheet perspective of states and therefore describe the appropriate change this physical situation is represented by fig. 3. The of state of the D-brane background. They can be target space picture is the following. At time t<0a thought of as describing the recoil of the assembly closed string state propagates towards the spacetime of D-particles in an impulse approximation, in which string defect which is fixed in space. At t =0it it starts moving as a whole only at time X0 =0. { i j } interacts with the defect by splitting and attaching The collection of constant matrices Yab,Ucd now itself to the D-particle. Then at time t>0, the form the set of coupling constants for the worldsheet instantaneous interaction causes a transfer of energy σ-model (2.3). from the string state to the defect such that the D- particle recoils with some non-zero velocity. For the Galilean Invariance and Worldsheet Loga- Galilean-boosted multi-D-particle system, the recoil rithmic Conformal Algebra is described by taking the deformation of the σ-model action in (2.3) to be of the form [5] The recoil operators (2.5) possess a very important property. They lead to a deformation of the free i 0 i 0 i 0 Yab(X ) = lim `sYab C(X )+Uab D(X ) →0+ σ-model action in (2.3) which is not conformally- (2.4) invariant, but rather defines a logarithmic conformal

5 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

field theory [18]. Logarithmic conformal field theo- with time t. The corresponding β-functions for the ries lie on the border between conformal field theo- worldsheet renormalization group flow are ries and generic two-dimensional renormalizable field dY dU β ≡ i =∆ Y +` U ,β ≡ i =∆ U theories. They contain logarithmic scaling violations Yi dt i s i Ui dt i in their correlation functions on the worldsheet. In (2.11) the present case, this can be seen by computing the pair correlators of the fields (2.5) which give [17] 3. Quantization of Moduli Space D E D E b C(z) C(0) =0 , C(z)D(0) = , z∆ We shall now begin describing the basic steps towards D E 2 b`s the quantization of the σ-model couplings represent- D(z) D(0) = log z (2.7) z∆ ing the collective degrees of freedom of the assembly of D-particles. where ||2 `2 ∆ = − s (2.8) 2 Liouville-dressed Renormalization Group is the conformal dimension of the recoil operators. Flows The constant b is fixed by the leading logarithmic di- We shall first need to identify the time variable of our vergence of the conformal blocks of the theory. Note system. Note that for finite , (2.8) shows that the that (2.8) vanishes as → 0, so that the logarithmic operators (2.5) lead to a relevant deformation of the worldsheet divergences in (2.7) cancel the modular free σ-model. The deformation becomes marginal in annulus divergences discussed above. An essential the limit → 0. When the field theory lies away from ingredient for this cancellation is the identification criticality we must dress the model by Liouville the- [17] − ory [19] in order to restore conformal invariance at 2 = −2`2 log Λ (2.9) s the quantum level. The worldsheet zero mode of the which relates the target space regularization parame- Liouville field can then be identified with the local ter to the worldsheet ultraviolet cutoff scale Λ (the worldsheet regularization scale. Thus the Liouville minus sign in (2.9) is due to the Minkowski signature field is interpreted as the target space time [20], and of X0). from the discussion of the previous section, it coin- 0 Logarithmic conformal field theories are charac- cides with the temporal embedding field X .This means that the incorporation of the regulated oper- terized by the fact that their Virasoro generator L0 is not diagonalizable, but rather admits a Jordan cell ators (2.5, 2.6) can be thought of as the appropriate { i j } structure. Here the operators (2.5) form the basis of dressing of the bare coupling constants Yab,Ucd of a2×2 Jordan block and they appear in the spec- the σ-model [4, 5]. trum of the two-dimensional quantum field theory In general, the Liouville dynamics ensures the as a consequence of the zero modes that arise from possibility of canonical quantization in the moduli the breaking of the target space translational symme- space of σ-model couplings through a set of proper- try by the topological defects. The mixing between ties known as Helmholtz conditions [5, 7]. The Li- C and D under a conformal transformation of the ouville field φ is defined by identifying the conformal worldsheet can be seen explicitly by considering a equivalence class of the metric γαβ of Σ,

finite-size scale transformation (2/`sQ)φ γαβ =e bγαβ (3.1) 0 − Λ → Λ =Λe t/`s (2.10) where bγαβ is a fixed fiducial worldsheet metric and Q is related to the central charge of the corresponding Using (2.9) it follows that the operators (2.5) are two-dimensional quantum gravity. Then the Liou- changed according to D → D + t` C and C → s ville dressing of the deformation of a free σ-model ac- C . Thus in order to maintain scale-invariance of the tion S , which is characterized by a set of vertex op- theory (2.3) the coupling constants must transform 0 erators {V } with corresponding coupling constants under (2.10) as [3, 17] Y i → Y i + U it and U i → U i, I {gI }, is described by the action which are just the Galilean transformation laws for Z i i the positions Y and velocities U . Thus a finite-size (L) 2 I S = S0 + d zg (φ)VI+ scale transformation of the worldsheet is equivalent σ Z Σ Z to a Galilean transformation of the moduli space of 1 2 Q 2 (2) −2 dz∂φ∂φ¯ − d zφR − σ-model couplings, with the parameter identified 2 2 2`s Σ 2`s Σ

6 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

I Q b The action (3.4) is the appropriate non-abelian ver- 2 dsφK (3.2) 2`s ∂Σ sion of (the T -dual of) the worldsheet action (2.1) describing the dynamics of a single D-brane, and where R(2) is the scalar curvature of Σ and K is the as such it represents the abelianization of the non- extrinsic curvature at the boundary ∂Σ(K=2for abelian D-particle dynamics. a disc). From (3.1) it follows that the Liouville zero mode is φ0 = −`sQ log Λ. Microscopically, quan- The Zamolodchikov metric ij j I 2h i i tum fluctuations in the dressed variables g (φ)are Gab;cd =2NΛ Vab(X;0)Vcd(X;0) M induced by summing over all worldsheettopologies, in on the moduli space controls the dynamics of the analogy with the Coleman approach [6] to wormhole D-particles, where the vacuum expectation value is calculus and the quantization of coupling constants taken with respect to the partition function (3.3). in quantum gravity [7]. This two-point function can be evaluated to leading orders in σ-model perturbation theory using the log- D-particle Dynamics on Moduli Space arithmic conformal algebra (2.7) and the propagator of the auxilliary fields to give [5] We are now ready to present the formalism for multi- 2 D-particle dynamics. In the following we shall, for ij 4¯gs ij ⊗ Gab;cd = 2 [η IN IN + simplicity, concentrate only on the case of the con- `s 2 stituent motion of the particles, i.e. we will subtract g¯s ⊗ ¯ i ¯ j ¯ j ¯ i cm [IN (U U + U U )+ out their center of mass degree of freedom Yi = 36 1 U¯i ⊗U¯j +U¯j ⊗U¯i + N tr Yi. This means that the effective gauge symmetry of the D-particle system is now SU(N). To ¯i¯j ¯j ¯i 6 (U U +U U )⊗IN]]db;ca + O(¯gs) (3.6) write the partition function (2.3) in the form of a local deformation of the free σ-model action S0[X], where IN is the identity operator of SU(N)andwe we need to disentangle the path-ordering in the Wil- have introduced the renormalized coupling constants son loop operator. This is done by considering a set i i g¯s = gs/`s and U¯ = U /`s. From the renormaliza- of corresponding abelianized vertex operators which tion group equations (2.11) it follows that the renor- are defined using an auxilliary field formalism for the malized velocity operator in target space is truly i SU(N)-invariant theory [5, 12, 15, 21]. We introduce marginal, dU¯ = 0, which ensures uniform motion of ¯ dt a set of complex auxilliary fields ξa(s),ξb(s)which the D-branes. It can also be shown that the renor- live on the boundary ∂Σ of the worldsheet and whose malized string couplingg ¯s is time-independent [5]. If h¯ 0 i 0 − propagator is ξa(s)ξb(s ) = δabθ(s s). The parti- we further define the position renormalization Y¯ i = tion function (2.3) can then be written as Y i/` , then the β-function equations (2.11) coin- Z Z s cide with the equations of motion of the D-particles, − 1 i S0[X] ¯ ¯ ¯ i ZN [Y ]= DX e DξDξξc(0) ξc(1) i.e. dY = U¯ . Note that the Zamolodchikov met- N dt Z h ric (3.6) is a complicated function of the D-brane 1 d × exp − ds ξ¯ (s) ξ (s)− dynamical parameters. It therefore represents the a ds a 0 appropriate effective target space geometry of the D- i igs ¯ ab 0 dX (s) particles and, as we will see, it naturally encodes the 2 ξa(s) Yi (X (s)) ξb(s) `s ds short-distance properties of the D-particle spacetime. i (3.3) The canonical momentum Pab for the D-particle dynamics on moduli space can also be determined per- in the static gauge A0 = 0. If we leave the integration turbatively in the σ-model (3.3) by noting that the over auxilliary fields in (3.3) until the very end, then Schrödinger representation of the Heisenberg alge- the partition function is expressed as a functional bra (1.6) implies that it is the one-point function of i i integral involving the local action the deformation vertex operators, Pab = Vab(X;0) . I A long and tedious calculation using the three-point ab 0 i S = S0[X]+ ds Yi (X (s)) Vab(X; s) (3.4) correlation functions of the logarithmic pair [5] gives ∂Σ

2 2 where the deformation is described by the set of ver- i 8¯gs ¯i g¯s ¯ 2 ¯i Pab = [U + (Uj U + tex operators `s 6 ¯ ¯i ¯j ¯i ¯2 O 6 ij ¯˙ cd i UjU U + U Uj )]ba + g¯s = `s Gab;cd Yj i −igs dX (s) ¯ Vab(X; s)= 2 ξa(s)ξb(s) (3.5) `s ds (3.7)

7 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

which, as expected for the uniform D-particle motion The definition (3.11) comes from the normalization here, coincides with the contravariantized velocity on of the Liouville field kinetic term in (3.2) appropri- M. ate to a Robertson-Walker spacetime geometry [20]. We can now write down an associated effective Its expression in (3.11) holds near any fixed point in target space Lagrangian which is defined in terms of moduli space and is valid in the usual regime of appli- the standard non-linear σ-model on M, cability of worldsheet σ-model perturbation theory.

ab cd L − `s ¯˙ ij ¯˙ M = 2 Yi Gab;cd Yj + ... (3.8) The Genus Expansion where the dots denote potential terms involving the We shall now describe the process of coupling con- Zamolodchikov C-function [8] plus additional terms stant quantization via the sum over worldsheet topolo- which depend on the choice of renormalization scheme. gies for the model (2.3). The key point in the non- The Lagrangian (3.8) is readily seen to coincide with abelian case is that the sum over genera and the aux- O 4 the expansion to (¯gs) of the symmetrized form of illiary field representation of the Wilson loop opera- the non-abelian Born-Infeld action for the D-brane tor in (3.3) commute, allowing one to write dynamics [16, 22], X X∞ Z q − (h) 1 2 2 S0 [X] × LNBI = tr Sym det [ηMN IN +`sg¯s FMN] ZN [Y ]= DX e `sg¯s M,N genera h=0 P (3.9) I 1 igs 0 M where Sym(M1,...,Mn)= Mπ1···Mπn n! π∈Sn tr P exp 2 AM (X ) dX = `s is the symmetrized matrix product and the compo- ∂Σh Z X∞ Z nents of the dimensionally reduced field strength ten- 1 1 ˙ g¯s DξDξ¯ ξ¯c(0) ξc(1) × DX × sor are given by F0i = 2 Y¯i and Fij = 4 [Y¯i, Y¯j ]. `s `s N In the abelian reduction to the case of a single D- I h=0 particle, the Lagrangian (3.9) reduces to the usual − (h) − ab 0 i exp S0 [X] ds Yi (X ) Vab(X; s) one describing the free relativistic motion of a mas- ∂Σh sive particle. The leading order F 2 term in the ex- (3.13) pansion of (3.9) is just the usual Yang-Mills Lagrangian. (h) The formalism described here thereby represents a where S0 [X]isthefreeσ-model action defined on highly non-trivial application of the theory of loga- agenushRiemann surface Σh with h + 1 bound- rithmic operators. aries (so that ∂Σh is a disjoint union of h+1 circles). Therefore, if we again leave the auxilliary field inte- Finally, we come to the definition of the target grations until the very end, then we can exploit the space time. The flat worldsheet Zamolodchikov C- abelianization of the non-abelian dynamics in (3.13) theorem [8] can be expressed as to study the topological expansion. ∂C −C = −e t/`s βab Gij βcd (3.10) The latter quantity can be described precisely ∂t i ab;cd j in the pinched approximation (fig. 1). There are where the running central charge C[Y ] ∼ Q2 is the two sorts of modular divergences which dominate Zamolodchikov C-function and the exponential fac- this truncation of the string genus expansion. The tor in (3.10) comes from the extrinsic curvature term leading ones are of the form (log δ)2 and arise from in the Liouville-dressed action (3.2). The non-linear the logarithmic nature of the deformation. It can be differential equation (3.10) can be solved for small shown [5] that these divergences are cancelled by re- velocities (extreme non-relativistic motion) to give quiring that the velocities of the D-particles in the the physical target space time coordinate [5] scattering of string matter off them change accord- s ¯ab − 1 ab Z ingto∆Ui = (ki +kf )iδ ,whereMs =1/g¯s`s √ t Ms 2¯gst 2 2 2 ¯ 2 is the BPS mass of the string solitons and k de- T ≡ C t ' √ U(U¯) dτ e2(τ −t )¯gsU(U)/`s i,f `s 0 note the initial and final momenta in the scatter- (3.11) ing process. Thus the leading divergences of the where we have introduced the velocity-dependent in- genus expansion are cancelled by imposing momen- variant function tum conservation in scattering processes involving string matter. Note that this result only controls the g¯2 U(U¯ )=trU¯2+ s tr 2U¯ 2U¯ 2 + U¯ U¯ U¯ iU¯ j + O g¯4 i 36 i j i j s dynamics of the constituent D-branes themselves and (3.12) not the open string excitations connecting them. It

8 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

therefore tells us nothing about short-distance (non- This leads to a very nice dynamical and geometrical commutative) spacetime structure. picture of short distance spacetime structure. This latter property of the moduli space comes For this, we employ a Born-Oppenheimer ap- from examining the sub-leading divergences, which proximation to the D-particle interactions to sep- are of the form log δ and are associated with the van- arate the diagonal D-particle coordinates from the ishing conformal dimension of the logarithmic opera- off-diagonal parts of the adjoint Higgs fields repre- tors. The regularization of these singularities induces senting the short open string excitations connecting quantum fluctuations of the D-particle collective co- them. This approximation is valid in the limit of ordinates and leads to short-distance uncertainties. small velocities [9] which corresponds to a configu- In the pinched approximation represented by fig. 1, ration of well-separated branes. In the free string the effect of the dilute gas of wormholes is to expo- limit gs 1, we may diagonalize the configuration nentiate the bilocal operator inserted on the bound- fields simultaneously in the static gauge by a time- ary of the disc Σ = Σ . This leads to a change of the independent gauge transformation, 0 action (3.4) given by ¯ i i i −1 ∈ II Y =Ωdiag y(1),...,y(N) Ω , Ω SU(N) 2 g 0 0 (3.16) ∆S' s log δ ds ds V i (X; s) Gab;cd V j (X; s ) ab ij cd where the eigenvalues yi ∈ R are the positions 2 ∂Σ (a) (3.14) of the D-particles which move at constant velocities i i This bilocal interaction term can be written as a local u(a) = dy(a)/dt. The unitary transformation (3.16) worldsheet effective action by using standard tricks of diagonalizes the Zamolodchikov metric (3.6) in its wormhole calculus [6] and introducing wormhole pa- su(N) ⊗ su(N) indices and we have ab M rameters ρi on the moduli space which linearize 2 2 ij 4¯gs ⊗ ij ⊗ g¯s U ij (3.14) via a functional Gaussian integral transforma- G = 2 (Ω Ω)[η IN IN + + `s 36 tion. The net result of the summation over genera in 4 −1 O g¯s ](Ω ⊗ Ω) (3.17) the pinched approximation is therefore X Z where 1 ab ij cd Z [Y ] ' Dρ exp − ρ G ρ ij i j i j i j N 2 i ab;cd j U =(2u u +2u u +u u + M 2Γ ab;cd (a) (a) (b) (b) (a) (b) genera Z j i u u(b))δad δbc (3.18) − (a) × DX e S0[X] × I It now remains to diagonalize the operator (3.18) igs 0 0 i in its 9 × 9 spacetime indices i, j. The situation is tr P exp 2 Yi(X )+ρi(X ) dX (3.15) `s ∂Σ very simple when a = b, as then the eigenvalues of (3.18) are given by where the width of the Gaussian wormhole√ distri- 1 2 2 9 bution function in (3.15) is given by Γ = gs log δ. λaa =6~u(a) ,λaa = ...=λaa = 0 (3.19) Eq. (3.15) shows that the genus expansion induces √ The orthogonal matrix Oaa which diagonalizes the statistical fluctuations ∆Y ab = g log δρab of the i s i Zamolodchikov metric in this case is simply the 9 × coordinates Y ab of the assembly of D-particles. As i 9 identity matrix upon rotation to the coordinate discussed in section 1, it is this property that will al- system in which the first direction is spanned by the low us to probe short-distance spacetime structure in i normalized velocity vector u /|~u(a)|. We shall refer terms of the geometry and the dynamics on moduli (a) to this frame as the “string frame” as it represents space. the one-dimensional coordinate system relative to the single open string excitation that starts and ends on Diagonalization of Moduli Space the same D-particle a (fig. 4). The situation is far more complicated for a =6 b because now the string To be able to write down a set of position uncer- interactions between a given pair of D-particles a, b tainties for each direction of target space and of the also play a role. In this case the eigenvalues are SU(N) group manifold, we shall need to diagonalize 1,2 2 2 · the bilinear form of the wormhole parameter distri- λab = ~u(a) + ~u(b) + ~u(a) ~u(b) bution in (3.15). This requires the diagonalization of 2 2 2 [(~u(a) + ~u(b) + ~u(a) · ~u(b)) + the inverse of the Zamolodchikov metric. The diago- 2 2 2 2 2 2 [(~u(a) · ~u(b)) + ~u ~u +2~u(a) ·~u(b)(~u +~u )] nalization of the moduli space M reveals the precise (a) (b) (a) (b) ]1/2 ~u 2 ~u 2 − (~u · ~u )2 manner in which the string interactions between D- (a) (b) (a) (b) 3 9 particles induce short-distance non-commutativity. λab = ...=λab = 0 (3.20)

9 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

Yi If we assume that the velocity vectors ~u(a) and ~u(b) configuration fields ab encode the information about are linearly independent, then they span a two di- the string interactions between D-particles. Using mensional space which we refer to as the string plane. the results of the previous section they lead to the The increase in dimension of this frame owes to the statistical variances increase in degrees of freedom of the open string † ∆Y˜ i ∆Y˜i = which now stretches between two different branes ab ab (fig. 4). In this coordinate system, the 9×9 orthogo- † ji ki Yab Yab Oab Oab j k = nal diagonalization matrix Oab is of a block diagonal conn form consisting of the 7 × 7 identity matrix I7×7 and 2 2 `s Γ − g¯s i O 4 a2×2matrix∆2×2 2 1 λab(u)+ g¯s (4.1) 2¯gs 36 1 ∆ × 0 O = × 2 2 where the average denotes the connected correlation ab 2/9 0 I7×7 ~u(a) + B(u)~u(b) function with respect to the probability distribution 2 (3.21) function P[ρ] in (3.15). Note thatg ¯s < 0 owing to the AB Minkowski signature of the target space time which with ∆2×2 = ,where CD is proportional to −2 < 0. A= ~u +B(u) ~u cos θ , (a) (b) ab The variances (4.1) can now be used to deter- B = −B(u) ~u sin θ , ( b) ab mine a set of uncertainty relations for the D-particle C = B(u) ~u sin θ ,and (b) ab spacetime coordinates. From (3.19)–(3.22) it follows D= ~u +B(u) ~u cos θ , θ is the angle be- (a) (b) ab ab that the resulting uncertainties will depend non triv- tween the velocity vectors and ially on the kinematical invariants of the D-brane · 2 2 2 motion. This energy dependence is a quantum deco- B(u)=[ ~u(a) ~u(b) +~u(a)~u(b) + herence effect which can be understood from a gen- 2~u ·~u ~u2 +~u2 −λ1 ~u · ~u ] × (a) (b) (a) (b) ab (a) (b) eralization of the Heisenberg microscope whereby we 2 · · 2 [2~u(a) ~u(a) ~u(b) +2 ~u(a) ~u(b) + scatter a low-energy closed string state off the assem- 4 − 1 2 −1 bly of D-particles. At time t<0 we send a closed 2~u(a) λab ~u(a)] (3.22) string towards the spacetime defects which are fixed It is in this way that the Zamolodchikov metric on M in space. As the closed string hits D0-brane a at naturally captures the geometry of the string inter- t = 0, it can split into two open strings, according actions among the D-branes and illustrates the com- to the closed-to-open string amplitude formalism of plexity change between the dynamics of a single D- [23], whose other ends can then either attach back to particle on its own (a = b) and the interactions of a particle a or to D-particle b. Due to this scattering multi-brane system (a =6 b). kinetic energy is transfered from the string state to the D-particles thereby setting them in motion, as depicted in fig. 4. u a b a Minimum Length Uncertainty u b For a single D-particle, from (3.19) and (4.1) we find the coordinate smearings

Yaa | |χ/2 1 | |2 2 ≥ ∆ i = g¯s `s 1+ 12 g¯s ~u(a) δi,1 + ... Figure 4: The string frame representation of D- χ/2 |g¯s| `s (4.2) particles a and b moving at velocities ~u(a) and ~u(b) in target space. where the exponent χ ≥ 0 is deﬁned through the relation which cancels the modular divergences of the genus expansion with the tree-level ultraviolet divergences according to the Fischler-Susskind mechanism χ −2 4. Quantum Uncertainty Relations [24] log δ =2|g¯s| . It may be ﬁxed upon consid- eration of more complicated processes, such as brane The desired position coordinates in which the bilin- exchanges between the D-particles. It is natural to ˜ i ear form of (3.15) is diagonal are given by Yab = have χ>0 since the modular divergences are induced ji ∗−1 ¯ ≡ ji Yab ¯ Oab[Ω YjΩ]ba Oab j (Y ), where the complex by string interactions. We can use this freedom to

10 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

ﬁx χ by the requirement that the minimum bound spacetime structure based on non-commutative ge- in (4.2) coincides with the 11-dimensional Planck ometry [25] which utilizes the algebra of observables 2 length `P, i.e. χ = 3 . Then the smearings (4.2) of the quantum string theory. According to (4.3) the coincide with standard predictions [9] based on the indeterminancies (4.4) probe much deeper into the non-relativistic scattering of two D-particles of mass exotic short-distance structure than the usual quan-

Ms with impact parameter of order |∆Y|.Notethat tum fluctuation relations. Its energy dependence sig- in the string frame (i = 1), the quantum fluctuations nifies the fact that when the D-particles recoil upon exhibit the decoherence effects discussed above. impact with a closed string probe they store information, through the open string degrees of freedom Non-commutative Spacetime Algebra of stretched between them, which prevents independent Observables position measurements for the D-particles. This leads to correlated spatial uncertainties which depend on The feature unique to the multi-D-particle system the scattering content, i.e. on the kinetic energies of comes from examining (4.1) for a =6 b which demon- the non-relativistic particles. Only when there is no strates explictly the non-commutativity of the D- recoil (~u(a) = ~u(b) = ~0) can one measure simultane- particle spacetime. Using (3.20) and (3.21), we ob- ously the positions of two D-particles. tain from this expression two equations in two un- knowns. Outside of the string frame (i>2) one Non-commutative Heisenberg Algebra Yab finds the same minimal lengths for ∆ i as in (4.2). ab Adding these two equations gives smearings ∆Yi , We will now describe the quantization of the phase i =1,2, in the string frame analogous to (4.2) that space of the multi-D-particle system. The canonical depend on the center of mass kinetic energy and mo- quantization condition (1.6) on moduli space leads mentum transfer of the scattering of D-particles a to the Heisenberg uncertainty principle and b [5]. Subtracting the two equations gives an ex- ∆Y¯ ab ∆P j ≥ 1 ¯h δj δa δb (4.5) pression for the connected correlation function of the i cd 2 M i c d two coordinate fields of the string plane. This can in The Planck constanth ¯M can be determined by inter- turn be written as an uncertainty relation using the preting (3.15) as a minimal uncertainty wavepacket Schwarz inequality on moduli space [4] and thereby saturating the lower i bound in (4.5). Since the canonical momentum Pab ∆Yab ∆Yab ≥ Y ab Yab ≥ 1 2 1 2 is implicitly represented as an operator on M (see conn ab ab (1.6)), the effects of the genus expansion on it are Re Y 1 Y2 (4.3) conn already taken into account and we may therefore where the right-hand side of (4.3) is found to be compute its variance directly from the worldsheet σ- model on a tree-level disc topology. Using the two-

ab ab point function (3.6) and the one-point function (3.7), Re Y 1 Y2 = conn we ﬁnd χ+2 2 2 |g¯s| `s ~u(a) + B(u)~u(b) Xab(u) 2 , i 2 ii − i 2 4¯gs ∆Pab = Gab;ab Pab = 2 δab + 144B(u) ~u(a) sin θab ~u(a) + B(u) ~u(b) cos θab `s 2 h i for a =6 b (4.4) 2 2 2¯gs ¯ i − ¯ i 2 2δab U 287 Uba + ...(4.6) 9`s ba Here Xab(u) is a complicated function of the D-particle Performing a Galilean boost to a comoving target velocities ~u(a) and ~u(b) and their scattering angle θab ¯ i (see [5] for details). The right-hand side of (4.4) van- space frame, i.e. setting U = 0 in (4.6), and using ishes for zero recoil velocities. the minimum length (4.2) in (4.5) then determines the Planck constant as The relation (4.4) gives a non-trivial correlation 1+χ/2 among diﬀerent spatial directions of the target space ¯hM =4|g¯s| (4.7) and represents a new form of non-commutative spacetime uncertainty relation. It yields the desired tran- Thus, the Planck constant in the present formalism sition from Lie algebraic non-commutativity to a gen- is proportional to the string coupling constant, which M uine spacetime non-commutativity, in which the spa- owes to the fact that the quantization of here is tial coordinates are no longer independent random induced by string interactions. variables due to their string interactions. This is pre- Note that to leading order the operator (3.7) co- i i cisely the form of the description of short-distance incides with the spacetime momentum p = MsU¯ .

11 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

However, stringy effects give corrections to this op- quantum mechanics applied to strings. It also fol- i i 2 i 3 erator of the form P = p + O(¯gs)(p ) . Iterating lows from very basic worldsheet conformal symme- the Heisenberg algebra (1.6) using this identification try arguments and it gives a natural representation along with (3.7) gives the string modified phase space of the s-t duality of perturbative string amplitudes. commutation relations [5] In the present case this uncertainty relation follows hh ii directly from the phase space uncertainty principle b ab bj j a b Y i , pcd = i ¯h M ( δ i δ c δ d + and it shows that there is a duality between short and large distance phenomena in string theory. However, 1 |g¯ |2`¯2[δj (δa [pb2]b + δb [pb2]a +[pb]a[pbk]b)+ 96 s s i c kd d k c k c d the choice χ = 0 gives a minimum length (4.2) which a j b b j a b j a δ pbi,pb +δ pbi,pb +[pbi] [pb ] + is much larger in general than the 11-dimensional c d d c d c a j b Planck scale. The ambiguities here follow from the [pbi] [pb ] ]+...) (4.8) c d fact that the physical target space (Liouville) time 2 coordinate T is not the same as the longitudinal to leading orders, where `¯s = `s/|g¯s| is the (time independent) 0-brane scale. For a = b = c = d worldline coordinate of a D-particle, but is rather and i = j, (4.8) yields the standard string-modified a collective time coordinate of the system of parti- Heisenberg uncertainty principle (1.1) [1] for a sin- cles which is induced by all of the string interactions gle recoiling D-particle [3, 17]. However, for the among them. We can nevertheless match our results off-diagonal degrees of freedom, (4.8) takes into ac- with those of 11-dimensional supergravity by multi- count of the string interactions among the D-particles plying the definition (3.11) by an overall factor of | |−χ/2 and represents the phase space version of the non- g¯s , which then implies that the target space commutative correlators that we obtained above. It propagation time for weakly-interacting D-particles would be interesting to study the representation the- is very long. ory of the algebra (4.8) and thereby determine properties of the non-commutative spacetime algebra of observables implied by the relations (4.3) and (4.4). Triple Uncertainty Relations

The commutation relation (4.9) for a =6 b illustrates Space–time Uncertainty Principle the eﬀects of the string interactions on the space– The target space time T in the “physical” frame is time duality relation. Using the canonical minimal given by (3.11). Upon summing over all worldsheet uncertainty (4.5) and rescaling the time coordinate T genera, the promotion of the couplings Y¯ and U¯ to as described above, we arrive at the triple uncertainty operators implies that T becomes an operator Tb.To relations leading orders in the string coupling constant expansion, we may replace the velocity operators in (3.12) | |χ 3 ¯ ab 2 g¯s `s ∆Yi ∆T ≥ √ ,a=6 b (4.11) by momentum operators according to (3.7), as de- 2 E scribed above. Using (1.6) and the present Born- Oppenheimer approximation to expand the function This uncertainty principle implies that the high-energy (3.12) as a power series in U¯ /u 1, a =6 b,we ab c scattering of D-particles can probe distances much arrive at the space–time quantum commutators [5] smaller than the characteristic length scale in (4.11), " # 2 2 hh ii 2 bab which for χ = 3 is `P`s. Triple uncertainty rela- i` ¯hM ` P Yb ab, Tb = s δab + 1 − δab s √j + ... tions of the sort (4.11) but involving only `3 have i 2|g¯ | 4|g¯ | P s s E been suggested based on the holographic principle of P (4.9) M-theory [9]. The existence of a limiting velocity to leading orders, where E = |g¯ |2 (U¯ i )2 is the s a aa |~ua| < 1 for the non-relativistic D-particle motion total kinetic energy of the constituent D-particles. implies a lower bound on (4.11), so that using the Using (4.7) we thus obtain the space–time uncer- 2 minimum spatial extensions (4.2) and setting χ = 3 tainty principle in (4.11), we arrive at the characteristic temporal length ¯ aa χ/2 2 ∆Yi ∆T ≥|g¯s| `s (4.10) −1/3 ∆T ≥|g¯s| `s (4.12) When χ = 0 the indeterminancy relation (4.10) coincides with the standard one [9] which can be de- which also agrees with the standard result based on rived from the energy-time uncertainty principle of D-particle kinematics [9].

12 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

5. Outlook: Potential Experimental surements in the transverse directions to the probe, Tests of Spacetime Non Commu- so that it is not immediately clear how to incorpo- tativity rate non-commutative correlations such as (4.4) into these analyses. A recent proposal [28] which is intimately re- We have seen that a perturbative worldsheet formal- lated to the ideas of this article is that fluctuations ism for systems of D-branes yields results which are in spacetime geometry on the scale of the Planck time consistent with the standard target space dynam- −44 τP ∼ 10 s may be detectable by atom interferom- ics, and which also describes interesting new short- eters. In analogy with Brownian motion, whereby distance structures, such as non-commutative spatial measurements on a macroscopic scale can be used coordinates that lead to a proper spacetime quanti- to determine quantities on an atomic scale, one can zation. The most dramatic feature of the uncertainty find a diffusion process which enables the determi- relations we exhibited in the previous section is their nation of quantities at the Planck scale by experi- dependence on the energy of the D-particle system. ments at an atomic scale. Spacetime fluctuations in- This fact distinguishes D-particle dynamics from or- duce diffusion in quantum amplitudes from which the dinary quantum mechanics, since it implies a bound value of τP can be measured and information about on the accuracy of length measurements which de- Planck scale dynamics can be extracted. The key pends entirely on the energy content of the system. feature of this analysis is an appropriate generaliza- Such a situation, whereby the accuracy with which tion of linear Markovian quantum state diffusion to one can measure a quantity depends on its size, is non-commuting fluctuation variables which span an a characteristic feature of decoherence in certain ap- isospin space of internal symmetries of the spacetime proaches to quantum gravity. In the present case that is distinct from the ordinary position space. The the presence of D-brane domain wall structures may canonical commuting fluctuations yield no effect in act as traps of low energy string states, thereby re- matter interferometers, but the decoherence effects sulting in a decoherent medium of quantum gravity resulting from non-commutative fluctuations lead to spacetime foam. The quantum coordinate fluctua- a suppression of the observed interference. The anal- tions, due to the open string excitations between D- ysis of [28] thus shows that the small numerical value particles, can lead to quantum decoherence for a low- of the Planck time does not on its own prevent ex- energy observer who cannot detect such recoil fluc- perimental access to Planck scale physics in the lab- tuations in the sub-Planckian spacetime structure. oratory. The resulting non-commutative metric is The short-distance physics described by non-abelian augmented into the isospin space which is attached D-particle dynamics in flat target spaces naturally to the original spacetime itself and not to the matter capture features of spacetime quantum gravity, and within it. Thus if we consider D-particles as being the construction outlined above therefore illuminates intrinsic topological defects of spacetime represent- the manner in which D-particle interactions probe ing short-distance singularities of quantum gravity, very short distances where the quantum nature of then the non-commutativity described in this paper gravity becomes important. may be related to the description of [28]. In [4] the It would be interesting to see if the non commu- relationship between D-brane recoil and diffusion in tativity of quantum spacetime is amenable in some open quantum systems is discussed. It would be in- way to experimental analysis. The foamy proper- teresting to explore the potential relationship with ties of the non-commutative structure may require D-particle spacetimes and those of [28] in more de- a reformulation of the phenomenological analyses of tail, and hence establish an experimental laboratory length measurements as probes of quantum gravity. for the Planck scale dynamics probed by D-branes. One such test is through neutral kaon systems [26] which are sensitive to the minimal length suppression 19 effects by the Planck mass scale MP ∼ 10 GeV, and Acknowledgements also to quantum gravity decoherence effects. A more recent suggestion is through γ-ray burst spectroscopy This work is based on talks given by r.j.s. at SUSY [27]. Such probes are cosmological in origin and are ’98, Oxford, England, July 11–17, 1998, and by n.e.m. sensitive to Planck scale energies through quantum at the Corfu Summer Institute 1998: TMR project gravity dispersion relations in which the velocity of Physics Beyond the Standard Model, Corfu, Greece, light depends on the photon energy. However, all September 15–18, 1998. We thank the organizers of these approaches do not incorporate length mea- of these conferences for their interest in our work

13 Corfu Summer Institute on Elementary Particle Physics, 1998 N.E. Mavromatosa and R.J. Szabob

and for providing a stimulating atmosphere for dis- [20] J. Ellis, N.E. Mavromatos and D.V. Nanopou- cussions. The work of n.e.m. is supported by a los, Phys. Lett. B293 (1992) 37; I.I. Kogan, in: P.P.A.R.C. (U.K.) Advanced Research Fellowship. Particles and Fields ’91, eds. D. Axen, D. Bry- man and M. Comyn (World Scientiﬁc, Singa- pore, 1992) 837. References [21] H. Dorn, Nucl. Phys. B494 (1997) 105. [1] G. Veneziano, Europhys. Lett. 2 (1986) 199; [22] A.A. Tseytlin, Nucl. Phys. B501 (1997) 41. D.J. Gross and P.F. Mende, Nucl. Phys. B303 [23] J.L. Cardy and D. Lewellen, Phys. Lett. B259 (1988) 407; D. Amati, M. Ciafaloni and G. (1991) 274. Veneziano, Phys. Lett. B216 (1989) 41. [24] W. Fischler and L. Susskind, Phys. Lett. B171 [2] J. Ellis, N.E. Mavromatos and D.V. Nanopou- (1986) 383; B173 (1986) 262. los, Int. J. Mod. Phys. A12 (1997) 2639. [25] F. Lizzi and R.J. Szabo, Commun. Math. Phys. [3] F. Lizzi and N.E. Mavromatos, Phys. Rev. D55 197 (1998) 667. (1997) 7859. [26] J. Ellis, J.S. Hagelin, D.V. Nanopoulos and M. [4] J. Ellis, N.E. Mavromatos and D.V. Nanopou- Srednicki, Nucl. Phys. B241 (1984) 381; J. Ellis, los, Int. J. Mod. Phys. A13 (1998) 1059. N.E. Mavromatos and D.V. Nanopoulos, Phys. [5] N.E. Mavromatos and R.J. Szabo, Phys. Rev. Lett. B293 (1992) 142; P. Huet and M.E. Pe- D59 (1999) 064023; hep-th/9808124, Phys. skin, Nucl. Phys. B434 (1995) 3; R. Adler et al., Rev. D in press. Phys. Lett. B364 (1995) 239; J. Ellis, J. Lopez, N.E. Mavromatos and D.V. Nanopoulos, Phys. [6] S. Coleman, Nucl. Phys. B310 (1988) 643. Rev. D53 (1996) 3846. [7] J. Ellis, N.E. Mavromatos and D.V. Nanopou- [27] G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, los, Mod. Phys. Lett. A10 (1995) 1685. D.V. Nanopoulos and S. Sarkar, Nature 393 [8] A.B. Zamolodchikov, JETP Lett. 43 (1986) 730; (1998) 763. Sov. J. Nucl. Phys. 46 (1987) 1090. [28] I.C. Percival and W.T. Strunz, Proc. R. Soc. [9] M. Li and T. Yoneya, Phys. Rev. Lett. 78 (1997) London A453 (1997) 431. 1219; hep-th/9806240, to appear in Chaos, Soli- tons and Fractals. [10] J. Dai, R.G. Leigh and J. Polchinski, Mod. Phys. Lett. A4 (1989) 2073; R.G. Leigh, Mod. Phys. Lett. A4 (1989) 2767. [11] J. Polchinski, Phys. Rev. Lett. 75 (1995) 184. [12] H. Dorn and H.-J. Otto, Phys. Lett. B381 (1996) 81. [13] Y. Lozano, Mod. Phys. Lett. A11 (1996) 2893; G. Amelino-Camelia and N.E. Mavromatos, Phys. Lett. B422 (1998) 101. [14] E. Witten, Nucl. Phys. B460 (1996) 335. [15] H. Dorn, J. High Energy Phys. 04 (1998) 013. [16] D. Brecher and M.J. Perry, Nucl. Phys B527 (1998) 121. [17] I.I. Kogan, N.E. Mavromatos and J.F. Wheater, Phys. Lett. B387 (1996) 483. [18] V. Gurarie, Nucl. Phys. B410 (1993) 535. [19] F. David, Mod. Phys. Lett. A3 (1988) 1651; J. Distler and H. Kawai, Nucl. Phys. B321 (1989) 509.