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

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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 ob- jects 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 space- time 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 fgI g [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[fg g]= DΦe , ton structures in string theory, emphasizing the Z I 2 I worldsheet σ-model approach to T -duality and LQFT[Φ; fg 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 fgI g (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.
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