Fuzzy Dimensions and Planck's Uncertainty Principle from p-brane Theory

Transcription

1 Fuzzy Dimensions and Planck's Uncertainty Principle from p-brane Theory Abstract The explicit form of the quantum propagator of a bosonic p-brane, previously obtained by the authors in the quenched-minisuperspace approximation, suggests the possibility of new transitions between p-branes of different dimensionality. For such "p-transitions" to take place, it is sufficient that the Lorentzian metric be replaced by a quadratic form on a Clifford manifold. This substitution has two far reaching consequences. On the one hand, it introduces into the theory a new source of fuzziness in the form of a quantum of resolution below which it is impossible to resolve the distance between two points; on the other hand, the introduction of the Clifford line element extends the usual relativity of motion to the case of relative dimensionalism of all p-branes that make up the spacetime manifold near the Planck scale.

2 A surface that appears smooth to the naked eye, looks rough when seen under a microscope. With increasing power of resolution, that seemingly 2-dimensional surface may reveal its thickness and eventually its atomic "graininess", at which point the very definition of dimensionality becomes fuzzy. It has long been conjectured that the same concept applies to the spacetime manifold: the short distance spacetime geometry and topology are subject to violent quantum fluctuations [1]. By "short distance", in this context, one usually means the Planck scale, and, by "violent", we mean that the quantum fluctuations in the geometry are of the same order of magnitude as the background metric, so that the smooth spacetime manifold of classical physics breaks down into a quantum foam of virtual gravitational bubbles [2] and other "extendons", or p-branes [3]. In the above picture, spacetime is not a primitive concept. Rather than being a pre-existing, albeit dynamical, background as in General Relativity, spacetime is an "emergent property", that is, a low energy, effective construct of the truly fundamental objects it is made of. Let us call "p-cells", as opposed to p-branes, the elementary simplexes that constitute the pre-geometric quanta of spacetime. p-cells are model manifolds that form the basis of algebraic topology, and can be defined independently of any spacetime background. Physical space-time emerges from a mapping, X, of p-cells into a target manifold. The image of a p-cell under the mapping X corresponds to a physical p-brane 1. Thus, a 0-cell, that is a point with no spatial dimensions, corresponds to a world-line in spacetime; a 1-cell, or line element, corresponds to a "spacetime pixel", or world-sheet element, and so on, all the way up to the maximal rank {D l)-brane enclosing an elementary spacetime D-volume. Note that the role of "time", in such a dynamical mapping, is played by the proper volume distance between two extremal configurations of the world history of the p-brane [4]. By analogy with a superconducting medium, which consists of a condensate of Cooper pairs, spacetime may be regarded as a sort of string or brane condensate [5], [6]. Near some critical scale, presumably the Planck energy, the spacetime manifold "evaporates" into its fundamental constituents, i.e., strings and branes of various dimensionality. In this extreme phase, spacetime consists of a "gas of branes", each subject to quantum fluctuations of its shape. The shape quantum mechanics of p-branes was discussed in detail in previous papers [7]. Keeping in mind the scenario outlined above, in this brief report we wish to introduce a new kind of quantum fluctuation. Specifically, we conjecture that, at the Planck scale, not only shapes and topology fluctuate, but the very dimensionality "p" of a p-brane becomes a quantum variable. This hypothesis is supported by an explicit calculation of the quantum propagator of a bosonic p-brane [8]. Let us recall that the world-history of an extended object is the mapping of a p-cell, i.e., a topological simplex of order "p", to a physical p-brane, and is usually encoded into an action of Dirac-Nambu-Goto type, or extensions of it [3]. From the action, in principle, one may deduce the Green function by evaluating the path integral over an appropriate set of p-brane histories [7]. In practice, the quantum propagator of a bosonic there and in the following, by convention, "p" stands for the order of the simplex, or the spatial dimension of its image, so that a p-brane is really a (p + l)-dimensional submanifold in spacetime.

3 p-brane cannot be computed exactly in closed form. Thus, in a recent paper [8] we borrowed the minisuperspace quantization scheme from Quantum Cosmology [9] and the quenching approximation from QCD [10] in order to derive a new form of the bosonic p-brane propagator. Combining those two techniques leads to an expression for the propagator that describes the boundary dynamics of the brane [7] rather than the spectrum of the bulk small oscillations. This new approximation includes both the center of mass quantum motion in target spacetime as well as the collective deformation of the brane as described by transitions between different (hyper-) volume quantum states. The latter quantum jumps account for the volume variations induced by local deformations of the brane shape, and generalize the "areal quantum dynamics", originally formulated by Eguchi in the case of strings [4], to the case of higher dimensional objects. The propagator calculated within the above approximation takes the following form in the Feynman parametrization scheme; while Kp (a (TQ ; s) stands for the "volume propagator" representing the transition amplitude between two brane states of different volume 2 : stands for the target spacetime image of the brane proper volume Vp and Mo is the product of the brane tension mp+i and the proper volume Vp. It may be helpful, before proceeding further with our discussion, to get a "feeling" of the quantum propagator by checking the consistency of our formulation against some well known cases of physical interest. The precise details of the derivation are in Ref. [8]. (4)

4 In the case of a closed string, p = 1 and the holographic coordinates (6) (7)

5 (1 1)

6 (12) (14) (13)

7 (15) (19) (20) (21)

8 (24) (25) (26) (27)

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