3 Relative Dimensionalism and Quantum Resolution

With hindsight, we may now regard the expression (20) as an effective geometry in which the memory of quantum effects is encoded into a power expansion of the metric in terms of the Planck length. Perhaps we should emphasize that this interpretation is not the result of idle speculation, but rests squarely on the recognition that the line element (19), extracted from the form of the $p$-brane propagator in the minisuperspace-quenched approximation, is a special case of the Clifford line element (20). This new perspective opens the way to the possibility of trans-dimensional mutations, that is, quantum transitions between objects with different dimensions, thus providing, in principle, a unified quantum picture of classically distinguishable $p$-branes.
In order to elaborate on this last point, which is the central message that we wish to communicate, let us take a step further and consider the mapping of the whole simplicial complex of elementary $p$-cells. The result is a superposition of $p$-branes, a state that we shall call a Planckion, to which we now wish to give a quantum mechanical meaning. For this purpose, we replace the various terms $l_{Pl}^{2p}$ with threshold functions $F(\, E - E_{(p)} \,)$ corresponding to each level of the poly-dimensional line element (20). The threshold functions are chosen to be sharply peaked around the resonance energy $E_{(p)}$, and rapidly decreasing away from the resonance value. In such a way we obtain an effective, energy dependent metric

$$d \Sigma ^{2} ( E ) = d y ^{\mu}\, d y _{\mu} + E_{(1)}^2\, F(\, ... ...E_{(1)} \,)\, d {Y} ^{\mu _{1} \mu _{2}}\, d {Y} _{\mu _{1} \mu _{2}} + \dots +$$

$$\qquad + E_{(p)}^2\, F(\, E - E_{(p)} \,)\, d {Y} ^{\mu _{1} \dots \mu _{p+1}}\, d {Y} _{\mu _{1} \dots \mu _{p+1}} + \dots\$$ (36)

In this effective line element, $E$ represents the energy of the test body used to probe the short distance structure of spacetime, and the threshold functions may be looked upon either as generalized ``metric components'' in the Clifford line element, or, as generalized ``Fourier components'' in the energy spectrum of spacetime. In the latter interpretation, it seems natural to order the set of threshold functions according to the energy required to resolve a component $p$-brane. Thus, we assume that ${E} _{(1)} > {E} _{(2)}> \dots > {E} _{(p)} > \dots > \dots$. If $E<< E_{(1)}$, only points are discernible, all the higher dimensional modes are unresolvable, and the effective metric (36) reduces to the usual Minkowski metric. Much more structure emerges when the energy $E$ is close to a resonance value, say $E_{(n)}$. In that case, the test body sees an effective background geometry composed of $n$-branes, each forever fluctuating about the center of mass as it evolves in Minkowski space.
The new quantum fluctuations anticipated at the beginning of this letter appear when the probe is not exactly tuned to the resonance energy. In this case a new source of ``fuzziness'' sets in because one no longer sees a brane with a definite dimension: the Planckion is a fuzzy ball of nested $p$-branes, so that the very value of ``$p$'' is blurred by quantum superposition of brane states of various dimensionality.
Finally, in the Planckian regime, one reaches the absolute minimum in spacetime resolution as dictated by the new uncertainty principle. Once that energy scale is reached, all brane channels are open, so that the Planckion consists of a quantum mechanical superposition of all possible $p$-definite states, with $0\le p\le D-1$. In this extreme phase, the vacuum consists of pure ``Planckian noise'' forever fluctuating in the double sense that, as smaller and smaller distance scales are probed, not only higher brane vibrational modes are excited , but new dimensions unfold as well.
Perhaps, the distinctive, and inspiring feature of this whole new scenario is the notion of dimensional democracy, reminiscent of the old bootstrap idea and particle democracy advocated in S-matrix theory. In the geometric revival of that idea, $p$-branes with diverse dimensions co-exist as equally fundamental building blocks, or ``spacetime quanta'' that are interchangeable under a generalized Lorentz transformation [15].