I would not want you to think that trying to get away from the Planck Scale is unique to my approach.
Here are other approaches.
Is there a way to combine “Quantum Minimum Length Structure (QMLS)”, M-Theory and LQG?
There is an approach “minisuperspace”.
http://arxiv.org/PS_cache/gr-qc/pdf/0703/0703057.pdf
Existence of generalized Kodama quantum states.
III. A new approach to finite, full quantum gravity.
Eyo Eyo Ita III
March 8, 2007
Also, check out Stefano Ansoldi to see more similarities when using “minisuperspace”.?
http://www-dft.ts.infn.it/~ansoldi/Research/PastResearchSummary.html
Past Research Activity of Stefano Ansoldi
In the first case, a great advantage of Eguchi's formulation of string dynamics is that it treats the extended object as a whole dynamical entity, without focusing on its constituents (points). The motion of the object can then be described in what is called loop space. In particular we worked on giving a path-integral formulation of string dynamics a la Eguchi [2], showing that the quantum dynamics can be described in terms of fractal properties of the world-sheet [6] and interpreting in terms of these aspects the small-scale space-time structure, whose microscopic constituents are branes, instead of points [7].
http://www-dft.ts.infn.it/~ansoldi/Research/LoopQuantumMechanics/HTML/index.html
Loop Quantum Mechanics and the Fractal Structure of Quantum Spacetime
5.1 Correspondence Principle, Uncertainty Principle and the Fractalization of Quantum Spacetime
If spacetime is a derived concept, then is seems natural to ask, ``what is the main property of the fuzzy stuff, let us call it quantum spacetime, that replaces the smoothness of the classical spacetime manifold, and what is the scale of distance at which the transition takes place?''. Remarkably, the celebrated Planck length represents a very near miss as far as the scale of distance is concerned. The new source of fuzziness comes from string theory, specifically from the introduction of the new fundamental constant which determines the tension of the string. Thus, at scales comparable to , spacetime becomes fuzzy, even in the absence of conventional quantum effects ( ). While the exact nature of this fuzziness is unclear, it manifests itself in a new form of Heisenberg's principle, which now depends on both and . Thus, in Witten's words, while ``a proper theoretical framework for the [new] uncertainty principle has not yet emerged, ...the natural framework of the [string] theory may eventually prove to be inherently quantum mechanical.''.
That new quantum mechanical framework may well constitute the core of the yet undiscovered -Theory, and the non perturbative functional quantum mechanics of string loops that we have developed in recent years may well represent a first step on the long road toward a matrix formulation of it. If this is the case, a challenging testing ground is provided by the central issue of the structure of quantum spacetime. This question was analyzed in Ref. [6] and we limit ourselves, in the remainder of this subsection, to a brief elaboration of the arguments presented there.
The main point to keep in mind, is the already mentioned analogy between ``loop quantum mechanics'' and the ordinary quantum mechanics of point particles. That analogy is especially evident in terms of the new areal variables, namely, the spacelike area enclosed by the string loop, given by Eq. (18), and the timelike, proper area of the string manifold, given by Eq. (4). With that choice of dynamical variables, the reparametrized formulation of the Schild action principle leads to the classical energy per unit length conservation . Then, the loop wave equation can be immediately written down by translating this conservation law in the quantum language through the Correspondence Principle
….we insist in maintaining the ``wholeness'' of the string and consider exact solutions in loop space, or adopt a minisuperspace approximation quantizing only one, or few oscillation modes, freezing all the other (infinite) ones. In the first case, it is possible to get exact ``free'' solutions, such as the plane wave.
The central result that follows from the above equations, is that the classical world-sheet of a string, a smooth manifold of topological dimension two, turns into a fractal object with Hausdorff dimension three as a consequence of the quantum areal fluctuations of the string loop [6].
Hence, quantum string dynamics can be described in terms of a fluctuating Riemannian -surface only when the observing apparatus is characterized by a low resolution power. As smaller and smaller areas are approached, the graininess of the world-sheet becomes manifest. Then a sort of de-compactification occurs, in the sense that the thickness of the string history comes into play, and the ``world-surface'' is literally fuzzy to the extent that its Hausdorff dimension can be anything between its topological value of two and its limiting fractal value of three.
5.2 Superconductivity and Quantum Spacetime
Quantum strings, or more generally branes of various kind, are currently viewed as the fundamental constituents of everything: not only every matter particle or gauge boson must be derived from the string vibration spectrum, but
spacetime itself is built out of them.
At the same time, the functional approach leads to a precise interpretation of the fuzziness of the underlying quantum spacetime in the following sense: when the resolution of the detecting apparatus is smaller than a particle DeBroglie wavelength, then the particle quantum trajectory behaves as a fractal curve of Hausdorff dimension . Similarly we have concluded on the basis of the ``shape uncertainty principle'' that the Hausdorff dimension of a quantum string world-sheet is , and that two distinct phases (smooth and fractal phase) exist above and below the loop DeBroglie area. Now, if particle world-lines and string world-sheets behave as fractal objects at small scales of distance, so does the world-history of a generic -brane including spacetime itself [19], and we are led to the general expectation that a new kind of fractal geometry may provide an effective dynamical arena for physical phenomena near the string or Planck scale in the same way that a smooth Riemannian geometry provides an effective dynamical arena or physical phenomena at large distance scales.
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I'll be trying to include these points in my blog.
jal
