From Osher Doctorow Ph.D.
Loop Quantum Gravity (LQG) keeps trying to make a comeback in arXiv,
this time in modified form via "Quantized four-dimensional gravity -
Chapters I and II (Solution to the kinematic constrains for the full
theory, and Conserved probability currents and the problem of time,"
Eyo Eyo Ita III, arXiv: 0706.2699 v1 [gr-qc] 19 Jun 2007.
The difficulty with Ita's paper, which is only part of Ita's intended
multiple-chapter work, is that so much of it is devoted to trying to
remedy defects of LQG, which almost requires "reinventing the wheel"
to proceed with.
No such difficulty occurs with the two main Probable Causation/
Influence (PI) methods of addition/subtraction and the Riccati
Differential Equation (the latter is quadratic and linear in the
"dependent" variable), and Ishwaree P. Neupane of U. Canterbury New
Zealand (who has 27 papers in arXiv) does basic quadratic analysis in
his "Natural quintessential inflation," arXiv: 0706.2654 v1 [hep-th]
19 Jun 2007, 4 pages (double-column). The inflaton potential is
quadratic in the Hubble parameter in this paper, and quadratics are
"all over the place" so to speak.
This reminds me that there are two ways to analyze Coulomb forces and
Newtonian gravitation in the 1/r^2 force dependence. One is to
convert this to subtraction via 1 - r^2 and then convert r^2 = r times
r to addition yielding r + r or 2r, so that 1/r^2 goes over to 1 -
2r. But the other is to retain the r^2 because it is quadratic as in
Riccati, but to convert 1/r^2 to p^2 where p is a proximity
("nearness") function rather than a distance ("farness") function. We
could define p as 1/r as p has just as much intuition as "nearness"
does relative to "farness" (r or d). As a matter of fact, Probable
Causation/Influence (PI) = 1 + y - x and its n-dimensional forms are
one-sided partial inverses of Euclidean-type distance-functions/
metrics as readers who've been following this thread undoubtedly know.
All this raises an arguably interesting question: why do so many
energy and force equations involve quadratic and/or linear functions,
and does this indicate that energy and force are both forms of
(Probable) Causation? It may be both simpler and more important to
answer this question than to try to rescue what remains of Loop
Quantum Gravity (LQG).
Osher Doctorow
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