THE HIDDEN MEANING OF PLANCK’S CONSTANT

A new cosmological paradigm based upon global discrete scale invariance proposes a radical revision in our understanding of atomic scale dynamics. The discrete self-similar scaling of this paradigm predicts a very large and discrete scale-dependence for gravitational coupling factors. This alternative gravitational scaling leads to revised values for the Planck mass, Planck length and Planck time, which appear to be more unified and physically comprehensible than is the case with the conventional Planck scale values. The fundamental interrelationship of the revised Planck mass, the corrected gravitational constant for atomic scale systems, the reduced Planck’s constant and the velocity of light suggests a hidden meaning for Planck’s constant. Within the context of the discrete fractal cosmological paradigm, Planck’s constant is revealed as the fundamental unit of gravitational action for atomic scale systems. Implications for atomic scale dynamics are briefly outlined.


The Hidden Meaning of Planck's Constant
"The next great awakening of the human intellect may well produce a method of understanding the qualitative content of the equations." Richard Feynman

THE ORIGIN OF PLANCK'S CONSTANT
Planck's constant entered physics in 1900 as a result of Max Planck's attempts to provide a theoretical explanation for the empirically discovered laws of blackbody radiation (Peacock, 2008). He found that Wien's heuristic approximation and existing observations could be reproduced if one adopted the concept that matter was a collection of discrete harmonic oscillators that obeyed an energy/frequency law of the form: for the emitted electromagnetic radiation. Since h has the dimensions of ML 2 /T which are the dimensions of action, i.e., energy multiplied by time, it was natural to think of h in terms of action principles. The implication of Planck's discovery of h was that the action of atoms is quantized and that h represents the fundamental unit of action for discrete atomic scale systems.
Planck's constant has become an integral component of modern atomic and subatomic physics, such that an understanding of the microcosm without h is virtually unthinkable. However, as pointed out by Peacock (2008), to this day physicists really have not had a convincing explanation for why action in the microcosm is quantized, nor why h has the specific quantitative value of 6.626 x 10 -27 erg sec. Here we will discuss the possibility that a discrete self-similar approach to modeling nature may offer a unique and deeper understanding of Planck's constant. 4

THE DISCRETE FRACTAL PARADIGM
The discrete fractal paradigm focuses on nature's fundamental organizational principles and symmetries, and is referred to as the Self-Similar Cosmological Paradigm (Oldershaw, 1989a,b). It emphasizes nature's hierarchical organization of systems from the smallest observable subatomic particles to the largest observable superclusters of galaxies. The new fractal paradigm also highlights the fact that nature's global hierarchy is highly stratified into discrete Scales, of which we can currently observe the Atomic, Stellar and Galactic Scales. A third important principle of the fractal paradigm is that the cosmological Scales are rigorously self-similar, such that for each class of objects or phenomena on a given Scale there is analogous class of objects or phenomenon every other cosmological Scale. The self-similar analogues from different Scales have rigorously analogous morphologies, kinematics and dynamics. When the general self-similarity among the discrete Scales is exact, the paradigm is referred to as Discrete Scale Relativity (Oldershaw, 2007) and nature's global spacetime geometry manifests a new universal dynamic symmetry principle: discrete scale invariance.
Based upon decades of studying the scaling relationships among analogue systems from the Atomic, Stellar and Galactic Scales (Oldershaw, 1989a,b), a close approximation to nature's actual Scale transformation equations for the length (L), time (T) and mass (M) parameters of analogue systems on neighboring cosmological Scales and -1 are as follows.

REVISED SCALING FOR GRAVITATION
Since the discrete self-similar scaling applies to all dimensional parameters, the Scale transformation equations also apply to dimensional "constants." Given the dimensionality of the gravitational constant, L 3 /MT 2 , the discrete fractal paradigm proposes that the gravitational coupling constants G scale as follows (Oldershaw, 2007).
where G 0 is the conventional Newtonian gravitational constant. Therefore the Atomic Scale The quantitative values for the conventional Planck scale parameters are listed in Table 1. When G -1 is substituted for G in Eqs. (6) -(8), as mandated by the discrete fractal paradigm, a radically different set of M, R and T values is generated. These revised Planck scale results are given in Table 2.

THE MEANING OF PLANCK'S CONSTANT
In trying to understand the meaning of h, we focus on Eq. (6) and make the assumption that M is not merely an approximate scale parameter, but rather that it is a fundamental constant of Atomic Scale dynamics. Given this assumption, M = (ħc/G -1 ) 1/2 is a much more rigorous interrelationship involving four of the fundamental Atomic Scale constants. We may rearrange Eq. (6) to give: Eq. (9) makes it explicit that h is primarily associated with Atomic Scale gravitational interactions. Within the context of the discrete self-similar paradigm, Planck's constant equals 2πG -1 M 2 /c and is the discrete unit of gravitational action for Atomic Scale systems. The concept that gravitational interactions dominate the dynamics within Atomic Scale systems is consistent with a recent potential advance in our understanding of the fine structure constant (Oldershaw, 2009). Within the context of the discrete fractal paradigm, the fine structure constant is identified as the ratio of the strengths of the unit electromagnetic and gravitational interactions within Atomic Scale systems. Therefore within Atomic Scale systems gravitational interactions generally are stronger than electromagnetic interactions by a factor of α -1 , or 137.036.
Since all cosmological Scales are rigorously self-similar to one another, there must be a separate set of M ,R and T values for each cosmological Scale, and their respective values are governed by the discrete Scale transformation equations (2) -(4), when measured relative to some fixed set of dimensional units (Oldershaw, 2007). These Planck scale sets define the "bottom", i.e., the most fundamental unit level, of the hadronic subhierarchy that characterizes each cosmological Scale. When we substitute ħ = G -1 M 2 /c into Eq. (7) we get: which is highly reminiscent of the standard Schwarzschild radius (R) equation for a non-rotating, uncharged black hole, and differs from R only by a factor of 2. This result is consistent with a recent finding that Atomic Scale hadrons, such as the proton and the alpha particle, can be modeled as Kerr-Newman or Schwarzschild black holes if G -1 is adopted as the appropriate gravitational coupling factor within hadrons (Oldershaw, 2010). One can also substitute ħ = G -1 M 2 /c into Eq. (8) and generate a new expression for T: It is somewhat ironic to think that for over 100 years the ubiquitous presence of h and ħ in the equations that govern atomic and subatomic physics has been thinly veiling the dominant influence of Atomic Scale gravitational interactions throughout the microcosm, while common knowledge proclaimed that gravitational interactions played only a trivial role in atomic physics.
In actuality, it appears that every time h or ħ is present in an Atomic Scale equation, we may 10 replace it with 2πG -1 M 2 /c or G -1 M 2 /c to reveal the true dominant influence of gravitation within the microcosm.

IMPLICATIONS FOR ATOMIC SCALE DYNAMICS
There are an enormous number of fundamental and secondary technical details regarding the physics and mathematics of the discrete self-similar paradigm that remain to be explored and resolved. Before our new understanding of Planck's constant can be fully implemented, a considerable amount of effort and insight must be applied to these technical issues. Here we must content ourselves with using the general principles of the discrete fractal paradigm and the results derived above to outline broadly the basic ways in which the new paradigm might alter our understanding of Atomic Scale dynamics. Below is a listing of the most important implications of defining h as the unit of gravitational action for Atomic Scale systems.
(a) Particles, Nucleons and Nuclei: If G -1 is the correct coupling factor for gravitational interactions within Atomic Scale systems, and h is the fundamental unit of gravitational action in the microcosm, then subatomic particles must be modeled as ultracompact gravitational objects. Currently the best available approximations for these particles are probably the Kerr-Newman, Schwarzschild and Reissner-Nordstrom black hole solutions of General Relativity. Unbound electrons might best be approximated as nearly structureless singularities, due to their substantial spin but relatively low mass, whereas hadrons would have event horizons and definite sizes on the order of their Schwarzschild radii. Presumably their radii would be more accurately determined via Kerr-Newman solutions which take charge, mass and rotational angular momentum into account.
Intriguing similarities between the physical characteristics of subatomic particles and black holes have been pointed out by several authors (Oldershaw, 2010). Subquantum Scale particles of relatively infinitesimal size, charge and mass (Oldershaw, 1989a,b). Schrodinger's "probability density", or 2 would have to be reinterpreted as the actual matter distribution (Barut, 1988) of the vast numbers of = -2 subquantum plasma particles. An atom in a very high Rydberg state would have a semiclassical electronic structure approximated by orbiting "particle-like" solutions (Kalinsky, Eberly, West and Stroud, 2003) of the Schrodinger equation. Atoms in the ground state and low energy states would have more wave-like electronic structures with subquantum plasma distributions characterized by the more familiar wavefunction shapes: spheroidal, toroidal, bipolar, etc. A recent paper (Oldershaw, 2008) demonstrating a high degree of self-similarity between the masses, sizes, shapes and frequency spectra of RR Lyrae variable stars and the masses, sizes, shapes and frequency spectra of excited helium 12 atoms undergoing single-level transitions between states with principal quantum numbers of 7 -10 lends credence to the idea that the physics of Atomic Scale systems and their Stellar Scale analogues might be rigorously self-similar. If this is the case, then being able to study the physics of analogues on radically different spatial and temporal scales should be of great benefit in developing unified models for stellar and atomic systems. distinction between hadrons and leptons appears to be whether they satisfy the constraint m 2 a 2 + q 2 or the constraint a 2 + q 2 > m 2 , respectively. Secondly, it appears that the Planck mass is very close to the unique mass for which m 2 = a 2 + q 2 for Atomic Scale systems. Therefore M probably does not represent the mass of an actual particle in nature, but rather it probably represents the "tipping point" mass that defines the lepton/hadron, or horizon-free/horizon-possessing, boundary for ultracompacts on a given cosmological Scale.
(b) Can Schrodinger's 2 be successfully reinterpreted as the density of Subquantum Scale plasma particles? Work along these lines was attempted by A. O. Barut (1988). Perhaps the new ideas introduced by the discrete fractal paradigm will contribute to the conceptual and analytical development of this research effort.
(c) By what mechanism does an ultracompact object such as an unbound electron, which is virtually a naked singularity, decompose into wavefunction-like plasma shell comprised of myriad Subquantum Scale particles when the electron becomes bound to a nucleus?
(d) If the discrete fractal paradigm heralds a new unified physics for all cosmological Scales, what is the best analytical framework for this unification? Would a simple combination of General Relativity, Electromagnetism, Wave Mechanics and Discrete Scale Relativity be sufficient, or is some alternative framework required, such as a discrete 5-dimensional Kaluza-Klein approach with the 5 th dimension related to discrete scale? Another possible framework might be a 4-dimensional spacetime whose fundamental global geometry has discrete conformal symmetry.
Clearly much work remains to be done before the discrete fractal paradigm evolves from the conceptual, empirical and scaling foundations of natural philosophy to mature mathematical physics. The general paradigm itself is singularly testable. The definitive predictions by which the discrete self-similar paradigm can be unambiguously tested concern the exact nature of the galactic dark matter (Oldershaw, 1987). Preliminary empirical results from persistently negative "WIMP" dark matter searches, and repeatedly positive stellar-mass microlensing observations, appear to be quite encouraging (Oldershaw, 2002).