Vacuum permeability

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This article is about the magnetic constant. For the analogous electric constant, see vacuum permittivity.

The physical constant μ₀, (pronounced "mu nought" or "mu zero"), commonly called the vacuum permeability, permeability of free space, permeability of vacuum, or magnetic constant, is the magnetic permeability in a classical vacuum. Vacuum permeability is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field production in a vacuum.

As of May 20, 2019, the vacuum permeability μ₀ will no longer be a defined constant (per the former definition of the SI ampere), but rather will need to be determined experimentally; The 2018 CODATA value is given below. It is proportional to the dimensionless fine-structure constant with no other dependencies.^[1][2][3]

μ₀ = 1.25663706212(19)×10⁻⁶ H/m

Before this, in the reference medium of classical vacuum, μ₀ had an exact defined value:^[4][5]

μ₀ = 4π×10⁻⁷ H/m = 1.2566370614…×10⁻⁶ N/A² (1 henry per metre ≡ newton per square ampere)

Permittivity

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A dielectric medium showing orientation of charged particles creating polarization effects. Such a medium can have a lower ratio of electric flux to charge (more permittivity) than empty space

In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a given medium. A charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

The SI unit for permittivity is farad per meter (F/m or F·m⁻¹).

The lowest possible permittivity is that of a vacuum.^{[citation needed]} Vacuum permittivity, sometimes called the electric constant, is represented by ε₀ and has a value of approximately 8.85×10⁻¹² F⋅m⁻¹.

The permittivity of a dielectric medium is often represented by the ratio of its absolute permittivity to the electric constant. This dimensionless quantity is called the medium's relative permittivity, sometimes also called "permittivity". Relative permittivity is also commonly referred to as the dielectric constant, a term which has been deprecated in physics and engineering^[1] as well as in chemistry.^[2]

${\displaystyle \kappa =\varepsilon _{r}={\frac {\varepsilon }{\varepsilon _{0}}}}$

By definition, a perfect vacuum has a relative permittivity of exactly 1. The difference in permittivity between a vacuum and air can often be considered negligible, as κ_air = 1.0006.

Relative permittivity is directly related to electric susceptibility (χ), which is a measure of how easily a dielectric polarizes in response to an electric field, given by

${\displaystyle \chi =\kappa -1}$

otherwise written as

${\displaystyle \varepsilon =\varepsilon _{\mathrm {r} }\varepsilon _{0}=(1+\chi )\varepsilon _{0}}$

Vacuum permittivity

The physical constant ε₀ (pronounced as "epsilon nought" or "epsilon zero"), commonly called the vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum, is an ideal, (baseline) physical constant, which is the value of the absolute dielectric permittivity of classical vacuum. It has the CODATA value (as of 2018)

ε₀ = 8.8541878128(13)×10⁻¹² F⋅m⁻¹ (farads per metre), with a relative uncertainty of 1.5×10⁻¹⁰.^[1]

It is the capability of the vacuum to permit electric field lines. This constant relates the units for electric charge to mechanical quantities such as length and force.^[2] For example, the force between two separated electric charges (in the vacuum of classical electromagnetism) is given by Coulomb's law:

${\displaystyle \ F_{\text{C}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}$

The value of the constant fraction, ${\displaystyle 1/4\pi \varepsilon _{0}}$, is approximately 9 × 10⁹ N⋅m²⋅C⁻², q₁ and q₂ are the charges, and r is the distance between them. Likewise, ε₀ appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation, and relate them to their sources.

Contents

• 1Value
  • 1.1Redefinition of the SI units
• 2Terminology
• 3Historical origin of the parameter ε₀
  • 3.1Rationalization of units
  • 3.2Determination of a value for ε₀
• 4Permittivity of real media
• 5See also
• 6Notes

Value[edit]

The value of ε₀ is defined by the formula^[3]

${\displaystyle \varepsilon _{0}={\frac {1}{\mu _{0}c^{2}}}}$

where c is the defined value for the speed of light in classical vacuum in SI units,^[4] and μ₀ is the parameter that international Standards Organizations call the "magnetic constant" (commonly called vacuum permeability). Since μ₀ has an approximate value 4π × 10⁻⁷ H/m,^[5] and c has the defined value 299792458 m⋅s⁻¹,^[6] it follows that ε₀ can be expressed numerically as

${\displaystyle \varepsilon _{0}={\frac {1}{(4\pi \times 10^{-7}\,{\textrm {N/A}}^{2})(299792458\,{\textrm {m/s}})^{2}}}={\frac {625000}{22468879468420441\pi }}\,{\textrm {F/m}}\approx 8.85418781762039\times 10^{-12}\,{\textrm {F}}{\cdot }{\textrm {m}}^{-1}}$ (or A²⋅s⁴⋅kg⁻¹⋅m⁻³ in SI base units, or C²⋅N⁻¹⋅m⁻² or C⋅V⁻¹⋅m⁻¹ using other SI coherent units).^[7][8]

The historical origins of the electric constant ε₀, and its value, are explained in more detail below.

Redefinition of the SI units[edit]

Main article: 2019 redefinition of the SI base units

The ampere was redefined by defining the elementary charge as an exact number of coulombs as from 20 May 2019,^[9] with the effect that the vacuum electric permeability no longer has an exactly determined value in SI units. The value of the electron charge became a numerically defined quantity, not measured, making μ₀ a measured quantity. Consequently, ε₀ is not exact. As before, it is defined by the equation ε₀ = 1/(μ₀c²), and is thus determined by the value of μ₀, the magnetic vacuum permeability which in turn is determined by the experimentally determined dimensionless fine-structure constant α:

${\displaystyle \varepsilon _{0}={\frac {1}{\mu _{0}c^{2}}}={\frac {e^{2}}{2\alpha hc}}\ ,}$

with e being the elementary charge, h being the Planck constant, and c being the speed of light in vacuum, each with exactly defined values. The relative uncertainty in the value of ε₀ is therefore the same as that for the dimensionless fine-structure constant, namely 1.5×10⁻¹⁰.^[10]

Terminology[edit]

Historically, the parameter ε₀ has been known by many different names. The terms "vacuum permittivity" or its variants, such as "permittivity in/of vacuum",^[11][12] "permittivity of empty space",^[13] or "permittivity of free space"^[14] are widespread. Standards Organizations worldwide now use "electric constant" as a uniform term for this quantity,^[7] and official standards documents have adopted the term (although they continue to list the older terms as synonyms).^[15][16] In the new SI system, the permittivity of vacuum will not be a constant anymore, but a measured quantity, related to the (measured) dimensionless fine structure constant.

Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity.^[17][18] However, in modern usage "dielectric constant" typically refers exclusively to a relative permittivity ε/ε₀ and even this usage is considered "obsolete" by some standards bodies in favor of relative static permittivity.^[16][19] Hence, the term "dielectric constant of vacuum" for the electric constant ε₀ is considered obsolete by most modern authors, although occasional examples of continuing usage can be found.

As for notation, the constant can be denoted by either ${\displaystyle \varepsilon _{0}\,}$ or ${\displaystyle \epsilon _{0}\,}$, using either of the common glyphs for the letter epsilon.

Historical origin of the parameter ε₀[edit]

As indicated above, the parameter ε₀ is a measurement-system constant. Its presence in the equations now used to define electromagnetic quantities is the result of the so-called "rationalization" process described below. But the method of allocating a value to it is a consequence of the result that Maxwell's equations predict that, in free space, electromagnetic waves move with the speed of light. Understanding why ε₀ has the value it does requires a brief understanding of the history.

Rationalization of units[edit]

The experiments of Coulomb and others showed that the force F between two equal point-like "amounts" of electricity, situated a distance r apart in free space, should be given by a formula that has the form

${\displaystyle F=k_{\text{e}}{\frac {Q^{2}}{r^{2}}},}$

where Q is a quantity that represents the amount of electricity present at each of the two points, and k_e is the Coulomb constant. If one is starting with no constraints, then the value of k_e may be chosen arbitrarily.^[20] For each different choice of k_e there is a different "interpretation" of Q: to avoid confusion, each different "interpretation" has to be allocated a distinctive name and symbol.

In one of the systems of equations and units agreed in the late 19th century, called the "centimetre–gram–second electrostatic system of units" (the cgs esu system), the constant k_e was taken equal to 1, and a quantity now called "gaussian electric charge" q_s was defined by the resulting equation

${\displaystyle F={\frac {{q_{\text{s}}}^{2}}{r^{2}}}.}$

The unit of gaussian charge, the statcoulomb, is such that two units, a distance of 1 centimetre apart, repel each other with a force equal to the cgs unit of force, the dyne. Thus the unit of gaussian charge can also be written 1 dyne^1/2 cm. "Gaussian electric charge" is not the same mathematical quantity as modern (MKS and subsequently the SI) electric charge and is not measured in coulombs.

The idea subsequently developed that it would be better, in situations of spherical geometry, to include a factor 4π in equations like Coulomb's law, and write it in the form:

${\displaystyle F=k'_{\text{e}}{\frac {{q'_{\text{s}}}^{2}}{4\pi r^{2}}}.}$

This idea is called "rationalization". The quantities q_s′ and k_e′ are not the same as those in the older convention. Putting k_e′ = 1 generates a unit of electricity of different size, but it still has the same dimensions as the cgs esu system.

The next step was to treat the quantity representing "amount of electricity" as a fundamental quantity in its own right, denoted by the symbol q, and to write Coulomb's Law in its modern form:

${\displaystyle \ F={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q^{2}}{r^{2}}}.}$

The system of equations thus generated is known as the rationalized metre–kilogram–second (rmks) equation system, or "metre–kilogram–second–ampere (mksa)" equation system. This is the system used to define the SI units.^[21] The new quantity q is given the name "rmks electric charge", or (nowadays) just "electric charge". Clearly, the quantity q_s used in the old cgs esu system is related to the new quantity q by

${\displaystyle \ q_{\text{s}}={\frac {q}{\sqrt {4\pi \varepsilon _{0}}}}.}$

Determination of a value for ε₀[edit]

One now adds the requirement that one wants force to be measured in newtons, distance in metres, and charge to be measured in the engineers' practical unit, the coulomb, which is defined as the charge accumulated when a current of 1 ampere flows for one second. This shows that the parameter ε₀ should be allocated the unit C²⋅N⁻¹⋅m⁻² (or equivalent units – in practice "farads per metre").

In order to establish the numerical value of ε₀, one makes use of the fact that if one uses the rationalized forms of Coulomb's law and Ampère's force law (and other ideas) to develop Maxwell's equations, then the relationship stated above is found to exist between ε₀, μ₀ and c₀. In principle, one has a choice of deciding whether to make the coulomb or the ampere the fundamental unit of electricity and magnetism. The decision was taken internationally to use the ampere. This means that the value of ε₀ is determined by the values of c₀ and μ₀, as stated above. For a brief explanation of how the value of μ₀ is decided, see the article about μ₀.

Permittivity of real media[edit]

By convention, the electric constant ε₀ appears in the relationship that defines the electric displacement field D in terms of the electric field E and classical electrical polarization density P of the medium. In general, this relationship has the form:

${\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} .}$

For a linear dielectric, P is assumed to be proportional to E, but a delayed response is permitted and a spatially non-local response, so one has:^[22]

${\displaystyle \mathbf {D} (\mathbf {r} ,\ t)=\int _{-\infty }^{t}dt'\int d^{3}\mathbf {r} '\ \varepsilon (\mathbf {r} ,\ t;\mathbf {r} ',\ t')\mathbf {E} (\mathbf {r} ',\ t').}$

In the event that nonlocality and delay of response are not important, the result is:

${\displaystyle \mathbf {D} =\varepsilon \mathbf {E} =\varepsilon _{\text{r}}\varepsilon _{0}\mathbf {E} }$

where ε is the permittivity and ε_r the relative static permittivity. In the vacuum of classical electromagnetism, the polarization P = 0, so ε_r = 1 and ε = ε₀.

See also

New Developments 2008

The Hidden Meaning of Planck's Constant [ March 2008]

 “The next great awakening of the human intellect may well produce a method of understanding the qualitative content of the equations.”

Richard Feynman  

1. 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.¹  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:

E = hν                                             (1)

for the emitted electromagnetic radiation.  Since h has the dimensions of ML²/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,¹ 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 the Self-Similar Cosmological Paradigm may offer a unique and deeper understanding of Planck’s constant. 

2. THE DISCRETE FRACTAL PARADIGM

The Self-Similar Cosmological Paradigm focuses on nature’s fundamental organizational principles and symmetries.  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 (# 12 of Selected Papers) and nature’s global spacetime geometry manifests a new universal symmetry principle: discrete scale invariance. 

3. REVISED SCALING FOR GRAVITATION

Since the discrete fractal scaling applies to all dimensional parameters, the Scale transformation equations also apply to dimensional “constants.”  Given the dimensionality of the gravitational constant, L³/MT², the discrete fractal paradigm proposes that the gravitational coupling constants G_Ψ scale as follows (# 12 of Selected Papers).

      G_Ψ = [Λ^1-D]^Ψ G₀ ,                                             (2)

where G₀ is the conventional Newtonian gravitational constant.  Therefore the Atomic Scale value G_-1 is Λ^2.174 times G₀ and equals ≅ 2.18 x 10³¹cm³/g sec². Now that we have the discrete fractal paradigm’s prediction for the appropriate G_Ψ value that applies in the case of Atomic Scale systems, we can derive a revised Planck mass, length and time, and compare the revised Planck scale with the conventional Planck scale. 

4. A REVISED PLANCK SCALE

In the early 1900s Max Planck realized that his newly discovered fundamental constant of the microcosm could be combined with the other known and apparently universal constants G and c to form a unique set of mass, length and time parameters that defined what has come to be known as the Planck scale.  The Planck mass (M), the Planck length (R) and the Planck time (T) are derived from the following relations.

M = (ħc/G)^1/2                                             (3)

R = (ħG/c³)^1/2                                             (4)

T = (ħG/c⁵)^1/2                                             (5)

The quantitative values for the conventional Planck scale parameters are listed in Table 1.

TABLE 1. CONVENTIONAL PLANCK SCALE

Parameter	Value	Counterpart in Nature
M	2.17 x 10-5 g	None observed
R	1.62 x 10-33 cm	?
T	5.43 x 10-44 sec	?

 

In the early 1900s it was not entirely clear what the Planck scale parameters corresponded to in nature since there was nothing observed at these values of M, R and T, and at that point no unambiguous theoretical interpretation was available.  Since that time, a better theoretical understanding of the Planck scale has emerged: the Planck scale parameters define the scale at which gravitation must be included in the dynamic modeling of the microcosm, i.e., the scale at which General Relativity and Quantum Electrodynamics both play major roles in the dynamics of the microcosm.

When G_-1 is substituted for G in Eqs. (3) - (5), 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.

TABLE 2. REVISED PLANCK SCALE

Parameter	Value	Counterpart in Nature
M	1.20 x 10-24 g	~ proton mass
R	2.93 x 10-14 cm	~ proton radius
T	9.81 x 10-25 sec	~ proton radius/c

 

Whereas the conventional set of Planck scale values constitutes a seemingly random collection of numbers that do not appear to correspond to anything observed in nature, the revised set of Planck scale values derived from the discrete fractal paradigm are self-consistent and are firmly linked to the scale of nature’s most fundamental baryon: the proton.  The value of the revised Planck mass is ≈ 0.72 times the mass of the proton, the revised Planck length is ≈ 0.4 times radius of the proton, and the revised Planck time is ≈ 0.4 times the proton radius divided by the velocity of light. 

5. THE MEANING OF PLANCK’S CONSTANT

In trying to understand the meaning of h, we focus on Eq. (3) 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. (3) to give:

ħ = G_-1M²/c  .                                             (6)

Eq. (6) makes it explicit that h is primarily associated with Atomic Scale gravitational interactions.  Within the context of the discrete fractal paradigm, Planck’s constant equals 2πG_-1M²/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 (July 2007 New Development: “Fine Structure Constant”).  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_Ψ, R_Ψ and T_Ψ values for each cosmological Scale, and their respective values are governed by the discrete Scale transformation equations of the SSCP, when measured relative to some fixed set of dimensional units (# 12 of Selected Papers).  These Planck scale sets define the “bottom”, or the most fundamental unit level, of the hadronic subhierarchy that characterizes each cosmological Scale.  When we substitute ħ = G_-1M²/c into Eq. (4) we get:

R = G_-1M/c² ,                                             (7)

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 (# 11 of Selected Papers).  One can also substitute ħ = G_-1M²/c into Eq. (5) and generate a new expression for T:

T = G_-1M/c³  .                                             (8)

We may also define a Planck velocity: R/T ≡ V = c. 

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 replace it with 2πG_-1M²/c or G_-1M²/c to reveal the true dominant influence of gravitation within the microcosm. 

6. IMPLICATIONS FOR ATOMIC SCALE DYNAMICS

There are an enormous number of fundamental and secondary technical details regarding the physics and mathematics of the discrete fractal 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 discrete fractal 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.

1. 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 definite sizes on the order of their Schwarzschild radii.  Presumably their radii would be more accurately determined via Kerr-Newman solutions which take charge and spin into account.  Intriguing similarities between the physical characteristics of subatomic particles and black holes have been pointed out by several authors, as noted in Paper # 11 of Selected Papers.
    
2. Neutral and Partially Ionized Atoms: Inside atoms the gravitational interaction is about 137.036 times stronger than the electromagnetic interaction and therefore the dynamics within atoms is dominated by gravitation.  Since the gravitational interactions among unbound particles, atoms and ions are 38 orders of magnitude weaker than their internal gravitational interactions, the overwhelmingly dominant interactions between unbound Atomic Scale systems are electromagnetic interactions.
    
3. Atomic and Stellar Wavefunctions: Assuming the discrete fractal paradigm is basically correct, when a proton and an electron make the transition from separate unbound particles to a single bound hydrogen atom, the virtually singular electron must decompose into a fluid-like plasma composed of enormous numbers of Ψ = -2 Subquantum Scale particles of relatively infinitesimal size, charge and mass.  Schrodinger’s “probability density”, or ψ² would have to be reinterpreted as the actual matter distribution² of the vast numbers of Ψ = -2 subquantum plasma particles.  An atom in a very high Rydberg state would have a semiclassical electronic structure approximated by orbiting particle-like solutions³ 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 (Sept 2005 New Development: “RR Lyrae Stars”) 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 atoms undergoing single-level transitions between states with principal quantum 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.
    
4. Quantum Mechanics for Atomic and Stellar Systems: A reinterpreted quantum mechanics wherein gravitation plays the dominant role for internal interactions, while electromagnetism plays the dominant role for external interactions among unbound systems, is conceivable.  This unified reinterpretation of quantum mechanics would be equally applicable to Atomic Scale systems, Stellar Scale systems, Galactic Scale systems, or systems on any other cosmological Scale.

The comments in this section provide only a speculative sketch of the basic implications for the revised Atomic Scale dynamics suggested by our new understanding of Planck’s constant.  No doubt many years of effort by the physics community will be required in order to develop the general principles of the discrete fractal paradigm into a rigorous and unified analytical theory, combining General Relativity and Schrodinger’s Wave Mechanics in a way that is consistent with Discrete Scale Relativity.  

7. SOME OPEN QUESTIONS

Finally, we will close the present discussion of Planck’s constant and the new Atomic Scale dynamics proposed by the discrete fractal paradigm with several questions for future study.

1. It is curious that M is close to the mass of the proton, but less by a factor of about (1/2π)^1/2.  Possibly this small disparity is due to the fact that two crucial energy sources are neglected: internal spin and electromagnetic interactions.  Discrete gravitational action may be the dominant action within Atomic Scale systems, but perhaps it is not the only form of action that applies.  Is it possible that when all forms of internal action are summed, then the combined action is quantitatively linked more exactly to the mass of the proton via an expanded version of Eq. (3)?
    
2. Can Schrodinger’s ψ² be successfully reinterpreted as the density of Subquantum Scale plasma particles?  Work along these lines was attempted by A. O. Barut.² Perhaps the new ideas introduced by the discrete fractal paradigm will contribute to the conceptual and analytical development of this research effort.
    
3. 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?
    
4. 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 eminently testable.  The definitive predictions by which the discrete fractal paradigm can be unambiguously tested concern the exact nature of the galactic dark matter.  Preliminary empirical results from relentlessly negative exotic particle searches and positive microlensing observations appear to be quite encouraging (# 5 of Selected Papers). 

REFERENCES

Peacock, K.A., The Quantum Revolution – A Historical Perspective, Greenwood Press, Westport, CT and London, 2008.
Barut, A.O., Foundations of Physics, 18(1), 95-105, 1988.
Kalinski, M., et al, Physical Review A, 67, 032503, 2003.

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