The answer to the first question is Yes: The d'Alembertian operator is invariant under global Lorentz transformations, that leave the phase velocity fixed, whatever the value of the phase velocity. The speed of light isn't special in that regard, nor are electromagnetic waves special, with respect to any other example of waves-at this level, at least.

The answer to the second question is No: Quantum fluctuations have Planck's constant as scale and don't make sense otherwise.

In 1905, Einstein presented his theory of special relativity, based on the constancy of the speed of light, C, simply to comply with the Michelson-Morley experiment for measuring the speed of light by interference.

In 2022 the author presented a concrete reformulation of Einstein's derivation of the theory of special relativity [1].

The author replaced Einstein's assumption of the constancy of the speed of light, which had no physical meaning, with his own hypothesis that C^2 is a universal constant, the conversion constant for mass and energy equivalence E=mC^2.

Furthermore, the author provided conclusive proof of Einstein's theory of special relativity without the need for the questionable light sphere.

A simple validation of the above study is that the parameter C^2 (C squared) is found in all relativistic formulas, but never in C alone.

The Lorentz transformation is a particularly notable example:

x′ = x/SQRT[1-v^2/C^2]

t′ = t.. SQRT[1-v^2/C^2]

and the mass-energy transformation is:

E = m C^2

*If we return to the Schrödinger PDE,

i h dΨ/dt)partial=-h²/2m . Nabla^2 Ψ+ V Ψ . . . . (1)

OR,

H^Ψ = EΨ. . . . . reduced form

The wave function Ψ itself is neither a scalar, nor a vector, nor a tensor. It has never been properly defined.

Furthermore, the Schrödinger PDE is not Lorentz invariant and is not compatible with the special theory of relativity, and is obviously less so with the general theory of relativity.

In 2022, 2023 [2,3], the author claimed to replace the classical Schrödinger equation 1 with its square Ψ^2, which is logical and physical.

Since Ψ^2 is needed to express the quantum energy density, the modern Schrödinger equation 2 must have the form of the famous energy density diffusion PDE:

dU/dt )partial = D. Nabla2U+S(U) . . . . (2)

Equation 2 can be easily solved numerically using the advanced artificial intelligence proposed by the author.

Let us now return to the Schrödinger PDE, reduced form H^Ψ = EΨ.

We find its solution energy eigenvalue equation with a total energy E=n h f, where n=1,2,3, . . .infinity.

Only h appears as an influencing parameter for the first levels, while for higher energies and n values, there is no notable difference between n and n+1.

In short, we are talking about or are confronted with relativity without relativity, and quantum mechanics without quantum mechanics, which is lacking nowadays especially in the West.

The West is losing the battle of relativity and quantum mechanics.

1-A rigorous reformulation of Einstein derivation of the special theory of relativity, February 2022

2-A new presentation of the Schrödinger partial differential equation , June 2024

3-A statistical numerical solution for the time- independent Schrödinger equation, November 2023

For me, no to both.

I think now adays you could dispute with GR, but not with SR.

You may compare newtonian practicity with that of GR for short distances, Newton wins.

However SR not good for cosmic distances. Clumsy when you get to force. Fine if no forces.

" Any theory has its own limitations"

It is unfortunate that the theories of special (SR) and general (GR), as well as the theory of quantum mechanics, are not yet rigorously defined.

The theories of special (SR) and general (GR) relativity are defined by their solution.

GR is defined by the Lorentz transformation:

x′ = x/SQRT[1-v^2/C^2]

t′ = t . SQRT[1-v^2/C^2] . . . . (1)

And GR is defined by its tensor solution:

Rμν - 1/2 g μν R = 8 Pi G T μν . . . . (2)

In equations 1 and 2, the underlying partial differential equation (PDE) is missing.

The Theory of Quantum Mechanics, Yesterday and Today:

Quantum mechanics was initially established and interpreted a century ago based on Schrödinger's partial differential equation (PDE),

i h dΨ/dt)partial=-h2/2m . Nabla^2 Ψ+ V Ψ . . . . (3)

OR,

H^Ψ = EΨ. . . . . (4)

reduced form

to understand the spectrum of the hydrogen atom.

Equations 3 and 4 are incomplete and, in many cases, yield an erroneous solution.

The only certainty is that equations 1, 2, 3, and 4 are valid for infinite free space.

You are partially exposing your own misconceptions.

Ie " gr defined through lorenz transformations"

TYPING ERROR

=============

GR [TO BE REPLACED BY SR] is defined by the Lorentz transformation:

SR:

x′ = x/SQRT[1-v^2/C^2]

t′ = t . SQRT[1-v^2/C^2] . . . . (1)

No

Ismail Abbas M. Abbas

I agree with you that c^2 is an electromagnetic constant for the simple reason that the energy making up the electron restmass oscillates in two perpendicular directions in each of which it reaches c as a maximum energy velocity. It is not conceivable that its rest mass energy, oscillating in stable standing mode, would move in one transverse direction without cyclically oscillate back in the other direction.

This is put in perspective in this 2024 paper, final instalment of the electromagnetic mechanics project:

Article From E=mc2 in normal space to E=m0cIcK in the complex config...

The same reasoning for Ψ^2

It is not conceivable that the phase wave energy would oscillate only on one side of its oscillating range. Put in perspective in this other 2024 paper that explains why the range of the real volume defined by Ψ^2 cannot be infinite as in standard QM:

Article Evolution From the Complex Plane to the Quaternion Coordinat...

Moreover, when Schrödinger established his wave equation from his interest for the phase wave concept that de Broglie proposed in his 1924 thesis, he was not aware that de Broglie had calculated an erroneous velocity for the phase wave, half the velocity of the electron that it was supposed to control in the Bohr atom idealized hydrogen atom.

Numerically established and easily confirmable with a simple pocket scientific calculator as critically analyzed in this third paper also published in 2024:

Article Critical Analysis of the Origins of Heisenberg's Uncertainty Principle

Adding and seconding to what André Michaud wrote, it is fully obvious from iSpace theory on exact value and integer geometry of constants of nature that c, e, h, … G, … are all fully interdependent - and not as thought by mainstream - independent constants.

These are in fact (basically) built up from 4 simple integer geometric base constants in the 4 physically ever required base units Volt, Ampere, Second and Meter or their iSpace-IQ quantum unit system based pedants iSpaceVolt, iSpaceAmpere, iSpaceSecond and iSpaceMeter, respective.

1) GoldenRatio iSpaceAmpere - the quantum of iSpaceAmpere (and Ampere!)

2) 1/6961 iSpaceSecond - nothing less than the iSpace predicted quantum of time (!)

3) 6 iSpaceVolt - the quantum of iSpaceVolt

4) GoldenRatio iSpaceMeter - the quantum of distance (!) - and not at all the same as some assumed only physical length entity (like Planck-length)

All combine simply first order multiplicative (!) to form (most but) all other constants like for example:

iQc = (GoldenRatio iSpaceMeter) / (6961 iSpaceSecond)

iQe = (GoldenRatio iSpaceAmpere) * (1/6961 iSpaceSecond)

and iQh, being numerically just 6 times iQe (of course one has to track the units properly with these quantum base values, like iSpace is able to do being a geometric exact value theory, giving much desired zero-unit-pressure (!) for the first time among all constants due to its model intrinsic zero-error-bar design.

So in short - NO - there is no physics at all without base of (exact, quantum iSpace calculable) constants of nature. See my RG home published papers on iSpace for details.

To Christian G. Wolf and Ismail Abbas M. Abbas

Hi Christian,

Adding to and seconding what you wrote about the interdependence of a set of fundamental constants, it turns out that the following set of fundamental constants inextricably define each other, just like in the case of Euler's identity (eiπ+1=0):

h=e2/2εoαc=e2μoc/2α

As mentioned in the paragraph following Equation (12) in this paper first published in 2017 in an astrophysics journal, and republished as a book chapter upon invitation by the editors in 2020:

Article Gravitation, Quantum Mechanics and the Least Action Electrom...

Whose first equality is mentioned as Equation (4) in this other paper as defining h, also first published in 2017 before being republished as a book chapter upon invitation in 2021:

Article The Last Challenge of Modern Physics: Perspective to concept...

In which Equations (2) and (3) explain how h emerges out of these electromagnetic equations, proof that h is not only a measured constant, but that it also naturally emerges from electromagnetic theory, unbeknownst apparently from the NIST people, since this definition of h could be found nowhere in the formal record, as mentioned in this paper.

Very interesting correlations between electromagnetic theory and Buckminster Fuller's Synergetics about the fine structure constant, the oscillation frequency of the electron rest mass energy in the hydrogen atom ground state orbital and Planck's constant, among other correlations, were discovered by Noel Coughlin, as he explains himself in 2 recent videos, as put in perspective in this preprint:

Preprint Correlation of the electromagnetic mechanics of elementary p...

Which is a definite sign that more research remain to be made to completely clarify what is happening at the fundamental magnitude level of physical reality.

Dear Ismail, please consider that the Michelson-Morley experiment is inconclusive, but luckily gave Einstein the correct hint. The experiment assumes that the Aether is a continuous and uniform flow. What if the Aether flows into the energetic bodies? Energy absorbing the Aether... By this, the Aether will be flowing into your Earth independently of its position in the solar system. Idem with a huge flow of Aether produced by the Sun´s mass. This idea will be in agreement with Schwarzschild's kinetic energy, which corresponds to time dilation and space contraction produced by gravity. Similar to Einstein's approach of spatial acceleration in his General Relativity. Hope this will inspire your research. Regards

I note that there is a serious objection to Einstein's assumed length contraction of ponderable bodies theoretically moving at relativistic velocities.

It is the fact that all ponderable bodies are made of almost empty atoms, that are themselves "systems of charged and massive elementary particles -- the only carriers of mass in the universe -- that are many orders of magnitude smaller than the atoms of which they constitute the inner structure stabilized in standing electromagnetic equilibrium, and that consequently, if any pondarable body was to contract lengthwise, it is physically impossible that it would not also contract in the other 2 dimensions, given that each atom not being solid, its internal charged structure cannot contract towards its center of presence in only one direction.

Put in perspective in the following paper:

Article Gravitation, Quantum Mechanics and the Least Action Electrom...

André Michaud While I agree to your arguments and overall direction, an equation given by you above is seen by most (and I tend to agree here) "just algebra" and hence not core physics (its self referencing).

iSpace G has show to be this relation (in its initial derivation 2015) to be based on a truly independent (hence "not-just-algebraic") identity:

G = (alpha)^10*(1/(e*R))^4*(1/(4*Pi*mu0)^2)*Fg

with Fg = iFg = quantum of gravitational force as defined by:

(Fg) / (FE) = (FE) / (FPlanck), so (iFg) = (iFE)^2 / (iFPlanck)

a simple square relation at first sight. However the result of iSpace iFg, while numerically extremely weak looking small value, can be and is exactly defined in iSpace theory, surprisingly (by the following simple relation):

iFg = 3 / (299792458*10^3)^4 [Volt*Ampere*Second*1/Meter] or [Newton]

EXACTLY.

To gain the required (addition geometric or otherwise) insight into any system of quantum equations, its is unbelievable helpful to switch to multiple unit systems, best iSpace-SI and iSpace-IQ where the integer geometric "netto effects" literally can be seen in the resulting output terms, as the physics behind the exact same iFg result value can be derived with this:

or in other words so while the force of an individual (!) quantum of gravitational force can be numerical to arbitrary digits be derived with above equation, only the following shows whats actually the proper physics behind:

iFg = (alphaG)^2 * (FPlank)

Likewise EXACT in iSpace theory, but now stemming from two physically well known entities like alphaG, the finestructure. constant of gravitation defined by many relations, the most known of which is electron mass squared over Planck mass squared ((me)^2 / (mPlanck)^2) and FPlanck being just the well known Planck-Force (typically defined by c^4/G).

Note while giving the exact same result, these definitions are not circular. The inner integer geometry of alphaG is (GoldenRatio^3*Pi^5) and hence in squared form identical to the iSpace derived integer geometric base of Newtonian gravitational constant G, which shows (GoldenRatio^6*Pi^10), in turn interestingly fully cancelling out in the microscopically small (SI) force value of about 3,7139704417...×10^-46 Newton (exactly, and in visible direct relation iSpace geometry - the 3 is Pi3 effectively and the 299792458^4 should be clear to where its comes from, in MKS/A SI units).

iSpace geometry - while not at all designed or aiming to be a form of ToE - seems to form the ultimate geometric core of any other scientific model whatsoever, which can best be understood if one makes themselves clear (whatever the own interests, theories, 4D, relativistic, QED/QCD approach or otherwise) is currently trying it from the wrong way up:

Its the - integer exact - physical constants of nature (as imho impressively shown with iSpace-IQ quantum geometric unit system) which build the core essence - and with this all constraints to obey from (!) - of the respective theory, and not the reverse out many (in the 1000+ range) have tried in vain in the past like String theory and many others.

ISpace does not allow for a 10^500 possibilities range like say String theory, it allows for exactly 1 (one!) - and that's it. Hence it must following Occam simply be the proper - correct - one.

Christian G. Wolf

I clearly understand your logic, and note that I do not dispute that G can be related exactly to a set of fundamental constants, simply due to the fact that I observed that all stable systems in the universe are related by mutually resonant frequencies in one way or another.

The reason why I could not integrate big G as belonging to the set of fundamental constants applicable at the subatomic level of interactions between the restricted set of charged and massive Elementary electromagnetic particles of which all atoms in the universe are made, and that are the only carriers of charge and mass in the universe, is that in big G, the mass of the Sun, the radius of the Earth's orbit and the time for the Earth to complete one revolution about the Sun are structurally embedded into astronomical constant "G".

As analyzed in this paper published in 2013:

Article Unifying All Classical Force Equations

Consequently, my conclusion was that big G turns out to be a proportionality constant applicable at the level of astronomical orbital systems.

Using standard "G" to calculate the force that would be acting on an electron in the hydrogen atom, amounts to calculate the force, that would apply to an electron, that would be moving on the same orbit as the Earth about the Sun, because the mass of the Sun, the radius of the Earth's orbit and the time for the Earth to complete one revolution about the Sun are directly embedded into astronomical constant "G"!

If you examine the numerical analysis carried out in the above referred paper, you will see how a proportionality constant Gp can easily be defined, different from big G that harmonizes with atomic orbital systems, which makes G not as a single fundamental constant, but a series of G(s) each valid for the series of existing magnitude levels, that lets possibly envision that a different G may be applicable at the galactic groups orbital systems, that I have not explored.

Note that defining this Gp that embeds the parameters consistent with atomic scale orbital systems was what allowed mathematically proving that all classical force equations can be derived from the fundamental F=ma acceleration equation and the reverse, including the Coulomb equation, actually, by means of the Coulomb equation.

The "Heisenberg-Born-Jordan matrix," the first consistent matrix formulation of quantum mechanics in 1925, is a bit strange.

It states that:

Ψ = √h

0 -i√1 0 0

i√1 0 -i√2 0 . .

0 i√2 0 -i√3 . . .

0 0 i√3 0 -i√4 . .

etc.

. . . Q1

It is clear that the Heisenberg matrix Q1 is the energy operator of the quantum system in the sense that:

E= ∭Ψ^2 (x,y,z,t) dx dy dz. where E is the total energy of the quantum system E (x,y,z,t) = ΨΨ* =Ψ^2. Equation Q1 suggests that the imaginary Heisenberg matrix describes the square root of the energy density of the mechanical system, not the energy density itself.

We suggest the following improvement, both physical and logical.

This improvement is the subject of this question.

We propose two major modifications to improve the matrix Q1 above:

i- Transform the matrix into a real matrix by removing the complex unit i and replacing Q1 with its square Q^2.

ii- Transform the matrix above into a sawtooth pattern, which is the most efficient for a quantum matrix.

We believe that this improvement actually constitutes a revival, or even the rise of matrix mechanics.

The new matrix Q^2 will be written as follows:

Equal to,0 -1^0.5      0      0      0      0      0      0      0      0      01^0.5      0 -2^0.5      0      0      0      0      0      0      0      00 2^0.5      0 -3^0.5      0      0      0      0      0      0      00      0 3^0.5      0 -4^0.5      0      0      0      0      0      00      0      0 4^0.5      0 -5^0.5      0      0      0      0      00      0      0      0 5^0.5      0 -6^0.5      0      0      0      00      0      0      0      0 6^0.5      0 -5^0.5      0      0      00      0      0      0      0      0 5^0.5      0 -4^0.5      0      00      0      0      0      0      0      0 4^0.5      0 -3^0.5      00      0      0      0      0      0      0      0 3^0.5      0 -2^0.50      0      0      0      0      0      0      0      0 2^0.5      0

Q=

-1     0   2^0.5       0       0       0       0       0       0       0     0

0    -3       0   6^0.5       0       0       0       0       0       0     0

2^0.5     0      -5       0 2*3^0.5       0       0       0       0       0     0

0 6^0.5       0      -7       0 2*5^0.5       0       0       0       0     0

0     0 2*3^0.5       0      -9       0 30^0.5       0       0       0     0

0     0       0 2*5^0.5       0     -11       0 30^0.5       0       0     0

0     0       0       0 30^0.5       0     -11       0 2*5^0.5       0     0

0     0       0       0       0 30^0.5       0      -9       0 2*3^0.5     0

0     0       0       0       0       0 2*5^0.5       0      -7       0 6^0.5

0     0       0       0       0       0       0 2*3^0.5       0      -5     0

0     0       0       0       0       0       0       0   6^0.5       0    -2

Note that the new matrix Q contains √2, √3, √5, etc. in addition to √10, all with 8-digit precision.

To be continued.