Reference Frame Transformations and Quantization
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Presented atPhysics Beyond Relativity 2019 conferencePrague, Czech Republic, October 20, 2019(invited presentation)
Reference Frame Transformationsand Quantization
Akira Kanda , Renata Wong and Mihai Prunescu Omega Mathematical Institute/University of Toronto, Toronto, Ontario, Canada Department of Computer Science and Technology, Nanjing University, Nanjing, China University of Bucharest, Bucharest, RomaniaE-mail: [email protected], [email protected], [email protected]
Abstract.
It has been said that Maxwell’s theory of electromagnetic field is relativistic asEinstein showed that these axioms of Maxwell are all Lorentz invariant. We investigate someissues regarding these results.“Mozart, You are a god, and do not even know it.” ... (Alexander Pushkin)“To punish my contempt of authority, I became an authority.” ... (Albert Einstein)
1. Transformations and dynamics
One of the biggest and most fundamental questions in relativity theory is what does it mean thatsomething is “relativistic”. Newton assumed the absolute reference frame and defined relativemotion as the difference of two absolute motion vectors to be observed from the outside of theabsolute frame. Galileo’s idea was to reject the absolute frame and associate a reference frameto each body and let each body observe the other bodies’ motion inside its own reference frame.This concept came with the restriction of “inertial reference frames”, by which are meant thereference frames that move with constant speed relative to each other inside each other. Thiscondition was added when it was discovered that accelerating reference frames do not sharethe same laws of physics. For example in the classical dynamics, by considering the mutuallyaccelerating frames, we violate the law of action-reaction. The condition of sharing the samelaws of physics was taken as fundamental for the theory of relativity and is called the “principleof relativity”.o represent the motion of inertial reference frames, Galilean relativity theory introduced anaturally associated “spacetime” coordinate transformation called Galilean transformation: t ′ = tx ′ = x − vt where v is the relative speed between the two inertial frames involved. With this, it was possibleto restate the principle of relativity as “The laws of physics must be invariant under the Galileantransformation.”Knowing that the adoption of inertial reference frames violates the third law (the law ofaction-reaction) of Newton’s dynamics, it seems a natural question to ask whether the secondlaw and the law of gravity are invariant under the Galilean transformation. Curiously, thisquestion was never considered. However, as we will present below, the answer is “yes” forGalilean transformation. F = m d xdt = ⇒ F = m d ( x − vt ) dt = m d xdt .F = GmM ( x m − x M ) = ⇒ F = GmM (( x m − vt ) − ( x M − vt )) = GmM ( x m − x M ) . Hence, in the case of Galilean transformation, the damage inflicted by relativization is limitedto the loss of the third law.
Remark (1)
By identical argument we can show that the Coulomb force law is invariantunder the Galilean transformation.
To the Galilean relativity theory Einstein added an extra axiom of the constancy of thespeed of light, which says that the speed of light is constant c in any inertial reference frames,in other words c + v = c .[11] The result is what we now call the special theory of relativity.The simplest and most effective refutation of this claim came from Anderton [1] of NaturalPhilosophy Alliance. Anderton incisively pointed out that “if c + v = c is true then c is nota speed.” We present the following argument to back up Anderton’s argument. Assume that v is the absolute speed of Michelson-Morley apparatus in the absolute space. Even if c + v isthe absolute speed of the light moving towards the mirror, the effect of this cancels out becausethe mirror is also moving with speed v in the absolute frame. On the same token, though thereflected light moves with speed c − v towards the emitter of light, as the emitter is moving withspeed v , the effect of v cancels out. So, we will never detect this v . This came from an incorrectinterpretation of Michelson-Morley experiment.Thus, Einstein derived the Lorentz transformation between two inertial reference frames.Before Einstein, Lorentz derived a coordinate transformation only between the absolute aetherframe and an observer’s frame moving in the absolute frame, assuming that the length ofa matter moving in the absolute frame contracts (also referred to as “Lorentz-FitzGeraldcontraction”).[21] Einstein generalized this result of Lorentz to the setting of arbitrary inertialreference frames without involving the absolute frame. It goes as follows: x ′ = ( x − vt ) p − ( v/c ) t ′ = 1 p − ( v/c ) (cid:16) t − vxc (cid:17) . With this, Einstein rewrote the principle of relativity to read that “laws of physics must beinvariant under the choice of inertial reference frames”, which later was taken as the invariancender the Lorentz transformations. Before this the principle of relativity meant that thelaws of physics must be invariant under the choice of inertial reference frames. (Though notbrought up explicitly, this implied that the laws of physics must be invariant under the Galileantransformations.)Being the extension of the Galilean relativity theory, this theory violates the third law ofdynamics. Interestingly, also the second law and the law of gravity fail to be invariant underthe Lorentz transformation, as we have F = m d xdt = ⇒ F = m d dt ( x − vt ) p − ( v/c ) = m d xdt .F = GmM ( x m − x M ) = ⇒ F = GmM ( ( x m − vt ) √ − ( v/c ) − ( x M − vt ) √ − ( v/c ) ) = GmM ( ( x m − x M ) √ − ( v/c ) ) = GmM ( x m − x M ) . This means that under Einsteinian relativisation all axioms of Newtonian dynamics except thefirst axiom of the Law of Inertia fail. The damage inflicted by relativisation is much bigger whenwe take Einsteinian relativisation over Galilean relativisation.
Remark (2)
Again, by identical argument we can show that Lorentz transformation fails toconserve the Coulomb’s laws.
2. Transformations and electromagnetism
One of the major reasons for the introduction of the Lorentz transformation was that Lorentzdiscovered that the electromagnetic wave equation of Maxwell is not invariant under theGalilean transformation. Lorentz discovered that a better coordinate transformation by himself,called Lorentz transformation, preserves Maxwell’s electromagnetic wave equation.[21] Einsteingeneralized this result by showing that under the Lorentz transformation all basic equations ofMaxwell’s electromagnetic field theory are invariant.[11] This became the vindication for theclaim that Maxwell’s electromagnetic field theory is relativistic in Einsteinian sense.Though wave equations are not conserved under the Galilean transformations, wave functionsare transformed into wave functions through Galilean transformations. So, it is not quite clear ifwe needed Lorentz transformations to begin with. If the issue of relativity is just the changing ofreference frames, then Galilean transformation is the most natural transformation representingthe choice of inertial reference frames. The dominating argument for supporting the Lorentztransformation is twofold:(i) Lorentz transformation maps electromagnetic wave equations into electromagnetic waveequations while Galilean transformations fail to do so. The right response to this is that wavemechanics and particle mechanics are entirely different theories covering entirely differentissues of physics. Galilean transformation came from the consideration of relativisingparticle kinematics. It is therefore totally expected that this transformation does not dealwith physical waves, which exist upon continuum wave mediums. In wave mechanics, it isnot particles that move in the direction of the wave. It is the localized vibration energyof the medium that moves. For this reason, wave mechanics has no naturally associatedmomentum, which is the product of speed and mass. The hypothetical momentum of wavescame from the relativistic theory of waves of de Broglie.[8](ii) Lorentz transformation transforms Maxwell’s electromagnetic equations into equivalentequations. We will discuss later the issue of whether Galilean transformations do the same.Maxwell reduced the entire electromagnetic theory to the following field plus currentequations: ∇ × E = − c ∂ H ∂t , ∇ · E = 4 πρ, ∇ · H = 0 , J = ρ v , ∇ × H = 4 πc J + 1 c ∂∂t E here c = √ ǫ µ . Here ρ is electric charge density and J is the conducting current createdby the charge density moving with speed v . Maxwell obtained the last equation, called the“Generalized Amp`ere’s Law”, under the assumption that the charges in J move with constantspeed v . This restriction was removed later.This introduction of J into the field theory is problematic. According to the definition ofelectric force field, charges placed in a field will not affect the field, meaning that the chargesplaced will not affect other charges that create the electric field.Ontologically, it is hard to understand J at this basic level. How is it possible that mutuallyrepelling electrons can form a coherent bundle? A possible answer says that this is possibleinside a conductor. But there is a vicious circularity here. Conductors are objects to be studiedin material science where we have to use the theory of electromagnetism. This problem somehowresembles the problem of tension rope in classical dynamics. The rope is supposed to be a mass-less entity that moves under acceleration. Then it must be the case that all such ropes will movewith infinite speed. How does it connect two bodies?Here is yet another consequence of the logical inconsistency discussed above of Maxwell’selectromagnetic field theory. Maxwell obtained the electromagnetic wave equation ∇ E = 1 c ∂ E ∂ t and ∇ H = 1 c ∂ H ∂ t . under the condition that J = 0 (meaning that there is no conducting current). From thiswave equation, he calculated the speed of electromagnetic wave in vacuum to be c = 1 / √ ε µ . Interestingly, Maxwell also showed that electromagnetic waves are created by acceleratingcharges. But do accelerating charges not constitute a current? According to the mathematicaldefinition of a current, even a singular moving charge is a current. In fact, radio engineerspostulate that a circular closed circuit conducting electrons inside produces electromagneticwave and one cycle of the electron corresponds to the cycle of the produced electromagneticwave. This is because electrons in the circuit are under centripetal acceleration.Lorentz found out that the Galilean transformation of the electromagnetic wave equationsdoes not lead to wave equations. This lead him to try his own Lorentz transformationinstead and he showed that the Lorentz transformation maps electromagnetic wave equationto electromagnetic wave equation.
Remark (3)
This claim by Lorentz is to be refuted later in
Section 6 however.
The above formed a base for the claim that Galilean transformation is invalid and shouldbe replaced by Lorentz transformation, despite all other problems that Lorentz transformationscreate, some of which we discussed in the previous section.
3. Transformation of waves v.s. transformation of wave equations
The argument that Galilean transformation is invalid and should be replaced by Lorentztransformation should be reconsidered due to the following legitimate argument: The waveequation in one dimension without sources for speed v is ∇ φ = 1 k ∂ φ∂ t . It is well known that when we Galilean transform this equation the result is not a wave equationanymore.The general solution of this wave equation is φ ( x, t ) = ψ + ( x − kt ) + ψ − ( x + kt ) . he first term is a wave propagating with speed + k , and the second one with − k . The Galileantransformation transforms this general solution to φ ( x, t ) = ψ + ( x − ( k + v ) t ) + ψ − ( x + ( k − v ) t ) . This is a superposition of two waves and so it is a wave. This wave has a different speed fromthe original wave. But is it not expected due to the Galilean transformation?We can support this argument using Fourier expansion too. A sinusoidal wave is transformedinto a sinusoidal wave. So, a Fourier expansion which describes a wave also transforms into aFourier series which represents a wave.However, when we Lorentz transform a wave function, we do not obtain a wave function ingeneral. For example it is known well that a sinusoidal wave will be transformed into a sinusoidalwave by Lorentz transformation if the wave amplitude is the solution of a wave equation whichis invariant under the Lorentz transformation.For this reason, relativistic theory of waves assumes that the wave equations are invariantunder the Lorentz transformation. Under this assumption, we have the invariance under Lorentztransformation of the phase of a plane wave k ′ · r ′ − ω ′ t ′ = k · r − ωt where k is the wave vector, r is a position vector and ω is the frequency. With this invariance,we obtain the following relativistic wave transformation equations: k ′ x = 1 p − ( v/c ) (cid:16) k x − v ωc (cid:17) k ′ y = k y k ′ z = k z ω ′ = 1 p − ( v/c ) ( ω − vk x ) . So it is clear that the theory of Lorentz transformation of waves is not so universal. It appliesonly to the waves that are invariant under the transformation. And it has been claimed thatelectromagnetic waves are examples of such waves. Clearly it has the same kind of deficiency asthe Galilean transformation of waves, if not worse.
Remark (4)
It is clear that the phase k · r − ωt is Galilean invariant for any wave. Remark (5)
We will show later in
Section [6] that, contrary to what has been claimedby Lorentz and Einstein, Lorentz transformations will not conserve any of wave equations,electromagnetic or non-electromagnetic.
De Broglie used this transformation of “relativistic” waves and Einstein’s relativistic theoryof photons to create a relativistic wave-particle duality. Observing an analogy between theabove discussed relativistic wave transformation and the relativistic transformation of energyand momentum: p ′ x = 1 p − ( v/c ) (cid:18) p x − v Ec (cid:19) p ′ y = k y p ′ z = k z E ′ = 1 p − ( v/c ) ( E − vp x ) . pplied to the relativistic theory of photons E = hf = pc = 0 / p = h/λ where λ is the wave length, de Broglie obtained the following relativistic equation for relativisticwaves: p = ℏ k E = ℏ ω. This is how the wave-particle duality of quantum mechanics was introduced.It was unfortunate that all of this was done without noticing the following contradictioncoming from the relativistic theory of photons: E = p ( cp ) + ( m ) c = cp = m vc q − (cid:0) vc (cid:1) = 00 cv = c hf = hf. The reason why electromagnetic wave equations are invariant under the Lorentztransformations is because Lorentz transformations are transformations obtained within thecontext of Maxwell’s electromagnetic field theory.Moreover, as we have shown above, Lorentz transformation fails to map axioms of Newton’sdynamics into axioms of Newton’s dynamics. What does this mean? A naturally expectedanswer would be that the concept of relativity is not applicable to dynamics as it violates thelaw of action-reaction. But as we have seen in the first section, Galilean transformation, whichalso is a coordinate transformation associated with inertial reference frames, conserves the secondlaw and the gravitational law. The only deficiency of the Galilean transformation we know of isthat it violates the third law of dynamics.This should mean that Lorentz transformations are more problematic than Galileantransformations though both of them are invalid as they are mathematical representations ofthe invalid concept of relatively moving reference frames which violates the third law.However, in practice, in most cases of dynamics around us, we can assume that one mass isway to large to be able to ignore the reaction on it from a smaller mass, and so, the conservationof the second law and gravitational law is enough to obtain a reasonable approximation. Theaction-reaction problem becomes important when we consider the astronomical scale massesunder gravitational acceleration.
Remark (6)
It is a tradition of pure mathematics and logic to make sure that it is properlyunderstood that a result must be always accompanied with restrictions imposed on the derivationof it. Coordinate transformations are applicable only to inertial frames. No accelerating framesshould be transformed. Yet in the discussion of transformation of waves we are completelyforgetting that physical waves come from acceleration. This type of errors appear in many placesin theoretical physics, unfortunately. For example the well tested concept of centrifugal forcecomes from the abusive use of a reference frame of an orbiting body.
4. Lorentz transformation and speed c What does it mean that Einstein proved that all axioms of Maxwell’s electromagnetic field theoryare Lorentz invariant? This transformation was built only for the electromagnetic field theoryby Lorentz as a solution to the problem that the Michelson-Morley experiment introduced intothe theory of electromagnetic fields. That c is the speed of light wave which is electromagneticwave in other reference frame gave an advantage. In other theories of physics where the speedof light is not the issue, it is hard to imagine that this c will play the role it played in theMaxwell’s electromagnetic field theory. Therefore, not surprisingly, Lorentz transformation failsto preserve the law of gravity and the second law of classical dynamics.axwell’s electromagnetic field theory is incomplete for a description of electrodynamics.Electrodynamics assumes classical dynamics as an underlying theory. This part was omitted byMaxwell and proponents.This problem surfaced explicitly in Lorentz force. Lorentz force, which depends also uponthe speed, contradicts the second law, which asserts that the force should be dependent uponacceleration, not speed.On the one hand the Galilean transformation preserves the second law and the gravitationallaw, and on the other hand Lorentz transformation fails to do so. This important fact has neverbeen noticed before.Einstein faced problems in showing the invariance of all axioms of Maxwell under the Lorentztransformation. Some of the axioms had to be translated into equivalent ones. This is ratherexpected. The problem is that J is not a force field. It is a different monstrosity that waschallenged when Maxwell put it in his axiomatic theory of electromagnetic fields. J is definedin terms of v . Lorentz transformation has a problem with transforming v , more specifically,relativistic addition of speeds. The argument goes as follows:Assume three inertial frames F F F
3. Let v and v ′ be the mutual speed between F F
2, and F F
3, respectively. As these speeds are used to define the gamma factor,they must be pre-relativistic speeds, i.e. classical speeds. So, v + v ′ is the mutual speed between F F L between F F x ′ = ( x − ( v + v ′ ) t ) p − ( v + v ′ ) /c , y ′ = y, z ′ = z, t ′ = (cid:16) t − ( v + v ′ ) xc (cid:17)p − ( v + v ′ ) /c . Let L and L ′ be Lorentz transformations from F F
2, and from F F x ′ = ( x − vt ) p − v /c , y ′ = y, z ′ = z, t ′ = (cid:0) t − vxc (cid:1)p − v /c .x ” = ( x ′ − v ′ t ′ ) p − ( v ′ ) /c , y ” = y ′ , z ” = z ′ , t ” = (cid:16) t ′ − v ′ x ′ c (cid:17)p − ( v ′ ) /c . respectively. It is clear that L = L ′ ◦ L where L ′ ◦ L is the mathematical composition of L and L ′ . It is contested by the mainstream that in the special theory of relativity calculating v + v ′ is the wrong thing to do. It is claimed that this addition should be replaced by the relativisticaddition (transformation) of speed v ⊕ v ′ = v − v ′ − vv ′ /c . This is to say that L should be x ” = ( x − ( v ⊕ v ′ ) t ) p − ( v + v ′ ) /c , y ” = y, z ” = z, t ” = (cid:16) t − ( v ⊕ v ′ ) xc (cid:17)p − ( v + v ′ ) /c . If so, then why is v + v ′ not replaced by v ⊕ v ′ in the gamma factor? Also we have a problem with( v ⊕ v ′ ) in the Lorentz transformation as Lorentz transformation is the agent that introducesthe concept of relativity and one cannot use already relativistic concept v ⊕ v ′ to define sucha transformation. This is a conceptual vicious circularity. The shear reason that this kind ofddition works for composing Lorentz transformations does not justify its use. Unfortunatelythis version of L still fails L = L ′ ◦ L. Mathematically, Lorentz transformations are lineartransformations from 4D space to itself. So, it is highly irregular that algebraic composition ofsuch transformations is not the desired transformation.Conceptually, ( v + v ′ ) is the classically measured speed. The reason why ( v ⊕ v ′ ) was introducedin the numerator part is because the classical addition ( v + v ′ ) does not work for relativisticaddition of speeds. As the relativistic addition ( v ⊕ v ′ ) is introduced by a relativistic argument,it is viciously circular to use this relativistic version in the gamma factor.Moreover, mathematically we have a problem too. It is naturally expected that x ” /t ” willserve as the observed speed v ⊕ v ′ . There is no way to prove that they are equivalent. Here is asimple calculation that leads to this conclusion. x ” t ” = ( x − ( v ⊕ v ′ ) t ) (cid:16) t − ( v ⊕ v ′ ) xc (cid:17) = c ( x − ( v ⊕ v ′ ) t ) c t − ( v ⊕ v ′ ) x = c x − c ( v ⊕ v ′ ) tc t − ( v ⊕ v ′ ) x. Note that even if we replace the classical v + v ′ in gamma factor with relativistic v ⊕ v ′ , theabove calculation holds.Now assume that c x − c ( v ⊕ v ′ ) tc t − ( v ⊕ v ′ ) x. = v ⊕ v ′ . Then we have c x − c ( v ⊕ v ′ ) t = ( v ⊕ v ′ )( c t − ( v ⊕ v ′ ) x.c x = 2 c ( v ⊕ v ′ ) t − ( v ⊕ v ′ ) xc x + ( v ⊕ v ′ ) x = 2 c ( v ⊕ v ′ ) txt = ( v ⊕ v ′ ) c + ( v ⊕ v ′ ) . There is no reason for this to be true.Empirically speaking, even though it could be possible to measure v, what about v ′ ? It isnext to impossible to measure v + v ′ , is it not? If v is the speed of a star A moving away fromus in distance and if v ′ is the speed of another star B moving away from A, then how can wemeasure the v ′ and so v + v ′ ?In case of axioms of electromagnetic fields, there is no v involved and this is why there wasno difficulty for the invariance of the field equations of Maxwell.Despite all of this, there are some positive results concerning the transformation of waveequations. Unlike Galilean transformation, Lorentz transformation maps electromagnetic waveequations into electromagnetic wave equations. The reason for this is straightforward. Thistransformation assumes that the constant c is the speed of light in any inertial reference frame. Remark (7)
Again, the above claim is false. As we will see later in
Section 6 , the claim thatLorentz transformations map wave equations into wave equations is false even for electromagneticwave equations.
5. Is Schr¨odinger’s wave equation relativistic?
De Broglie obtained the following relativistic transformation of a plane wave for a wave invariantunder the Lorentz transformation (we call it a “relativistic wave”): k ′ x = 1 p − ( v/c ) (cid:16) k x − v ωc (cid:17) , k ′ y = k y , k ′ z = k z , ω ′ = ( ω − νk x ) p − ( v/c ) here k = ( k x , k y , k z ) is the wave vector and ω is the frequency. We denote the wave number | k | by k . So, k = | k | . This restriction to relativistic waves is in place because otherwise the wavephase k · r − ωt would not be invariant under the Lorentz transformation. Using the analogybetween this and the momentum-energy transformation p ′ x = 1 p − ( v/c ) (cid:18) p x − v Ec (cid:19) , p ′ y = p y , p ′ z = p z , ω ′ = ( E − νp x ) p − ( v/c ) where p = ( p x , p y , p z ) is the momentum vector and E is the energy, de Broglie proposed thefollowing association between a particle and a wave (called matter wave): p = ℏ k E = ℏ ω where ℏ is a constant. It is called de Broglie relation. Though it resembles Einstein’s particle-wave duality E = hf = pc p = h/λ where λ is the wave length, there is a fundamental difference.There are several issues to be discussed.(i) Unlike the photon-light duality where the speed of photon and that of light are equal, thephase speed of matter wave and the speed of particle can be different.(ii) De Broglie further assumed that, associated with a particle with speed V , was a wave havingphase speed w = ω/k . This association requires further explanation.(iii) De Broglie also assumed that the energy in the wave traveled along with a group speed v g = dω/dk , which was identical to the particle’s speed V . Here it is not quite clear whatdid he mean by “energy” in the wave. The de Broglie relation above is only a hypothesisbased upon the above mentioned analogy of wave vector-frequency transformation andmomentum-energy transformation. Certainly this does not yield the concept of energy inthe wave.De Broglie assumed that c ( ω/c − k ) is invariant under the relativistic transformation asan analogy to the relativistic invariance of c ( E/c − p ). Thus, c ( ω/c − k ) = constant = C. From this, it follows that 2 ω/c dωdk − k = 0 . This leads to v g = dωdk = c kω . As the phase speed is w = ω/k, we have v g = c w . It now follows that either the phase speed w or the group speed v g could exceed c , but not both.We do not know what this means for the special theory of relativity, which asserts that nothingmoves faster than the speed c .All of this is relative to the analogy-based hypothesis that a particle with speed v has a wavedual called matter wave whose group speed is v g and whose phase speed is w = ω/k . A particlein motion carries energy and so it is expected that the wave dual of this particle also carriesnergy of the same amount if the energy conservation law is to be respected. But according towave mechanics, for a wave to carry energy it has to have wave medium. A concern we have isthat de Broglie’s wave is a mathematical wave that appears to have no wave medium, just likethat electromagnetic wave carries energy without having wave medium. We have already pointedout that electromagnetic field which carries electromagnetic waves is a fiction, a counter-factualmodality that plays no ontological role in physics. So, what happened to the energy issue of thematter wave? This is not the case with de Broglie waves.However, it is not clear why we have to choose Lorentz transformed version over Galileantransformed version. That the Galilean transformation of a wave function is a wave function,seems to suggest that the theory of Lorentz transformation of wave functions is rather self-serving. That Lorentz transformation came from time dilation and length contraction, whichcauses paradoxes (contradictions), seems to suggest that there are more fundamental thingsthat have to be reexamined in the theory of Lorentz transformations. Indeed, almost all wavesthat wave mechanics works on are not relativistic. The only familiar waves that are relativisticare the electromagnetic waves. But this is overshadowed by the fact that the electromagnetictheory, which gave birth to the electromagnetic waves, is not relativistic either. The most basicaxioms of this theory, the Coulomb’s laws, are not relativistic as we have established above.So, the claim that electromagnetic waves are relativistic strongly suggests that the theory ofelectromagnetism is inconsistent. Remark (8)
As we will see later in
Section 6 , the claim that Lorentz transformations mapwave equations into wave equations is false even for electromagnetic wave equations.5.2. Schr¨odinger’s wave equation
Schr¨odinger used Hamilton’s energy dynamics for the particle theory and applied de Broglie’spilot wave theory to produce a wave-particle duality that looks after the energy issue of deBroglie’s relation.[27][8]All waves propagated along the x -axis obey the following wave equation ∂ Ψ ∂x = 1 ω ∂ Ψ ∂t where Ψ( x, t ) is the wave function and ω is the wave speed.Here, we consider the wave function Ψ whose square yields the probability of locating aparticle at any point in the space. We consider only systems whose total energy E is constant andwhose particles move along the x − axis and are bound in space. Then the frequency associatedvia the de Broglie relation, which is hypothetical and relativistic, with the bound particle is alsoconstant, and we can take the wave function Ψ( x, t ) to beΨ( x, t ) = ψ ( x ) f ( t ) . As the frequency is assumed to be precisely defined, f ( t ) = cos 2 πνt. So, we have ∂ ψ∂x = − (cid:18) πλ (cid:19) ψ = − (cid:16) ph (cid:17) ψ where the wave length is λ = ω/ν and the momentum of the particle is p = h/λ. We take the particle of mass m to be interacting with its surroundings through a potential-energy function V ( x ). The total energy of the system is given by E = E k + V = p m + V here E k is the kinetic energy of the particle. Then we have p = 2 m ( E − V )and we have ℏ m ∂ ψ∂x + ( E − V ) ψ = 0 . So, − ℏ m ∂ ψ∂x + V ψ = E = i ℏ ∂ψ∂t . This equation is called (non-relativistic) Schr¨odinger wave equation as the energy equationinvolves non-relativistic mass m and it is not invariant under the Lorentz transformation. Thisdoes not mean that quantum mechanics is a non-relativistic theory however. The derivation ofSchr¨odinger wave equation involved de Broglie relation which is nothing but a relativistic theory.We observe an issue with the above argument. It is claimed that ∂ Ψ ∂x = 1 ω ∂ Ψ ∂t is the wave equation and − ℏ m ∂ ψ∂x + V ψ = E = i ℏ ∂ψ∂t . is given as its example. This obviously becomes a wave equation only when V = 0.As pointed out above, Schr¨odinger’s wave equation is in fact not a wave equation as it isirreconcilable with the classical equation for waves. A further observation of its form indicatessimilarities between Schr¨odinger’s equation and the diffusion equation used in describing densityfluctuations in materials due to diffusion. The diffusion equation is given as ∂φ ( x, t ) ∂t = ∇ D ( φ, x ) φ ( x, t )where φ ( x, t ) is the density of the diffusing material at position x and time t , and D ( φ, x ) is thediffusion coefficient for density φ at position x . When D is constant, the equation reduces to ∂φ ( x, t ) ∂t = D ∇ φ ( x, t ) . which is a partial differential equation with first derivative in time and second derivative inposition, just like the Schr¨odinger’s equation. This particular form of diffusion equation wasproposed by Fourier in 1822 to describe the heat distribution in a given region of a material overa particular time and hence is sometimes referred to as “heat equation”.[13]A crucial difference between the Schr¨odinger equation and the diffusion equation is that thecoefficient in the latter ( D ) is real, while in the former it is complex. Consider for instance theequation for a free particle: ∂ψ ( x, t ) ∂t = i ~ m ∇ ψ ( x, t ) . This difference makes the solutions to the diffusion equation decay with time (gradient), while thesolutions to the Schr¨odinger’s equation oscillate (wave). Note however that originally, before theBorn interpretation became common, Schr¨odinger attempted to interpret the wavefunction aselectronic charge distribution in space (with charge density at position x and time t proportionalto | ψ | ): “the charge of the electron is not concentrated in a point, but is spread out throughthe whole space [...] the charge is nevertheless restricted to a domain of, say, a few Angstroms,he wavefunction ψ practically vanishing at greater distance from the nucleus.”[27] This wouldsuggest his treatment of charge density as having a character of a radiation, indicating certaingradient properties.It is unfortunate that the name “wave equation” became the common name for Schr¨odinger’sequation, thereby confusing the classical wave equation with a formalism lacking grounding inontology. It appears that the Schr¨odinger equation is an attempt at merging the concept ofwave-particle duality with the treatment of electronic charges in terms of density distribution.Regarding the former, Schr¨odinger himself had reservations assuming the veracity of matterwaves, justifying the concept by stating that neglecting de Broglie’s waves leads to seriousdifficulties in atomic mechanics. Regarding the latter, Schr¨odinger himself noticed that thisinterpretation of wavefunction does not work for systems of multiple electrons.[28]Nevertheless, Schr¨odinger was aware of this problem and tried, unsuccessfully, to make hiswave equation relativistic. Later, Gordon, Klein and Dirac attempted to resolve this problemin the development of the quantum electrodynamics.With all of this, it is clear why Schr¨odinger failed to show that his wave equation for particlesis relativistic. To make the matter even worse, Schr¨odinger’s equation is not relativistic, andthus a quantization of such an equation is impossible because de Broglie’s quantization of wavesworked only for relativistic waves.Instead of relativising Schr¨odinger’s wave equation, Gordon and Klein quantized relativisticenergy-momentum equation of Einstein by replacing energy variable and momentum variablewith quantum energy operator and quantum momentum operator. This however does not makethe Schr¨odinger wave equation relativistic and therefore does not compensate for the deficiencystated above. Remark (9)
The energy-momentum relation is a consequence of the relativistic energyequation e = mc which is false. Here, m is the relativistic mass m / p − ( v/c ) which isobtained through a thought experiment that assumed that v is constant. With this Einsteinobtained relativistic second law P = mv. By taking a time derivative, Einstein then obtained therelativistic second law F = dP/dt = vdm/dt + mdv/dt. This lead him to conclude that e = mc . Unfortunately, v is a constant, which leads to e = 0 instead. As Einstein pointed out, if e = mc fails, the entire modern physics fails. Dirac took advantage of the Gordon-Klein equation and derived a relativistic theory ofelectrons which yielded the positron and opened a gate to quantum electrodynamics whichis considered the most successful theory of physics in history.
6. Are wave equations really relativistic?
Now we have reached the point where the question has to be asked whether the wave equationsreally represent waves that appear in physics correctly. Another question is whether the Lorentztransformation which seemingly maps electromagnetic wave equations to electromagnetic waveequations does so with ontological background.The Galilean transformation fails to conserve electromagnetic wave equations and the Lorentztransformation conserves electromagnetic wave equations. It seems to be the only reason whyLorentz transformation replaced Galilean transformation. The Galilean relativity theory wasrejected (except the faulty interpretation of the Michelson-Morley experiment) because it failedto map electromagnetic wave equations to electromagnetic wave equations. So, if can be safelysaid that as far as the wave theory is concerned, it was the failure to conserve electromagneticwave equations which dethroned the Galilean transformation.Here we have to ask whether the wave equations are the basic axioms of physical theories.Clearly not. They are the product of the basic axioms under certain circumstances. So, logicallythere is no convincing reason why such secondary equations must be conserved under coordinatetransformations.ut to make the argument more articulate, let us discuss the issue in a more general setting. ∂ψ ( x ′ , t ′ ) ∂x = ∂ψ ( x ′ , t ′ ) ∂x ′ ∂x ′ ∂x + ∂ψ ( x ′ , t ′ ) ∂t ′ ∂t ′ ∂x = ∂ψ ( x ′ , t ′ ) ∂x ′ ∂γ ( x − vt ) ∂x + ∂ψ ( x ′ , t ′ ) ∂t ′ ∂γ ( t − vxc ) ∂x = γ ∂ψ ( x ′ , t ′ ) ∂x ′ − γvc ∂ψ ( x ′ , t ′ ) ∂t ′ Similarly, ∂ψ ( x ′ , t ′ ) ∂t = − γv ∂ψ ( x ′ , t ′ ) ∂x ′ + γ ∂ψ ( x ′ , t ′ ) ∂t ′ Then, ∂ ψ ( x ′ , t ′ ) ∂x = (cid:18) γ ∂∂x ′ − γvc ∂∂t ′ (cid:19) (cid:18) γ ∂∂x ′ − γvc ∂∂t ′ (cid:19) = γ ∂ ∂x ′ − γ vc ∂ ∂x ′ ∂t ′ + γ v c ∂ ∂t ′ And similarly, ∂ ψ ( x ′ , t ′ ) ∂t = γ v ∂ ∂x ′ − γ v ∂ ∂x ′ ∂t ′ + γ ∂ ∂t ′ With this, the wave equation now becomes γ ∂ ∂x ′ − γ vc ∂ ∂x ′ ∂t ′ + γ v c ∂ ∂t ′ = 1 ω (cid:18) γ v ∂ ∂x ′ − γ v ∂ ∂x ′ ∂t ′ + γ ∂ ∂t ′ (cid:19) . This is valid only under the condition v = c = ω. The second equality comes from the fact that ω is the wave speed. The first equation implies that the frame speed is c which is not possiblein the special theory of relativity. This means that Einstein’s claim that the electromagneticwave equation is invariant under the Lorentz transformation is invalid. It is a well understoodfact that there is no reference frame for light at the pain of contradiction.We summarize the results thus far as follows: Conclusion (1)
Lorentz transformation fails to conserve all wave equations includingelectromagnetic wave equations.
Conclusion (2)
Lorentz transformation serves no imaginable purpose. It fails theconservation of the third law, that of the second law, that of gravitational law, that of Coulomb’slaw. Now we also know that it does not conserve even the electromagnetic wave equations. Thisremoves the claim that Einsteinian relativity theory is more appropriate than Galilean relativitytheory. Naturally, Lorentz transformation does not conserve wave functions either.
Conclusion (3)
All of this is a consequence of the wrong interpretation of Michelson-Morley’s experiment. As we demonstrated previously [18], the Michelson-Morley experimentshowed that one cannot detect v in c + v in the way we measure the speed of light. Hence itappears that the problems in modern physics started from the Michelson-Morley experiment. Galilean transformation conserves all of the basic laws and constructs of physics except thethird law and the wave equation. The only issue with this transformation is that it is basedupon the faulty concept of moving reference frames. To see the problem with moving referenceframes, assume a train runs on a track. When the tip of the train’s power pole touches theower line at point A, a spark occurs at A. An observer located in the train straight down fromthe tip A of the power pole will observe that this light comes straight down to him/her frompoint A. But as point A also is a stationary point of the power line, the observer will also seethat the light reaches him/her diagonally from point A on the power line.[17]Mathematically this problem can be explained as follows: in geometry one cannot moveany point in geometric spaces as doing so breaks the metric structure of geometric spaces. Onecannot move a point 5 to the position of point 3 and vice versa as this breaks the metric topologyof the real number line. This is why Newton did not attempt to move any geometric points.Instead he reduced a physical body to a point body and moved it inside a geometric space. Ifwe cannot move even a single point in a geometric space, how can we move the entire spaceitself inside another space? If we move a point 5 on a real line then what is the function thatdescribes such motion?Topologist Ren´e Thom pointed out there is no point in geometry. In geometry we must assumethat mysterious linear ordering among real points. This makes the geometry a continuum.Mathematical logician (the founder of model theory) Abraham Robinson expressed this in termsof infinitesimals. Points are all “glued” together by invisible infinitesimals. In the end, standardreal analysis and infinitesimal calculus do the same thing.
7. Relativistic transformations and 4D spacetime
A motion in the 3D space is a function f ( t ) = ( x, y, z ). This can be expressed as a line graph inthe 4D spacetime. If the speed of the motion is constant, the graph is a straight line, and if it isunder acceleration, then the graph is a curved line. When we apply a Galilean transformationto this graph, then the resulting graph is a translated line. f ( t ) = ( x − vt, y, z ) . However, when we apply Lorentz transformation to this graph, due to time dilation andlength contraction combined, the resulting graph becomes incomprehensible. So, the resultinggraph is unusable for the purpose of physics. In symbols, the resulting “motion” becomes thegraph of f p − ( v/c ) (cid:16) t − vxc (cid:17)! = p − ( v/c ) ( x − vt ) , y, z ! . This is expected as under the Lorentz transformation time and space coordinates areinterdependent.
8. Dirac’s aether theory
As can be seen in the vortex theory, which we will discuss in the next section, the wholephilosophy of aether theory is to “squeeze out” particles from a continuum. At the mostfundamental level, as Ren´e Thom pointed out, this is impossible as it destroys the topologyof the continuum. The difficulty the classical aether theory had is naturally expected becauseof the nature of continuum.Dirac was the first who managed to create this paradigm upon the quantum field which isthe quantization of classical field using the mathematical tool of Fourier expansion.[10] In thismethod, Dirac did not create a geometric point as the quanta. He created a finite approximationof infinite Fourier expansion as a particle. So, Dirac’s particles are infinitary objects describedby waves.This project started with a new theory of photons proposed by Planck, which Planck himselfdid not take seriously and presented as a purely mathematical convention arising from graphfitting as the last resort to resolve the mystery of the blackbody radiation. Planck presentedhe argument that if one accepts that the minimum energy carried by the electromagnetic waveis hf , where f is the frequency of the wave and h is a constant, which is now called the Planckconstant, the infamous blackbody radiation problem is resolved. So, Planck proposed that thelight wave of frequency f carries waves as nhf , where n is a natural number.Under the assumption that the speed v of light in vacuum without conducting current isconstant c , which came from Maxwell’s theory of electromagnetism, Einstein concluded that themass of Planck’s particle (photon) must be 0 to avoid the relativistic energy e = mc = m p − ( v/c ) c of the photon become undefined (or diverge) where m is the rest mass of the photon. With thisconvention, Einstein took the energy equation for the photon to become e = 0 / hf. Remark (10)
Einstein thought that since / is equivalent to x = 0 , and for the latter x canbe any number, / can be any number and he chose it to be hf. However the difference betweenthe two is such that the former involves division by , which is not allowed in mathematics, andthe latter does not involve it. As discussed above, this leads to yet another contradiction. The relativistic energy equation e = mc leads to the famous relativistic energy-momentum relation e = p ( cp ) + ( m c ) whichin turn leads to the following contradiction e = p ( cp ) + ( m ) c = cp = m vc q − (cid:0) vc (cid:1) = 00 c = c hf = hf. Logically speaking the real problem with the Planck-Einstein photon theory is that the issueof blackbody radiation was an empirical refutation of the classical electromagnetic field theoryof Maxwell. The convention Planck and Einstein presented, which turned out to be invalid as wehave shown, did not repair the deficiency of Maxwell’s theory. No change was made to Maxwell’stheory after the Planck’s proposal. So, these two mutually contradicting theories were combinedtogether to produce another theory that makes opportunistic choice. Namely, when it comes tomost of the classical part of electromagnetism, it uses the original Maxwell’s theory and when itcomes to the issue of the light waves, it chooses the Planck-Einstein addition, which contradictsMaxwell.Dirac does not appear to have known of this fatal error of Planck-Einstein. But he wasrightly unhappy with the ad hoc nature of the process of obtaining the equation e = 0 / hf. He concluded that obtaining photons from electromagnetic wave equation is the wrong thing todo. So instead, Dirac presented the photons through Fourier expansion of the vector potential.In this way, he managed to obtain a richer theory of photons. However, deducing photonsthrough Fourier expansion of vector potential lacks in ontology. Also, the quantization of thecharges and currents in the Maxwell’s theory remained to be reviewed.Dirac’s solution to the problem of correctly quantizing charges and particles was to rely uponthe Schr¨odinger wave equations. He found it impossible to quantize charges and particles asthey are already particles. According to the basic principle of wave-particle duality, namely deBroglie relation, quantum particles must come from waves. So, Dirac first converted particlessuch as charges into the Schr¨odinger wave equations. Instead of going through von Neumann’squantization, Dirac used Fourier expansion of the solutions of the wave equations to createquantized particles. Particle interactions were modeled through the interference of the wavequations derived from the particles. Through this process Dirac obtained more particle varietiesand more interesting operators on particles such as the annihilation operators.Unfortunately, as we have discussed in the section “Is Schr¨odinger’s wave equationrealtivistic?”, Schr¨odinger’s wave equations are not relativistic, meaning that they are notinvariant under the Lorentz transformation in general. This means that the claim of Diracthat his new theory of quantum electrodynamics is relativistic is false as his quantization usesSchr¨odinger’s wave equations. To make the matter even more confusing, Schr¨odinger’s waveequation was obtained by applying de Broglie relation to classical Hamiltonian theory of particlesand this relation is relativistic upon the assumption that the wave of de Broglie is relativistic(meaning Lorentz-invariant). Schr¨odinger’s wave equation is not relativistic because Schr¨odingermisunderstood what de Broglie did. Schr¨odinger used de Broglie relation to convert classicalHamiltonian energy equation into a wave function. De Broglie did not associate a particle to awave equation however. Indeed, what he did was the opposite. He associated a particle havingrelativistic energy and relativistic momentum with waves. His wave-particle duality is a oneway association. Moreover de Broglie had to assume that the wave equation in his theory hasto be relativistic, meaning that it is invariant under the Lorentz transformation. For such arelativistic wave equation, he associated a particle with energy and momentum.In this way, the wave-particle duality of Schr¨odinger’s is fundamentally flawed, putting theinvalidity of relativity theory aside.After all, as the relativity theory is inconsistent, there is no point in considering whetherDirac’s theory is relativistic or not.Moreover, Dirac quantized electromagnetic fields, which are not physical reality but amodality, through Fourier expansion to obtain photons. This makes Dirac’s photons sufferfrom the same conceptual obscurity as Planck-Einstein’s photons, which are also the product ofquantizing (in a different way) electromagnetic waves, which are modal waves.Regarding Feynman’s quantum electrodynamics, despite some improvements such as leavingHamiltonians behind and moving into the Lagrangian, this theory did not resolve the problemassociated with Schr¨odinger’s wave equations discussed just above.Gordon-Klein’s quantization of invalid relativistic energy-momentum equation of Einsteindoes not offer any solution to this fundamental problem that Schr¨odinger’s wave equation isnot relativistic. Replacing relativistic energy variable and relativistic momentum variable withenergy operator and momentum operator in the faulty relativistic energy-momentum relation isnot what a quantization should entail.Below we posit some major questions regarding how quantum physics led to quantumelectrodynamics.(i) There are many concepts of quantization. Namely, Planck-Einstein quantization ofelectromagnetic waves, de Broglie’s quantization of associating relativistic waves witha particle with momentum and energy through analogy between the transformation ofrelativistic waves and transformation of energy-momentum, Schr¨odinger’s quantizationof converting classical particle equations into wave equations using de Broglie’srelation, Dirac’s quantization of electromagnetic fields through Fourier expansion, Dirac’squantization of particles expressed as Schr¨odinger wave equations through Fourierexpansion, Gordon-Klein quantization of Einstein’s energy-momentum relation, etc. Yetthere is no study of how they are related.(ii) De Broglie’s quantization, which plays key role in many quantizations as listed above, is notproperly understood. This quantization works only for relativistic waves that are invariantunder the Lorentz transformation. As the momentum-energy of de Broglie particle is relatedto the transformation of relativistic waves only through analogy, we cannot find a way toobtain a wave that is relativistic and represents the original particle. As Schr¨odinger’s waveas created using this obscure de Broglie relation, the validity of it is questionable. Thismakes Dirac’s quantization of Schr¨odinger’s wave questionable as well.(iii) When it comes to the Gordon-Klein equation, which is accepted as an alternative to thefailed attempt of making Schr¨odinger’s wave mechanics relativistic, this is an attempt toquantize a relativistic relation in an unprecedented way. Does replacing classical variableswith corresponding Hermitian operators make the classical theory quantum?(iv) It is important to ask these questions instead of experimentally try to verify the predictionsof these incoherent theories where the core discussion is based only upon analogy andthe wrong assumption that Schr¨odinger’s wave equations are relativistic. On the top ofit, as quantum theory is inherently probabilistic, its experimental verification is highlycompromised. The verification is done as the statistical calculation of standard deviation.So, the claimed accuracy of the expensive experiments on particles is verified in the sameway we evaluate the accuracy of the prediction of the life span of automobiles.Regarding (iii) above: The Gordon-Klein equation does not conserve probability, theconservation of which is a major requirement imposed by the usual interpretation of quantummechanics. Quantum mechanics interprets the square of the modulus of a wave function’samplitudes as probabilities. For that reason, Schr¨odinger’s equation was made to make sure thatthe coefficients of wave functions were normalized at every point in time. This unfortunately isnot the case for the Klein-Gordon equation. It cannot therefore be seen as a valid replacementfor a relativistic version of Schr¨odinger’s equation. In order to conserve probability, a timeevolution equation needs to satisfy the following condition with regards to a wave function
Z (cid:12)(cid:12) ψ ( x, t ) (cid:12)(cid:12) dx = 1Furthermore, as the conservation must hold at any point in time, it has to be independent of timeevolution. This is to say that the Gordon-Klein equation must satisfy the following equation aswell ∂∂t Z (cid:12)(cid:12) ψ ( x, t ) (cid:12)(cid:12) dx = 0 . Now consider the Gordon-Klein equation1 c ℏ ∂ ∂t ψ ( x, t ) = ( ℏ ∇ − m c ) ψ ( x, t ) . Since it involves the second derivative with regards to time, it is clear that the first derivativeterm in the probability conservation expression will in general not disappear. Hence, theexpression will not produce the required value 0 and therefore the Gordon-Klein equation clearlydoes not describe the probability wave that the Schr¨odinger equation describes.The most important issue is that relativity theory as per Einstein is false and there is nopoint in trying to make classical theories relativistic. Classical theories such as the theory ofelectromagnetism have their own problems. Relativity theory is a wrong answer to the problemof classical electromagnetic theory. Considering the fact that the theory of relativity came fromthe wrong interpretation of the Michelson-Morley experiment, the entire quantum theory mustbe reevaluated.
9. Aether theory
Classical aether theory proposed by Descartes is yet another example of a continuum mediumproducing atoms (particles) through type lowering. Here a vortex, which is a substructure of theniversal medium aether, is supposed to be the atom which will induce the inter-atomic forces.We do not know how far we can push this idea forward as from the start this idea leads to acontradiction. Here is a brief discussion on the basic idea of Descartes on aether theory:(i)
Proposition : “No empty space can exist, therefore space must be filled with matter.” Descartes is saying that there is no such thing as a geometric space like e.g. the 3D spacethen. As Newton made it clear, no matter what we place in a geometric space, the spaceitself is a geometric space. Otherwise we cannot even define a motion which is, as Newtonsaid, a function from time to space. The other way of nailing down the problem is thatby saying “space must be filled with matter” Descartes is already assuming that space is acontainer that can be empty. Yet he is claiming that such thing does not exist.(ii)
Proposition : “Each part of this matter is inclined to move in straight paths, but becauseparts are close to each other, their interaction makes every part make circular motion.Each part making this circular motion is called vortices. They are often called ‘atoms’.Descartes also assumes that rough matter resists the circular movement more strongly thanfine matter.” It appears that just this claim requires a massive amount of physics. Thisrequires a fully developed and articulate theory of fluid. It is clear that the theory of fluidshould be something much more advanced than particle-based dynamics. In fact, we havea serious problem with the transition from particle dynamics to fluid dynamics. Indeed, itis becoming increasingly clearer that fluid dynamics is a very different theory from particledynamics. So, it appears that before we venture into aether theory we must have a solidunderstanding of what fluid mechanics is about. We certainly do not have a clear notion offluid dynamics and its relation with particle dynamics yet.(iii)
Proposition : “Due to centrifugal force, matter tends towards the outer edges of the vortex,which causes a condensation of this matter there. The rough matter cannot follow thismovement due to its greater inertia—so, due to the pressure of the condensed outer matter,those parts will be pushed into the center of the vortex. This inward pressure is gravity.” There is no such thing as centrifugal force. This is why this force is called a fictitious force.This is a good example of how the violation of the principle of relativity of Galileo occurswhen we consider reference frames under acceleration. This is why we do not allow referenceframes under acceleration. The effect of the so-called centrifugal force appears only when weconsider an object floating inside a container that is rotating about a centre of the rotation.This body tries to stay where it is when the centripetal force pulls the container down. Ifa body is fixed to the body of the orbiting container, it will not feel any centripetal force.After all this kind of situation is not theorizable as the classical particle dynamics does notallow us to consider things like orbiting container that has an inside space. To be precise,every object must be a point object in classical dynamics.Upon the ideas of Descartes, Huygens presented a more articulate vortex theory. Thefollowing is a short discussion on his work:(i)
Proposition : “
Huygens assumed that the free moving aether particles are pushed back atthe outer borders of the vortex and causing a greater concentration of fine matter at theouter borders of the vortices. This causes the fine matter press the rough matter into thecenter of the vortex.”
It is not clear how this distribution of the fine matter (aetherparticles) will occur. This argument requires a full theory of particle-based fluid mechanics.It requires a very advanced theory to explicate this process. More fundamentally, due tothe very concept of fluid, particle-based fluid is untenable. Particles and fluid cannot beunified. The former is discrete and the latter is continuous. As the space is a continuum,no matter how densely we pack the particles, we still have empty spaces in between thepacked particles. The only aether we can think of must be continuum fluid.ii)
Proposition : “According to Huygens the centrifugal force is equal to the force that acts in theopposite direction of the centripetal force.” Again, this claim needs a fully developedfluid theory. Huygens’ definitions of centripetal force and centrifugal force are differentfrom standard Newtonian version. Newton’s version is simple and clear. There is no suchthing as a centrifugal force. It is a misunderstanding of the fact that a free body inside acontainer will remain where it is despite the motion of the container under acceleration. So,there is no such thing as a centrifugal force. This “force” arose when Newton’s successorsmisunderstood Newton’s theory and included the reference frame.(iii)
Proposition : “Huygens also assumed that “bodies”, whatever they may be, must consistmostly of “empty space” so that the aether can penetrate them.” Huygens assumed thatthere is no such thing as empty space. Moreover, there is no definition of what a body is.(iv)
Proposition : “He further concluded that the aether moves much faster than the fallingbodies.” A more fundamental question is: what is causing the motion of aether (aetherparticles) such as fine matters and rough matters? The same question can be asked aboutthe issue of the motion of bodies.(v)
Proposition : “His theory could not explicate Newton’s law of gravity, the inverse squarelaw. Huygens tried to deal with this problem by assuming that the speed of the aether issmaller in greater distance.” Again, the same problem as above. We do not know whatthe speed of aether is until we learn what causes the motion of aether.The overall judgment on Huygen’s aether theory is that it failed to explicate the dynamicsof aether itself. It appears to be something even more complex than what we know asfluid dynamics. Fluid dynamics is a derivative of Newton’s dynamics that came with greatcompromises. Fluid dynamics is continuum dynamics as fluid is a continuum. The compromisesmade include, for example, that “pressure” is a highly questionable derivative of Newton’s forceas a vector. In dynamics, force is applicable only to a point matter because force is a pointedarrow. This concept was extended from a point to an area or to a volume, going backward of thedirection Newton took to make physics possible, which is to reduce a continuum body to a pointbody. This compromise and Newton’s mechanics combined created fluid dynamics. Therefore itis difficult to imagine that a theory of aether can be framed without using Newtonian mechanicsas it was the case with fluid dynamics.
We have discussed the difficulty in producing continuum dynamics (fluid dynamics) from particledynamics of Newton. Aether theory starts with a super fluid structure called aether and then,from the aether, induces particles called atoms. This is yet another example of type loweringtaking place in theoretical physics as the fluid is treated as a continuum made from points. So,the trouble associated with the type lowering manifests itself in any aether theory.Maxwell was compelled to reject his aether theory and accept the field theory for the theory ofelectromagnetism of Heaviside and Hertz as a shortcut to resolving the problem of nonlinearity.The problem here is that the theory of electromagnetic fields is not ontological as the conceptof a force field is not reality. It is counterfactual modality, as we discussed above. Moreover, itwas not understood that the force field theory violates the law of action-reaction and that it isa modal theory. Newton was aware of this and did not use the concept of a force field.As we showed, Lorentz transformation does not conserve electromagnetic wave equationsand so by proof by contradiction we can conclude that it does not conserve all of the Maxwell’saxioms, contrary to the claim by Einstein that all axioms of Maxwell are Lorentz-invariant. Ifall axioms of Maxwell were relativistic as Einstein attempted to prove, then the electromagneticwave equations must also be relativistic. This is to say that as Maxwell’s electromagnetic waveequations are not relativistic, the theory of electromagnetic field as per Maxwell is not relativistic. his is consistent with Einstein’s failure to prove that Lorentz transformations conserve allaxioms of Maxwell.Now let us go back to the aether theory which Maxwell tried to build in order to interpretaxiomatic electromagnetic field theory. Maxwell was reluctant to accept Heaviside’s andHertz’s axiomatic approach of compiling experimental lab results as the vector equations ofelectromagnetic fields. From our view point, Maxwell was correct in this reluctance as weunderstand that force fields violate the third law of Newton. Record shows that even afteraccepting the axiomatic force field approach in producing his axioms of electromagnetic filedtheory, Maxwell still was attempting to push forward with his aether theory. One of the reasonsfor this was that his theory of electromagnetic waves required a medium as all waves of physicsneed a medium, while the axiomatic theory does not provide it.There are some factors that made it very difficult for this project of Maxwell’s to succeed.In order to succeed, we have to consider at least the following issues:(i) Electromagnetic force must respect the law of action-reaction. This is in conflict withthe electromagnetic force fields which violate the law of action-reaction. Maxwell’sfield equations did not produce a solution here. This means that the right ontologicalelectromagnetism theory, if any, shall not agree with the description of Maxwell’selectromagnetic field axioms, which ignores the third law.(ii) Exactly the same argument applies regarding modality. As discussed above, force fieldsare not ontological reality. They are all counterfactual modalities. However, the desiredontological theory of electromagnetism cannot be a modality of any kind.(iii) All of the above is to question whether the desired aether theory, if any, would be a modaltheory or not. The answer naturally is “no”. A modal theory does not describe physicalreality. The concept of force fields must be rejected from this point of view. There is nosuch thing as an electromagnetic wave as there is no such an ontological reality as a modalwave. The correct view of what we call “electromagnetic wave” is the transmission at adistance of the vibration of electromagnetic force to a location where there is a charge.There is no transmission in between. This is to say that in reality there is no such thing aselectromagnetic waves.All of this suggests that trying to find an aether model for Maxwell’s electromagnetic fieldequations is futile. Instead, we should focus on developing the unjustly abandoned Gauss-Weber’s action at a distance theory of electromagnetism [31] which does not use problematicfield equations. The reason why Newton’s dynamics is a little less problematic is because it isnot a force field-based theory.
Remark (11)
Electromagnetic force depends on the speed of charge in either the Maxwell-Lorentz formalism (known as the Lorentz force) or the Gauss-Weber formalism. This makeselectrodynamics inconsistent as it violates the second law, which is a most important axiom of anydynamics. Therefore it is not quite clear how we can put gravitational aether and electromagneticaether together.
From logical perspective, it is clear that an aether theory, if any, would be more complexthan the theory that the given aether theory attempts to explicate. So, attempts to lay out anaether theoretic foundation of a physical theory will tend to be viciously circular. And even ifnot, there is no obvious way to verify such a meta-theory theoretically and empirically. Onemay say that we can do that through an empirical verification of the theory which the givenaether theory is to lay foundation for. Then it is nothing but vicious circularity.
10. Type lowering in mathematics
The problem of type lowering which we discussed in the preceding in the context of theoreticalphysics also appeared in mathematics in more acute forms. We will discuss some of them here.
Church developed a symbolic reductional calculus called λ -calculus that described the theoryof applying a function of one variable to another such function. By defining natural numbersas a special collection of such symbolic functions, Church simulated universal Turing machineshowing that his calculus has the same computational power as that of Turing machines. Butsince its invention, this formal calculus needed a proof that this reductional calculus is consistent.Dana Scott presented an interpretation of this symbolic calculus by considering a set equation D = [ D → D ]where the right hand side represents the set of all functions from the set D to itself. It is a wellknow fact that for any set D , [ D → D ] is larger than D . So, Scott introduced a complete latticestructure and restricted the elements of [ D → D ] to order continuous functions. In this way hemanage to cut down the size of [ D → D ] and establish a complete order isomorphism betweenthe left hand side and the right hand side of the above equation, presenting the “first model of λ -calculus”. This success came with a price. Now we identify a natural number with an elementof D which is infinitary. Therefore in Scott’s calculus we cannot decide if two natural numbersare equal. In Scott’s model, if a term is reducible to another term, semantically, there are manyterms that do not syntactically represent natural numbers but we cannot find that out using thesyntactic reduction of the calculus. Logicians call this kind of natural numbers recursive naturalnumbers. Consider the solution of the following set equation S = [ S → T ]where T is the truth value set { true, f alse } . Unfortunately this equation has no solution as theright hand side again is larger than the left hand side. Russel presented this problem as thefamous set as one (left side of the equation) and set as many (right hand side of the equation)paradox. This tells us that the claim that a set can be fully described by its characteristicfunction is not correct. This is yet another paradox of set theory. The method Scott developedto solve D = [ D → D ] gives us a solution as the collection of order continuous functions. But it isa well known fact that sets as characteristic functions in mathematics are not “order continuous”though they are “order monotonic”. So, we have to solve the equation as the collection of ordermonotonic functions from S to T . Apostoli and Kanda [3] found a solution as a set of monotonicfunctions from S to T . In this way the authors obtained the first consistent universal set theorywhich has the set universe S . But this set theory, called CFG, has a drawback. CFG cannotidentify two sets on the basis of the “extensional identity” which says that two sets are equal ifand only if they are made of exactly the same member sets. The so-called axiom of extensionalityfails. It is replaced by the axiom of indiscernibility which says that two sets are equal if and onlyif they belong to exactly the same members of S . The loss of extensional membership relationmakes it unusable in the mathematics for working mathematicians. This is yet another price wepay for type lowering. Recursion theory is a branch of mathematical logic developed by G”odel in which we definecomputable partial functions of natural numbers as functional programs over natural numbers.Using Turing machines, G¨odel showed how to calculate a natural number which uniquelyrepresents a functional program as a natural number. This process is called G¨odel numberingof partial recursive functions. This process certainly is a type lowering from the type ofomputable functions to natural numbers. Here each computable function will be represented byinfinitely many natural numbers each of which represents a functional program that computesthe computable function. This implies that there are infinitely many recursive programs foreach computable function. However, at the pain of contradiction, given two natural numbers,we can not computationally decide if the functions by these two numbers are the same or not.So, we loose the identity of computable functions.
11. Type lifting in mathematics and particle-based physics
Understanding all of these fundamental difficulties that the top down approach creates,mathematicians took the bottom up approach as a better methodology for building mathematicaltheories. A good example is the development of the theory of real numbers. It goes as follows:(i) Natural numbers: closed under the operations + and × . (ii) Integers: closed under the operations + , × and − . (iii) Rational numbers are precisely the fractions n/m of integers where m = 0 : closed under+ , − , × and ÷ . They are precisely the repeating infinite decimals.(iv) Irrational numbers are non repeating infinite decimals.(v) Real numbers are precisely the collection of all rational numbers and irrational numbers.Real numbers are also closed under + , − , × , ÷ . Moreover, they are closed under boundedlimit.From this definition of real numbers we can prove that the real numbers are a “boundedcomplete ordered field.” This is because the algebra ( R , + , − , × , ÷ , ≤ ) is an ordered field withthe linear ordering ≤ and is closed under bounded limit. Here R is the set of all real numbers.The mentioned closure under operations properties can readily be proved except the boundedlimit which requires a little bit of work.This is a simplest way of developing the theory of real numbers so that we can make it intoa calculus (mathematical analysis). One cannot rely upon intuitive, simplistic understanding ofreal numbers to develop calculus by replacing the concept of limit by a geometric intuition.This type lifting (or bottom up) approach is based upon the same philosophy of atomism inphysics where the most basic physical entities are atoms and from atoms we build more complexphysical entities. References [1] Anderton R 2007 private communication[2] Apostoli P and Kanda A 2006 Inconsistency of a Formal Theory, private circulation with Fefferman[3] Apostoli P and Kanda A 2009 Alternative Set Theories,
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