View source for Limitation of Applicability of Einstein's Energy-Momentum Relationship

{{Infobox paper
| title = Limitation of Applicability of Einstein's Energy-Momentum Relationship
| url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_5204.pdf Link to paper]
| author = [[Koshun Suto]]
| keywords = [[Special Theory of Relativity]], [[Einstein?fs energy-momentum relationship]], [[Klein-Gordon equation]], [[Dirac equation.]]
| published = 2005
| journal = [[General Science Journal]]
| num_pages = 12
}}

'''Read the full paper''' [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_5204.pdf here]

==Abstract==

When a particle moves through macroscopic space, for an isolated system, as its velocity increases, the kinetic energy and hence total energy of the particle will increase. However, according to classical quantum theory, when the momentum and kinetic energy of an electron inside a hydrogen atom increases, total energy decreases. From this truth, it is evident that the equation for Einstein's energy-momentum relationship does not hold true inside a hydrogen atom.

==Overview==

Koshun Suto's paper argues that Einstein's relation ''E''<sup>2</sup> = ''c''<sup>2</sup>''p''<sup>2</sup> + ''E''<sub>0</sub><sup>2</sup> is a statement about isolated particles in free space and fails inside a bound system. His observation is that for a free particle, increasing momentum increases total energy, whereas for an [[Electron|electron]] falling to a lower level of a [[Hydrogen Atom|hydrogen atom]] the momentum and kinetic energy increase while the total energy ''decreases''. He concludes that the sign of the momentum term must be reversed for a bound electron, and proposes

: (''E''<sub>0</sub> + ''E''<sub>n</sub>)<sup>2</sup> + ''c''<sup>2</sup>''p''<sub>n</sub><sup>2</sup> = ''E''<sub>0</sub><sup>2</sup>&nbsp;&nbsp;(''n'' = 1, 2, ···, ''E''<sub>n</sub> < 0),

where ''E''<sub>n</sub> is the usual (negative) Bohr level and ''E''<sub>0</sub> + ''E''<sub>n</sub> is what he calls the electron's total energy "defined in absolute terms," measured from the free electron at rest rather than from zero at infinity.

Having obtained this relation, Suto quantises it as Einstein's relation is quantised to give the Klein–Gordon equation, and follows [[Paul Dirac]]'s route of factorising the second-order operator. He finds a set of 4&times;4 coefficient matrices differing from Dirac's, and offers them not as a refutation but as "another form of Dirac's equation." The paper is thus a claim about the ''domain'' of a relativistic identity rather than an attack on relativity as such: Suto explicitly says he does not disagree with quantum mechanics.

==The argument==

===Why the standard derivation does not carry over===

Suto begins from the textbook route to ''E''<sup>2</sup> = ''c''<sup>2</sup>''p''<sup>2</sup> + ''E''<sub>0</sub><sup>2</sup> (he cites A. P. French's ''Special Relativity''), whose key step is d''E'' = ''v'' d''p''. That step relies on the work–energy theorem d''K'' = ''F'' d''x'' = (d''p''/d''t'') d''x'' = ''v'' d''p'', together with the assumption that the total energy and the kinetic energy increase together, d''E'' = d''K''.

Inside the atom, he argues, the second premise fails. If the potential energy of a hydrogen atom falls by &Delta;''V''(''r''), energy conservation gives &minus;&Delta;''V''(''r'') = &Delta;''K'' + &#295;&omega;: half the released potential energy raises the electron's kinetic energy and half leaves the atom as a [[Photon|photon]]. Hence &Delta;''K'' = &minus;&Delta;''V''(''r'')/2 and &Delta;''E'' = &Delta;''V''(''r'')/2, so

: d''E'' = &minus;d''K'', and therefore &minus;d''E'' = ''v'' d''p''.

Appendix B supplies the classical backing from the circular Bohr orbit: ''mv''<sup>2</sup>/''r'' = ''e''<sup>2</sup>/4&pi;&epsilon;<sub>0</sub>''r''<sup>2</sup> gives ''mv''<sup>2</sup>/2 = ''e''<sup>2</sup>/8&pi;&epsilon;<sub>0</sub>''r'' = &minus;''V''(''r'')/2, so ''E'' = ''K'' + ''V'' = &minus;''K'' = ''V''/2.

===Integrating the reversed relation===

Combining ''p'' = ''mv'' with ''m'' = ''E''/''c''<sup>2</sup> gives ''E'' = ''c''<sup>2</sup>''p''/''v''. Multiplying this by &minus;d''E'' = ''v'' d''p'' yields ''E'' d''E'' = &minus;''c''<sup>2</sup>''p'' d''p'', which integrates to ''E''<sup>2</sup> = &minus;''c''<sup>2</sup>''p''<sup>2</sup> + const. Suto notes that the constant "should normally be determined through experimentation," but takes it, "from the analogy" with Einstein's relation, to be ''E''<sub>0</sub><sup>2</sup>:

: ''E''<sup>2</sup> + ''c''<sup>2</sup>''p''<sup>2</sup> = ''E''<sub>0</sub><sup>2</sup>.

He then argues that the ''E'' appearing here must be an absolute quantity including the rest energy. The conventional Bohr energy ''E''<sub>n</sub> = &minus;(1/''n''<sup>2</sup>)(''m''<sub>e</sub>''e''<sup>4</sup>/2(4&pi;&epsilon;<sub>0</sub>&#295;)<sup>2</sup>) is measured from zero at infinite separation and is negative; but an electron at rest at infinity "should have rest mass energy ''E''<sub>0</sub>." He therefore defines ''E''<sub>ab,''n''</sub> = ''E''<sub>0</sub> + ''E''<sub>n</sub> and arrives at the paper's headline result, equation (4.4).

===Quantisation and the coefficient matrices===

Section 5 applies the substitutions ''E'' &rarr; i&#295;&part;/&part;''t'', '''''p''''' &rarr; &minus;i&#295;&nabla;. Applied to Einstein's relation these give the Klein–Gordon equation; applied to Suto's relation they give the same wave operator with the sign of the spatial derivatives reversed. Following Dirac, he writes a first-order equation with unknown coefficients &alpha;<sub>''i''</sub> and &beta;, squares the operator, and matches. The conditions he obtains are the familiar anticommutation relations &alpha;<sub>''i''</sub>&alpha;<sub>''j''</sub> + &alpha;<sub>''j''</sub>&alpha;<sub>''i''</sub> = 0, &alpha;<sub>''i''</sub>&beta; + &beta;&alpha;<sub>''i''</sub> = 0, &beta;<sup>2</sup> = 1 — but with &alpha;<sub>''i''</sub><sup>2</sup> = &minus;1 in place of Dirac's &alpha;<sub>''i''</sub><sup>2</sup> = +1. He exhibits a 4&times;4 solution (his equation 5.8) differing from Dirac's standard set (5.9) by factors of i, and a four-component wave function, and declines to discuss the significance of the conditions further.

===Appendix C===

The final appendix takes up Gasiorowicz's relativistic scalar treatment of the bound electron, the operator version of (''E'' &minus; ''V'')<sup>2</sup> = ''c''<sup>2</sup>''p''<sup>2</sup> + ''E''<sub>0</sub><sup>2</sup>. Suto notes that if ''E'' is read as the conventional bound-state energy, then ''E'' &minus; ''V'' = (''K'' + ''V'') &minus; ''V'' = ''K'', which would require ''K''<sup>2</sup> > ''E''<sub>0</sub><sup>2</sup> — an inequality that "should normally not be possible." Reading ''E'' instead as ''E''<sub>0</sub> &minus; ''K'' repairs it and returns (''E''<sub>0</sub> + ''K'')<sup>2</sup> = ''c''<sup>2</sup>''p''<sup>2</sup> + ''E''<sub>0</sub><sup>2</sup>, which he offers as "strong evidence to validate" his absolute definition of total energy.

==Assessment==

The paper is careful, modest in tone, and makes its assumptions visible — including the one it cannot justify, where the constant of integration is fixed "from the analogy" rather than from anything derived. The physical observation that opens it is correct and worth stating: for a Coulomb-bound electron the virial theorem gives ''E'' = &minus;''K'', so tighter binding really does mean more momentum and less total energy, and a reader who imports the free-particle intuition will get the sign wrong. Appendix B's derivation of that fact is textbook-correct.

The arithmetic also works. Putting the ground state into the paper's own equation (4.4), with ''E''<sub>0</sub> = 510999 eV and ''E''<sub>1</sub> = &minus;13.606 eV, gives ''cp''<sub>1</sub> = &radic;(''E''<sub>0</sub><sup>2</sup> &minus; (''E''<sub>0</sub>+''E''<sub>1</sub>)<sup>2</sup>) = 3729 eV, against the Bohr value ''E''<sub>0</sub>&alpha; = ''m''<sub>e</sub>''c''<sup>2</sup>/137.036 = 3728 eV — agreement to about one part in 10<sup>4</sup>. The relation therefore does reproduce the Bohr momentum.

But that agreement is not evidence for the equation, because it is an identity. Expanding (''E''<sub>0</sub>+''E''<sub>n</sub>)<sup>2</sup> + ''c''<sup>2</sup>''p''<sub>n</sub><sup>2</sup> = ''E''<sub>0</sub><sup>2</sup> gives ''c''<sup>2</sup>''p''<sub>n</sub><sup>2</sup> = &minus;2''E''<sub>0</sub>''E''<sub>n</sub> &minus; ''E''<sub>n</sub><sup>2</sup>, i.e. ''p''<sub>n</sub><sup>2</sup>/2''m''<sub>e</sub> = |''E''<sub>n</sub>| &minus; ''E''<sub>n</sub><sup>2</sup>/2''E''<sub>0</sub>. To leading order this is exactly the virial statement ''K'' = &minus;''E''<sub>n</sub> that Appendix B started from, dressed in relativistic notation. The whole content of the new equation, apart from a term of order (''E''<sub>n</sub>/''E''<sub>0</sub>)<sup>2</sup>, is the non-relativistic ''p''<sup>2</sup> = 2''m''''K'' — put in at the start and recovered at the end.

Worse, the paper gives that second-order term in two mutually contradictory forms. Equation (4.4) yields ''c''<sup>2</sup>''p''<sup>2</sup> = 2''E''<sub>0</sub>''K'' &minus; ''K''<sup>2</sup>. Appendix C's equation, offered as confirmation of the same scheme, is (''E''<sub>0</sub> + ''K'')<sup>2</sup> = ''c''<sup>2</sup>''p''<sup>2</sup> + ''E''<sub>0</sub><sup>2</sup>, which yields ''c''<sup>2</sup>''p''<sup>2</sup> = 2''E''<sub>0</sub>''K'' + ''K''<sup>2</sup>. The two differ in the sign of the only term that distinguishes the proposal from ordinary Bohr theory, and the appendix presented as corroboration in fact reproduces Einstein's relation unchanged, with the standard total energy ''E''<sub>0</sub> + ''K''. The internal case for the reversed sign therefore does not close.

The derivation has a further equivocation. d''K'' = ''v'' d''p'' is the work–energy theorem along a mechanical trajectory; d''E'' = &minus;d''K'' is a relation between two ''different'' stationary states connected by the emission of a photon. Treating the latter as a differential along a continuous path in (''E'', ''p'') space, and integrating it, silently converts a discrete radiative cascade into a smooth mechanical process. Nothing in the paper justifies that step, and it is the step that produces the reversed sign.

Set against measurement, the proposal is under-determined rather than wrong: equation (4.4) contains only the principal quantum number ''n'', so it assigns one energy to each shell and predicts no fine structure at all. The [[Dirac Equation]] with a Coulomb potential, by contrast, gives the ''n'',''j'' dependence that matches the observed 2P<sub>3/2</sub>–2P<sub>1/2</sub> splitting of 10 969 MHz in hydrogen, and the residual 1057 MHz Lamb shift between 2S<sub>1/2</sub> and 2P<sub>1/2</sub> is the classic confirmation of [[Quantum Electrodynamics]]. A relation with no [[Angular Momentum|angular-momentum]] label cannot address either. The framing question is also arguably a category error: Einstein's relation connects a free particle's energy and momentum, and the standard treatment of a bound electron does not apply it to the bound-state energy but embeds the potential in the wave equation, precisely as Appendix C's Gasiorowicz form does.

Finally, the quantised version carries a cost the paper does not weigh. Requiring &alpha;<sub>''i''</sub><sup>2</sup> = &minus;1 means the &alpha;<sub>''i''</sub> cannot be Hermitian, so the resulting Hamiltonian is not Hermitian; energies need not be real and probability need not be conserved. And ''E''<sup>2</sup> + ''c''<sup>2</sup>''p''<sup>2</sup> = ''E''<sub>0</sub><sup>2</sup> has no real solutions for ''cp'' > ''E''<sub>0</sub>, imposing a hard ceiling ''p'' &le; ''m''<sub>e</sub>''c'' on any electron the equation can describe. Calling the result "another form of Dirac's equation" understates how much has changed. The paper is honest and readable, and its opening observation about the sign of d''E'' inside an atom is sound; the construction built on it recovers a known identity and contradicts itself on the one point where it says something new.

==See also==

* [[Koshun Suto]]
* [[Hydrogen Atom]]
* [[Electron]]
* [[Dirac Equation]]
* [[Paul Dirac]]
* [[Niels Bohr]]
* [[Atom]]
* [[Mass]]
* [[Spin]]
* [[Fine Structure Constant]]
* [[Quantum mechanics]]
* [[Quantum Electrodynamics]]
* [[Relativity]]

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