The Lagrange Function
Greg Oganessian, a Physicist from the Tata Institute of Fundamental Research, India, asked the question: “Has anyone succeeded in applying AD to quantum mechanics, and obtaining an analogue to well known Dirac equation in special relativity quantum mechanics? Also, does AD allow field-theoretical formulation in the guise of action principle? Can you write the Lagrangian for AD?” As will be shown, the problem is about what Lagrangian we are referring to. The complete answer is the following:
The Dirac equation comes originally from the Balmer equation and Bohr application of classic mechanics to the Hydrogen atom, suggested by Rutherford, which we now know as the Bohr atom, who introduced the notion of “quantification” created by Planck.
Sommerfeld applied the SR equations to the Bohr atom, Ulembeck and Goudsmit introduces the magnetic-electron and later another physicist introduces the wave mechanics, and so on, and so on, for 70 years and dozen of physicists in turn.
The success of QED is symbolized by the greater precision that the Dirac equation offers to explain fine structure, or Alpha constant, introduced by Sommerfeld in the above mentioned application. But many “hypotheses”, “theories” or “empirical corrections” were needed to reach the precision that the equation actually provides!
Many years ago, Carezani tried to apply AD equations to the Bohr atom, but AD yielded a more complicated equation and it was difficult to separate variables to make the integration possible. This is possible in the Sommerfeld case using the Cauchy method of the residues. Carezani and other collaborators did not find the solution and Carezani abandoned the effort to look for experimental proofs for his AD theory.
When Sommerfeld needed to find the momentum in polar coordinates, as two variables to be equalized to two quantum numbers, he could not define the momentum as the partial derivative of the SR KE (Kinetic Energy) equation because this “will not yield the momentum equation” and the variable separation.
Is the solution to use the Lagrangian from the classic mechanics? That’s impossible, we will have in the same situation. What can be done?
Sommerfeld needed to “create” a Lagrangian. That is what Sommerfeld did and he wrote the Lagrangian as
L = – mo c^2 ( 1 – B^2)^1/2 – [U]
This yields the solution sought by Sommerfeld. If we really tell the truth, this Lagrangian is an invented solution. This equation, mo c^2 (1 – B^2)^1/2, is the AD equation of mass variation (In the form of energy variation). Sommerfeld did not know this, of course, but this makes complete sense. The photon emission by atoms is an AD process of energy absorption with further decay!
This is revolutionary, but right now, not important.
The important thing is that the Lagrange function was created (invented) by Sommerfeld. We can invent Lagrangians as we need them. This was done in the case of Weak Interaction even though it is a very strange Lagrangian.
Lii = gm/2^1/2 [jdl(x)]*[jdl(x)]
What happens in AD when applying the creation method? (invention method). The Lagrange function could be
L = mo c^2/ (1 – B^2)^1/2
Taking the partial derivative we have
pr = 4 mo r (1-B^2)^1/2
pt = 4 mo r^2 t (1-B^2)^1/2
(t = angle theta)
in polar coordinates.
The AD momentum starting from the SR mass variation (As energy variation).
I do not know if Carezani started with the same equation, because it is possible to start directly with a momentum polar equation, as variable separation, and without using a Lagrangian!
This is all I know. Whether AD needs to invent another Lagrange function to use in other applications in the future is unknown to me, but we are free to do so.
Please, do not make an erroneous interpretation of what I am telling you. We do not reject the idea of creating something empirical to explain a Natural phenomenon when we don’t have the correct theory that explains it. But we need to be careful with the consequences. There is no fantastic theory in doing so. Many physicists are left with the idea that all of this is a powerful theory created by God, Einstein, or Olympus. Physicists are looking the Weak Interaction or the Standard Model as something HOLY. There is nothing HOLY in this. It is an invention to cover SR’s failure to explain the phenomenon.