429(3): The Spin Orbit Hamiltonian in m Space
Thanks again. The method of UFT428 can indeed be adopted for the energy levels of the Dirac atom:
E sub nj = E sub n ( 1 + (alpha / n) squared (n / ( j + 1/2) – 3 /4)
where E sub n are the energy levels of UFT428. We have for 2P sub 1/2
n = 1 , L = 1, S = – 1/2, J = 1/2
and for 2S sub 1/2
n = 1, L = 0, S = 1/2, J = 1/2
So in the Dirac atom there is no difference between the energy levels of 2P sub 1/2 and 2S sub 1/2 in the Dirac atom, but in m theory there is a difference as shown in UFT428, which calculated E sub n and found a splitting due to m theory as you point out. This is indeed the Lamb shift, and a triumph of m theory.
It would be very interesting to compute the spin orbit splitting of the 2P level with quantum relativistic m theory. As described in Atkins third edition the spin orbit energy level in the Dirac theory is:
E sub SO = (1/2) h c zeta (J(J+1) – L(L+1) – S(S+1))
We have for 2P sub 3/2: n = 1, L = 1, S = 1/2, J = 3/2, and for 2P sub 1/2: n = 1, L = 1, S = -1/2, J = 1/2. For hydrogen the splitting is 0.365 wavenumbers, and the theory of UFT429 can be used to find the effect of m (r) on this splitting. That would show that fine structure in atoms and molecules depends on the m space, another major discovery. The calculation depends on computing the expectation value < 1 / ( r cubed m(r) power half) >. For constant m(r) this is just a simple correction: 1 / (m(r) power half)( < 1 / r cubed>). This would make a very interesting section 3.
429(3): The Spin Orbit Hamiltonian in m Space
The Lamb shift could be computed as in UFT 428 via the total energy. The m function gives a splitting between the 2s and 2p states. That would be a non-relativistic approach without spin-orbit splitting. It would be better to compute the difference between 2S1/2 and 2P1/2 as in the note. It is not clear to me how to handle the quantum number factor
J*(J+1)-L*(L+1)-S*(S+1)
correctly. for 2P1/2 we have
L=0
S=1/2
J=L+S=1/2
which gives zero for the above factor. Only for 2P3/2 we obtain 1. So the modification of the expectation value <1/r^3> has no effect on 2P1/2.
Horst
Am 24.01.2019 um 07:08 schrieb Myron Evans:
429(3): The Spin Orbit Hamiltonian in m Space
Thanks to both colleagues! Prof. Sean Carroll’s notes are free online, Google "Spacetime and Geometry: an Introduction to General Relativity". They were developed from graduate lectures at Harvard, Chicago, UCSB, and now at Caltech. Carroll was an early supporter of ECE theory in an e mail to the late Prof. John B. Hart. The first three chapters of Carroll deal with geometry. Cartan geometry is briefly described at the end of chapter three. I developed all the proofs by Carroll in great detail in the UFT papers, so I gave complete proofs in all detail of the Maurer Cartan structure relations; the Cartan identity, and in the famous UFT88 extended the second Bianchi identity with torsion, developing it into the Jacobi Cartan Evans (JCE) identity of UFT313. The latter paper was the start of ECE2 and now m theory. Carroll also shows how the Einstein field equation is based directly on the 1902 second Bianchi identity, which was inferred in an era when torsion was not known. The corrected Bianchi identity is the JCE identity and this is completely different from the 1902 second Bianch identity. So the Einstein field equation is completely wrong because it was developed when torsion was unknown. When Cartan inferred torsion in Paris in the early twenties, he pointed it out to Einstein in a series of letters, but Einstein did not change his field equation of 1915. Starting in 2003, torsion has been incorporated into the whole of physics in the now famous Einstein Cartan Evans unified field theory (ECE). Part of this theory (the B(3) field) has been nominated several times for a Nobel Prize, and was recognized by a Civil List Pension in 2005 together with over fifty nominations for the major prizes. The m theory is described by Carroll in his chapter seven. It is based on the most general spherically symmetric space. The geometry of the space gives rise to a new source of energy. This is proven experimentally in the radiative corrections (for example the anomalous g factor of elementary particles and the Lamb shift). The interest in ECE is currently running at three million hits a year. Many thanks to the colleagues at AIAS / UPITEC, especially to Horst Eckardt for many original ideas and advances of his own.
Exactly GJE!!! A very profound set of ideas. Can’t wait to get to Carroll ch. 7 then re-read the notes.
Stephen C. Bannister, Ph.D. Assistant Professor, Economics Director, MIAGE Associate Director, Economic Evaluation Unit, Macroeconomics University of Utah Open calendar at: https://bit.ly/2vFymyYOn 1/23/2019 7:42 AM, Myron Evans wrote:
429(3): The Spin Orbit Hamiltonian in m Space
Agreed with GJE. There is enough material for a few essays, the extra energy from the m space was found to be the same using the Euler Lagrange and Hamilton methods. The m theory so far is the theory with general m and n, with the simplifying assumption m = n = function of r. If m is not equal to n then many new results will be inferred at the expense of using two parameters instead of one. I will write up UFT429 now and then rework UFT418 to UFT429 with m not equal to n.
429(3): The Spin Orbit Hamiltonian in m Space
Key statement of course is that m space acts as a source of energy that is completing missing in the standard model. Almost time for another classic essay?
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