Christoffel Symbols and Chiral Properties of the Space-Time Geometry for the Atomic Electron States

Quantum electron states, in the case of an improved Dirac equation, are linked to the Christoffel symbols of the connection of space-time geometry. Each solution of the wave equation, in the case of the hydrogen atom induces a connection which is completely calculated. This allows us to discover the global and chiral properties of the space-time connection, with spin 2.


Introduction
After L. de Broglie's discovery of the quantum wave [1], Dirac formulated his relativistic wave equation for the electron [2], correcting the non-relativistic Schrödinger equation, and conserving the probability density linked to the wave. The main success of this wave equation was its application to the case of the hydrogen atom: all the expected quantum numbers were obtained, as well as the true number of states, and the true energy levels [3]. Moreover, the Dirac equation explained the spin 1/2 of the electron. The main problem was the presence of negative energies which were then accounted for as due to charge conjugation.
Following de Broglie's ideas on the necessity of non-linearity to unify quantum physics and gravitation, an improved Dirac equation was studied [4]- [43]. This relativistic wave equation was extended to a wave equation of all fermions and anti-fermions of the first generation, as described in "Developing a Theory of Everything" [41]. The resolution of the wave equation in the case of the hy-drogen atom was completed in [42]. The aim of the present work is the use of these solutions as physical examples of the differential geometry linked to the quantum wave. The calculation of the Christoffel symbols was previously made only in the case of plane waves: geometry of space-time was characterized by a not null torsion and a null curvature. The torsion was linked to the mass term of the electron. The case of the electron in the hydrogen atom is much more complicated indeed, and also much more interesting because it will allow us to encounter the geometric aspect of chirality in quantum mechanics.

Tensorial Densities without Derivative
Early on, the Dirac theory encountered 16 of the 36 tensorial densities that may be computed without derivatives from the electron wave. This is easy to see with the Pauli algebra [5] [7] [8]. From the wave of the electron, ( ) , we get four space-time vectors (16 densities): And we get 20 densities (2 with 0 S and 6 for each k S ) as the components of: : The previous equalities use the three well-known Pauli matrices 1 σ , 2 σ , 3 σ and we let 0 : 1 σ = , identifying real numbers and scalar matrices. The 16 densities of the early theory were the components of the probability current where β is the Yvon-Takabayasi angle. We just encountered the right ( ξ ) and left (η ) parts of the wave. We also need:  .
Now we use the similitude D such as: We have And we get: Therefore the coefficients of the connection satisfy: ( )( ) 1 ; .
By using the D similitude we get: The Christoffel symbols of this connection satisfy (see [41] 4.1): For the complete calculation of these coefficients of the connection we need the following quantities: With the improved wave equation of the electron, we obtained in D.4 of [41] the following symbols: , 1, 2,3, 1, 2,3, .

Improved Dirac Wave Equation
The improvement of the Dirac equation was first presented in the frame of the Clifford algebra of space-time used by Hestenes, who considered the µ γ matrices of the Dirac theory as a basis of space-time [44] [45]. Read now in 3 Cl the Lagrangian density of the Dirac equation is: where 21 2 1 3 : i σ σ σ σ = = − , A is the electromagnetic potential space-time vector and X is the real part of X. The improved wave equation is obtained by simplifying the Lagrangian density as: 21: .
The improved equation which comes from this simplified Lagrangian density reads: This non-linear wave equation has the Dirac equation as linear approximation when the Yvon-Takabayasi β angle is null or negligible: The mass term of the improved Equation (32) comes from Lochak's theory of the magnetic monopole [46]- [53] in the particular case where the Dirac Lagrangian is the linear approximation of the simplified Lagrangian density. The first improvement of this simplification is the resolution of the problem of the negative energies, because the positron is no longer associated with non-physical negative energy (see for instance [41] 1.5.3 and 1.5.6). Another improvement is the partial decoupling of the left and right spinors, the wave equation also reads: The momentum-energy of the electron v qA m + is the sum of an electromagnetic part qA and an inertial part v m . Hence the improved equation may be generalized to all fermions, it is compatible with the entire gauge group of the Standard Model, , and it is also compatible with the relativistic invariance of general relativity. This is the reason for the appearance of the Christoffel symbols that we will calculate from the solutions of the improved equation in the case of the hydrogen atom.

Resolution in the Case of the Hydrogen Atom
The Dirac equation was solved as early as 1928 by the mathematician C. G. Darwin using the previous resolution of the non-relativistic equation for an electron with spin found by Pauli. This method used kinetic momentum operators, which is ill-suited to the resolution of a non-linear equation like (32). Fortunately another method exists, found more recently by H. Krüger [54], who discovered a very astute method of separation of variables in spherical coordinates. This uses: We use the following notations: : e e ; : sin , . sin H. Krüger obtained the remarkable identity: which with: gives also: . sin Then the Yvon-Takabayasi angle depends neither on the time nor on the ϕ angle. It depends only on r and θ . Therefore the separation of variables can begin similarly for both the Dirac equation and the improved equation. We have: For the hydrogen atom we have: where α is the fine structure constant. We have: Also the improved Equation (32) becomes: while the Dirac equation gives: Now we let: ; .
Therefore the improved equation reads: Conjugating the equations containing the conjugates we obtain the system: Next we let:   Then using a κ constant satisfying: the (62) system becomes:

Christoffel Symbols
We need to calculate ( ) XX X X σ = = and since: we get:

Terms with Index 0 and 3
For the 0 σ case we have: cos . sin Next we have: Using (63) and (64) we obtain: We let: These functions 1 1 2 1 2 1 2 , , , , , , r p p q q s s depend on r and θ , with values in  .
We get: We then get: This gives: We let: ; , 2 Using (72) we get: Next we have the following, with the radial and angular functions previously defined in (61): And we get, using (77) and (78): We then get:

Calculation of the Currents
We have: We then let, for any space-time vector  : We then have: This allows us a simplification of the scalar product. We get: † 1 1 2 2 1 2 Sum and difference of 0 d and 3 d are simple, which will be useful. We get:

Calculation of 12 Christoffel Symbols
We have: We also get the symbols: We obtain also: Then we have: So we get: This gives: Similarly we have: We calculate now: This gives:

Calculation of 16 Christoffel Symbols
We finally have all the pieces to finish the calculation of the Christoffel symbols. We encountered in (21) to (26) left and right terms: This allows to calculate the four ( ) We obtain: The separation between real and imaginary parts of this equality gives: Now we need: We let:  The same calculation must be made for the right terms: The separation between real and imaginary parts of this equality gives: Now we need (with .

Torsion and Symmetric Part of the Connection
The 64 Christoffel symbols may be calculated from the 4 7 28 × = independent terms, using the 36 relations described in (16), (17) and (18): The torsion tensor is usually defined as:

( )
Rodichev [55] studied the torsion in the frame of a Euclidean geometry with torsion. The present study acts in the frame of a space-time manifold, not Euclidean, so we get very different properties. In space-time the torsion tensor has 24 independent components while the connection contains 28 independent Christoffel symbols. We obtain these 28 symbols from the 4 8 where we have: These vectors allow us to obtain all Christoffel symbols as scalar products: The chiral structure of the connection appears here, from the fact that three definitions only act with indexes 0 and 3 while four definitions act with the left and right vectors. Moreover only the symbols containing the three indexes 1, 2, 0, contain the mass term 2mρ . This chirality is also linked to the electric gauge transformation, which acts everywhere in quantum mechanics, even in the non-relativistic case. It induces a rotation in the 1-2 plane, in the direction from 1 to 2: the rotation transforming 1 into 2 transforms 2 into −1. This partially remains in non-relativistic quantum mechanics, where the conservation of the probability density still acts.
The previous calculation must be completed by the examination of the different cases corresponding to the different quantum numbers characterizing the electron states. This will be carried out in the second part of this work. These states are different first from the sign of the κ number. This number is present in the previous calculation, not only by the value of E which contains 2 κ , but also directly in the r κ terms, and also in the λ terms: the values of λ are . It then happens that space-time turns more rapidly than the wave (or the wave turns less rapidly than the space time). Space-time geometry defined by the Christoffel symbols is animated not only by waves with a 2ζ phase, but also by waves with a 4ζ phase. This kind of phase was first encountered in general relativity as waves with spin 2. Our calculation shows that it is linked to the quantum wave of the electron, as suspected by Feynman [56].
Since the term of C′ with rank 0 is null from the n factor: The last term of this sum is null, as it is a difference of two equal terms. Then the sum contains one term less and we obtain: And we have: n n a a n a a a n a n a a a n a n a a which implies: Therefore we have: ( ) Consider now the case 0 κ < that means κ κ = − . Still for 0 λ > we have