An Extended Theory of General Relativity Unifying the Matter-Antimatter States

This paper follows one of our earlier publications detailing an extended theory of General Relativity (GR) with two types of curvatures. In this framework, we first extend the Riemannian spinor theory in the formalism of A.Lichnerowicz. We then suggest that the non vanishing covariant derivative of the metric tensor linked to the extra curvature of our extended GR theory, be related to the trace of a specific Hermitean matrix. In the Riemannian theory, this matrix reduces to the classical gamma matrix that is precisely designed to distinguish the spin one half fields from its anti Fermionic counterpart. Our extended General Relativity theory therefore appears as unifying the Fermion-anti Fermion field states, and would legitimate the hidden medium concept introduced by Louis de Broglie in his attempt to explain the matter-antimatter symmetry.


General
We recall here the main results of the Extended Theory of General Relativity (EGR theory) which is published in [Marquet. P, 2009].
Unlikein the Riemannian theory, we consider a four dimensional manifold M endowed with two types of curvatures implying a semi-affine connection.
(space-time latin indices run here from a = b = 1, 2, 3, 4) On the manifold M referred to a natural basis {e a }, it is known that the connection coefficients  c ab can be generally decomposed as follows ab } are the Christoffel symbols of the second kind.K c ab is the contorsion tensor which includes the torsion tensor ab)S = ½ g ce (D b g ae + D a g be -D e g ab ) is the segmental connection formed with the general covariant derivatives (denoted by D), of the metric tensor: This latter connection characterizes a particular property of the manifold M related to a specific type of curvature, known as the segmental curvature.
In a dual basis θ a defined on M, to any parallel transported vector along a closed path, can be then associated the following forms We postulate that the manifold M is torsion free with 40 general symmetric connection coefficients Γ a bc = { a bc } + (Γ a bc ) S (1.0)The components R e acd of the general curvature tensor are assumed to have the same form as the ones of the Riemann tensor we define the tensor density from which we build the invariant density Following a suggestion by Einstein, we consider the connection coefficients (1.0) as the true field variables, so that for a variation δΓ a bc , the action principle δ S = ∫ H d 4 x = 0 (1.4) be written ] d 4 x = 0 and integrating by parts, we obtain δ∫[(R bc ∂R bc / ∂Γ a de ) -∂ f ((R bc ∂ R bc )/∂(∂ f Γ a de ))] δΓ a de + ∫∂ f [R bc ∂ R bc δΓ a de / ∂(∂ f Γ a de )] d 4 x = 0 (1.5)If the variationsΓ a de are zero on the integration boundary, the last divergence integral does not contribute, so the condition (1.5) reduces to The stationary principle for symmetrical Γ a bc , leads to the eulerian equations Q a From the expression (1.1), we next calculate the derivatives (1.12)

Connection coefficients
In order to determine the exact form of the connection, we first decompose R bc into the metric tensor density G bc = (-g) 1/2 g bc and two further parts E bc + A bc , (A bc being antisymmetric): .13) the quantity between brackets represents the Riemann Ricci tensor density so that when A bc = 0, (1.13) reduces as it should, to (1.13)bis.
Consistency of our theory thus leads to impose the following constraint we next set with J a = (-g) 1/2 J a , Let us now check whether the condition G bc ', c = 0 re-instates a Riemann connection whereby the general curvature tensor R ab (1.1) would reduce to the riemann Ricci tensor R ab .
By contracting (1.12) on c and a, and taking into account (1.13)ter, one finds If we now insert (1.15) into (1.12), the conditions (1.8) eventually read Dividing by (-g) 1/2, one obtains (1.17) so, multiplying through by g bc, taking account of g ba g ca = δ c b , d g = gg bc dg bc = -gg bc dg bc , we infer Γ e ae = ∂ a ln (g) 1/2 + 1/3 J a (1.18) Substituting this relation into (1.17) and multiplying by g bd g fc (noting that dg ed =g ec g bd dg bc) , we eventually find  .19)shows that when J a = 0, D a g bc = 0 and G ab ', b = 0 From (1.18) the condition J a = 0 implies [Γ b ae ] J = 0, so we actually proved that in this case, the generalized curvature tensor R ab (1.1) reduces to the Riemann Ricci tensor R ab .

The EGR Curvatures
1.4.1 The EGR fourth rank curvature tensor From the connection the EGR curvature tensor can be inferred as with the definition (1.21) we have The relation (1.24) eventually leads to the contracted tensor we then have once more the splitting (1.34) 1.5 The EGR field equations

The EGR Einstein tensor
Here the subscript EGR is omitted for clarity purpose.
Unlike the Riemann curvature tensor, the EGR curvature tensor is no longer antisymmetric on indices pair cd (1.35)bis Raising the indice d in (1.32 )bis and contracting on d and b as well as on e and c, we obtain we next replace R cd ca by its value taken from (1.35)bis, and we eventually find ) which is formally the generalized conservation law for the tensor (re-instating the subscript EGR) (1.39) We call it the EGR Einstein Tensor.
In the Riemannian regime, this tensor obviously reduces to the usual Einstein tensor G da = R da -1/2g da R within the classical GR field equations.

The EGR field equations
With the equations (1.39) wecan form the EGR source free field equations [G da ] EGR = 0 which are the tensor transcription of the tensor density equations and whose conservation law is expressed by It is now easy to determine the expression of the tensor which is the conserved Einstein 's tensor G ab as inferred from the Bianchi 's identities in classical GR When a massive (antisymmetric) tensor T ab () is present on the right hand side, the field equations become (: Einsteins 's constant, with c = 1): (1.41)In the EGR theory, the mass density ' is now sligthly modified by its own gravity field.
[ paper to appear in the A.Zelmanov Journal, vol. 3 ].

General
The EGR theory exhibits a non vanishing covariant derivative of the fundamental metric tensor.
How, can we relate this non metricity to unify the spin 1/2 field symmetry ?
In classical spinor theory, consider the classical γ 5 matrix which includes all other Dirac gamma matrices  a . In the massless case, the sign of γ 5 distinguishes either one of the anti/Fermionic field.
 Amazingly, for the massive field, the Dirac equation does not require the matrix γ 5 for describing the electron /positon symmetry: one can thus wonder whether the current theory is lacking some consistency.
We may avoid this last ambiguity by achieving a full united theory in the light of our GR extension.
To this effect, we suggest to introduce a 4 x 4 Hermitean matrix that precisely generalizes the Riemannnian gamma matrix γ 5 .This particular matrix is related to the non metric part of the EGR connection.
In the massless situation, we will consider the neutrino (Fermionic) particle which is assumed to have zero mass.
For clarity of the text, the mathematics are here kept to a minimum and for deeper analyses, the reader is invited to refer to the suggested bibliography.

Space-time orientation
Let M be a C ∞ differentiable 4-manifold which is referred to general frames which are themselves elements of a principal fiber space denoted by E(M) .
In the Riemannian scheme, this space is denoted by E(V4) and admits a structural group: the homogeneous (or full) Lorentz group denoted by L(4) [Richtmyer. R.D. 1981].
With respect to orthonormal frames y = {e 4 , e α } ( = 1, 2, 3), the metric can be written on an open neighborhood of the Riemann manifold V4 as where the θ a are the Pffafian forms in the considered region.
The full (homogeneous) Lorentz group L(4) then includes 4 components:  The identity component denoted by L 0 (4), which is the Lorentz connected group.
To the matrix F = (F b' c )  L(4), can be assigned the relevant component of the Lorentz group in the following way: The global signature ε F of the matrix F is: (+/-)1 according to the sign of det F.
The time signature t F of the matrix F is: (+/-)1 according to the sign of F 4' 4 .The spatial signature s F of F can be obtained by the product ε F t F .
The four components of L(4) may thus be characterized by the following signatures The global orientation of the manifold V4 is then a pseudoscalar denoted by ε, whose square is 1, and which is defined with respect to the frames y of E(V4), as the component As a result, for two global orientations on V4, ε and ε', we have at most two total orientations ε and -ε.
Using the pfaffian forms θ a , we are now able to define the orientable volume element η as The manifold considered here is always understood to be globally hyperbolic.

The Dirac gamma matrices
Let now R be the real numbers set on which the vector space is defined.
This vector space is spanned by the 16 matrices where I is the unit matrix, and the γ a are the Dirac matrices.[Dirac. P.A.M., 1931] (small latin indices a = b = 1,2.3,4) These matrices are formed with the Pauli matrices σ k -1 ) which are Hermitean and anticommuting: We recall here further properties of these matrices: The matrix γ 4 is Hermitean whereas the γ k are non-Hermitean.
The expression γ aA B ∂ a is known as the Dirac operator and it is customary to omit the spinor indices A, B by simply writing (2.4) so that we have the usual representation Fixing now a metric tensor g ab , we write the SR relation (2.2) as which is verified in the standard representation of real gamma matrices, with the following elements (2.7) We note that

The EGR spinor concept
From E(M), we define a principal fiber space denoted by S(M) whose each point z represents a general spinor frame: at this stage, it is essential to understand that Spin (4) is not here the structural group.
M is therefore a vector space of 4 x 4 matrices with complex elements, which is not acted upon by the Spin (4) group.
Let us denote by  the canonical projection of S(M) onto M , and p the projection of S(M) onto E(M), so that a tensor of M is referred to its frame by y = p z.
The contravariant 1-spinor  at x  M , is defined as a mapping z   (z) of  -1 (x) onto M.
The covariant 1-spinor  at x , is a mapping z   (z) of  -1 (x) onto the space M' dual to M.
The contravariant 1-spinors  form a vector space Sx on the complex numbers, whereas the covariant 1-spinors  form the vector space S'x dual to Sx.
Let B EGR be the automorphism into Sx and defined as follows We have thus introduced a specific 4 x 4 Hermitean matrix denoted by (γ 5 ) EGR and which generally satisfies We now postulate that the gamma matrix (γ 5 ) EGR is related to the metric EGR covariant derivative D c g ab ≠ 0, which in the EGR theory, is linked to the 4-vector J a that spans the non metric part of the EGR global connection ( d ab ).Therefore, we set where k is a real positive constant.This is fully legitimate since the skew components of the Hermitean matrix (γ 5 ) EGR are always real.
The exact form of the hermitean matrice (γ 5 ) EGR is of no immediate importance as long as it satisfies the above relations.

The EGR spinor connection
Within a neighborhood V of E(M), we define a connection 1-form  that is represented by either of the two matrices ( a b ) EGR or ( ab ) EGR and whose elements are linear forms.The matrix defines the spinor connection corresponding to .
The elements of N EGR are given by the local 1-forms with respect to the frames in V, The corresponding coefficients of N EGR are written (2.15)

The EGR covariant derivatives of the tensor-spinors
Inspection shows that the EGR absolute differential of the gamma matrices are given by The EGR covariant differential is still denoted by D.
Let us remark that this differential is generally not zero.
With (2.15) it can also be shown that the EGR covariant derivative of a spinor and for the covariant 1-spinor  By tensor product, one obtains the components of the EGR covariant derivative of any arbitrary tensor-spinor.

Introducing now the EGR Dirac operators W and W ' as
for a massive spin 1/2 field, the EGR Dirac equations are written where the mass m 0 is now slightly modified according to the EGR concept.

2.5
The EGR Lichnerowicz operator

Fundamental identities
For the following demonstrations, we need some important identities, so we first evaluate the EGR vector-spinor T of type ( 1 1 ) (contravariant/covariant), omitting the EGR subscript.
one obtains the identity from which can be inferred Omitting the subscript EGR for N, the matrix of Ω S is composed of local 2-forms expressed by We call EGR curvature tensor-spinor P, related to the EGR spinor connection, the tensor-spinor antisymmetric on c and h, whose components in a dual basis, are represented by From the expression of N EGR , we infer therefore, in accordance with equation (1.03), P verifies the analogue of the Bianchi's identity One can show that the EGR spinor connection curvature is related to the global curvature by where  a b is the Riemannian curvature 2-form in the dual basis Let us now denote by the Laplacian acting on the contravariant 1-spinor .
This second order operator Δ EGR can be explicitly written if we define the Laplacian of a contravariant 1-spinor (resp.covariant) as and expanding the Ricci identity yields so taking into account (2.7) and (2.27), also due to the symmetry of P A B , cd , one finally obtains from (2.24) and (2.32) In both cases, we have the same operator with the EGR curvature scalar: The operator (2.37) is the EGR version of the Lichnerowicz operator, (or Lichnerowicz-Weitzenböck formula) which is classically used to express the Dirac second order propagation equation.

A specific gamma matrix
At the Riemannian approximation, we postulate that (γ 5 ) EGR  γ 5 , M  V4 which is endowed with the time orientation t, thus we must find back the classical results of the Riemannian spinor theory.
In this case, at the point x on the manifold V4, the vector space consisting of all forms and antisymmetric tensors is S isomorphic to the space of all 2-spinors of type ( 11 ).
Among all complex valued p-forms  (p) one can show that  only one form such that  = S  (3.0) On V4 endowed with the global orientation ε, is defined the real volume element form (2.1), to which corresponds by S the ( 11 ) spinor S η.
We now define the matrix isomorphic to η as and displaying the components, we can show that On an open neighbourhood of V4 we may express this matrix with respect to spinorial indices as Since we have adopted the standard representation for the gamma matrices, this spinor is purely imaginary and satisfies These relations clearly appear as a particular Riemannian case of the EGR relations (2.11) and (2.12).
The contravariant 1-spinor , has also now its covariant 1-spinor counterpart which we write as and which is classically known as the Dirac conjugate.(see next § 3.2) Conversely, any covariant 1-spinor  is now the image of the contravariant 1-spinor t β .
Reducing to the Riemannian scheme leads to J a = k tr ( 5 ) EGR = 0, which thereby re-instates the metricity of the classical spinor theory for which holds the known condition tr γ 5 = 0 (3.8)

The Charge conjugation and the Adjoint operation
There exists an antilinear mapping C of Sx onto itself, which maps a 1-spinor  to another 1-spinor such as C is known as the charge conjugate operation.
In particular, the charge conjugate of the covariant 1-spinor  is The relation (2.6) results from the identity (u a γ a) ² = -(u a u b g ab ) I where u a  C (complex numbers), thus from (2.8) we find Introducing now the Hermitean matrix we derive from (3.9) the important relation By defining an antilinear mapping A of Sx onto S'x, as we have the Dirac adjoint operation A.

The fundamental spinor 2-form
We now consider a contravariant 1-spinor  which satisfies C = * On the other hand A = t ̃ β, so we infer The Dirac adjoint operator and the charge conjugation anticommute on the 1-spinors.
We define now a covariant spinor 2-form denoted by ω, which allows to identify either the contravariant or covariant 1-spinor.
This form can be regarded as the analogue of the fundamental tensor g defined on the riemannian manifold V4.
Consider again the contravariant 1-spinor  to which corresponds the covariant 1-spinor AC .
ω then represents an isomorphism (linear mapping of Sx onto S'x), that is a covariant 2-spinor whose components are ω AB , so that and which is antisymmetric.
A fundamental vector-spinor γ of type ( 1 1 ) can be contracted by applying ω to yield a covariant vector 2-spinor which is symmetric on spinor indices Unlike the EGR picture (2.16), the Riemannian covariant differential of the gamma matrices is known to be zero Let us now summarize the results obtained above a) In the canonical spinor connection, the absolute differential of the fundamental tensor-spinor γ is zero.
b) The charge conjugation and the Dirac adjoint operation commute with the covariant derivative.
c) The covariant derivative of the spinor 2-form ω is zero.
In the EGR theory, it is obvious that none of these conditions are fulfilled, in particular the condition c) which like in Riemannian GR, reflects here the metricity condition of the classical spinor theory.

The double spinor formulation
At the Riemannian approximation, B EGR  B, and now verifies B ² = Identity, (  ).
Therefore, B admits two eigenvalues: + /-1, so we may decompose Sx into a sum (Sx) +  (Sx) -of the eigen subspaces of B.
An element of (Sx) + is a spinor of positive type denoted by  + .
An element of (Sx) -is a spinor of negative type denoted by  -.
We check that Furthermore, if   Sx, we have

Riemannian Dirac operations
Let us now express the Dirac operators as The charge conjugation C commutes with the covariant derivative and therefore The Dirac adjoint operator A also commutes with the covariant derivative and we can show that From these two relations, we infer ω Z '  = Z ' ω 

The Dirac current vector density
The Dirac Lagrangian for a massive fermion field is shown to be An alternative expression is given by Since these forms differ only by a divergence which vanishes at infinity, they generate the same action and correspond to the same physics.
Following Noether 's theorem, we now apply the invariance rule to L D (3.26) upon the global transformations (where U is a positive scalar).
 → e i U ,  →  e -i U for linear transformations of , the corresponding Lagrangian variation is Thus we have defined the Dirac current vector density (j a ) D which is a true vector.
To prove this, we write (j a ) D with the aid of (3.12) (j a ) D = i t ̃ β γ a  by applying the usual adjoint operation (j a ) D * =i t γ̃ a β and taking into account γ̃ a β = -β γ a , we eventually find which concludes the demonstration.

Special features of the Dirac current
In the massless case m 0 = 0, the Dirac current density can be split up in 2 parts.
Let us write the first part as From the relation (2.6), we find By virtue of the Dirac equations Z  = m 0  and Z ' = m 0 , we eventually get The Dirac current density j a can therefore be decomposed as with the (field) spin current and the (field) orbital current A similar result can be found in [Bogoliubov. N.N., 1959].

The free field Dirac equation
Consider the contravariant 1-spinor .Since the covariant derivative of the matrix γ 5 is zero, we have The Dirac operators act on the type of positive (resp.negative)spinors, whereas the spinor Laplacian Δ preserves either type.
The Dirac massive spin ½ field may be described by a contravariant 1-spinor  that satisfies the free field equation (no external interacting field).which is interpreted as the positon equation wherein the anti-electron appears when  C is substituted to .

3.5
The spin canonical energy-momentum tensor

The massive field
Let L ξ be the Lie derivative operation with respect to an arbitrary vector field ξ.
Our aim here is to determine the expression of an energy momentum tensor which follows from the general varied action according to the procedure detailed in [Landau. L., 1962]  (so-called gravitinos ' field).
In such a perspective, it would fully agree with the theory suggested by Louis de Broglie postulating the existence of a hidden thermostat [De Broglie. L, 1968] [Vigier. J.P., 1954], which we explicited and generalized in the strict Riemannian geometry [Marquet.P., 2009]- [ Marquet.P., 2010].

=
 d  e ac - c  e ad +  e ac  f fdof the Ricci like tensor formed with the global Γ a bc R bc = ∂ a Γ a bc -∂ c Γ a ba + Γ d bc Γ a da - bc (the ', or D are the covariant derivatives constructed with the global connection  defined in (1.0))The condition (1.8) explicitly yields R abcd γ a γ b γ c γ d = -2 R ab g ab I (2.24) 2.5.2The EGR curvature tensor-spinor By perfect analogy with the classical 1-forms theory, we may define the 2-form Ω S referred in any bases which are elements of E(M), as the EGR spinor connection curvature 2-form [Ω S ] EGR = d N EGR + ½ [ N EGR , N EGR ] (2.25) .16) we may apply the charge conjugation to B  CB = γ 5 * = -γ 5 * = γ 5  C = -BC hence CB = -BC , and CB' = -B' C (3.17) Applying now the Dirac adjoint operation A to B , we have AB  = t  ̃ γ̃ 5 β = t ̃ γ 5 β = -t ̃ β γ 5 = - γ 5 = Bthe following relation holds CB  + = C  + which according to (3.19) yields B (C + ) = -C  + C maps (Sx) -onto (Sx) + and vice versa.
from which we expect to infer a current density (j a ) D through a classical action variationδ S D = -∫  a (j a ) D δU η = 0 (3.28)where we have set(j a) D = i γ a  (3.29)If  is a solution of the field equations δ S D vanishes for any δU, so  a (j a ) D = 0 (3.30)

(
conventional manner, the Dirac equation for a free particule with rest mass m 0, reads [ γ a  a -m 0 ] Dirac field interacting with an electromagnetic field Let us consider here the lagrangian for a charged Dirac massive field coupled with a potential A aL (, A a ) = L ( ) + L (A a ) -e  A a (j a ) D (3.41)the coupling constant e is here taken as a negative charge.(e.g.electron)Taking into acccount the expression of the Dirac current density(j a ) D = i ( γ a ) (3.42)we shall evaluate the variation of the lagrangian L (, A a ).After a simple but lengthy calculation, we obtain (omitting D in j )δ L (, A a ) = [ δ (γ a ( aie A a )m 0  ) ] + [(-( a + ie A a ) γ am 0  ) δ ] + [- b ( b A a - a A b ) -e j a ]δA a + divergence term.(3.43) Extremizing the action defined from L (, A a ), we get two field equations γ a ( aie A a )  = m 0  (3.44) -( a + ie A a )  γ a = m 0  (3.45) and δ d A = e j (3.46)Under the C operation, the following transformations take place  →  C , j →j Applying the Dirac adjoint operation A, the first equation becomes γ a ( a ie  a  C = m 0  C (3.47) and where δ S = ∫ ξ b (T a b ); a (-g) 1/2 d 4 x = 0 (3.48)We start by evaluating the Lie derivative of the lagrangianL D L ξ L D = ξ a  a L D (3.49)Using in particular the identitiesγ b ( a  b - b  a )  = -¼ R abcd γ b γ c γ d  = -1/2R ab γ b  ( a  b - b  a )  γ b = ¼ R abcd  γ c γ d  b = -1/2R ab  γ bwe eventually find after a ferocious algebraL ξ L D = ½ξ a  b [( a )γ b  -γ b  a ] + ξ a [ a (Z -m 0 ) +  a (Z' -m 0 ) ° ] (3.50)The last two terms cancel out for the Dirac solutions.On the other hand,L ξ L D = ξ a  a L D = ξ a  b [δ b a L D ] hence, applying (3.48) the equations (3.50) reduce to a conservation law  b [ ½ (( a  ) γ b  - γ b  a )δ b a L D ] a  ) γ b  - γ b  a  )δ b a L D ] (3.52)With the Dirac solutions (3.38) and (3.39), inspection shows that the term δ a b L D vanishes, so the spin canonical energy-momentum tensor can be finally writtenT ab = ½ [ ( a  ) γ b  - γ b ( a  ) ] (3.53)This tensor is still antisymmetric which agreeably copes with the framework of the EGR theory The spin canonical energy momentum tensor (which is shown to be real), may be symmetrized by adding the divergence of another antisymmetric tensor (T ab ) sym. = T ab + ½  c S cba S cba = ¼  (σ cb γ a + γ a σ cb )  is called the moment spin tensor.
19) Interchanging a and b, then a and c, one obtains two more equations of type (1.19), which are here denoted by (1.19)bis and (1.19)ter (virtual equations).