THE ROLE OF ORTHOGONAL AND SYMPLECTIC CLIFFORD ALGEBRAS IN QUANTUM FIELD THEORY
Matej Pavšič J. Stefan Institute, Ljubljana, Slovenia
Contents
- Introduction - Spaces with orthogonal and symplectic forms - Heisenberg equations as equation of motion for basis vectors - Point particle in `superspace’ Description of fields Orthogonal case Symplectic case Discussion: Prospects for quantum gravity - Conclusion Introduction
Persisting problem of quantum gravity and the unification of interactions. A need to reformulate the conceptual foundations of physics and to employ a more evolved mathematical formalism.
In this talk I will consider Clifford algebras which provide very promising tools for description and generalization of geometry and physics.
Orthogonal Clifford algebra
The inner, symmetric , product of basis vectors 1 γ a gives the orthogonal metric, . γ a γ· γ γ γ γb≡2 ( ab + ba ) = g ab gab 1 The outer, antisymmetric , product of basis vectors γ γ a∧ γ γ γ γ b ≡2 ( ab − b a ) gives the basis bivector.
Symplectic Clifford algebra The inner, antisymmetric , product of basis vectors q 1 a gives the symplectic metric, .J ab qab∧q ≡2 ( q ab qqq− b a ) = J ab 1 The outer, symmetric , product of basis vectors qab·q ≡ 2 (q ab qqq+ b a ) gives a symplectic bivector. The generators of an orthogonal Clifford algebra can be transformed into a basis in which they behave as fermionic creation and annihilation operators. The generators of a symplectic Clifford algebra behave as bosonic creation and annihilation operators. We will show how both kinds of operators can be united into a single structure so that they form a basis of a `superspace’. We will consider an action for a point particle in such superspace.
Instead of finite dimensional spaces, we can consider infinite dimensional spaces. Then we have description of a field theory in terms of fermionic and bosonic creation and annihilation operators. The latter operators can be considered as being related to the basis vectors of the corresponding infinite dimensional space. Spaces with orthogonal an symplectic forms
I. Orthogonal case (,)ab= ( γ γabab γ γ , ) = a a (,) bagb bab = g a bg abg ab a= a γ
(γ γa , b ) g= g ab metric
For a basis we can take generators of the Basis vectors orthogonal Clifford algebra: 1 γ γ γ γ γ (,)γ γ =g = ( + ) =· = g abg ab2 ab ba a b ab Vectors are Clifford numbers: 1 (,)a b= ( ab + ba ) = a ·b g 2
Inner product of vectors a and b II. Symplectic case
ab a bab (,')(zzJ = zqza ,')q bJ = zqqz (,)' a b J = zJzab ' a z= z q a symplectic metric (qa , q b ) J = J ab
Symplectic basis vectors For a symplectic basis we can take generators of the symplectic Clifford algebra: 1 (,)qq= J = ( qq −q q ) = q ∧ q = J a b J ab2 ab b a a b a b Vectors are now symplectic Clifford numbers: 1 (,')zz= ( zzzz '− ') = z ∧ z ' J 2
Inner product of symplectic vectors z and z’ Explicit notation with coordinates and momenta za = ( x , p ) a ()x () p Symplectic vector z= zqa = xq + pq
a b a 1 b (,')zzJ= zqq (,)ab J z ' = z 2 (q abbaq− qq) z ' a b = z Jab z ' 0 η ν ν ν ν J = =(xp ' − px ') η ν ab −η ν 0
1 Relations give 2 [qa , q b ] = J ab
1()()xx 1 ()( pp ) ν 2[qq ν , ]0,= 2 [ qq , ]0 = 1 (x )( p ) Heisenberg commutation 2 [ νq ν , q ] =η relations Poisson bracket (symplectic case) ∂f ∂ g {f , g } ≡ J ab PB ∂za ∂ z b By introducing the symplectic basis vectors, we can rewrite the above expression as Symplectic metric
1 ∂fa ∂ g b ∂fab ∂ g [a q, b q ] = aJ b 0 −η ν 2 ∂zz∂ ∂ z ∂ z J ab = η ν 0 c d If we take fz=, gz =
0 η ν 1 c d cd J ab = [q , q ] = J −η ν 0 2
These are the Heisenberg commutation c d relations for `operators’ and .q q
a a a q are thus `quantized’ phase space coordinates z . z are real coordinates Poisson bracket (orthogonal case) ∂f ∂ g {f , g } ≡ g ab PB ∂ λλ a ∂ b By introducing the basis vectors we can rewrite the above expression as Orthogonal metric
1 ∂fa ∂ g b ∂f a b ∂ g γγ , = g η ν 0 2 ∂ λλ a∂b ∂ λλ a ∂ b g ab = ν 0 η c d If we take f λ=λ , g =
η ν 0 1cd 1 cd dc cd gab = {γ γ γ γ , γγ } ≡ (+ ) = g 0 η ν 2 2
These are the anticommutation relations for c d `operators’ and .γ γ
a a a γ are thus `quantized’ .λ λ are real anticommuting coordinates Representation of operators
I. Orthogonal Clifford algebra η 0 1 ν gab = γ γ γγ γab·≡ ( ab + ba ) = g ab 2 0 η ν In even dimensions we can write:
γ γ γ a =( , ), = 0,1,2,3 We can introduce Witt basis: 1 γ γ θ = (+ i ) 2 1 γ γ θ = (− i ) 2 1 θθ θ θ θ θ θ η θ ·= ( += ),·0,·0 = = ν ν ν ν ν ν ν ν 2
γ ,γ , θ , θ can be represented: 1) as 4 x 4 matrices, 2) in terms of Grassmann coordinates: ν ∂ θ ξ θ →2 , → 2 θ η θ = ν ∂ξ ξ ξ ν ξ ν + = 0 II. Symplectic Clifford algebra 1 qa∧≡ q b( qqqq ab − ba ) = J ab 0 η ν 2 J ab = −η ν 0 We can write: ()x () p qa =( q , q ), = 0,1,2,3
1 q()x∧ q () p= (q ()() xp q − q ()() pxq ) =η ν ν ν 2 ν ()xx () () pp () q ν∧ qν = 0,q ∧ q = 0 Then the operators cannot ()x () p can be represented: be cast into Hermitian form q , q 1) as 4 x 4 matrices, 2) in terms of commuting coordinates:
()x () p ∂ (x) ν (x ) q →2x , q → 2 q=η q ∂x ν xx ν ν − xx = 0 Heisenberg equations as equations of motion for basis vectors Let us now consider the action za = ( x , p ) I=1 dzJzτ (ɺa b + zKz a b ) H= 1 zKza b 2 ∫ ab ab 2 ab δ za ∂H zɺa= J ab ∂zb
a Let us consider trajectories as componez (τ ) nts of an infinite dimensional vector: τq τ a()τ ∧ q b (') = J ab ()(') zzq= a(τ ) ≡ dz τ τ τ a () q () a(τ ) ∫ a =J ab δ τ τ( − ′ ) τ ɺa()τ a () ɺ zq τ a()τ = − zq a () and write the action in the form τ τ 1 τ a()τ ba (') 1 () b (') H 1 a b ɺ = 2 τdz τ τ () Kzab () IzJ=2 τ ab()(') τ z + 2 zK ab ()(') z ∫ 1 τ a(τ ) b ( ') = z Ka( τ ) b ( ') z The equations of motion are: 2 K= K δ τ τ( − ′ ) ∂H ab( τ ) ( ') ab τ τ τ τ zqɺa()τ =− zqqJ a ()ɺ = ab ()(') =− qKz a () b (') τ τ a()()()τ a a ∂zb(τ ') a( τ ) b ( ') a(τ ) a τ τ b(τ ')ɺ a ( ) b ( ') q≡ q (τ ) zq τ b(τ ')= qK a ( ) b ( ') z
b(τ ) This equation holds for any z ɺ a(τ ) τqb(τ ')= q K a ( ) b ( ') Equations of motion for operators qa(τ )
Kab( τ ) ( ') = K ab δ τ τ( − ′ )
ɺ a Kab= K ba qb τ()τ = q () K ab ˆ 1 a b H= 2 qKqab Symplectic inner product ˆ b [qHa , ] = Kq ab is given by commutator ɺ ˆ qa= [ q a , H ] Heisenberg equations of motion
We see that the basis vectors of phase space satisfy the Heisenberg equations of motion for `quantum’ operators qa(τ ) ≡ q a (τ ) τ τ b(τ ') a ( ) b ( ') ɺ a(τ ) a zq τ b(τ ')= qK a ( ) b ( ') z z≡ z (τ )
b(τ ) This equation hold for any z ɺ a(τ ) τqb(τ ')= q K a ( ) b ( ') Equations of motion for operators qa(τ ) K= K δ τ τ( − ′ ) ab( τ ) ( ') ab a(τ ) That the operator equations hold for any z means ɺ a that coordinates and momenta are undetermined.Kab= K ba qb τ()τ = q () K ab ˆ 1 a b H= 2 qKqab b Symplectic inner product [qH ,ˆ ] = Kqa az τ τ ab()τ = ( x (), p ())is given by commutator ɺ ˆ qa= [ q a , H ] Heisenberg equations of motion
We see that the basis vectors of phase space satisfy the Heisenberg equations of motion forClassical solution Any trajectory `quantum’ operators Point particle action in `superspace’
We introduce the generalized vector space whose elements are: A a a a a A z= ( z ,λ ), z= ( x , x ) , λλ λ = ( , ) Z= z q A qA= (qa ,γ a ), qa = (q , q ), γγ a = ( ,γ )
coordinates basis elements symplectic orthogonal part J ab 0 part 〈qAq B 〉= S G AB = 0 gab 0 η ν qa ∧ qb = J ab = symplectic metric −η ν 0
η ν 0 γ a ·γ b =gab = orthogonal metric 0 η ν Let us consider a particle moving in such space. Its worldline is given by: zA= Z A (τ )
parameter on the worldline ) Example of a possible action Example of a possible action za = ( z , z ) 1A B 1 A B J abɺ0 A a. ɺ a I= τ d τ 〈ZqZAqB 〉 S = d ZGZ AB 2∫GAB = , 2 ∫ z = (,) z λ a a bɺ a0 ɺ g a λλ λ = ( , ) I= dτ ( zJzɺ ɺ + λ g λ )ab ∫ ab ab J 0 = 0,1,2,3 G=ab , zA = (,) z a. λ a J= − J AB g = g Since ,ab ba Since ,0 gab ab ba this term differs this term differs from zero if from zero if J= − J are Grassmann1 aɺ b aare commutingɺ b ab ba I=2 d τ (zJzab + λ g ab ) coordinates.∫ coordinates. gab= g ba
In this term, za In this term, λ a are commuting are anticommuting coordinates. (Grassmann) coordinates.
Canonical momenta: ∂L p( z) = = 1 J zb , a∂zɺa 2 ab
(λ ) ∂L 1 a pa=ɺa = 2 g a bλ ∂λ I=1 d τ (zJzaɺ b + λ a g ɺ b ) 2 ∫ ab ab
0 η ν η ν 0 J ab = , gab = −η ν 0 0 η ν ɺ I=1 d ητ ( xɺ ν ν ν x ν − xɺ λη η λ λ η λ x +ɺ + ) 2 ∫ ν ν ν ν
ν ν x , x are commuting [xx , ]= 0,[ xx , ] = 0 λ λ , are Grassmannian νλ { λλ λ ν , }0,{= , }0 =
Canonical momenta: ∂L ∂ L p()x == η 1 η xpν , ()x ==− 1 x ν ∂xɺ 2 ν ∂ xɺ 2 ν
(λ )∂L1 ν (λ ) ∂ L 1 ν p ==ɺ ν η λ 2 η λ ν , p == ɺ 2 ∂λ ∂λ Quantization ()x () x Operators xp, → xpˆ , ˆ λ λ λ λ ,p()→ ˆ , pˆ () Similar relations hold where (x ) for barred quantities.. [,]xˆ ν p ˆ ν = i δ , ν ()x () x [,]0,[xxˆˆ= pp ˆˆ ν , ]0 =
λ ˆ λ ()ˆ () δ λ δ ν{,λ ν ν piˆν }= ,{, pi ˆ } = λ λ ˆ ν ˆ () () {,}0,λ λ = {pˆ ν , p ˆ }0 = a Altogether, we have z= ( z , z ) a( z) a( z ) a zp,a→ zpˆ , ˆ a λλ λ = ( , ) a λ (λ ) ˆ a ( ) λλ ,pa→ , pˆ a = 0,1,2,3 where the operators satisfy
a( z ) a [zˆ , pˆb] = i δ b Commutators ˆa(λ ) a {λ , pˆb } = i δ b Anticommutators a( z ) a [zˆ , pˆb ] = i δ b ˆa(λ ) a δ{λ ,pˆb } = i b (z ) 1 b pa= 2 J ab z , (λ ) 1 a pa= 2 g ab λ 1 a b ab 2 [zˆ , z ˆ ] = iJ 1 ˆa ˆ b ab 2 {λ λ , } = i g
But we see that the above operator equations are just the relations for the basis vectors of the orthogonal and symplectic Clifford algebra, provided that we identify: zˆa = ( q , iq ) γλ ˆa = ( ,i )
We se that `quantization’ is in fact the replacements a a of the coordinates with the correspondiz , λ ng basis vectors. The only difference is in the factor i in front of q 1 a b aɺ b I= d τ (zJzɺ + λ g λ ) 2 ∫ ab ab 1 a baɺ b =dτ 〈 z q q zɺ +λ γ λγ 〉 qa = ( q , q ), 2 ∫ ab ab S γ γγ a = ( , )
Basis vectors, entering the action, are `quantum operators’, apart from the i in the relations zˆa =( q ,i q ), γ γ λ ˆa = ( , i ) qa = ( q , q ), 1 a b aɺ b I= d τ (zJzɺ + λ g ) 2 ∫ ab ab γ γγ a = ( , ) =〈1 dτ za q q zɺ ba +λ γ γ λ ɺ b 〉 2 ∫ ab ab S λ λ ξ λ λλ ′a = (,), ′′ ′ ≡=−1 (i ) 1 a b a ɺ b 2 = λ dτ (zJzɺ + λ′ g ′ ′ ) 2 ∫ ab ab ′ 1 λ λ ξ λ ≡ =2 ( + i ) 0 η ν ɺ ɺν ν ξ ξ η η ξ + g '=γ γ′ · ′ = ν ν ab a b η ν 0 The above action is not complete. An additional term is needed.
I= 1 dZGZτ Aɺ B J 0 2 ∫ AB G = ab AB 0 g ɺ B ɺ B B C replace with covariant derivative ab ZA→Z + C Z I=1 dZGτ A( Zɺ B + AZ BC ) Generalized Bars action 2 ∫ AB C (invariant under dependent τ rotations of )Z A In particular, this term gives: Mass comes from βα λ p p+ p extra dimensions
A Lagrange multipliers (contained in )A C Upon quantization, the classical constraint λ p = 0 becomes the Dirac equation:
ˆ λ pˆ Ψ = 0 where .λ γˆ =
Ψ can be represented α 1) as a column ψ 2) as a function ψ ξ (x , )
λ γˆ = can be represented 1) as matrices ∂ 2) as ξ + ∂ξ We also have which can be represe γλ ˆ = i nted 1) as matrices ∂ 2) as i(ξ − ) ∂ξ 1 1 θ η θ ·≡ {,} = , From γ γ θ =( + i ) ν ν ν 2 2 1 θ θθ θ ·= 0, · = 0 γ γ θ (i ) ν ν =2 − we can build up spinors by taking a `vacuum’
= ∏θ which satisfies θ = 0 = 0,1,2,3 and acting on it by `creation’ operators .θ So we obtain a `Fock space’ basis for spinors: ɶ θ θ θ θ ϑ Aɶ = (1 , νρ νρσ ν , , , ) A = 1,2,...,16
in terms of which any state can be expanded:
Aɶ Ψ = ψ ϑ ɶ ∑ A
With operators defined above, we can cθ θ , onstruct the spinors as the elements of a minimal left ideal of .Cℓ(8) =1 ( − i ) λ λ ξ 2 Taking all possible vacua, such as λ λ ξ =1 ( + i ) =θ θ θ θ ...... ,r = 0,1,2,3,4 2 12rr+ 12 r + n we obtain the Fock space basis for the whole Cl (8) . Description of fields Orthogonal case
Ψ = ψ i( x ) h ℝ3 ℝ 1,3 i(x ) i=1,2; x ∈ or x ∈ particular case h· h = ρ ix( ) jx ( ') i(x ) j ( x ') metric δ δρ i( x ) j ( x ') = ij (x − x ')
New basis: 1 h()x=2 ( h 1() x + ih 2() x ) Witt basis 1 h*(x )=2 ( h 1( x ) − ih 2( x ) ) δρ ≡(x − x ') (x ) *( x ) (x )*( x ') h(x )· h *( x ')= ρ ( xx )*( ') Ψ = ψψ h(x ) + h *(x ) ρρ (xx )*( ')= *( xx ) ( ') hh(xx )· ( ')= h *( x ) · h *( x ') = 0 Fermionic commutation Scalar product: relations (x ) *( xx ') *( ) ( x ') 〈ΨΨ〉=Sψ ψ ψ ρ ( xx )*( ') + *( xx ) ( ')
(x ) ψ h(x ) →| Ψ〉 Both vectors bring the same *(x ) information about the state ψψ h*(x ) → 〈 |
〈ΨΨ〉=| ψ ψ ψ ψ ρ ψ *(x )hh · ( xx ') = *( ) ( x ') = dxxx * ()() *(xx ) ( ') *( xx )( ') ∫ Vacuum h = 0 = ∏ h*(x ) *(x ) x (x ) The second part of disappears Ψ =ψ h (x ) Ψ (x ) *( x ) Ψ = ψψ h(x ) + h *( x )
Let us consider a more general case:
()x ()(') x x Ψ =+( ψψ ψ 0h ()x + h ()(') x h x + ... ) This state is the infinite dimensional space analog of the spinor as an element of a left ideal of Clifford algebra
Reversed state:
‡ ‡† ‡ * *()x *()(') x x ()Ψ = Ψ= ψ (ψ ψ 0 + h *()x + h *()*(') x h x + ...) Then (Ψ ) ‡ Ψ = ‡ Ψ ‡ Ψ ‡ *(x ) ( x ') ‡ *( xx ) ( ') ψ ψ δ = ψ ψ h*(x ) h ( x ') += ... 2 (x ) ( x ') + ...
2δ(xx )( ')− h ( x ') h *( x ) This acting on vacuum gives 0 Vacuum h = 0 = ∏ h*(x ) *(x ) x (x ) The second part of disappears Ψ =ψ h (x ) Ψ (x ) *( x ) Ψ = ψψ h(x ) + h *( x )
Let us consider a more general case:
()x ()(') x x Ψ =+( ψψ ψ 0h ()x + h ()(') x h x + ... ) This state is the infinite dimensional space analog of the spinor as an element of a left ideal of Clifford algebra
We obtain a non vanishing form is we take the reversed state ‡ ‡ *(x ) ( x ') 〈 ΨΨ 〉S =ψ ψ δ ( x ) ( x ') + ... ‡ ‡‡ ‡ * *()x *()(') x x ()Ψ = Ψ= ψ (ψ ψ 0 + h *()x + h *()*(') x h x + ...) scalar part Then (Ψ ) ‡ Ψ = ‡ Ψ ‡ Ψ ‡ *(x ) ( x ') ‡ *( xx ) ( ') ψ ψ δ = ψ ψ h*(x ) h ( x ') += ... 2 (x ) ( x ') + ...
2δ(xx )( ')− h ( x ') h *( x ) This acting on vacuum gives 0 Symplectic case
3 * ɺ x ∈ℝ Schroedinger field I= ∫ τ d φ τ d x i (, φτ x ) (, x ) − H x ∈ℝ1,3 ∂L * Stueckelberg field ɺ = Π = iφ ∂φ I τ= dτ dx φ φ Πφ ɺ −H = d d x 1 ( Π ɺ −Π−ɺ ) H ∫ ∫ 2
φφ ix()= ( () x ,Π () x ) , i = 1,2
ɺix( ) jx ( ') 0 δ(x )( x ') IdJ= φτ φ (i( x ) j ( x ') − H ) J = ∫ i( x ) j ( x ') −δ(x )( x ') 0 Symplectic vector: ix() 1() x 2() x δδ ≡(x − x ') Φ = φφ kix()=k 1() x + φ k 2() x (x )( x ') ≡φ ()xk + Π () x k φ()xΠ () x Symplectic inner product: Bosonic commutation relations kix()∧ k j(x ') = J ijx ()(') x
basis vectors metric Symplectic case
3 * ɺ x ∈ℝ Schroedinger or scalar field I= ∫ τ d φ τ d x i (, φτ x ) (, x ) − H x ∈ℝ1,3 ∂L * Stueckelberg field ɺ = Π = iφ ∂φ I τ= dτ dx φ φ Πφ ɺ −H = d d x 1 ( Π ɺ −Π−ɺ ) H ∫ ∫ 2
φφ ix()= ( () x ,Π () x ) , i = 1,2
ɺix( ) jx ( ') 0 δ(x )( x ') IdJ= φτ φ (i( x ) j ( x ') − H ) J = ∫ i( x ) j ( x ') −δ(x )( x ') 0 Symplectic vector: ix() 1() x 2() x H φ= φ ix( ) K jx ( ') δδ ≡(x − x ') Φ = φφ kix()=k 1() x + φ k 2() x i( x ) j ( x ') (x )( x ') ≡φ ()xk + Π () x k φ()xΠ () x 1 r Ki( x ) j ( x ') = −∂ ∂r + V( x ) δ ( xxg − ') ij , Symplectic inner product: 2m 0 1 kix()∧ k j(x ') = J ijx ()(') x gij = 1 0 2 r basis vectors metric (m + ∂ ∂r )δ (x − x ') 0 Ki( x ) j (x ') = , 0δ (x− x ' ) Poisson bracket ∂f ∂ g 0 − δ ix() ix () ixjx () (' ) i( x ) j ( x ') (x )( x ') {}f φ(φ ),( g ) = ix() J jx (' ) J = PB ∂φ ∂ φ δ(x )( x ') 0 In particular: f φ=φ kx( ''), g = lx ( ''')
{ φφ kx( ''), lx ( ''')} =J kxlx ( '') ( ''')= k kx ( '') ∧ k l(x ''') PB ≡ 1 kk (x '') , k l( x ''') 2
The Poisson bracket of two classical fields is equal to the symplectic metric.
On the other hand, the symplectic metric is equal to the wedge product of basis vectors.
In fact, the basis vectors are quantum operators, and satisfy the quantum canonical commutation relations: 1 [kxkx (), (')]=δ ( xx − ') ˆ 2 φ Π φ()x= kφ (), x or ˆ ˆ ˆ 1 δ [(),φ xΠ (')] x = ixx ( − ') Π()x = iq2 Π () x Discussion: Prospects for quantum gravity
`Matter’ configuration
i a system of point particles X M A compact notation X for a configuration a (system) of branes X ξ( ) etc.
Basis vectors hM
Metric hM ·hN =ηMN
In Witt basis we have annihilation and creation operators: + − hM, h M + | Ψ〉=(wavepacket profile)( ∏ hM ) | 0 〉 M Expectation value
〈ΨhM Ψ〉≡〈1 h M 〉 Induced metric
〈hM 〉〈· h M 〉= g MN I expect that in general we will obtain an induced metric with non vanishing curvature.
Since spacetime is a subspace of a configuration space, we will also obtain the metric of spacetime.
Curved spacetime metric originates from quantum configurations of many `particle’ systems. Conclusion
An action for a physical system can be written in the phase space form, and it contains either the symplectic or the orthogonal form (or both).
The corresponding basis vectors satisfy either the fermionic anticommutation relations or the bosonic commutation relations, and satisfy the Heisenberg equations of motion.
Quantum operators are just the basis vectors of the phase space action.
The fact that basis vectors on the one hand are quantum operators, and on the other hand they give metric, can be exploited in the development of quantum gravity.
According to Feynman it is necessary to know several different representations of the same physics.
We have pointed out how `quantization’ can be seen from yet another perspective. The idea that basis vectors are quantum operators can be found in a book M. Pavšič: The Landscape of Theoretical Physics: A Global view; From Point Particles to the Brane World and Beyond, in Search of a Unifying Principle (Kluwer Academic, 2001) where the orthogonal and symplectic cases are discussed. Very promising is the description of gravity in terms of the Clifford algebra equivalent of the tetrad field which simplifies calculations significantly.
Some other related publications: Class. Quant. Grav. 20, 2697 2714 (2003); gr qc/0111092 Kaluza Klein theory without extra dimensions: Curved Clifford space, Phys. Lett. B614, 85 95 (2005); hep th/0412255 Spin gauge theory of gravity in Clifford space: A Realization of Kaluza Klein theory in 4 dimensional spacetime, Int. J. Mod. Phys. A21, 5905 5956 (2006); gr qc/0507053 Beyond the relativistic point particle: A reciprocally invariant system, Phys. Lett. B 680, 526 532 (2009) On the relativity in configurations space: A renewed physics in sight, 0912.3669 [gr qc] Space inversion of spinors revisited: A possible explanation of chiral behavior in weak interactions, Phys. Lett. B 692, 212 217 (2010)