New Approach to the Positron Equation by Means of the Algebraic Structure of the Ideals

Starting from the electron Lagrangian density and expanding it with a more general idempotent than the one used by Hestenes, we achieve to show how the algebraic structure of the ideals draws a distinction between electrons and positrons without any need of the charge conjugation. A remarkable fact, consequence of the formalism used, is evidenced by the magnetization density discriminating the sign of the charge of the particles involved. David Hestenes showed in 1966 [1] how to geometrize the Dirac equation. One procedure to do this -although it was not the original one adopted by himis to start with the Dirac Lagrangian for the electron and to consider the spinor involved in it as an element belonging to an algebraic ideal, this kind of elements are called algebraic spinors. It is possible to do it because this procedure leads to the same value of the electron Lagrangian density. The idempotent used to convert the original Dirac spinor into an algebraic one is ∗ Also at Laboratori de F́ısica Matemàtica, Societat Catalana de F́ısica, IEC, Catalunya. Advances in Applied Clifford Algebras 12 No. 2, 183-188 (2002) 184 New Approach to the Positron Equation... D. Miralles, J. M. Parra, J. Vaz Jr. fixed in a successful attempt to recover the Dirac results. In this paper we propose a more general approach giving arbitrariness to the selection of the idempotent. As an algebraic consequence of the formalism it is possible to make a distinction between electron and positron fields even without electromagnetic coupling. We start with the Lagrangian density of the Dirac field LD = i 2 ∂μφ− ∂μφ̄γφ)− q φ̄Aμγφ−mφ̄φ (1) where the second term is the electromagnetic coupling, being the electron charge q negative. The spinor φ ∈ C can be included in C(4), for instance φ −→ Ψ =   ψ1 0 0 0 ψ2 0 0 0 ψ3 0 0 0 ψ4 0 0 0   (2) Now Ψ belongs to the minimal left ideal C(4)f, where f is a matrix idempotent. Different isomorphisms exist between matrix and Clifford algebras [2], one of them C(4) Cl1,3 ⊗ C allows us to write (1) using the complexified Clifford algebra of space-time. Given z ∈ C(4) and z ∈ Cl1,3⊗C related by the previous isomorphism is well-known [3] that tr z = 4 < z >0, where the projection < z >0 is over the scalar part of z. Furthermore, the Hermitian conjugation of z is e0z̃∗e0 where ∗ is the complex conjugation and ̃the main antiautomorphism in Cl1,3 ⊗ C. After some little effort we can write the Lagrangian density in Cl1,3 ⊗ C as LD = −4 Im[< eμ∂μψe0ψ̃∗ >0]− 4 q < Aμeμψe0ψ̃∗ >0 −4m < ψe0ψ̃∗ >0 (3) where the projector Im takes the imaginary part and ψ = ψf ∈ Cl1,3⊗C. ψ, f are related to Ψ, f respectively by means of the above mentioned isomorphism. From this point taking f = 12 (1 + e0) 1 2 (1 + ie12) it is possible to obtain the Dirac-Hestenes equation ∂φoe21 − qAφo −mφoe0 = 0 where ∂ = e∂μ, A = Aμe (4) being φo an operational spinor [4] belonging to Cl 1,3 where the positive sign determines the Cl1,3 even subalgebra. Let us study the consequences of a more Advances in Applied Clifford Algebras 12, No. 2 (2002) 185 general idempotent, with this aim we propose the next one fg = 1 2 (1 + e−β0e5e0) 1 2 (1 +R0ie12) (5) where e5 ≡ e0123 is the volume element and R0 ≡ e23 δ2 e31e23 δ2 both in Cl1,3 ⊗ C. The new paremeters β0, θ and δ are totally arbitrary and they do not have any relation with the spinor. Knowing that e−β0e5e0f = R0ie12f = f it is not difficult to see that ψ = φf , where φ is even and real. Now a new Lagrangian density can be written in Cl1,3 LD(R0, β0) =< ∂φe210R0φ̃ >0 −q < φe0φ̃A >0 −m < φe−β0e5 φ̃ >0, φ = φ(R0, β0) ∈ Cl 1,3 (6) It is important to note that the dependence of φ = φ(R0, β0) on the selected idempotent and the explicit emergence of R0 and e−β0e5 involves different Lagrangians. Now using the Euler-Lagrange equations in the Clifford formalism [5] we have ∂φe21R̃0 − qAφ−mφe0e−β0e5 = 0 φ ∈ Cl 1,3 (7) This expression depends on the idempotent which we are working with. There are four interesting cases: (i). R̃0 = e−β0e5 = 1, (ii). R̃0 = e−β0e5 = −1 (iii). R̃0 = 1, e−β0e5 = −1 (iv.) R̃0 = −1, e−β0e5 = 1 (8) In the first one we recover the equation (4) where φo = φ(1, 1). For the second case we have ∂φpe21 + qAφp −mφpe0 = 0 (9) where φp = φ(−1,−1). This one corresponds to the positron equation. In fact this can be proved doing the charge conjugation φ = φ e01 = φp on the DiracHestenes equation. The last two cases, (iii) and (iv), are exactly as the previous ones but here the spin is turned. For an intermediate case it seems not to have physical account perhaps, in other context it can make possible to understand some change of ideal as an inner continuous symmetry. 186 New Approach to the Positron Equation... D. Miralles, J. M. Parra, J. Vaz Jr. For the Tetrode tensor (energy-momentum tensor) in the free case (A = 0) we have Tμν = i 2 (φ̄γν∂μφ− ∂μφ̄γνφ) Following our former approach this tensor can be written in Cl1,3 as Tμν =< eν∂μφe210R0φ̃ >0, φ = φ(R0, β0) ∈ Cl 1,3 (10) Again this expression depends on the chosen ideal, but given a solution the tensor is now just based on R0. Although (4) and (9) are the same equations for the free case, the Tetrode tensor is different. For a free particle propagating into the z axis with a linear moment p and an energy E the solutions of (4) and (9) are φ1 = N(1− p E +m e03)ee12(Et−pz) (11) φ2 = N(e31 − p E +m e01)ee12(Et−pz) (12) φ3 = N( p E +m − e03)ee21(Et−pz) (13) φ4 = N( p E +m e13 − e01)ee21(Et−pz) (14) where N is the normalization factor. The first case in (8) corresponds to the Dirac approach, here the Tetrode tensor shows that φ1, φ2 have positive energy and φ3, φ4 have negative energy as is expected. Due to the dependency of the Tetrode tensor just on R0 an unknown result is obtained for the second case in (8): the energy for the solutions φ3, φ4 is positive and it is negative for φ1, φ2, oppositely to the first case. Even without electromagnetic coupling the solutions for particles and antiparticles are different, this is reflected as we will see immediately, not only in the ideal algebraic structure but in the canonical decomposition of the operational spinors. There is a canonical decomposition for φ ∈ Cl 1,3 in the form φ = ρe β 2 5R where ρ ∈ R, R ∈ Spin+(1, 3) and e β2 e5 is a duality rotation, where β is Advances in Applied Clifford Algebras 12, No. 2 (2002) 187 called the Yvon-Takabayasi angle [6, 7] or β-angle . The previous solutions present different angles; β = 0 for φ1, φ2 and β = π for φ3, φ4. This makes it impossible to connect this two sets of solutions through a transformation belonging to Spin+(1, 3). The expression for the spin depending on the chosen ideal is S = φe−β0e5R0e21φ−1 (15) and it has the same expression for the first two cases in (8), S = φe21φ−1. Moreover, using the canonical form S = ρ 1 2 e β 2 Re21ρ − 2 e− β 2 e5R−1 = Re21R̃ (16) Therefore S does not depend on β-angle either. In the same way it is easy to see that the magnetization density M does not depend on the selected ideal. M can be then written as M = q 2m φe21φ̃ = q 2m ρeRe21R̃ (17) Now the free solutions with positive energy for the first and second case in (8), i.e. electron and positron solutions, exhibit the corresponding sign for the magnetization density without having to add it ad hoc due to the β-angle and the algebraic structure of the ideals. These results suggest to consider the algebraic ideals as a promising tool for the mathematical description of the fundamental particles. In a following work we will implement this approach to the inner spaces formed by electrons and neutrinos. Acknowledgements JV is grateful to FAPESP and CNPq for partial financial support. JMP and DM acknowledge support from the Spanish Ministry of Science and Technology contract No. BFM2000-0604 and 2000SGR/23 from the DGR of the Generalitat de Catalunya. 1 It does not have to be confused with β0 188 New Approach to the Positron Equation... D. Miralles, J. M. Parra, J. Vaz Jr.