Hestenes spinor

But, and that was entirely new, D. Hestenes replaces, in a strict equivalence, the Dirac spinor geometrical object t/j that we call for its applications in Quantum Mechanics a Hestenes spinor, an element of the even sub-algebra Cl+(M) of Cl(M). As a consequence, the set without any proper structure of the Dirac spinors is replaced by a ring, and also the matrices and the spinors are unified in a same structure, implying only geometrical elements of the Minkowksi real space M. 

We denote ip, that we call a Hestenes spinor, as a biquaternion when it is applied to a wave function expressed by a Dirac spinor E in Quantum Mechanics. 

Note. In what follows, the identification of the Hamilton quaternion q and the Hestenes spinor tjj with the Pauli spinor and the Dirac spinor P is based on the articles [28,44]. Perhaps, it is not the shortest one, but it allows a step-by-step conversion of P into vice versa when P is expressed by means of its four complex components. 

The clarity comes from the real form given by D. Hestenes to the electron wave function that replaces, in a strict equivalence, the Dirac spinor. This form is directly inscribed in the frame of the geometry of the Minkwoski space in which the experiments are necessarily placed. The simplicity derives from the unification of the language used to describe the mathematical objects of the theory and the data of the experiments. 

Abstract. This chapter is a recall of the properties of the long-range part of the electromagnetic fields created by time-periodic currents, as they may be observed in particular in the Zeeman effect. The aim of this part is also to place the vector frame of these observations, that is, one of the spherical coordinates, which is in the center of the presentation in the real formalism of the relativistic central potential problem. This frame is the one in which are expressed the Dirac probability current, associated with a state and with the transition between two states. But it is to notice that, as a specificity of the real formalism, the form given by Hestenes to the wave function of the electron, strictly equivalent to the Dirac spinor, may be presented, in the case of central potential, as a combination of the vectors of this frame. 

Abstract. This chapter is a recall of the algebraic tool used in the real formalism and the passage from the Dirac equation of the electron in the complex spinor formalism to the real form given by Hestenes to this equation. It is completed by the first paragraph of the Appendix of Part I, which allows a step-by-step traduction of the complex formalism to the real one and vice versa. 

More surprising is the fact that the stucture of all the objects, quarks, and leptons, defined by means of the Dirac spinor-in the form given by D. Hestenes-which fill this f-space, also appears as algebraically priviledged. 

Let us mention that biquaternionic solutions of the central potential problem had been already achieved by Sommerfeld (see [54]), but in the frame of the complex formalism, the Dirac spinor being expressed by means of the Dirac matrices. Nevertheless, though, in particular, some ambiguities related to the role of the imaginary number t/—1 in the complex formalism had been removed in this work, the use of the pure real formalism of Hestenes brings noticeable simplifications and above all the entire geometrical clarification of the theory of the electron. 

Abstract. This chapter concerns a presentation of the Darwin solutions of the Dirac equation, in the Hestenes form of this equation, for the central potential problem. The passage from this presentation to that of complex spinor is entirely explicited. The nonrelativistic Pauli and Schrodinger theories are deduced as approximations of the Dirac theory. 

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