Particle-antiparticle duality from an extra timelike dimension

Abstract

It is a well known fact that the usual complex structure on the real Clifford Algebra (CA) of Minkowski spacetime can be obtained by adding an extra time-like dimension, instead of the usual complexification of the algebra. In this article we explore the consequences of this approach and reinterpret known results in this new context.We observe that Dirac particles and antiparticles at rest can be interpreted as eigenstates of the generator of rotations in the plane formed by the two time-like coordinates and find an effective finite scale for the extra dimension when no EM fields are present (without postulating compactness). In the case of non-vanishing EM fields, we find a gauge condition to preserve such a scale. *

Complexifying the spacetime algebra by means of an extra timelike dimension: Pin, Spin and algebraic spinors

“…Similar isomorphism regarding This way of complexifying the CA, also lead us to think about the nature of complex numbers in the Dirac theory, which in this case is linked to the existence of an extra timelike dimension. In a recent article [1] we observed that pure particle and antiparticle solutions in the extra-dimension approach are eigenvectors of the generators of rotations in the plane of two times.…”

“…Of course, this answer may be indeed too simple: as was noted by Hestenes in his foundational papers [7,8,9], it is possible to work only with the real Clifford algebra by replacing algebraic spinors with operator spinors 1 and the Dirac equation with the Dirac-Hestenes equation. However, if we want to recover the algebraic spinor appearing in Eq.…”

“…However, if we want to recover the algebraic spinor appearing in Eq. (1), it is necessary to work in the complex spacetime algebra 2 .…”

“…If we accept to work with the algebraic spinors appearing in the Dirac equation (1), it is necessary to consider the complex Clifford algebra Cl 1,3 (C). Regarding this aspect, a known fact about Cl 1,3 (C) is that it is isomorphic to the real Clifford algebra Cl 2,3 (R).…”

“…In the present article we extend this research by analyzing the properties of the Clifford algebra Cl 2,3 (R), the space of algebraic spinors in this algebra and the inner products that can be defined on spinors. We construct an injective mapping from Pin (1,3) to Spin (2,3) and find parity and time reversal operators in Spin (2,3). We also recover the usual complex structure of complex spinors in Cl 1,3 (C) by means of the Clifford conjugation inner product in the real algebra Cl 2,3 (R).…”