This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications We show how the definition of a “geometric product” of vectors in 2-and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods Reflections and rotations are analyzed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics), Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime This is demonstrated by examples from electromagnetism In the course of this purely classical exposition many surprising results are obtained—results which are usually thought to belong to the preserve of quantum theory We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics

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Imaginary numbers are not real—The geometric algebra of spacetime

Fundamental experiments on the dual nature of atomic entities (photons, electrons, neutrons, etc.) can be interpreted in terms of “empty” waves not carrying energy and momentum. Similar points of view were advanced in famous papers by Einstein, de Broglie, Bohr, and Born. Recent proposals could lead to experimental tests of this idea, using low intensity photon beams, thanks to modern experimental apparatuses.

On the direct observability of quantum waves

We analyze the proof given by J. S. Bell of an inequality between mean values of measurement results which, according to him, would be characteristic of any local hidden-parameter theory. It is shown that Bell's proof is based upon a hypothesis already contained in von Neumann's famous theorem: It consists in the admission that hidden values of parameters must obey the same statistical laws as observed values. This hypothesis contradicts in advance well-known and certainly correct statistical relations in measurement results: One must therefore reject the type of theory considered by Bell, and his inequality has no general meaning.

Has Bell's inequality a general meaning for hidden-variable theories

An equation proposed by Levy, Perdew and Sahni (Phys. Rev. A 30:2745, 1984) is an orbital-free formulation of density functional theory. However, this equation describes a bosonic system. Here, we analyze on a very fundamental level, how this equation could be extended to yield a formulation for a general fermionic distribution of charge and spin. This analysis starts at the level of single electrons and with the question, how spin actually comes into a charge distribution in a non-relativistic model. To this end we present a space-time model of extended electrons, which is formulated in terms of geometric algebra. Wave properties of the electron are referred to mass density oscillations. We provide a comprehensive and non-statistical interpretation of wavefunctions, referring them to mass density components and internal field components. It is shown that these wavefunctions comply with the Schrodinger equation, for the free electron as well as for the electron in electrostatic and vector potentials. Spin-properties of the electron are referred to intrinsic field components and it is established that a measurement of spin in an external field yields exactly two possible results. However, it is also established that the spin of free electrons is isotropic, and that spin-dynamics of single electrons can be described by a modified Landau-Lifshitz equation. The model agrees with the results of standard theory concerning the hydrogen atom. Finally, we analyze many-electron systems and derive a set of coupled equations suitable to characterize the system without any reference to single electron states. The model is expected to have the greatest impact in condensed matter theory, where it allows to describe an N-electron system by a many-electron wavefunction Ψ of four, instead of 3N variables. The many-body aspect of a system is in this case encoded in a bivector potential.

Unconventional Approach to Orbital-Free Density Functional Theory Derived from a Model of Extended Electrons

The traditional standard quantum mechanics theory is unable to solve the spin–statistics problem, ie to justify the utterly important “Pauli Exclusion Principle” A complete and straightforward solution of the spin–statistics problem is presented on the basis of the “conformal quantum geometrodynamics” theory This theory provides a Weyl-gauge invariant formulation of the standard quantum mechanics and reproduces successfully all relevant quantum processes including the formulation of Dirac’s or Schrodinger’s equation, of Heisenberg’s uncertainty relations and of the nonlocal EPR correlations When the conformal quantum geometrodynamics is applied to a system made of many identical particles with spin, an additional constant property of all elementary particles enters naturally into play: the “intrinsic helicity” This property, not considered in the Standard Quantum Mechanics, determines the correct spin–statistics connection observed in Nature

Proof of the Spin–Statistics Theorem

Les idees qui m'avaient guidees en 1923–24 lors de la decouverte de la mecanique ondulatoire et celles que j'avais ensuite developpees jusqu'en 1928 etaient tout a fait differentes de l' Interpretation de cette decouverte qui fut ensuite proposee par l'Ecole de Copenhague et notamment par Max Born, Niels Bohr et Werner Heisenberg. A partir de 1928, n'ayant pu developper completement mes conceptions a ce sujet et constatant le succes et le caractere rigoureux des formalismes qui sont aujourd'hui connus sous les noms de mecanique quantique et de theorie quantique des champs, j'ai adopte et enseigne pendant plus de 20 ans l'interpretation de l'Ecole de Copenhague.

L De Broglie

Papers

La réinterpretation de la mécanique ondulatoire