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American Journal of Physics

A graphical representation of the Dirac algebra

David M. Goodmanson

Published Jul 1, 1996• 5 citations
Publisher

Abstract

The elements of the Dirac algebra are represented by sixteen 4×4 gamma matrices, each pair of which either commute or anticommute. This paper demonstrates a correspondence between the gamma matrices and the complete graph on six points, a correspondence that provides a visual picture of the structure of the Dirac algebra. The graph shows all commutation and anticommutation relations, and can be used to illustrate the structure of subalgebras and equivalence classes and the effect of similarity transformations. Since gamma matrices are the direct products of two Pauli spin matrices, they provide an appropriate way to describe a system of two spin-1/2 particles. Such multiparticle spin states are intimately connected with the theorems of John Bell. The graph is helpful in analyzing an important example of the Bell–Kochen–Specker theorem.

DOI
10.1119/1.18113

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The elements of the Dirac algebra are represented by sixteen 4×4 gamma matrices, each pair of which either commute or anticommute. This paper demonstrates a correspondence between the gamma matrices and the complete graph on six points, a correspondence that provides a visual picture of the structure of the Dirac algebra. The graph shows all commutation and anticommutation relations, and can be used to illustrate the structure of subalgebras and equivalence classes and the effect of similarity transformations. Since gamma matrices are the direct products of two Pauli spin matrices, they provide an appropriate way to describe a system of two spin-1/2 particles. Such multiparticle spin states are intimately connected with the theorems of John Bell. The graph is helpful in analyzing an important example of the Bell–Kochen–Specker theorem.
The link between a quantum spin 1/2 and its associated su(2) algebra of Pauli spin matrices with Clifford algebra and quaternions is well known. A pair of spins or qubits, which are important throughout the field of quantum information for describing logic gates and entangled states, has similarly an su(4) algebra. We develop connections between this algebra and its subalgebras with the projective plane of seven elements (also related to octonions) and other entities in projective geometry and design theory.
Abstract Chapter 1 opens with a brief review of some basic features of quantum mechanics, including the Schrödinger equation, linear and angular momentum and the theory of the hydrogenic atom: It also includes complete orthonormal sets of eigenfunctions, the translation operator, current, spin, equation of continuity, gauge transformation, determinant & permanent multiparticle energy eigenfunctions for noninteracting particles and the Pauli exclusion principle. Attention is then focused on Dirac bra-ket notation and complete sets of commuting observables. In this connection, representations and transformation among representations are discussed in detail for the Schrödinger system state vector and the eigenstates, as well as bra-ket matrix elements of operators. Finally, Schwinger’s interpretation of ket-bra matrix operator structures (Schwinger “Measurement Symbols”) in terms of annihilation and creation of systems in eigenstates is introduced.
In "Part I: Vector Analysis of Spinors", the author studied the geometry of two component spinors as points on the Riemann sphere in the geometric algebra of three dimensional Euclidean space. Here, these ideas are generalized to apply to four component Dirac spinors on the complex Riemann sphere in the complexified geometric algebra of spacetime, which includes Lorentz transformations. The development of generalized Pauli matrices eliminate the need for the traditional Dirac gamma matrices. We give the discrete probability distribution of measuring a spin 1/2 particle in an arbitrary spin state, assuming that it was prepared in a given state immediately prior to the measurement, independent of the inertial system in which measurements are made. The Fierz identities between the physical observables of a Dirac spinor are discussed.
In this chapter we study the application of geometric algebra to both nonrelativistic and relativistic quantum mechanics. We concentrate on the quantum theory of spin-1/2 particles, whose dynamics is described by the Pauli and Dirac equations. For interactions where spin and relativity are not important the dynamics reduces to that of the Schrödinger equation. There are many good textbooks describing this topic and we will make no attempt to cover it here. We assume, furthermore, that most readers have a basic understanding of quantum mechanics, and are familiar with the concepts of states and operators. Both the Pauli and Dirac matrices arise naturally as representations of the geometric algebras of space and spacetime. It is no surprise, then, that much of quantum theory finds a natural expression within geometric algebra. To achieve this, however, one must reconsider the standard interpretation of the quantum spin operators. Like much discussion of the interpretation of quantum theory, certain issues raised here are controversial. There is no question about the validity of our algebraic approach, however, and little doubt about its advantages. Whether the algebraic simplifications obtained here are indicative of a deeper structure embedded in quantum mechanics is an open question. In this chapter we only consider the quantum theory of single particles in background fields. Multiparticle systems are considered in the following chapter.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and ccc. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.