The generalized Dirac equation related to 7-component space-time with one time coordinate and six space coordinates has been introduced. Three 8-component Dirac equations have been derived from the same 256-dimensional Clifford-Dirac matrix algebra. Corresponding Clifford-Dirac algebra is considered in the Pauli-Dirac representation of $8 \times 8$ gamma matrices. It is proved that this matrix algebra over the field of real numbers has 256-dimensional basis and it is isomorphic to geometric $\textit{C}\ell^{\texttt{R}}$(1,7) algebra. The corresponding gamma matrix representation of 45-dimensional $\mathrm{SO}(1,9)$ algebra is derived and the way of its generalization to the $\mathrm{SO}(m,n)$ algebra is demonstrated. The Klein-Gordon equation in 7-component space-time is considered as well. The way of corresponding consideration of the Maxwell equations and of equations for an arbitrary spin is indicated.

Abstract We investigate the general properties of the dimensional reduction of the Dirac theory, formulated in a Minkowski spacetime with an arbitrary number of spatial dimensions. This is done by applying Hadamard’s method of descent, which consists in conceiving low-dimensional theories as a specialization of high-dimensional ones that are uniform along the additional space coordinate. We show that the Dirac equation reduces to either a single Dirac equation or two decoupled Dirac equations, depending on whether the higher-dimensional manifold has even or odd spatial dimensions, respectively. Furthermore, we construct and discuss an explicit hierarchy of representations in which this procedure becomes manifest and can easily be iterated.

particles. Gamma matrices were introduced by Paul Dirac in 1928. In the Dirac basis of the Dirac representation, the four contravariant gamma matrices are γ In mathematical physics, the gamma matrices, { γ 0 , γ 1 , γ 2 The "fifth matrix" γ 5 {\displaystyle \ \gamma ^{5}\ } is not a proper member of the main set of four; it is used for separating nominal left and right chiral representations. The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, for any signature of the metric. For example, the 2×2 Pauli matrices are a set of "gamma" matrices in three dimensional space with metric of Euclidean signature (3, 0). In five spacetime dimensions, the four gammas, above, together with the fifth gamma-matrix to be presented below generate the Clifford algebra.

Abstract The Dirac equation (DE) is one of the cornerstones of quantum physics. We prove in the present contribution that the notion of internal degrees of freedom of the electron represented by Dirac’s matrices 1,2 is superfluous. One can write down a coordinate-free manifestly covariant vector equation by direct quantization of the 4-momentum vector with modulus m : Pψ=mψ (no slash!), the spinor ψ taking care of the different vector grades at the two sides of the equation. Electron spin and all the standard DE properties emerge from this equation. In coordinate representation, the four orthonormal time-space frame vectors X µ appear instead of Dirac’s Y µ -matrices, the two sets obeying to the same Clifford algebra. The present formalism expands Hestenes’ spacetime algebra (STA) by adding a reflector vector x 5 that is defining for the C (particle-antiparticle) symmetry and the CPT symmetry of DE, as well as for left- and right-handed rotors and spinors. In 3D it transforms a parity-odd vector x into a parity-even vector σ=x 5 x and vice versa. STA augmented by the reflector will be referred to as STAR, which operates on a real vector space of same dimension as the equivalent real dimension of Dirac’s complex 4×4 matrices. There are no matrices in STAR and the complex character springs from the signature and dimension of spacetime-reflection. This appears most clearly by first showing that STAR comprises two isomorphic subspaces, one for generators of polar vectors and boosts and the other for generators of axial vectors and rotors, including Pauli spin vectors. These then help to discuss the symmetries, probability current, transformation properties and nonrelativistic approximation of STAR DE. We prove here that the information from γ-matrices is contained in spacetime-reflection, which makes the matrices redundant. Therefore, it becomes relevant to reexamine those areas of quantum physics that take the γ-matrices and their generalizations as fundamental.

size of the Dirac spinor corresponding to the dimensionality of the gamma matrices. While the Dirac equation always exists, since every dimension admits In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-⁠1/2⁠ massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of spe w…