Dirac algebra: Spatial inversion of cliffors via the unit time vector

A cliffor in Clifford algebra is given by

,

which is a sum of a scalar, a vector, an imaginary vector (bivector), and an imaginary scalar (trivector).

Let be the unit time vector in the Clifford algebra , also known as the Dirac algebra with the signature. With this vector , we can define the spatial inverse of the cliffor in Eq. (1) as

.

Distributing the time vector in each component of on the left side of the equation and using the orthonormality axiom, we arrive at

Thus, the spatial inversion operation only affects vectors and imaginary numbers; the scalars and imaginary vectors remain unchanged.

The algebraic definition of spatial inversion in Eq. (3) lets us easily obtain two theorems:

That is, the spatial inverse of a sum (product) of two cliffors is the sum (product) of their individual spatial inverses. Furthermore, if is a scalar, then

,

where we used the expansion of the exponential of in terms of circular or hyperbolic cosine and sine functions.

Electromagnetic energy-momentum equation without tensors: a geometric algebra approach

Quirino M. Sugon Jr., Daniel J. McNamara

(Submitted on 9 Jul 2008)

Abstract: In this paper, we define energy-momentum density as a product of the complex vector electromagnetic field and its complex conjugate. We derive an equation for the spacetime derivative of the energy-momentum density. We show that the scalar and vector parts of this equation are the differential conservation laws for energy and momentum, and the imaginary vector part is a relation for the curl of the Poynting vector. We can show that the spacetime derivative of this energy-momentum equation is a wave equation. Our formalism is Dirac-Pauli-Hestenes algebra in the framework of Clifford (Geometric) algebra Cl_{4,0}.

Comments: 5 pages, no figures.
Subjects: Classical Physics (physics.class-ph)
Cite as: arXiv:0807.1382v1 [physics.class-ph]
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