Paravector

The name paravector is used for the sum of a scalar and a vector in any Clifford algebra (Clifford algebra is also known as Geometric algebra in the physics community.)

This name was given by J. G. Maks, Doctoral Dissertation, Technische Universiteit Delft (Netherlands), 1989.

The complete algebra of paravectors along with corresponding higher grade generalizations, all in the context of the Euclidean space of three dimensions, is an alternative approach to the Spacetime algebra (STA) introduced by Hestenes. This alternative algebra is called Algebra of physical space (APS).

Fundamental axiom

For Euclidean spaces, the fundamental axiom indicates that the product of a vector with itself is the scalar valueof the length squared (positive)

:$mathbf\{v\}\; mathbf\{v\}\; =\; mathbf\{v\}cdot\; mathbf\{v\}$

Writing :$mathbf\{v\}\; =\; mathbf\{u\}\; +\; mathbf\{w\},$

and introducing this into the expression of the fundamental axiom

:$(mathbf\{u\}\; +\; mathbf\{w\})^2=\; mathbf\{u\}\; mathbf\{u\}\; +mathbf\{u\}\; mathbf\{w\}\; +\; mathbf\{w\}\; mathbf\{u\}\; +mathbf\{w\}\; mathbf\{w\},$

we get the following expression after appealing to the fundamental axiom again

:$mathbf\{u\}\; cdot\; mathbf\{u\}\; +2\; mathbf\{u\}\; cdot\; mathbf\{w\}\; +mathbf\{w\}\; cdot\; mathbf\{w\}\; =\; mathbf\{u\}\; cdot\; mathbf\{u\}\; +mathbf\{u\}\; mathbf\{w\}\; +\; mathbf\{w\}\; mathbf\{u\}\; +mathbf\{w\}\; cdot\; mathbf\{w\},$

which allows toidentify the scalar product of two vectors as

:$mathbf\{u\}\; cdot\; mathbf\{w\}\; =\; frac\{1\}\{2\}left(\; mathbf\{u\}\; mathbf\{w\}\; +\; mathbf\{w\}\; mathbf\{u\}\; ight).$

As an important consequence we conclude that two orthogonal vectors (with zeroscalar product) anticommute

:$mathbf\{u\}\; mathbf\{w\}\; +\; mathbf\{w\}\; mathbf\{u\}\; =\; 0$

The Three-dimensional Euclidean space

The following list represents an instance of a complete basis for the $Cl\_3$space,

$\{\; 1\; ,\; \{\; mathbf\{e\}\_1,mathbf\{e\}\_2,mathbf\{e\}\_3\; \}\; ,\; \{\; mathbf\{e\}\_\{23\},mathbf\{e\}\_\{31\},mathbf\{e\}\_\{12\}\; \}\; ,\; mathbf\{e\}\_\{123\}\; \},$

which forms an eight-dimensional space, where the multiple indices indicate the product of the respective basis vectors, for example

$mathbf\{e\}\_\{23\}\; =\; mathbf\{e\}\_2\; mathbf\{e\}\_3\; .$

The grade of a basis element is defined in terms of the vector multiplicity such that

*Note.- The volume element is invariant under the bar conjugation.

pecial subspaces

Four special subspaces can be defined in the $Cl\_3$ spacebased on their symmetries under the reversion and bar conjugations.

* Scalar Space: Invariant under bar conjugation.
* Vector Space: Changes sign under bar conjugation.
* Real Space: Invariant under reversion conjugation.
* Imaginary Space: Changes sign under reversion conjugation.

Given that $p$ is a general Clifford number, the complementary scalar and vector parts of $p$ are given by symmetric and antisymmetric combinations with the Clifford conjugation

$langle\; p\; angle\_S\; =\; frac\{1\}\{2\}(p\; +\; overline\{p\}),$

$langle\; p\; angle\_V\; =\; frac\{1\}\{2\}(p\; -\; overline\{p\})$.

Note that trivectors fall into the scalar category.

In similar way, the complementary Real and Imaginary parts of $p$ are givenby symmetric and antisymmetric combinations with the Reversion conjugation

$langle\; p\; angle\_R\; =\; frac\{1\}\{2\}(p\; +\; p^dagger),$

$langle\; p\; angle\_I\; =\; frac\{1\}\{2\}(p\; -\; p^dagger)$.

The following table summarizes the grades of the respective subspace

from which the null basis elements become

A general Clifford number in 3D can be written as

:$Psi\; =\; psi\_1^dagger\; P\_3\; -\; psi\_2^dagger\; P\_3\; mathbf\{e\}\_1\; +\; psi\_3\; P\_3\; +\; psi\_4\; mathbf\{e\}\_1\; P\_3$

In this way, the matrix representation becomes

:

If this Clifford number $Psi$ is used to represent a spinor,there is a one-to-one relation with the column spinor in the Weyl representationsuch that

:

or in the Pauli-Dirac representation

:

In the non-relativistic limit only the even grade elements survive in the case of particles and only the odd grade elements in the case of anti-particles. This means that the non-relativistic spinor can be described using only the first column of the 2x2 matrix representation:

:

The matrix representation for higher dimensions of a Euclidean space can be constructed in terms of the tensor matrix product of the Pauli matrices resulting incomplex matrices of dimension $2^n$. The 4D representationcould be taken as

The 7D representation could be taken as

Lie algebras

Clifford algebras can be used to represent any classical Lie algebra.In general it is possible to identify Lie algebras of compact groups by using anti-Hermitian elements, which can be extended to non-compact groups by adding Hermitian elements.

The bivectors of and n-dimensional Euclidean space are Hermitian elements and can be used to represent the $spin(n)$ Lie algebra.

The bivectors of the three-dimensional Euclidean space form the $spin(3)$ Lie algebra, which is isomorphicto the $su(2)$ Lie algebra. This accidental isomorphism allows to picture a geometric interpretation of thestates of the two dimensional Hilbert space by using the Bloch sphere. One of those systems is the spin 1/2 particle.

The $spin(3)$ Lie algebra can be extended by adding the three unitary vectors to form a Lie algebra isomorphicto the $SL(2,C)$ Lie algebra, which is the double cover of the Lorentz group $SO(3,1)$. This isomorphism allows the possibility to develop a formalism of special relativity based on $SL(2,C)$, which is carried outin the form of the algebra of physical space.

There is only one additional accidental isomorphism between a spin Lie algebra and a $su(N)$ Lie algebra. This is the isomorphism between $spin(6)$ and $su(4)$.

Another interesting isomorphism exists between $spin(5)$ and $sp(4)$. So, the $sp(4)$ Lie algebra can be used to generate the $USp(4)$ group. Despite that this group is smaller than the $SU(4)$ group, it is seen to be enough to span the four-dimensional Hilbert space.

ee also

* Algebra of physical space
* Dirac equation in the algebra of physical space

References

Textbooks

* Baylis, William (2002). "Electrodynamics: A Modern Geometric Approach" (2th ed.). Birkhäuser. ISBN 0-8176-4025-8

* [H1999] David Hestenes: New Foundations for Classical Mechanics (Second Edition). ISBN 0-7923-5514-8, Kluwer Academic Publishers (1999)

* Chris Doran and Antony Lasenby, Geometric Algebra for Physicists, Cambridge, 2003

* Baylis, William, Clifford (Geometric) Algebras With Applications in Physics, Mathematics, and Engineering, Birkhauser (1999)

Articles

* William E. Baylis, "Relativity in Introductory Physics", Can. J. Phys. 82 (11), 853--873 (2004). (ArXiv:physics/0406158)

* C. Doran, D. Hestenes, F. Sommen and N. Van Acker, "Lie groups and spin groups", J. Math. Phys. 34 (8), 1993

* R. Cabrera, W. E. Baylis, C. Rangan, "Sufficient condition for the coherent control of n-qubit systems" , Phys. Rev. A, 76 , 033401, 2007