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

The fundamental axiom indicates that the product of a vector with itself is the scalar value of the length squared

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

Writing

$\mathbf {v} =\mathbf {u} +\mathbf {w}$

the scalar product of two vectors is found to be the symmetric product

$\mathbf {u} \cdot \mathbf {w} ={\frac {1}{2}}\left(\mathbf {u} \mathbf {w} +\mathbf {w} \mathbf {u} \right),$

where, it is clear that two orthogonal vectors anticommute.

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

Grade	Type	Basis element/s
0	Unitary real scalar	$1$
1	Vector	$\{\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\}$
2	Bivector	$\{\mathbf {e} _{23},\mathbf {e} _{31},\mathbf {e} _{12}\}$
3	Trivector volume element	$\mathbf {e} _{123}$

According to the fundamental axiom, two different basis vectors anticommute,

$\mathbf {e} _{i}\mathbf {e} _{j}+\mathbf {e} _{j}\mathbf {e} _{i}=2\delta _{ij}$

or in other words,

$\mathbf {e} _{i}\mathbf {e} _{j}=-\mathbf {e} _{j}\mathbf {e} _{i}\,\,;i\neq j$

This means that the volume element $\mathbf {e} _{123}$ squares to $-1$

$\mathbf {e} _{123}^{2}=\mathbf {e} _{1}\mathbf {e} _{2}\mathbf {e} _{3}\mathbf {e} _{1}\mathbf {e} _{2}\mathbf {e} _{3}=\mathbf {e} _{2}\mathbf {e} _{3}\mathbf {e} _{2}\mathbf {e} _{3}=-\mathbf {e} _{3}\mathbf {e} _{3}=-1.$

Moreover, the volume element $\mathbf {e} _{123}$ commutes with any other element of the $Cl(3)$ algebra, so that it can be identified with the complex number $i$, whenever there is no danger of confusion. In fact, the volume element $\mathbf {e} _{123}$ along with the real scalar forms an algebra isomorphic to the standard complex algebra.

Paravectors

The corresponding paravector basis that combines a real scalar and vectors is

$\{1,\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\}$,

which forms a four-dimensional linear space. The paravector space in the three-dimensional Euclidean space $Cl_{3}$ can be used to represent the space-time of Special Relativity.

It is convenient to write the unit scalar as $1=\mathbf {e} _{0}$, so that the complete basis can be written in a compact form as

$\{\mathbf {e} _{\mu }\},$

where the Greek indices such as $\mu$ run from $0$ to $3$.

Antiautomorphism

Reversion conjugation

The Reversion antiautomorphism is denoted by $\dagger$. The action of this conjugation is to reverse the order of the geometric product (product between Clifford numbers in general).

$(AB)^{\dagger }=B^{\dagger }A^{\dagger }$,

where vectors and real scalar numbers are invariant under reversion, for example:

$\mathbf {a} ^{\dagger }=\mathbf {a}$

$1^{\dagger }=1$

The reversion conjugation applied to each basis element is given below

Element	Reversion conjugation
$1$	$1$
$\mathbf {e} _{1}$	$\mathbf {e} _{1}$
$\mathbf {e} _{2}$	$\mathbf {e} _{2}$
$\mathbf {e} _{3}$	$\mathbf {e} _{3}$
$\mathbf {e} _{12}$	-$\mathbf {e} _{12}$
$\mathbf {e} _{23}$	-$\mathbf {e} _{23}$
$\mathbf {e} _{31}$	-$\mathbf {e} _{31}$
$\mathbf {e} _{123}$	-$\mathbf {e} _{123}$

Clifford conjugation

The Clifford Conjugation is denoted by a bar over the object ${\bar {}}$. This conjugation is also called bar conjugation.

The action of the Clifford conjugation is to reverse the sign of the vectors, maintaining the sign of the real scalar numbers, for example

${\bar {\mathbf {a} }}=-\mathbf {a}$

${\bar {1}}=1$

As antiautomorphism, the Clifford conjugation is distributed as

${\overline {AB}}={\overline {B}}\,\,{\overline {A}}$

The bar conjugation applied to each basis element is given below

Element	Bar conjugation
$1$	$1$
$\mathbf {e} _{1}$	-$\mathbf {e} _{1}$
$\mathbf {e} _{2}$	-$\mathbf {e} _{2}$
$\mathbf {e} _{3}$	-$\mathbf {e} _{3}$
$\mathbf {e} _{12}$	-$\mathbf {e} _{12}$
$\mathbf {e} _{23}$	-$\mathbf {e} _{23}$
$\mathbf {e} _{31}$	-$\mathbf {e} _{31}$
$\mathbf {e} _{123}$	$\mathbf {e} _{123}$

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

Special subspaces

Four special subspaces can be defined in the $Cl_{3}$ space based 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

$\langle p\rangle _{S}={\frac {1}{2}}(p+{\overline {p}}),$

$\langle p\rangle _{V}={\frac {1}{2}}(p-{\overline {p}})$.

Note that trivectors fall into the scalar category.

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

$\langle p\rangle _{R}={\frac {1}{2}}(p+p^{\dagger }),$

$\langle p\rangle _{I}={\frac {1}{2}}(p-p^{\dagger })$.

The following table summarizes the grades of the respective subspace

	Real	Imaginary
Scalar	0	3
Vector	1	2

• Remark: The term "Imaginary" is used in the context of the $Cl_{3}$ algebra and does not imply the introduction of the standard complex numbers in any form.

Closed Subspaces respect to the product

There are two subspaces that are closed respect to the product. They are the scalar space and the even space that are isomorphic with the well known algebras of complex numbers and quaternions.

• The scalar space made of grades 0 and 3 is isomorphic with the standard algebra of complex numbers with the identification of

$\mathbf {e} _{123}=i$

• The even space, made of elements of grades 0 and 2, is isomorphic with the algebra of quaternions with the identification of

$\mathbf {e} _{12}=i$

$\mathbf {e} _{23}=j$

$\mathbf {e} _{31}=k$

Scalar Product

Given two paravectors $u$ and $v$, the generalization of the scalar product is

$\langle u{\bar {v}}\rangle _{S}.$

The magnitude square of a paravector $u$ is

$\langle u{\bar {u}}\rangle _{S},$

which is not positive-definite and can be equal to zero even if the paravector is not equal to zero.

It is very suggestive that the paravector space automatically obeys the metric of the Minkowski space because $\eta _{\mu \nu }=\langle \mathbf {e} _{\mu }{\bar {\mathbf {e} }}_{\nu }\rangle _{S}$

and in particular:

$\eta _{00}=\langle \mathbf {e} _{0}{\bar {\mathbf {e} }}_{0}\rangle =\langle 1(1)\rangle _{S}=1,$

$\eta _{11}=\langle \mathbf {e} _{1}{\bar {\mathbf {e} }}_{1}\rangle =\langle \mathbf {e} _{1}(-\mathbf {e} _{1})\rangle _{S}=-1,$

$\eta _{01}=\langle \mathbf {e} _{0}{\bar {\mathbf {e} }}_{1}\rangle =\langle 1(-\mathbf {e} _{1})\rangle _{S}=0.$

Biparavectors

Given two paravectors $u$ and $v$, the biparavector B is defined as:

$B=\langle u{\bar {v}}\rangle _{V}$.

The biparavector basis can be written as

$\{\langle \mathbf {e} _{\mu }{\bar {\mathbf {e} }}_{\nu }\rangle _{V}\},$

which contains six independent terms.

Triparavectors

Given three paravectors $u$, $v$ and $w$, the triparavector T is defined as:

$T=\langle u{\bar {v}}w\rangle _{I}$.

The triparavector basis can be written as

$\{\langle \mathbf {e} _{\mu }{\bar {\mathbf {e} }}_{\nu }\mathbf {e} _{\lambda }\rangle _{I}\},$

but there are only four independent triparavectors, so it can be reduced to

$\{i\mathbf {e} _{\rho }\}$.

Pseudoscalar

The pseudoscalar basis is $\{\langle \mathbf {e} _{\mu }{\bar {\mathbf {e} }}_{\nu }\mathbf {e} _{\lambda }{\bar {\mathbf {e} }}_{\rho }\rangle _{IS}\},$

but a calculation reveals that it contains only a single term. This term is the volume element $i=\mathbf {e} _{1}\mathbf {e} _{2}\mathbf {e} _{3}$.

The four grades, taken in combination of pairs generate the paravector, biparavector and triparavector spaces as shown in the next table

	1	3
0	Paravector	Scalar
2	Biparavector	Triparavector

The pseudoscalar basis element is not part of the previous table because is single graded.

Null Paravectors as Projectors

Null paravectors are elements that are not necessarily zero but have magnitude identical to zero. For a null paravector $p$, this property necessarily implies the following identity

$p{\bar {p}}=0.$

In the context of Special Relativity they are also called lightlike paravectors.

Projectors are null paravectors of the form

$P_{\mathbf {k} }={\frac {1}{2}}(1+{\hat {\mathbf {k} }}),$

where ${\hat {\mathbf {k} }}$ is a unit vector.

A projector $P_{\mathbf {k} }$ of this form has a complementary projector ${\bar {P}}_{\mathbf {k} }$

${\bar {P}}_{\mathbf {k} }={\frac {1}{2}}(1-{\hat {\mathbf {k} }}),$

such that

$P_{\mathbf {k} }+{\bar {P}}_{\mathbf {k} }=1$

As projectors, they are idempotent

$P_{\mathbf {k} }=P_{\mathbf {k} }P_{\mathbf {k} }=P_{\mathbf {k} }P_{\mathbf {k} }P_{\mathbf {k} }=...$

and the projection of one on the other is zero due to the fact that they are null paravectors

$P_{\mathbf {k} }{\bar {P}}_{\mathbf {k} }=0.$

The associated unit vector of the projector can be extracted as

${\hat {\mathbf {k} }}=P_{\mathbf {\mathbf {k} } }-{\bar {P}}_{\mathbf {k} },$

this means that ${\hat {\mathbf {k} }}$ is an operator with eigenfunctions $P_{\mathbf {\mathbf {k} } }$ and ${\bar {P}}_{\mathbf {\mathbf {k} } }$, with respective eigenvalues $1$ and $-1$.

From the previous result, the following identity is valid assuming that $f({\hat {\mathbf {k} }})$ is analytic around zero

$f({\hat {\mathbf {k} }})=f(1)P_{\mathbf {k} }+f(-1){\bar {P}}_{\mathbf {k} }.$

This gives origin to the pacwoman property, such that the following identities are satisfied

$f({\hat {\mathbf {k} }})P_{\mathbf {k} }=f(1)P_{\mathbf {k} },$

$f({\hat {\mathbf {k} }}){\bar {P}}_{\mathbf {k} }=f(-1){\bar {P}}_{\mathbf {k} }.$

Null Basis for the paravector space

A basis of elements, each one of them null, can be constructed for the complete $Cl_{3}$ space. The basis of interest is the following

$\{{\bar {P}}_{3},P_{3}\mathbf {e} _{1},P_{3},\mathbf {e} _{1}P_{3}\}$

so that an arbitrary paravector

$p=p^{0}\mathbf {e} _{0}+p^{1}\mathbf {e} _{1}+p^{2}\mathbf {e} _{2}+p^{3}\mathbf {e} _{3}$

can be written as

$p=(p^{0}+p^{3})P_{3}+(p^{0}-p^{3}){\bar {P}}_{3}+(p^{1}+ip^{2})\mathbf {e} _{1}P_{3}+(p^{1}-ip^{2})P_{3}\mathbf {e} _{1}$

This representation is useful for some systems that are naturally expressed in terms of the light cone variables that are the coefficients of $P_{3}$ and ${\bar {P}}_{3}$ respectively.

Every expression in the paravector space can be written in terms of the null basis. A paravector $p$ is in general parametrized by two real scalars numbers $\{u,v\}$ and a general scalar number $w$ (including scalar and pseudoscalar numbers)

$p=u{\bar {P}}_{3}+vP_{3}+w\mathbf {e} _{1}P_{3}+w^{*}P_{3}\mathbf {e} _{1}$

the paragradient in the null basis is

$\partial =2P_{3}\partial _{u}+2{\bar {P}}_{3}\partial _{v}-2\mathbf {e} _{1}P_{3}\partial _{w^{*}}-2P_{3}\mathbf {e} _{1}\partial _{w}$

Matrix Representation

The algebra of the $Cl(3)$ space is isomorphic to the Pauli matrix algebra such that

	Matrix Representation 3D	Explicit matrix
$\mathbf {e} _{0}$	$\sigma _{0}^{}$	${\begin{pmatrix}1&&0\\0&&1\end{pmatrix}}$
$\mathbf {e} _{1}$	$\sigma _{1}^{}$	${\begin{pmatrix}0&&1\\1&&0\end{pmatrix}}$
$\mathbf {e} _{2}$	$\sigma _{2}^{}$	${\begin{pmatrix}0&&-i\\i&&0\end{pmatrix}}$
$\mathbf {e} _{3}$	$\sigma _{3}^{}$	${\begin{pmatrix}1&&0\\0&&-1\end{pmatrix}}$

from which, the null basis elements become ${P_{3}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}}\,;{\bar {P}}_{3}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}\,;{P_{3}}\mathbf {e} _{1}={\begin{pmatrix}0&1\\0&0\end{pmatrix}}\,;\mathbf {e} _{1}{P}_{3}={\begin{pmatrix}0&0\\1&0\end{pmatrix}}.$

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

$\Psi ={\begin{pmatrix}\psi _{3}&-\psi _{2}^{\dagger }\\\psi _{4}&\psi _{1}^{\dagger }\end{pmatrix}},$

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 representation such that

$\Psi ^{W}\rightarrow {\begin{pmatrix}\psi _{1}\\\psi _{2}\\\psi _{3}\\\psi _{4}\end{pmatrix}}$

or in the Pauli-Dirac representation

$\Psi ^{PD}\rightarrow {\begin{pmatrix}\psi _{3}+\psi _{1}\\\psi _{4}+\psi _{2}\\\psi _{3}-\psi _{1}\\\psi _{4}-\psi _{2}\end{pmatrix}}$

In the non-relativistic limit only the ven grade elements survives 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

$\Psi _{Non-Relativistic}=\Psi P_{3}={\begin{pmatrix}\psi _{1}&0\\\psi _{2}&0\end{pmatrix}}$

The matrix representation for higher dimensions of an Euclidean space can be constructed in terms of the tensor matrix product of the Pauli matrices resulting in complex matrices of dimension $2^{n}$. The 4D representation could be taken as

	Matrix Representation 4D
$\mathbf {e} _{1}$	$\sigma _{3}\otimes \sigma _{1}$
$\mathbf {e} _{2}$	$\sigma _{3}\otimes \sigma _{2}$
$\mathbf {e} _{3}$	$\sigma _{3}\otimes \sigma _{3}$
$\mathbf {e} _{4}$	$-\sigma _{2}\otimes \sigma _{0}$

The 7D representation could be taken as

	Matrix Representation 7D
$\mathbf {e} _{1}$	$\sigma _{0}\otimes \sigma _{3}\otimes \sigma _{1}$
$\mathbf {e} _{2}$	$\sigma _{0}\otimes \sigma _{3}\otimes \sigma _{2}$
$\mathbf {e} _{3}$	$\sigma _{0}\otimes \sigma _{3}\otimes \sigma _{3}$
$\mathbf {e} _{4}$	$\sigma _{0}\otimes \sigma _{2}\otimes \sigma _{0}$
$\mathbf {e} _{5}$	$\sigma _{3}\otimes \sigma _{1}\otimes \sigma _{0}$
$\mathbf {e} _{6}$	$\sigma _{1}\otimes \sigma _{1}\otimes \sigma _{0}$
$\mathbf {e} _{7}$	$\sigma _{2}\otimes \sigma _{1}\otimes \sigma _{0}$

Lie algebras

Some Lie algebras appear naturally including the $spin(n)$ Lie algebra from which all clasical Lie algebras can be constructed.

In fact, the $spin(n)$ Lie algebra can be constructed from the bivector representation of the bivector space in the N Euclidean space.

The special linear Lie algebra $sl(2)$ generated by the biparavector space in 3D including vectors and bivectors is important for special relativity. This happens because the associated Lie group $SL(2)$ plays the role of the double covering of the Lorentz group $SO(3,1)$

See 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

• Baylis, William (2002). Relativity in Introductory Physics, Can. J. Phys. 82 (11), 853--873 (2004). (ArXiv:physics/0406158)