paravector

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

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

Writing

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

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 to

identify the scalar product of two vectors as

:$\backslash mathbf\{u\}\; \backslash cdot\; \backslash mathbf\{w\}\; =$

\frac{1}{2}\left( \mathbf{u} \mathbf{w} + \mathbf{w} \mathbf{u} \right).

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

:$$

\mathbf{u} \mathbf{w} + \mathbf{w} \mathbf{u} = 0

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

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

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

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

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

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 $\backslash mathbf\{e\}\_\{123\}$ squares to $-1$

:$\backslash mathbf\{e\}\_\{123\}^2\; =$

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

basis as

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

:$\backslash \{\; 1\; ,\; \backslash mathbf\{e\}\_1,\backslash mathbf\{e\}\_2,\backslash mathbf\{e\}\_3\; \backslash \}$,

which forms a four-dimensional linear space. The paravector space in the three-dimensional Euclidean space $C\backslash ell\_3$ can be used to represent the space-time of special relativity as expressed in the algebra of physical space (APS).

It is convenient to write the unit scalar as $1=\backslash mathbf\{e\}\_0$, so that

the complete basis can be written in a compact form as

:$\backslash \{\; \backslash mathbf\{e\}\_\backslash mu\; \backslash \},$

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

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

:$(AB)^\backslash dagger\; =\; B^\backslash dagger\; A^\backslash dagger$,

where vectors and real scalar numbers are invariant under

reversion conjugation and are said to be real, for example:

:$\backslash mathbf\{a\}^\backslash dagger\; =\; \backslash mathbf\{a\}$

:$1^\backslash dagger\; =\; 1$

On the other hand, the trivector and bivectors change sign under reversion

conjugation and are said to be purely imaginary. The reversion conjugation applied to each basis element is given

below

The Clifford Conjugation is denoted by a bar over the object

$\backslash bar\{\; \}$. This conjugation is also called bar conjugation.

Clifford conjugation is the combined action of grade involution and reversion.

The action of the Clifford conjugation on a paravector is to reverse the sign of the

vectors, maintaining the sign of the real scalar numbers, for example

:$\backslash bar\{\backslash mathbf\{a\}\}\; =\; -\backslash mathbf\{a\}$

:$\backslash bar\{1\}\; =\; 1$

This is due to both scalars and vectors being invariant to reversion ( it is impossible

to reverse the order of one or no things ) and scalars are of zero order and so are of

even grade whilst vectors are of odd grade and so undergo a sign change under grade involution.

As antiautomorphism, the Clifford conjugation is distributed as

:$\backslash overline\{AB\}\; =\; \backslash overline\{B\}\; \backslash ,\backslash ,\; \backslash overline\{A\}$

The bar conjugation applied to each basis element is given

below

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

The grade automorphism

:$$

\overline{A B}^\dagger = \overline{A}^\dagger \overline{B}^\dagger

is defined as the inversion of the sign of odd-grade multivectors, while maintaining the even-grade multivectors invariant:

Four special subspaces can be defined in the $C\backslash ell\_3$ space

based on their symmetries under the reversion and Clifford conjugation

• Scalar subspace: Invariant under Clifford conjugation.
• Vector subspace: Reverses sign under Clifford conjugation.
• Real subspace: Invariant under reversion conjugation.
• Imaginary subspace: Reverses sign under reversion conjugation.

Given $p$ as 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})

.

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)

.

It is possible to define four intersections, listed below

:$$

\langle p \rangle_{RS} = \langle p \rangle_{SR} \equiv \langle \langle p \rangle_R \rangle_S

:$$

\langle p \rangle_{RV} = \langle p \rangle_{VR} \equiv \langle \langle p \rangle_R \rangle_V

:$$

\langle p \rangle_{IV} = \langle p \rangle_{VI} \equiv \langle \langle p \rangle_I \rangle_V

:$$

\langle p \rangle_{IS} = \langle p \rangle_{SI} \equiv \langle \langle p \rangle_I \rangle_S

The following table summarizes the grades of the respective subspaces, where for example,

the grade 0 can be seen as the intersection of the Real and Scalar subspaces

• Remark: The term "Imaginary" is used in the context of the $C\backslash ell\_3$ algebra and does not imply the introduction of the standard complex numbers in any form.

There are two subspaces that are closed with 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
• :$\backslash mathbf\{e\}\_\{123\}\; =\; i.$
• The even space, made of elements of grades 0 and 2, is isomorphic with the algebra of quaternions with the identification of
• :$-\backslash mathbf\{e\}\_\{23\}\; =\; i$
• :$-\backslash mathbf\{e\}\_\{31\}\; =\; j$
• :$-\backslash mathbf\{e\}\_\{12\}\; =\; k.$

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

:$\backslash langle\; u\; \backslash bar\{v\}\; \backslash rangle\_S.$

The magnitude square of a paravector $u$ is

:$\backslash langle\; u\; \backslash bar\{u\}\; \backslash rangle\_S,$

which is not a definite bilinear form 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.

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

defined as:

:$B\; =\; \backslash langle\; u\; \backslash bar\{v\}\; \backslash rangle\_V$.

The biparavector basis can be written as

:$\backslash \{\; \backslash langle\; \backslash mathbf\{e\}\_\backslash mu\; \backslash bar\{\backslash mathbf\{e\}\}\_\backslash nu\; \backslash rangle\_V\; \backslash \},$

which contains six independent elements, including real and imaginary terms.

Three real elements (vectors) as

:$\backslash langle\; \backslash mathbf\{e\}\_0\; \backslash bar\{\backslash mathbf\{e\}\}\_k\; \backslash rangle\_V\; =\; -\backslash mathbf\{e\}\_k\; ,$

and three imaginary elements (bivectors) as

:$\backslash langle\; \backslash mathbf\{e\}\_j\; \backslash bar\{\backslash mathbf\{e\}\}\_k\; \backslash rangle\_V\; =\; -\backslash mathbf\{e\}\_\{jk\}$

where $j,k$ run from 1 to 3.

In the Algebra of physical space,

the electromagnetic field is expressed as a biparavector as

:$$

F = \mathbf{E} + i \mathbf{B}^{\,},

where both the electric and magnetic fields are real vectors

:$\backslash mathbf\{E\}^\backslash dagger\; =\; \backslash mathbf\{E\}$

:$\backslash mathbf\{B\}^\backslash dagger\; =\; \backslash mathbf\{B\}$

and $i$ represents the pseudoscalar volume element.

Another example of biparavector is the representation of the space-time rotation rate that can be expressed as

:$$

W = i \theta^j \mathbf{e}_j + \eta^j \mathbf{e}_j,

with three ordinary rotation angle variables $\backslash theta^j$ and three rapidities $\backslash eta^j$.

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

defined as:

:$T\; =\; \backslash langle\; u\; \backslash bar\{v\}\; w\; \backslash rangle\_I$.

The triparavector basis can be written as

:$\backslash \{\; \backslash langle\; \backslash mathbf\{e\}\_\backslash mu\; \backslash bar\{\backslash mathbf\{e\}\}\_\backslash nu\; \backslash mathbf\{e\}\_\{\backslash lambda\}\; \backslash rangle\_I\; \backslash \},$

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

:$\backslash \{\; i\; \backslash mathbf\{e\}\_\{\backslash rho\}\; \backslash \}$.

The pseudoscalar basis is

:$\backslash \{\; \backslash langle\; \backslash mathbf\{e\}\_\backslash mu\; \backslash bar\{\backslash mathbf\{e\}\}\_\backslash nu\; \backslash mathbf\{e\}\_\{\backslash lambda\}$

\bar{\mathbf{e}}_{\rho}\rangle_{IS} \},

but a calculation reveals that it contains only a single term. This term is the volume element $i\; =\; \backslash mathbf\{e\}\_1\; \backslash mathbf\{e\}\_2\; \backslash mathbf\{e\}\_3$.

The four grades, taken in combination of pairs generate the paravector, biparavector and triparavector spaces as shown in the next table, where for example, we see that the paravector is made of grades 0 and 1

The paragradient operator is the generalization of the gradient operator in the paravector space. The paragradient in the standard paravector basis is

:$$

\partial = \mathbf{e}_0 \partial_0 - \mathbf{e}_1 \partial_1 - \mathbf{e}_2 \partial_2 - \mathbf{e}_3 \partial_3,

which allows one to write the d'Alembert operator as

:$$

\square = \langle \bar{\partial} \partial \rangle_S = \langle \partial \bar{\partial} \rangle_S

The standard gradient operator can be defined naturally as

:$$

\nabla = \mathbf{e}_1 \partial_1 + \mathbf{e}_2 \partial_2 + \mathbf{e}_3 \partial_3,

so that the paragradient can be written as

:$$

\partial = \partial_0 - \nabla,

where $\backslash mathbf\{e\}\_0\; =\; 1$.

The application of the paragradient operator must be done carefully, always respecting its non-commutative nature. For example, a widely used derivative is

:$$

\partial e^{ f(x) \mathbf{e}_3 } =

(\partial f(x)) e^{ f(x) \mathbf{e}_3 } \mathbf{e}_3,

where $f(x)$ is a scalar function of the coordinates.

The paragradient is an operator that always acts from the left if the function is a scalar function. However, if the function is not scalar, the paragradient can act from the right as well. For example, the following expression is expanded as

:$(L\; \backslash partial)\; =$

\mathbf{e}_0 \partial_0 L + (\partial_1 L) \mathbf{e}_1 +

(\partial_2 L)\mathbf{e}_2 + (\partial_3 L) \mathbf{e}_3

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

:$p\; \backslash bar\{p\}\; =\; 0.$

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

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

:$$

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

such that

:$P\_\{\backslash mathbf\; k\}\; +\; \backslash bar\{P\}\_\{\backslash mathbf\; k\}\; =\; 1$

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 because they are null paravectors

:$P\_\{\backslash mathbf\; k\}\; \backslash bar\{P\}\_\{\backslash mathbf\; k\}\; =\; 0.$

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 $\backslash hat\{\backslash mathbf\{k\}\}$ is an operator

with eigenfunctions $P\_\backslash mathbf\{\backslash mathbf\{k\}\}$ and

$\backslash bar\{P\}\_\backslash mathbf\{\backslash mathbf\{k\}\}$, with respective eigenvalues

$1$ and $-1$.

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

A basis of elements, each one of them null, can be constructed for the complete

$C\backslash ell\_3$ space. The basis of interest is the following

:$\backslash \{\; \backslash bar\{P\}\_3,\; P\_3\; \backslash mathbf\{e\}\_1,\; P\_3,\; \backslash mathbf\{e\}\_1\; P\_3\; \backslash \}$

so that an arbitrary paravector

:$p\; =\; p^0\; \backslash mathbf\{e\}\_0\; +\; p^1\; \backslash mathbf\{e\}\_1\; +\; p^2\; \backslash mathbf\{e\}\_2\; +\; p^3\; \backslash mathbf\{e\}\_3$

can be written as

:$p\; =\; (p^0+p^3)P\_3\; +\; (p^0\; -\; p^3)\backslash bar\{P\}\_3\; +\; (p^1+ip^2)\backslash mathbf\{e\}\_1\; P\_3\; +\; (p^1-ip^2)P\_3\; \backslash mathbf\{e\}\_1$

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

$\backslash bar\{P\}\_3$ 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

$\backslash \{\; u\; ,\; v\; \backslash \}$ and a general scalar number $w$ (including scalar and pseudoscalar numbers)

:$p\; =\; u\; \backslash bar\{P\}\_3\; +\; v\; P\_3\; +\; w\; \backslash mathbf\{e\}\_1\; P\_3\; +\; w^\{\backslash dagger\}P\_3\; \backslash mathbf\{e\}\_1$

the paragradient in the null basis is

:$\backslash partial\; =\; 2P\_3\; \backslash partial\_u\; +\; 2\backslash bar\{P\}\_3\; \backslash partial\_v\; -$

2\mathbf{e}_1 P_3 \partial_{w^{\dagger}} - 2 P_3 \mathbf{e}_1 \partial_w

An n-dimensional Euclidean space allows the existence of multivectors of grade n (n-vectors). The dimension of the vector space is evidently equal to n and a simple combinatorial analysis shows that the dimension of the bivector space is $\backslash begin\{pmatrix\}\; n\; \backslash \backslash \; 2\; \backslash end\{pmatrix\}$. In general, the dimension of the multivector space of grade m is $\backslash begin\{pmatrix\}\; n\; \backslash \backslash \; m\; \backslash end\{pmatrix\}$ and the dimension of the whole Clifford algebra $C\backslash ell(n)$ is $2^n$.

A given multivector with homogeneous grade is either invariant or changes sign under the action of the reversion conjugation $\backslash dagger$. The elements that remain invariant are defined as Hermitian and those that change sign are defined as anti-Hermitian. Grades can thus be classified as follows:

The algebra of the $C\backslash ell(3)$ space is isomorphic to the Pauli matrix algebra such that

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_{11} P_3 - \psi_{12} P_3 \mathbf{e}_1 + \psi_{21} \mathbf{e}_1 P_3 +

\psi_{22} \bar{P}_3,

where the coefficients $\backslash psi\_\{jk\}$ are scalar elements (including pseudoscalars). The indexes were chosen such that the representation of this Clifford number in terms of the Pauli matrices is

:$$

\Psi \rightarrow

\begin{pmatrix}

\psi_{11} & \psi_{12} \\ \psi_{21} & \psi_{22}

\end{pmatrix}

The reversion conjugation is translated into the Hermitian conjugation and the bar conjugation is translated into the following matrix:

:$$

\bar{\Psi} \rightarrow

\begin{pmatrix}

\psi_{22} & -\psi_{12} \\ -\psi_{21} & \psi_{11}

\end{pmatrix},

such that the scalar part is translated as

:$$

\langle \Psi \rangle_S \rightarrow

\frac{ \psi_{11} + \psi_{22} }{2}\begin{pmatrix}

1 & 0 \\ 0 & 1

\end{pmatrix} = \frac{Tr[\psi]}{2} \mathbf{1}_{2\times 2}

The rest of the subspaces are translated as

:$$

\langle \Psi \rangle_V \rightarrow

\begin{pmatrix}

0 & \psi_{12} \\ \psi_{21} & 0

\end{pmatrix}

:$$

\langle \Psi \rangle_R \rightarrow

\frac{1}{2}

\begin{pmatrix}

\psi_{11}+\psi_{11}^* & \psi_{12}+\psi_{21}^* \\

\psi_{21}+\psi_{12}^* & \psi_{22}+\psi_{22}^*

\end{pmatrix}

:$$

\langle \Psi \rangle_I \rightarrow

\frac{1}{2}

\begin{pmatrix}

\psi_{11}-\psi_{11}^* & \psi_{12}-\psi_{21}^* \\

\psi_{21}-\psi_{12}^* & \psi_{22}-\psi_{22}^*

\end{pmatrix}

The matrix representation of a Euclidean space in higher dimensions can be constructed in terms of the Kronecker product of the Pauli matrices, resulting in complex matrices of dimension $2^n$. The 4D representation could be taken as

The 7D representation could be taken as

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 an n-dimensional Euclidean space are Hermitian elements and can be used to represent the $\backslash mathrm\{spin\}(n)$ Lie algebra.

The bivectors of the three-dimensional Euclidean space form the $\backslash mathrm\{spin\}(3)$ Lie algebra, which is isomorphic

to the $\backslash mathrm\{su\}(2)$ Lie algebra. This accidental isomorphism allows to picture a geometric interpretation of the

states of the two dimensional Hilbert space by using the Bloch sphere. One of those systems is the spin 1/2 particle.

The $\backslash mathrm\{spin\}(3)$ Lie algebra can be extended by adding the three unitary vectors to form a Lie algebra isomorphic

to the $\backslash mathrm\{SL\}(2,C)$ Lie algebra, which is the double cover of the Lorentz group $\backslash mathrm\{SO\}(3,1)$. This isomorphism

allows the possibility to develop a formalism of special relativity based on $\backslash mathrm\{SL\}(2,C)$, which is carried out

in the form of the algebra of physical space.

There is only one additional accidental isomorphism between a spin Lie algebra and a $\backslash mathrm\{su\}(N)$ Lie algebra. This

is the isomorphism between $\backslash mathrm\{spin\}(6)$ and $\backslash mathrm\{su\}(4)$.

Another interesting isomorphism exists between $\backslash mathrm\{spin\}(5)$ and $\backslash mathrm\{sp\}(4)$. So, the

$\backslash mathrm\{sp\}(4)$ Lie algebra can be used to generate the $USp(4)$ group. Despite that this group

is smaller than the $\backslash mathrm\{SU\}(4)$ group, it is seen to be enough to span the four-dimensional Hilbert space.

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

{{reflist}}

• Baylis, William (2002). Electrodynamics: A Modern Geometric Approach (2nd ed.). Birkhäuser. {{ISBN|0-8176-4025-8}}
• Baylis, William, Clifford (Geometric) Algebras With Applications in Physics, Mathematics, and Engineering, Birkhauser (1999)
• [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

• {{cite journal | last=Baylis | first=W E | title=Relativity in introductory physics | journal=Canadian Journal of Physics | publisher=Canadian Science Publishing | volume=82 | issue=11 | date=2004-11-01 | issn=0008-4204 | doi=10.1139/p04-058 | pages=853–873|arxiv=physics/0406158| bibcode=2004CaJPh..82..853B | s2cid=35027499 }}
• {{cite journal | last1=Doran | first1=C. | last2=Hestenes | first2=D. | last3=Sommen | first3=F. | last4=Van Acker | first4=N. | title=Lie groups as spin groups | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=34 | issue=8 | year=1993 | issn=0022-2488 | doi=10.1063/1.530050 | pages=3642–3669| bibcode=1993JMP....34.3642D }}
• {{cite journal | last1=Cabrera | first1=R. | last2=Rangan | first2=C. | last3=Baylis | first3=W. E. | title=Sufficient condition for the coherent control of n-qubit systems | journal=Physical Review A | publisher=American Physical Society (APS) | volume=76 | issue=3 | date=2007-09-04 | issn=1050-2947 | doi=10.1103/physreva.76.033401 | page=033401|arxiv=quant-ph/0703220| bibcode=2007PhRvA..76c3401C | s2cid=45060566 }}
• {{cite journal | last1=Vaz | first1=Jayme | last2=Mann | first2=Stephen | title=Paravectors and the Geometry of 3D Euclidean Space | journal=Advances in Applied Clifford Algebras | publisher=Springer Science and Business Media LLC | volume=28 | issue=5 | year=2018 | issn=0188-7009 | doi=10.1007/s00006-018-0916-1 | page=99|arxiv=1810.09389| s2cid=253600966 }}

{{Algebra of Physical Space}}

Category:Multilinear algebra

Category:Clifford algebras

Category:Geometric algebra