Paravector

The name paravector is used for the combination of a scalar and a vector in any Clifford algebra, known as geometric algebra among physicists.

This name was given by J. G. Maks in a doctoral dissertation at Technische Universiteit Delft, Netherlands, in 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 David 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 value of the length squared (positive)

v

v

=

v

⋅

v

{\displaystyle \mathbf {v} \mathbf {v} =\mathbf {v} \cdot \mathbf {v} }

Writing

v

=

u

+

w

,

{\displaystyle \mathbf {v} =\mathbf {u} +\mathbf {w} ,}

and introducing this into the expression of the fundamental axiom

(

u

+

w

)

2

=

u

u

+

u

w

+

w

u

+

w

w

,

{\displaystyle (\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

u

⋅

u

+

2

u

⋅

w

+

w

⋅

w

=

u

⋅

u

+

u

w

+

w

u

+

w

⋅

w

,

{\displaystyle \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

u

⋅

w

=

1

2

(

u

w

+

w

u

)

.

{\displaystyle \mathbf {u} \cdot \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

u

w

+

w

u

=

0

{\displaystyle \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

C

ℓ

3

{\displaystyle C\ell _{3}}

space,

{

1

,

{

e

1

,

e

2

,

e

3

}

,

{

e

23

,

e

31

,

e

12

}

,

e

123

}

,

{\displaystyle \{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

e

23

=

e

2

e

3

.

{\displaystyle \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

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

e

i

e

j

+

e

j

e

i

=

2

δ

i

j

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

or in other words,

e

i

e

j

=

−

e

j

e

i

;

i

≠

j

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

This means that the volume element

e

123

{\displaystyle \mathbf {e} _{123}}

squares to

−

1

{\displaystyle -1}

e

123

2

=

e

1

e

2

e

3

e

1

e

2

e

3

=

e

2

e

3

e

2

e

3

=

−

e

3

e

3

=

−

1.

{\displaystyle \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

e

123

{\displaystyle \mathbf {e} _{123}}

commutes with any other element of the

C

ℓ

(

3

)

{\displaystyle C\ell (3)}

algebra, so that it can be identified with the complex number

i

{\displaystyle i}

, whenever there is no danger of confusion. In fact, the volume element

e

123

{\displaystyle \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

Paravectors

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

{

1

,

e

1

,

e

2

,

e

3

}

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

,

which forms a four-dimensional linear space.