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 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)
Writing 
and introducing this into the expression of the fundamental axiom
we get the following expression after appealing to the fundamental axiom again
which allows to
identify the scalar product of two vectors as
As an important consequence we conclude that two orthogonal vectors (with zero scalar product) anticommute
The Three-dimensional Euclidean space.
The following list represents an instance of a complete basis for the formula_7space,
formula_8
which forms an eight-dimensional space, where the multiple indices indicate the product of the respective basis vectors, for example
formula_9
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, 
or in other words,
This means that the volume element formula_12 squares to formula_13
Moreover, the volume element formula_15 commutes with any other element of the formula_16 algebra, so that it can be identified with the complex number formula_17, whenever there is no danger of confusion. In fact, the volume element formula_15 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
formula_19,
which forms a four-dimensional linear space. The paravector space in the three-dimensional Euclidean space formula_7 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 formula_21, so that
the complete basis can be written in a compact form as
formula_22
where the Greek indices such as formula_23 run from formula_24 to formula_25.
Antiautomorphism.
Reversion conjugation.
The Reversion antiautomorphism is denoted by formula_26. The action of this conjugation is to reverse the order of the geometric product (product between Clifford numbers in general).
formula_27,
where vectors and real scalar numbers are invariant under 
reversion conjugation and are said to be real, for example:
formula_28
formula_29
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
Clifford conjugation.
The Clifford Conjugation is denoted by a bar over the object 
formula_30. 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
formula_31
formula_32
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
formula_33
The bar conjugation applied to each basis element is given
below
Grade automorphism.
The grade automorphism
formula_34
is defined as the composite action of both the reversion conjugation and Clifford conjugation and has the effect to invert the sign of odd-grade multivectors, while maintaining the even-grade multivectors invariant:
Invariant subspaces according to the conjugations.
Four special subspaces can be defined in the formula_7 space
based on their symmetries under the reversion and Clifford conjugation
Given formula_36 as a general Clifford number, the complementary scalar and vector parts of formula_36 are given by 
symmetric and antisymmetric combinations with the Clifford conjugation
formula_38
formula_39.
In similar way, the complementary Real and Imaginary parts of formula_40 are given
by symmetric and antisymmetric combinations with the Reversion conjugation
formula_41
formula_42.
It is possible to define four intersections, listed below
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
Closed Subspaces with respect to the product.
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.
Scalar Product.
Given two paravectors formula_52 and formula_53, the generalization of the scalar product is
formula_54
The magnitude square of a paravector formula_52 is
formula_56
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
formula_57
and in particular:
formula_58
formula_59
formula_60
Biparavectors.
Given two paravectors formula_52 and formula_53, the biparavector B is
defined as:
formula_63.
The biparavector basis can be written as
formula_64
which contains six independent elements, including real and imaginary terms.
Three real elements (vectors) as 
and three imaginary elements (bivectors) as
where formula_67 run from 1 to 3.
In the Algebra of physical space,
the electromagnetic field is expressed as a biparavector as
where both the electric and magnetic fields are real vectors
and formula_71 represents the pseudoscalar volume element.
Another example of biparavector is the representation of the space-time rotation rate that can be expressed as
with three ordinary rotation angle variables formula_73 and three rapidities formula_74.
Triparavectors.
Given three paravectors formula_52, formula_53 and formula_77, the triparavector T is
defined as:
formula_78.
The triparavector basis can be written as
formula_79
but there are only four independent triparavectors, so it can be reduced to
formula_80.
Pseudoscalar.
The pseudoscalar basis is
formula_81
but a calculation reveals that it contains only a single term. This term is the volume element formula_82.
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
Paragradient.
The paragradient operator is the generalization of the gradient operator in the paravector space. The paragradient in the standard paravector basis is
which allows one to write the d'Alembert operator as
The standard gradient operator can be defined naturally as
so that the paragradient can be written as
where formula_87.
The application of the paragradient operator must be done carefully, always respecting its non-commutative nature. For example, a widely used derivative is 
where formula_89 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
Null Paravectors as Projectors.
Null paravectors are elements that are not necessarily zero but have magnitude identical to zero. For a null paravector formula_40, this property necessarily implies the following identity
formula_92
In the context of Special Relativity they are also called lightlike paravectors.
Projectors are null paravectors of the form
formula_93
where formula_94 is a unit vector.
A projector formula_95 of this form has a complementary projector formula_96
formula_97
such that
formula_98
As projectors, they are idempotent
formula_99
and the projection of one on the other is zero because they are 
null paravectors
formula_100
The associated unit vector of the projector can be extracted as
formula_101
this means that formula_102 is an operator
with eigenfunctions formula_103 and 
formula_104, with respective eigenvalues 
formula_105 and formula_13.
From the previous result, the following identity is valid assuming that formula_107 is analytic around zero
formula_108
This gives origin to the pacwoman property, such that the following identities are satisfied
formula_109
formula_110
Null Basis for the paravector space.
A basis of elements, each one of them null, can be constructed for the complete 
formula_7 space. The basis of interest is the following
formula_112
so that an arbitrary paravector
formula_113
can be written as
formula_114
This representation is useful for some systems that are naturally expressed in terms of the
light cone variables that are the coefficients of formula_115 and 
formula_116 respectively.
Every expression in the paravector space can be written in terms of the null basis. A paravector formula_40 is in general parametrized by two real scalars numbers 
formula_118 and a general scalar number formula_119 (including scalar and pseudoscalar numbers)
formula_120
the paragradient in the null basis is
formula_121
Higher Dimensions.
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 formula_122. In general, the dimension of the multivector space of grade m is formula_123 and the dimension of the whole Clifford algebra formula_124 is formula_125.
A given multivector with homogeneous grade is either invariant or changes sign under the action of the reversion conjugation formula_126. 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:
Matrix Representation.
The algebra of the formula_16 space is isomorphic to the Pauli matrix algebra such that
from which the null basis elements become
formula_128
A general Clifford number in 3D can be written as
where the coefficients formula_130 are scalar elements (including pseudoscalars). The indexes were chosen such that the representation of this Clifford number in terms of the Pauli matrices is
Conjugations.
The reversion conjugation is translated into the Hermitian conjugation and the bar conjugation is translated into the following matrix:
formula_132
such that the scalar part is translated as
The rest of the subspaces are translated as
Higher Dimensions.
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 formula_137. The 4D representation could 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 an n-dimensional Euclidean space are Hermitian elements and can be used to represent the formula_138 Lie algebra.
The bivectors of the three-dimensional Euclidean space form the formula_139 Lie algebra, which is isomorphic
to the formula_140 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 formula_139 Lie algebra can be extended by adding the three unitary vectors to form a Lie algebra isomorphic
to the formula_142 Lie algebra, which is the double cover of the Lorentz group formula_143. This isomorphism 
allows the possibility to develop a formalism of special relativity based on formula_142, 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 formula_145 Lie algebra. This 
is the isomorphism between formula_146 and formula_147.
Another interesting isomorphism exists between formula_148 and formula_149. So, the 
formula_149 Lie algebra can be used to generate the formula_151 group. Despite that this group 
is smaller than the formula_152 group, it is seen to be enough to span the four-dimensional Hilbert space.