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 
space,
which forms an eight-dimensional space, where the multiple indices indicate the product of the respective basis vectors, for example
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 | Vector |  |
| 2 | Bivector |  |
| 3 | Trivector volume element |  |
According to the fundamental axiom, two different basis vectors anticommute,
or in other words,
This means that the volume element squares to 
Moreover, the volume element commutes with any other element of the 
algebra, so that it can be identified with the complex number 
, whenever there is no danger of confusion. In fact, the volume element 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
| Grade | Type | Basis element/s |
|---|---|---|
| 0 | Unitary real scalar | |
| 1 | Vector | |
| 2 | Bivector |
|
| 3 | Trivector volume element |
|
Paravectors
The corresponding paravector basis that combines a real scalar and vectors is
,
which forms a four-dimensional linear space. The paravector space in the three-dimensional Euclidean space 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 
, so that
the complete basis can be written in a compact form as
where the Greek indices such as 
run from 
to 
.
Antiautomorphism
Reversion conjugation
The Reversion antiautomorphism is denoted by 
. The action of this conjugation is to reverse the order of the geometric product (product between Clifford numbers in general).
,
where vectors and real scalar numbers are invariant under reversion conjugation and are said to be real, for example:
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
| Element | Reversion conjugation |
|---|---|
| | |
 | |
 | |
 | |
 |  |
 |  |
 |  |
| |  |
Clifford conjugation
The Clifford Conjugation is denoted by a bar over the object
. 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
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
The bar conjugation applied to each basis element is given below
| Element | Bar conjugation |
|---|---|
| | |
| |  |
| |  |
| |  |
| | |
| | |
| | |
| | |
- Note.- The volume element is invariant under the bar conjugation.
Grade automorphism
The grade automorphism
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:
| Element | Grade involution |
|---|---|
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
Invariant subspaces according to the conjugations
Four special subspaces can be defined in the 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 
as a general Clifford number, the complementary scalar and vector parts of are given by
symmetric and antisymmetric combinations with the Clifford conjugation
.
In similar way, the complementary Real and Imaginary parts of are given
by symmetric and antisymmetric combinations with the Reversion conjugation
.
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
| Real | Imaginary | |
|---|---|---|
| Scalar | 0 | 3 |
| Vector | 1 | 2 |
- Remark: The term "Imaginary" is used in the context of the 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
- The even space, made of elements of grades 0 and 2, is isomorphic with the algebra of quaternions with the identification of
Scalar Product
Given two paravectors 
and 
, the generalization of the scalar product is
The magnitude square of a paravector is
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
and in particular:
Biparavectors
Given two paravectors and , the biparavector B is
defined as:
.
The biparavector basis can be written as
which contains six independent elements, including real and imaginary terms. Three real elements (vectors) as
and three imaginary elements (bivectors) as
where 
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 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 
and three rapidities 
.
Triparavectors
Given three paravectors , and 
, the triparavector T is
defined as:
.
The triparavector basis can be written as
but there are only four independent triparavectors, so it can be reduced to
.
Pseudoscalar
The pseudoscalar basis is
but a calculation reveals that it contains only a single term. This term is the volume element 
.
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
| 1 | 3 | |
|---|---|---|
| 0 | Paravector | Scalar/Pseudoscalar |
| 2 | Biparavector | Triparavector |
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 
.
The application of the paragradient operator must be done carefully, always respecting its non-commutative nature. For example, a widely used derivative is
where 
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 , this property necessarily implies the following identity
In the context of Special Relativity they are also called lightlike paravectors.
Projectors are null paravectors of the form
where 
is a unit vector.
A projector 
of this form has a complementary projector 
such that
As projectors, they are idempotent
and the projection of one on the other is zero because they are null paravectors
The associated unit vector of the projector can be extracted as
this means that 
is an operator
with eigenfunctions 
and
, with respective eigenvalues
and .
From the previous result, the following identity is valid assuming that 
is analytic around zero
This gives origin to the pacwoman property, such that the following identities are satisfied
Null Basis for the paravector space
A basis of elements, each one of them null, can be constructed for the complete
space. The basis of interest is the following
so that an arbitrary paravector
can be written as
This representation is useful for some systems that are naturally expressed in terms of the
light cone variables that are the coefficients of 
and
respectively.
Every expression in the paravector space can be written in terms of the null basis. A paravector is in general parametrized by two real scalars numbers
and a general scalar number (including scalar and pseudoscalar numbers)
the paragradient in the null basis is
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 
. In general, the dimension of the multivector space of grade m is 
and the dimension of the whole Clifford algebra 
is 
.
A given multivector with homogeneous grade is either invariant or changes sign under the action of the reversion conjugation . 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:
| Grade | Classification |
|---|---|
| | Hermitian |
| | Hermitian |
 | Anti-Hermitian |
| | Anti-Hermitian |
 | Hermitian |
 | Hermitian |
 | Anti-Hermitian |
 | Anti-Hermitian |
 | |
Matrix Representation
The algebra of the space is isomorphic to the Pauli matrix algebra such that
| Matrix Representation 3D | Explicit matrix | |
|---|---|---|
 |  |
|
 |  |
|
 |  |
|
 |  |
|
from which the null basis elements become
A general Clifford number in 3D can be written as
where the coefficients 
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:
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}](../I/m/f96b36bdc81bc3dd529caf048dd2ec48.png)
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 . The 4D representation could be taken as
| Matrix Representation 4D | |
|---|---|
| |
|
| |
|
| |
|
 |
|
The 7D representation could be taken as
| Matrix Representation 7D | |
|---|---|
| |
|
| |
|
| |
|
| |
|
 |
|
 |
|
 |
|
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 
Lie algebra.
The bivectors of the three-dimensional Euclidean space form the 
Lie algebra, which is isomorphic
to the 
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 Lie algebra can be extended by adding the three unitary vectors to form a Lie algebra isomorphic
to the 
Lie algebra, which is the double cover of the Lorentz group 
. This isomorphism
allows the possibility to develop a formalism of special relativity based on , 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 
Lie algebra. This
is the isomorphism between 
and 
.
Another interesting isomorphism exists between 
and 
. So, the
Lie algebra can be used to generate the 
group. Despite that this group
is smaller than the 
group, it is seen to be enough to span the four-dimensional Hilbert space.
See also
References
Textbooks
- 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
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