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.
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 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 According to the fundamental axiom, two different basis vectors anticommute, or in other words, This means that the volume element
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 which forms a four-dimensional linear space.
can be used to represent the space-time of special relativity as expressed in the algebra of physical space (APS).
, so that the complete basis can be written in a compact form as where the Greek indices such as
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.
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 The grade automorphism 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
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 There are two subspaces that are closed with respect to the product.
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, the following expression is expanded as Null paravectors are elements that are not necessarily zero but have magnitude identical to zero.
, this property necessarily implies the following identity In the context of Special Relativity they are also called lightlike paravectors.
is analytic around zero This gives origin to the pacwoman property, such that the following identities are satisfied A basis of elements, each one of them null, can be constructed for the complete
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
Every expression in the paravector space can be written in terms of the null basis.
(including scalar and pseudoscalar numbers) the paragradient in the null basis is An n-dimensional Euclidean space allows the existence of multivectors of grade n (n-vectors).
A given multivector with homogeneous grade is either invariant or changes sign under the action of the reversion conjugation
from which the null basis elements become A general Clifford number in 3D can be written as where the coefficients
The indexes were chosen such that the representation of this Clifford number in terms of the Pauli matrices is 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 The rest of the subspaces are translated as 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
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
This accidental isomorphism allows to picture a geometric interpretation of the states of the two dimensional Hilbert space by using the Bloch sphere.
This isomorphism allows the possibility to develop a formalism of special relativity based on