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)
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
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
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
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
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:
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.