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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).
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
Unitary real scalar
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
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
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
Note.- The volume element is invariant under the bar conjugation.
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.
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
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 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.
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
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
The standard gradient operator can be defined naturally as
so that the paragradient can be written as
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
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
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)
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:
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
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.