Wikipedia / Paravector
Wikipedia / Paravector
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
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 thatAccording to the fundamental axiom, two different basis vectors anticommute,
or in other words,
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
,
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
.
Antiautomorphism
Reversion conjugation
The Reversion antiautomorphism. 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
. 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 inversion of the sign of odd-grade multivectors, while maintaining the even-grade multivectors invariant:
Invariant subspaces according to the conjugations
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.
are given by symmetric and antisymmetric combinations with the Clifford conjugation
.
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
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
Scalar product
, the generalization of the scalar product is
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
, 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
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
represents the pseudoscalar volume element.
Another example of biparavector is the representation of the space-time rotation rate that can be expressed as
and three rapidities.
Triparavectors
, 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
.
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
.
The application of the paragradient operator must be done carefully, always respecting its non-commutative nature. For example, a widely used derivative is
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
, 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
is a unit vector.
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
.
is analytic around zero
This gives origin to the pacwoman property, such that the following identities are satisfied
Null basis for the paravector space
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 respectively.
(including scalar and pseudoscalar numbers)
the paragradient in the null basis is
Higher dimensions
.
. 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
space is isomorphic to the Pauli matrix algebra such thatfrom which the null basis elements become
A general Clifford number in 3D can be written as
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
The rest of the subspaces are translated as
Higher dimensions
. The 4D representation could be taken asThe 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.
Lie algebra.
The bivectors of the three-dimensional Euclidean space form Lie algebra, which is isomorphic 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.
, which is carried out in the form of the algebra of physical space.
.
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.
Baylis, William, Clifford (Geometric) Algebras With Applications in Physics, Mathematics, and Engineering, Birkhauser (1999)
[H1999] David Hestenes: New Foundations for Classical Mechanics (Second Edition). , Kluwer Academic Publishers (1999)
Chris Doran and Antony Lasenby, Geometric Algebra for Physicists, Cambridge, 2003
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