Paravector

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 C ℓ 3 space,

{ 1 , { e 1 , e 2 , e 3 } , { e 23 , e 31 , e 12 } , e 123 } ,

which forms an eight-dimensional space, where the multiple indices indicate the product of the respective basis vectors, for example

e 23 = e 2 e 3 .

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 e 123 squares to − 1

Moreover, the volume element e 123 commutes with any other element of the C ℓ ( 3 ) algebra, so that it can be identified with the complex number i , whenever there is no danger of confusion. In fact, the volume element e 123 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

{ 1 , e 1 , e 2 , e 3 } ,

which forms a four-dimensional linear space. The paravector space in the three-dimensional Euclidean space C ℓ 3 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 1 = e 0 , so that the complete basis can be written in a compact form as

{ e μ } ,

where the Greek indices such as μ run from 0 to 3 .

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).

( A B ) † = B † A † ,

where vectors and real scalar numbers are invariant under reversion conjugation and are said to be real, for example:

a † = a

1 † = 1

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

a ¯ = − a

1 ¯ = 1

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

A B ¯ = B ¯ A ¯

The bar conjugation applied to each basis element is given below

Grade automorphism

The grade automorphism A B ¯ † = A ¯ † B ¯ † 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 C ℓ 3 space based on their symmetries under the reversion and Clifford conjugation

Given p as a general Clifford number, the complementary scalar and vector parts of p are given by symmetric and antisymmetric combinations with the Clifford conjugation

⟨ p ⟩ S = 1 2 ( p + p ¯ ) ,

⟨ p ⟩ V = 1 2 ( p − p ¯ ) .

In similar way, the complementary Real and Imaginary parts of p are given by symmetric and antisymmetric combinations with the Reversion conjugation

⟨ p ⟩ R = 1 2 ( p + p † ) ,

⟨ p ⟩ I = 1 2 ( p − p † ) .

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

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.

Scalar Product

Given two paravectors u and v , the generalization of the scalar product is

⟨ u v ¯ ⟩ S .

The magnitude square of a paravector u is

⟨ u u ¯ ⟩ S ,

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 η μ ν = ⟨ e μ e ¯ ν ⟩ S

and in particular:

η 00 = ⟨ e 0 e ¯ 0 ⟩ = ⟨ 1 ( 1 ) ⟩ S = 1 ,

η 11 = ⟨ e 1 e ¯ 1 ⟩ = ⟨ e 1 ( − e 1 ) ⟩ S = − 1 ,

η 01 = ⟨ e 0 e ¯ 1 ⟩ = ⟨ 1 ( − e 1 ) ⟩ S = 0.

Biparavectors

Given two paravectors u and v , the biparavector B is defined as:

B = ⟨ u v ¯ ⟩ V .

The biparavector basis can be written as

{ ⟨ e μ e ¯ ν ⟩ V } ,

which contains six independent elements, including real and imaginary terms. Three real elements (vectors) as

and three imaginary elements (bivectors) as

where j , k 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 i 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 θ j and three rapidities η j .

Triparavectors

Given three paravectors u , v and w , the triparavector T is defined as:

T = ⟨ u v ¯ w ⟩ I .

The triparavector basis can be written as

{ ⟨ e μ e ¯ ν e λ ⟩ I } ,

but there are only four independent triparavectors, so it can be reduced to

{ i e ρ } .

Pseudoscalar

The pseudoscalar basis is { ⟨ e μ e ¯ ν e λ e ¯ ρ ⟩ I S } ,

but a calculation reveals that it contains only a single term. This term is the volume element i = e 1 e 2 e 3 .

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

where e 0 = 1 .

The application of the paragradient operator must be done carefully, always respecting its non-commutative nature. For example, a widely used derivative is

where f ( x ) 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 p , this property necessarily implies the following identity

p p ¯ = 0.

In the context of Special Relativity they are also called lightlike paravectors.

Projectors are null paravectors of the form

P k = 1 2 ( 1 + k ^ ) ,

where k ^ is a unit vector.

A projector P k of this form has a complementary projector P ¯ k

P ¯ k = 1 2 ( 1 − k ^ ) ,

such that

P k + P ¯ k = 1

As projectors, they are idempotent

P k = P k P k = P k P k P k = . . .

and the projection of one on the other is zero because they are null paravectors

P k P ¯ k = 0.

The associated unit vector of the projector can be extracted as

k ^ = P k − P ¯ k ,

this means that k ^ is an operator with eigenfunctions P k and P ¯ k , with respective eigenvalues 1 and − 1 .

From the previous result, the following identity is valid assuming that f ( k ^ ) is analytic around zero

f ( k ^ ) = f ( 1 ) P k + f ( − 1 ) P ¯ k .

This gives origin to the pacwoman property, such that the following identities are satisfied

f ( k ^ ) P k = f ( 1 ) P k ,

f ( k ^ ) P ¯ k = f ( − 1 ) P ¯ k .

Null Basis for the paravector space

A basis of elements, each one of them null, can be constructed for the complete C ℓ 3 space. The basis of interest is the following

{ P ¯ 3 , P 3 e 1 , P 3 , e 1 P 3 }

so that an arbitrary paravector

p = p 0 e 0 + p 1 e 1 + p 2 e 2 + p 3 e 3

can be written as

p = ( p 0 + p 3 ) P 3 + ( p 0 − p 3 ) P ¯ 3 + ( p 1 + i p 2 ) e 1 P 3 + ( p 1 − i p 2 ) P 3 e 1

This representation is useful for some systems that are naturally expressed in terms of the light cone variables that are the coefficients of P 3 and P ¯ 3 respectively.

Every expression in the paravector space can be written in terms of the null basis. A paravector p is in general parametrized by two real scalars numbers { u , v } and a general scalar number w (including scalar and pseudoscalar numbers)

p = u P ¯ 3 + v P 3 + w e 1 P 3 + w † P 3 e 1

the paragradient in the null basis is

∂ = 2 P 3 ∂ u + 2 P ¯ 3 ∂ v − 2 e 1 P 3 ∂ w † − 2 P 3 e 1 ∂ w

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 ( n 2 ) . In general, the dimension of the multivector space of grade m is ( n m ) and the dimension of the whole Clifford algebra C ℓ ( n ) is 2 n .

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:

Matrix Representation

The algebra of the C ℓ ( 3 ) space is isomorphic to the Pauli matrix algebra such that

from which the null basis elements become P 3 = ( 1 0 0 0 ) ; P ¯ 3 = ( 0 0 0 1 ) ; P 3 e 1 = ( 0 1 0 0 ) ; e 1 P 3 = ( 0 0 1 0 ) .

A general Clifford number in 3D can be written as

where the coefficients ψ j k 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: Ψ ¯ → ( ψ 22 − ψ 12 − ψ 21 ψ 11 ) , such that the scalar part is translated as

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 2 n . The 4D representation could be taken as

The 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.

The bivectors of an n-dimensional Euclidean space are Hermitian elements and can be used to represent the s p i n ( n ) Lie algebra.

The bivectors of the three-dimensional Euclidean space form the s p i n ( 3 ) Lie algebra, which is isomorphic to the s u ( 2 ) 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 s p i n ( 3 ) Lie algebra can be extended by adding the three unitary vectors to form a Lie algebra isomorphic to the S L ( 2 , C ) Lie algebra, which is the double cover of the Lorentz group S O ( 3 , 1 ) . This isomorphism allows the possibility to develop a formalism of special relativity based on S L ( 2 , C ) , 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 s u ( N ) Lie algebra. This is the isomorphism between s p i n ( 6 ) and s u ( 4 ) .

Another interesting isomorphism exists between s p i n ( 5 ) and s p ( 4 ) . So, the s p ( 4 ) Lie algebra can be used to generate the U S p ( 4 ) group. Despite that this group is smaller than the S U ( 4 ) group, it is seen to be enough to span the four-dimensional Hilbert space.