Paravector

Last updated

The name paravector is used for the combination of a scalar and a vector in any Clifford algebra, known as geometric algebra among physicists.

Contents

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

GradeTypeBasis element/s
0Unitary real scalar
1Vector
2Bivector
3Trivector 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 basis as

GradeTypeBasis element/s
0Unitary real scalar
1Vector
2Bivector

3Trivector volume element

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

Element Reversion conjugation

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

Element Bar conjugation
  • 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:

Element Grade involution

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

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

RealImaginary
Scalar03
Vector12

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.

Scalar product

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

The magnitude square of a paravector 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

Given two paravectors and , 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

where 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 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 and three rapidities .

Triparavectors

Given three paravectors , and , 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

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

13
0ParavectorScalar/Pseudoscalar
2BiparavectorTriparavector

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 .

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

Null paravectors as projectors

Null paravectors are elements that are not necessarily zero but have magnitude identical to zero. For a null paravector , 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

where is a unit vector.

A projector of this form has a complementary projector

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

this means that is an operator with eigenfunctions and , with respective eigenvalues and .

From the previous result, the following identity is valid assuming that is analytic around zero

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

Null basis for the paravector space

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 respectively.

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)

the paragradient in the null basis is

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 . 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:

Grade Classification
Hermitian
Hermitian
Anti-Hermitian
Anti-Hermitian
Hermitian
Hermitian
Anti-Hermitian
Anti-Hermitian

Matrix representation

The algebra of the space is isomorphic to the Pauli matrix algebra such that

Matrix representation 3DExplicit matrix

from which the null basis elements become

A general Clifford number in 3D can be written as

where the coefficients 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 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

Matrix representation 4D

The 7D representation could be taken as

Matrix representation 7D

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 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.

See also

Related Research Articles

Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread.

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way. It has become vital in the building of the Standard Model.

In quantum mechanics, a density matrix is a matrix that describes an ensemble of physical systems as quantum states. It allows for the calculation of the probabilities of the outcomes of any measurements performed upon the systems of the ensemble using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed ensembles. Mixed ensembles arise in quantum mechanics in two different situations:

  1. when the preparation of the systems lead to numerous pure states in the ensemble, and thus one must deal with the statistics of possible preparations, and
  2. when one wants to describe a physical system that is entangled with another, without describing their combined state; this case is typical for a system interacting with some environment. In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement.

In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-symmetry is an abbreviation of the phrase "charge conjugation symmetry", and is used in discussions of the symmetry of physical laws under charge-conjugation. Other important discrete symmetries are P-symmetry (parity) and T-symmetry.

An operator is a function over a space of physical states onto another space of states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

<span class="mw-page-title-main">Probability amplitude</span> Complex number whose squared absolute value is a probability

In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square of the modulus of this quantity represents a probability density.

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).

<span class="mw-page-title-main">Bloch sphere</span> Geometrical representation of the pure state space of a two-level quantum mechanical system

In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.

<span class="mw-page-title-main">Two-state quantum system</span> Simple quantum mechanical system

In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector.

In mathematics, the Fubini–Study metric is a Kähler metric on a complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.

The following are important identities involving derivatives and integrals in vector calculus.

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) to physics. Spacetime algebra provides a "unified, coordinate-free formulation for all of relativistic physics, including the Dirac equation, Maxwell equation and General Relativity" and "reduces the mathematical divide between classical, quantum and relativistic physics."

<span class="mw-page-title-main">Helium atom</span> Atom of helium

A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with two neutrons, depending on the isotope, held together by the strong force. Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be used to estimate the ground state energy and wavefunction of the atom. Historically, the first such helium spectrum calculation was done by Albrecht Unsöld in 1927. Its success was considered to be one of the earliest signs of validity of Schrödinger's wave mechanics.

In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a quantum-mechanical prediction for the system represented by the state. Knowledge of the quantum state, and the quantum mechanical rules for the system's evolution in time, exhausts all that can be known about a quantum system.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

<span class="mw-page-title-main">Weyl equation</span> Relativistic wave equation describing massless fermions

In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions.

In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.

References

    Textbooks

    Articles