In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the algebra of physical space (APS) is the use of the
Clifford or
geometric algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
Cl
3,0(R) of the three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
as a model for (3+1)-dimensional
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
, representing a point in spacetime via a
paravector
The name paravector is used for the sum 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 ...
(3-dimensional vector plus a 1-dimensional scalar).
The Clifford algebra Cl
3,0(R) has a
faithful representation In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group on a vector space is a linear representation in which different elements of are represented by distinct linear mapp ...
, generated by
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
, on the
spin representation
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equ ...
C
2; further, Cl
3,0(R) is isomorphic to the even subalgebra Cl(R) of the Clifford algebra Cl
3,1(R).
APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum mechanics.
APS should not be confused with
spacetime algebra
In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra . According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of speci ...
(STA), which concerns the
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
Cl
1,3(R) of the four-dimensional
Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inert ...
.
Special relativity
Spacetime position paravector
In APS, the
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
position is represented as the
paravector
The name paravector is used for the sum 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 ...
where the time is given by the scalar part , and e
1, e
2, e
3 are the
standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the c ...
for position space. Throughout, units such that are used, called
natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a Coherence (units of measurement), coherent unit of a quantity. For e ...
. In the
Pauli matrix
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in c ...
representation, the unit basis vectors are replaced by the Pauli matrices and the scalar part by the identity matrix. This means that the Pauli matrix representation of the space-time position is
Lorentz transformations and rotors
The restricted Lorentz transformations that preserve the direction of time and include rotations and boosts can be performed by an exponentiation of the spacetime rotation
biparavector ''W''
In the matrix representation, the Lorentz rotor is seen to form an instance of the SL(2,C) group (
special linear group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the genera ...
of degree 2 over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s), which is the double cover of the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
. The unimodularity of the Lorentz rotor is translated in the following condition in terms of the product of the Lorentz rotor with its Clifford conjugation
This Lorentz rotor can be always decomposed in two factors, one
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...
, and the other
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigroup ...
, such that
The unitary element ''R'' is called a
rotor
Rotor may refer to:
Science and technology
Engineering
*Rotor (electric), the non-stationary part of an alternator or electric motor, operating with a stationary element so called the stator
* Helicopter rotor, the rotary wing(s) of a rotorcraft ...
because this encodes rotations, and the Hermitian element ''B'' encodes boosts.
Four-velocity paravector
The
four-velocity
In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
, also called proper velocity, is defined as the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the spacetime position paravector with respect to
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
''τ'':
This expression can be brought to a more compact form by defining the ordinary velocity as
and recalling the definition of the
gamma factor
The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
:
so that the proper velocity is more compactly:
The proper velocity is a positive
unimodular paravector, which implies the following condition in terms of the
Clifford conjugation
The proper velocity transforms under the action of the Lorentz rotor ''L'' as
Four-momentum paravector
The
four-momentum
In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
in APS can be obtained by multiplying the proper velocity with the mass as
with the
mass shell
In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called "on the mass shell" or simply more often on shell; while those that do not are called "off the mass shell" ...
condition translated into
Classical electrodynamics
The electromagnetic field, potential, and current
The
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
is represented as a bi-paravector ''F'':
with the Hermitian part representing the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
''E'' and the anti-Hermitian part representing the
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
''B''. In the standard Pauli matrix representation, the electromagnetic field is:
The source of the field ''F'' is the electromagnetic
four-current
In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of ''four-dimensional spa ...
:
where the scalar part equals the
electric charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
''ρ'', and the vector part the
electric current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional are ...
j. Introducing the
electromagnetic potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. Whe ...
paravector
The name paravector is used for the sum 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 ...
defined as:
in which the scalar part equals the
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
''ϕ'', and the vector part the
magnetic potential Magnetic potential may refer to:
* Magnetic vector potential, the vector whose curl is equal to the magnetic B field
* Magnetic scalar potential
Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism analogous to electr ...
A. The electromagnetic field is then also:
The field can be split into electric
and magnetic
components.
Where
and ''F'' is invariant under a
gauge transformation
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
of the form
where
is a
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
.
The electromagnetic field is
covariant under Lorentz transformations according to the law
Maxwell's equations and the Lorentz force
The
Maxwell equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
Th ...
can be expressed in a single equation:
where the overbar represents the
Clifford conjugation.
The
Lorentz force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
equation takes the form
Electromagnetic Lagrangian
The electromagnetic
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
is
which is a real scalar invariant.
Relativistic quantum mechanics
The
Dirac equation
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- massive particles, called "Dirac part ...
, for an electrically
charged particle
In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, ...
of mass ''m'' and charge ''e'', takes the form:
where e
3 is an arbitrary unitary vector, and ''A'' is the electromagnetic paravector potential as above. The
electromagnetic interaction
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
has been included via
minimal coupling In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between field theory (physics), fields which involves only the electric charge, charge distribution and not higher multipole moments of the charge distribution. ...
in terms of the potential ''A''.
Classical spinor
The
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
of the Lorentz rotor that is consistent with the Lorentz force is
such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest
which can be integrated to find the space-time trajectory
with the additional use of
See also
*
Paravector
The name paravector is used for the sum 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 ...
*
Multivector
In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors (a ...
*
wikibooks:Physics in the Language of Geometric Algebra. An Approach with the Algebra of Physical Space
*
Dirac equation in the algebra of physical space
In physics, the algebra of physical space (APS) is the use of the Clifford algebra, 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 ...
*
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
References
Textbooks
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Articles
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{{Industrial and applied mathematics
Mathematical physics
Geometric algebra
Clifford algebras
Special relativity
Electromagnetism