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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 ...
and
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Lorentz group is the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of all
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s of
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 ...
, the classical and
quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity ( physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizat ...
setting for all (non-gravitational)
physical phenomena Physical may refer to: *Physical examination In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally cons ...
. The Lorentz group is named for the
Dutch Dutch commonly refers to: * Something of, from, or related to the Netherlands * Dutch people () * Dutch language () Dutch may also refer to: Places * Dutch, West Virginia, a community in the United States * Pennsylvania Dutch Country People E ...
physicist
Hendrik Lorentz Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorent ...
. For example, the following laws, equations, and theories respect Lorentz symmetry: * The kinematical laws of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
*
Maxwell's field 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 ...
in the theory of
electromagnetism 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 o ...
* 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 par ...
in the theory of the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
* The
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
of particle physics The Lorentz group expresses the fundamental
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
of space and time of all known fundamental laws of nature. In small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as special relativity.


Basic properties

The Lorentz group is a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
—the group of all
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
of
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 ...
. Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is an isotropy subgroup of the
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the homogeneous Lorentz group while the Poincaré group is sometimes called the ''inhomogeneous Lorentz group''. Lorentz transformations are examples of
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s; general isometries of Minkowski spacetime are
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...
s. Mathematically, the Lorentz group may be described as the indefinite orthogonal group O(1,3), the matrix Lie group that preserves the
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
: (t, x, y, z) \mapsto t^2 - x^2 - y^2 - z^2 on \mathbb^4 (The vector space equipped with this quadratic form is sometimes written \mathbb^). This quadratic form is, when put on matrix form (see classical orthogonal group), interpreted in physics as the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
of Minkowski spacetime. The Lorentz group is a six-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
al noncompact non-abelian real Lie group that is not
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
. The four connected components are not
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
. The
identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity compo ...
(i.e., the component containing the identity element) of the Lorentz group is itself a group, and is often called the restricted Lorentz group, and is denoted SO+(1,3). The restricted Lorentz group consists of those Lorentz transformations that preserve both the orientation of space and the direction of time. Its
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
has order 2, and its universal cover, the indefinite spin group Spin(1,3), is isomorphic to both the
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 ge ...
SL(2, C) and to the symplectic group Sp(2, C). These isomorphisms allow the Lorentz group to act on a large number of mathematical structures important to physics, most notably the
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s. Thus, in
relativistic quantum mechanics In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light  ...
and in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, it is very common to call SL(2, C) the Lorentz group, with the understanding that SO+(1,3) is a specific representation (the vector representation) of it. The biquaternions, popular in
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 ...
, are also isomorphic to SL(2, C). The restricted Lorentz group also arises as the point symmetry group of a certain
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
.


Connected components

Because it is a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
, the Lorentz group O(1,3) is both a group and admits a topological description as a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
. As a manifold, it has four connected components. Intuitively, this means that it consists of four topologically separated pieces. The four connected components can be categorized by two transformation properties its elements have: * Some elements are reversed under time-inverting Lorentz transformations, for example, a future-pointing timelike vector would be inverted to a past-pointing vector * Some elements have orientation reversed by improper Lorentz transformations, for example, certain
vierbein The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independen ...
(tetrads) Lorentz transformations that preserve the direction of time are called . The subgroup of orthochronous transformations is often denoted O+(1, 3). Those that preserve orientation are called proper, and as linear transformations they have determinant +1. (The improper Lorentz transformations have determinant −1.) The subgroup of proper Lorentz transformations is denoted SO(1, 3). The subgroup of all Lorentz transformations preserving both orientation and direction of time is called the proper, orthochronous Lorentz group or restricted Lorentz group, and is denoted by SO+(1, 3). (Note that some authors refer to SO(1,3) or even O(1,3) when they actually mean SO+(1, 3).) The set of the four connected components can be given a group structure as the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
O(1, 3)/SO+(1, 3), which is isomorphic to the
Klein four-group In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one ...
. Every element in O(1,3) can be written as the
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
of a proper, orthochronous transformation and an element of the
discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
: where ''P'' and ''T'' are the
parity Parity may refer to: * Parity (computing) ** Parity bit in computing, sets the parity of data for the purpose of error detection ** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
and time reversal operators: : ''P'' = diag(1, −1, −1, −1) : ''T'' = diag(−1, 1, 1, 1). Thus an arbitrary Lorentz transformation can be specified as a proper, orthochronous Lorentz transformation along with a further two bits of information, which pick out one of the four connected components. This pattern is typical of finite-dimensional Lie groups.


Restricted Lorentz group

The restricted Lorentz group \text^+(1,3) is the
identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity compo ...
of the Lorentz group, which means that it consists of all Lorentz transformations that can be connected to the identity by a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
curve lying in the group. The restricted Lorentz group is a connected
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of the full Lorentz group with the same dimension, in this case with dimension six. The restricted Lorentz group is generated by ordinary spatial rotations and
Lorentz boost In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...
s (which are rotations in a hyperbolic space that includes a time-like direction). Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation (specified by 3 real parameters) and a boost (also specified by 3 real parameters), it takes 6 real parameters to specify an arbitrary proper orthochronous Lorentz transformation. This is one way to understand why the restricted Lorentz group is six-dimensional. (See also the Lie algebra of the Lorentz group.) The set of all rotations forms a
Lie subgroup In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
isomorphic to the ordinary
rotation group SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is ...
. The set of all boosts, however, does ''not'' form a subgroup, since composing two boosts does not, in general, result in another boost. (Rather, a pair of non-colinear boosts is equivalent to a boost and a rotation, and this relates to Thomas rotation.) A boost in some direction, or a rotation about some axis, generates a
one-parameter subgroup In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism :\varphi : \mathbb \rightarrow G from the real line \mathbb (as an additive group) to some other topological group G. If \varphi i ...
.


Surfaces of transitivity

If a group acts on a space , then a surface is a surface of transitivity if is invariant under (i.e., ) and for any two points there is a such that . By definition of the Lorentz group, it preserves the quadratic form :Q(x) = x_0^2 - x_1^2 - x_2^2 - x_3^2. The surfaces of transitivity of the orthochronous Lorentz group , acting on flat
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 differ ...
\mathbb^ are the following: * is the upper branch of a
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
of two sheets. Points on this sheet are separated from the origin by a future time-like vector. * is the lower branch of this hyperboloid. Points on this sheet are the past time-like vectors. * is the upper branch of the
light cone In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
, the future light cone. * is the lower branch of the light cone, the past light cone. * is a hyperboloid of one sheet. Points on this sheet are
space-like 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 dif ...
separated from the origin. *The origin . These surfaces are , so the images are not faithful, but they are faithful for the corresponding facts about . For the full Lorentz group, the surfaces of transitivity are only four since the transformation takes an upper branch of a hyperboloid (cone) to a lower one and vice versa.


As symmetric spaces

An equivalent way to formulate the above surfaces of transitivity is as a symmetric space in the sense of Lie theory. For example, the upper sheet of the hyperboloid can be written as the quotient space \text^+(1,3)/\text(3), due to the
orbit-stabilizer theorem In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphis ...
. Furthermore, this upper sheet also provides a model for three-dimensional hyperbolic space.


Representations of the Lorentz group

These observations constitute a good starting point for finding all infinite-dimensional unitary representations of the Lorentz group, in fact, of the Poincaré group, using the method of
induced representation In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represe ...
s. One begins with a "standard vector", one for each surface of transitivity, and then ask which subgroup preserves these vectors. These subgroups are called little groups by physicists. The problem is then essentially reduced to the easier problem of finding representations of the little groups. For example, a standard vector in one of the hyperbolas of two sheets could be suitably chosen as . For each , the vector pierces exactly one sheet. In this case the little group is , the rotation group, all of whose representations are known. The precise infinite-dimensional unitary representation under which a particle transforms is part of its classification. Not all representations can correspond to physical particles (as far as is known). Standard vectors on the one-sheeted hyperbolas would correspond to
tachyon A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...
s. Particles on the light cone are
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s, and more hypothetically,
graviton In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathem ...
s. The "particle" corresponding to the origin is the vacuum.


Homomorphisms and isomorphisms

Several other groups are either homomorphic or isomorphic to the restricted Lorentz group SO+(1, 3). These homomorphisms play a key role in explaining various phenomena in physics. * The
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 ge ...
SL(2, C) is a double covering of the restricted Lorentz group. This relationship is widely used to express the
Lorentz invariance In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation In physics, the Lorentz transformations are a six-parameter famil ...
of 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 par ...
and the covariance of
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s. In other words, the (restricted) Lorentz group is isomorphic to SL(2, C)/\mathbb * The symplectic group Sp(2, C) is isomorphic to SL(2, C); it is used to construct Weyl spinors, as well as to explain how spinors can have a mass. * The
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a ...
Spin(1, 3) is isomorphic to SL(2, C); it is used to explain spin and spinors in terms of 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 hyperco ...
, thus making it clear how to generalize the Lorentz group to general settings in
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, including theories of
supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
and
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
. * The restricted Lorentz group is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the
projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
PSL(2, C) which is, in turn, isomorphic to the Möbius group, the
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
of
conformal geometry In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
on the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
. This relationship is central to the classification of the subgroups of the Lorentz group according to an earlier classification scheme developed for the Möbius group.


The Weyl representation

The Weyl representation or spinor map is a pair of
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s from SL(2,C) to SO+(1, 3). They form a matched pair under
parity Parity may refer to: * Parity (computing) ** Parity bit in computing, sets the parity of data for the purpose of error detection ** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
transformations, corresponding to left and right
chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
spinors. One may define an action of SL(2,C) on Minkowski spacetime by writing a point of spacetime as a two-by-two
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...
in the form : \overline = \begin ct + z & x - iy \\ x + iy & ct - z \end = ct 1\!\!1 + x\sigma_x + y\sigma_y + z\sigma_z = ct 1\!\!1 + \vec \cdot \vec in terms of
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 ...
. This presentation, the Weyl presentation, satisfies : \det\, \overline = (ct)^2 - x^2 - y^2 - z^2. Therefore, one has identified the space of Hermitian matrices (which is four-dimensional, as a ''real'' vector space) with Minkowski spacetime, in such a way that the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime. An element S\in \operatorname(2,\mathbb) acts on the space of Hermitian matrices via : \overline \mapsto S \overline S^\dagger ~, where S^\dagger is the
Hermitian transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex co ...
of S. This action preserves the determinant and so SL(2,C) acts on Minkowski spacetime by (linear) isometries. The parity-inverted form of the above is : X = ct 1\!\!1 - \vec \cdot \vec which transforms as : X \mapsto \left(S^\right)^\dagger X S^ That this is the correct transformation follows by noting that :\overline X = \left(c^2 t^2 - \vec\cdot\vec\right) 1\!\!1 = \left(c^2 t^2 - x^2 - y^2 - z^2\right) 1\!\!1 remains invariant under the above pair of transformations. These maps are
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
, and
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
of either map is the two element subgroup ±''I''. By the first isomorphism theorem, the quotient group PSL(2, C) = SL(2, C) / is isomorphic to SO+(1, 3). The parity map swaps these two coverings. It corresponds to Hermitian conjugation being an automorphism of \operatorname(2, \mathbb). These two distinct coverings corresponds to the two distinct
chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
actions of the Lorentz group on
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s. The non-overlined form corresponds to right-handed spinors transforming as \psi_R\mapsto S\psi_R~, while the overline form corresponds to left-handed spinors transforming as \psi_L\mapsto \left(S^\dagger\right)^ \psi_L~. It is important to observe that this pair of coverings does ''not'' survive quantization; when quantized, this leads to the peculiar phenomenon of the chiral anomaly. The classical (i.e., non-quantized) symmetries of the Lorentz group are broken by quantization; this is the content of the
Atiyah–Singer index theorem In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the sp ...
.


Notational conventions

In physics, it is conventional to denote a Lorentz transformation \Lambda \in \operatorname^+(1, 3) as _\nu~, thus showing the matrix with spacetime indexes \mu, \nu = 0, 1, 2, 3. A four-vector can be created from the Pauli matrices in two different ways: as \sigma^\mu = (I, \vec\sigma) and as ^\mu = \left(I, -\vec\sigma\right)~. The two forms are related by a parity transformation. Note that _\mu = \sigma^\mu~. Given a Lorentz transformation x^\mu \mapsto x^ = _\nu x^\nu ~, the double-covering of the orthochronous Lorentz group by S \in \operatorname(2,\mathbb) given above can be written as :x^_\mu = _\mu _\nu x^\nu = S x^\nu _\nu S^\dagger Dropping the x^\mu this takes the form :_\mu _\nu = S _\nu S^\dagger The parity conjugate form is :\sigma_\mu _\nu = \left(S^\right)^\dagger \sigma_\nu S^


Proof

That the above is the correct form for indexed notation is not immediately obvious, partly because, when working in indexed notation, it is quite easy to accidentally confuse a Lorentz transform with its inverse, or its transpose. This confusion arises due to the identity \eta\Lambda^\textsf\eta = \Lambda^ being difficult to recognize when written in indexed form. Lorentz transforms are ''not'' tensors under Lorentz transformations! Thus a direct proof of this identity is useful, for establishing its correctness. It can be demonstrated by starting with the identity :\omega \sigma^k \omega^ = -\left(\sigma^k\right)^\textsf = -\left(\sigma^k\right)^* where k = 1, 2, 3 so that the above are just the usual Pauli matrices, and (\cdot)^\textsf is the matrix transpose, and (\cdot)^* is complex conjugation. The matrix \omega is :\omega = i\sigma_2 = \begin 0 & 1 \\ -1 & 0 \end Written as the four-vector, the relationship is :\sigma^\textsf_\mu = \sigma^*_\mu =\omega \overline_\mu \omega^ This transforms as :\begin \sigma^\textsf_\mu _\nu &= \omega \overline_\mu \omega^ _\nu \\ &= \omega S\; \overline_\nu\, S^\dagger \omega^ \\ &= \left(\omega S \omega^\right)\, \left(\omega \overline_\nu \omega^\right)\, \left(\omega S^\dagger \omega^\right) \\ &= \left(S^\right)^\textsf\, \sigma^\textsf_\nu\, \left(S^\right)^* \end Taking one more transpose, one gets :\sigma_\mu _\nu = \left(S^\right)^\dagger \sigma_\nu S^


The symplectic group

The symplectic group Sp(2, C) is isomorphic to SL(2, C). This isomorphism is constructed so as to preserve a symplectic bilinear form on \mathbb^2, that is, to leave the form invariant under Lorentz transformations. This may be articulated as follows. The symplectic group is defined as :\operatorname(2,\mathbb) = \left\ where :\omega = i\sigma_2 = \begin 0 & 1 \\ -1 & 0 \end Other common notations are \omega = \epsilon for this element; sometimes J is used, but this invites confusion with the idea of
almost complex structure In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex ...
s, which are not the same, as they transform differently. Given a pair of Weyl spinors (two-component spinors) :u = \begin u_1 \\ u_2 \end ~,\quad v = \begin v_1 \\ v_2 \end the invariant bilinear form is conventionally written as :\langle u,v\rangle = - \langle v, u\rangle = u_1 v_2 - u_2 v_1 = u^\textsf\omega v This form is invariant under the Lorentz group, so that for S\in \operatorname(2,\mathbb) one has :\langle Su, Sv\rangle = \langle u,v\rangle This defines a kind of "scalar product" of spinors, and is commonly used to defined a Lorentz-invariant
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
term in Lagrangians. There are several notable properties to be called out that are important to physics. One is that \omega^2 = -1 and so \omega^ = \omega^\textsf = \omega^\dagger = -\omega The defining relation can be written as :\omega S^\textsf \omega^ = S^ which closely resembles the defining relation for the Lorentz group :\eta \Lambda^\textsf \eta^ = \Lambda^ where \eta = \operatorname (+1, -1, -1, -1) is the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
for
Minkowski space 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 iner ...
and of course, \Lambda \in \operatorname(1, 3) as before.


Covering groups

Since is simply connected, it is the
universal covering group In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. ...
of the restricted Lorentz group . By restriction, there is a homomorphism . Here, the special unitary group SU(2), which is isomorphic to the group of unit norm
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s, is also simply connected, so it is the covering group of the rotation group SO(3). Each of these covering maps are twofold covers in the sense that precisely two elements of the covering group map to each element of the quotient. One often says that the restricted Lorentz group and the rotation group are doubly connected. This means that the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
of the each group is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the two-element
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
Z2. Twofold coverings are characteristic of
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a ...
s. Indeed, in addition to the double coverings : Spin+(1, 3) = SL(2, C) → SO+(1, 3) : Spin(3) = SU(2) → SO(3) we have the double coverings : Pin(1, 3) → O(1, 3) : Spin(1, 3) → SO(1, 3) : Spin+(1, 2) = SU(1, 1) → SO(1, 2) These spinorial double coverings are constructed from
Clifford algebras 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 hyp ...
.


Topology

The left and right groups in the double covering : SU(2) → SO(3) are deformation retracts of the left and right groups, respectively, in the double covering : SL(2, C) → SO+(1, 3). But the homogeneous space SO+(1, 3)/SO(3) is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to
hyperbolic 3-space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
H3, so we have exhibited the restricted Lorentz group as a principal fiber bundle with fibers SO(3) and base H3. Since the latter is homeomorphic to R3, while SO(3) is homeomorphic to three-dimensional real projective space RP3, we see that the restricted Lorentz group is ''locally'' homeomorphic to the product of RP3 with R3. Since the base space is contractible, this can be extended to a global homeomorphism.


Conjugacy classes

Because the restricted Lorentz group SO+(1, 3) is isomorphic to the Möbius group PSL(2, C), its conjugacy classes also fall into five classes: * Elliptic transformations * Hyperbolic transformations * Loxodromic transformations * Parabolic transformations * The trivial identity transformation In the article on
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
s, it is explained how this classification arises by considering the fixed points of Möbius transformations in their action on the Riemann sphere, which corresponds here to
null Null may refer to: Science, technology, and mathematics Computing * Null (SQL) (or NULL), a special marker and keyword in SQL indicating that something has no value * Null character, the zero-valued ASCII character, also designated by , often use ...
eigenspace In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of restricted Lorentz transformations in their action on Minkowski spacetime. An example of each type is given in the subsections below, along with the effect of the
one-parameter subgroup In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism :\varphi : \mathbb \rightarrow G from the real line \mathbb (as an additive group) to some other topological group G. If \varphi i ...
it generates (e.g., on the appearance of the night sky). The Möbius transformations are the conformal transformations of the Riemann sphere (or celestial sphere). Then conjugating with an arbitrary element of SL(2,C) obtains the following examples of arbitrary elliptic, hyperbolic, loxodromic, and parabolic (restricted) Lorentz transformations, respectively. The effect on the flow lines of the corresponding one-parameter subgroups is to transform the pattern seen in the examples by some conformal transformation. For example, an elliptic Lorentz transformation can have any two distinct fixed points on the celestial sphere, but points still flow along circular arcs from one fixed point toward the other. The other cases are similar.


Elliptic

An elliptic element of SL(2, C) is : P_1 = \begin \exp\left(\frac\theta\right) & 0 \\ 0 & \exp\left(-\frac\theta\right) \end and has fixed points = 0, ∞. Writing the action as and collecting terms, the spinor map converts this to the (restricted) Lorentz transformation : Q_1 = \begin 1 & 0 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) & 0 \\ 0 & \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end = \exp\left(\theta\begin 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end\right) ~. This transformation then represents a rotation about the axis, exp(). The one-parameter subgroup it generates is obtained by taking to be a real variable, the rotation angle, instead of a constant. The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same two fixed points, the North and South poles. The transformations move all other points around latitude circles so that this group yields a continuous counter-clockwise rotation about the axis as increases. The ''angle doubling'' evident in the spinor map is a characteristic feature of ''spinorial double coverings''.


Hyperbolic

A hyperbolic element of SL(2,C) is : P_2 = \begin \exp\left(\frac\right) & 0 \\ 0 & \exp\left(-\frac\right) \end and has fixed points = 0, ∞. Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this Möbius transformation is a dilation from the origin. The spinor map converts this to the Lorentz transformation : Q_2 = \begin \cosh(\eta) & 0 & 0 & \sinh(\eta) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sinh(\eta) & 0 & 0 & \cosh(\eta) \end = \exp \left( \eta \begin 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end\right)~. This transformation represents a boost along the axis with
rapidity In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with d ...
. The one-parameter subgroup it generates is obtained by taking to be a real variable, instead of a constant. The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same fixed points (the North and South poles), and they move all other points along
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
s away from the South pole and toward the North pole.


Loxodromic

A loxodromic element of SL(2, C) is : P_3 = P_2 P_1 = P_1 P_2 = \begin \exp \left(\frac(\eta + i\theta)\right) & 0 \\ 0 & \exp \left(-\frac(\eta + i\theta)\right) \end and has fixed points = 0, ∞. The spinor map converts this to the Lorentz transformation : Q_3 = Q_2 Q_1 = Q_1 Q_2. The one-parameter subgroup this generates is obtained by replacing ''η'' + i''θ'' with any real multiple of this complex constant. (If ''η'', ''θ'' vary independently, then a ''two-dimensional'' abelian subgroup is obtained, consisting of simultaneous rotations about the axis and boosts along the -axis; in contrast, the ''one-dimensional'' subgroup discussed here consists of those elements of this two-dimensional subgroup such that the rapidity of the boost and angle of the rotation have a ''fixed ratio''.) The corresponding continuous transformations of the celestial sphere (excepting the identity) all share the same two fixed points (the North and South poles). They move all other points away from the South pole and toward the North pole (or vice versa), along a family of curves called loxodromes. Each loxodrome spirals infinitely often around each pole.


Parabolic

A parabolic element of SL(2, C) is : P_4 = \begin 1 & \alpha \\ 0 & 1 \end and has the single fixed point = ∞ on the Riemann sphere. Under stereographic projection, it appears as an ordinary
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
along the
real axis In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
. The spinor map converts this to the matrix (representing a Lorentz transformation) :\begin Q_4 &= \begin 1 + \frac\vert\alpha\vert^2 & \operatorname(\alpha) & \operatorname(\alpha) & -\frac\vert\alpha\vert^2 \\ \operatorname(\alpha) & 1 & 0 & -\operatorname(\alpha) \\ -\operatorname(\alpha) & 0 & 1 & \operatorname(\alpha) \\ \frac\vert\alpha\vert^2 & \operatorname(\alpha) & \operatorname(\alpha) & 1 - \frac\vert\alpha\vert^2 \end \\ pt &= \exp \begin 0 & \operatorname(\alpha) & \operatorname(\alpha) & 0 \\ \operatorname(\alpha) & 0 & 0 & -\operatorname(\alpha) \\ -\operatorname(\alpha) & 0 & 0 & \operatorname(\alpha) \\ 0 & \operatorname(\alpha) & \operatorname(\alpha) & 0 \end ~. \end This generates a two-parameter abelian subgroup, which is obtained by considering a complex variable rather than a constant. The corresponding continuous transformations of the celestial sphere (except for the identity transformation) move points along a family of circles that are all tangent at the North pole to a certain
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
. All points other than the North pole itself move along these circles. Parabolic Lorentz transformations are often called null rotations. Since these are likely to be the least familiar of the four types of nonidentity Lorentz transformations (elliptic, hyperbolic, loxodromic, parabolic), it is illustrated here how to determine the effect of an example of a parabolic Lorentz transformation on Minkowski spacetime. The matrix given above yields the transformation : \begin t \\ x \\ y \\ z \end \rightarrow \begin t \\ x \\ y \\ z \end + \operatorname(\alpha) \; \begin x \\ t - z \\ 0 \\ x \end + \operatorname(\alpha) \; \begin y \\ 0 \\ z - t \\ y \end + \frac \; \begin t - z \\ 0 \\ 0 \\ t - z \end. Now, without loss of generality, pick . Differentiating this transformation with respect to the now real group parameter and evaluating at produces the corresponding vector field (first order linear partial differential operator), : x\, \left( \partial_t + \partial_z \right) + (t - z)\, \partial_x. Apply this to a function , and demand that it stays invariant; i.e., it is annihilated by this transformation. The solution of the resulting first order linear partial differential equation can be expressed in the form : f(t, x, y, z) = F\left(y,\, t - z,\, t^2 - x^2 - z^2\right), where is an ''arbitrary'' smooth function. The arguments of give three ''rational invariants'' describing how points (events) move under this parabolic transformation, as they themselves do not move, : y = c_1, ~~~~ t-z = c_2, ~~~~ t^2 - x^2 - z^2 = c_3. Choosing real values for the constants on the right hand sides yields three conditions, and thus specifies a curve in Minkowski spacetime. This curve is an orbit of the transformation. The form of the rational invariants shows that these flowlines (orbits) have a simple description: suppressing the inessential coordinate , each orbit is the intersection of a ''null plane'', , with a ''hyperboloid'', . The case 3 = 0 has the hyperboloid degenerate to a light cone with the orbits becoming parabolas lying in corresponding null planes. A particular null line lying on the light cone is left ''invariant''; this corresponds to the unique (double) fixed point on the Riemann sphere mentioned above. The other null lines through the origin are "swung around the cone" by the transformation. Following the motion of one such null line as increases corresponds to following the motion of a point along one of the circular flow lines on the celestial sphere, as described above. A choice instead, produces similar orbits, now with the roles of and interchanged. Parabolic transformations lead to the gauge symmetry of massless particles (like
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s) with helicity , , ≥ 1. In the above explicit example, a massless particle moving in the direction, so with 4-momentum , is not affected at all by the -boost and -rotation combination defined below, in the "little group" of its motion. This is evident from the explicit transformation law discussed: like any light-like vector, ''P'' itself is now invariant; i.e., all traces or effects of have disappeared. 1 = 2 = 3 = 0, in the special case discussed. (The other similar generator, as well as it and ''z'' comprise altogether the little group of the light-like vector, isomorphic to (2).)


Appearance of the night sky

This isomorphism has the consequence that Möbius transformations of the Riemann sphere represent the way that Lorentz transformations change the appearance of the night sky, as seen by an observer who is maneuvering at relativistic velocities relative to the "fixed stars". Suppose the "fixed stars" live in Minkowski spacetime and are modeled by points on the celestial sphere. Then a given point on the celestial sphere can be associated with , a complex number that corresponds to the point on the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
, and can be identified with a
null vector In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms an ...
(a light-like vector) in Minkowski space : \begin u^2 + v^2 + 1 \\ 2u \\ -2v \\ u^2 + v^2 - 1 \end or, in the Weyl representation (the spinor map), the Hermitian matrix : N = 2\begin u^2 + v^2 & u + iv \\ u - iv & 1 \end. The set of real scalar multiples of this null vector, called a ''null line'' through the origin, represents a ''line of sight'' from an observer at a particular place and time (an arbitrary event we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars. Then the points of the
celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphe ...
(equivalently, lines of sight) are identified with certain Hermitian matrices.


Projective geometry and different views of the 2-sphere

This picture emerges cleanly in the language of projective geometry. The (restricted) Lorentz group acts on the projective celestial sphere. This is the space of non-zero null vectors with t>0 under the given quotient for projective spaces: (t,x,y,z)\sim (t',x',y',z') if (t',x',y',z') = (\lambda t, \lambda x, \lambda y, \lambda z) for \lambda > 0. This is referred to as the celestial sphere as this allows us to rescale the time coordinate t to 1 after acting using a Lorentz transformation, ensuring the space-like part sits on the unit sphere. From the Möbius side, \text(2,\mathbb) acts on complex projective space \mathbb^1, which can be shown to be diffeomorphic to the 2-sphere - this is sometimes referred to as the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
. The quotient on projective space leads to a quotient on the group \text(2,\mathbb). Finally, these two can be linked together by using the complex projective vector to construct a null-vector. If \xi is a \mathbb^1 projective vector, it can be tensored with its Hermitian conjugate to produce a 2\times 2 Hermitian matrix. From elsewhere in this article we know this space of matrices can be viewed as 4-vectors. The space of matrices coming from turning each projective vector in the Riemann sphere into a matrix is known as the Bloch sphere.


Lie algebra

As with any Lie group, a useful way to study many aspects of the Lorentz group is via its
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
. Since the Lorentz group \text(1,3) is a matrix Lie group, its corresponding Lie algebra \mathfrak(1,3) is a matrix Lie algebra, which may be computed as :\mathfrak(1,3) = \left\. If \eta is the diagonal matrix with diagonal entries (1, -1, -1, -1), then the Lie algebra \mathfrak(1,3) consists of 4\times 4 matrices X such that Proposition 3.25 :\eta X \eta = -X^\textsf. Explicitly, \mathfrak(1, 3) consists of 4\times 4 matrices of the form :\begin 0 & a & b & c \\ a & 0 & d & e \\ b & -d & 0 & f \\ c & -e & -f & 0 \end, where a, b, c, d, e, f are arbitrary real numbers. This Lie algebra is six dimensional. The subalgebra of \mathfrak(1, 3) consisting of elements in which a, b, and c equal to zero is isomorphic to \mathfrak(3). The full Lorentz group \text(1, 3), the proper Lorentz group \text(1, 3) and the proper orthochronous Lorentz group \mathrm^+(1, 3) (the component connected to the identity) all have the same Lie algebra, which is typically denoted \mathfrak(1,3). Since the identity component of the Lorentz group is isomorphic to a finite quotient of \text(2, \mathbb) (see the section above on the connection of the Lorentz group to the Möbius group), the Lie algebra of the Lorentz group is isomorphic to the Lie algebra \mathfrak(2, \mathbb). As a complex Lie algebra \mathfrak(2, \mathbb) is three dimensional, but is six dimensional when viewed as a real Lie algebra.


Commutation relations of the Lorentz algebra

The standard basis matrices can be indexed as M^ where \mu,\nu take values in \. These arise from taking only one of a,b,\cdots, f to be one, and others zero, in turn. The components can be written as :(M^)_ = \delta^\mu_\rho\delta^\nu_\sigma - \delta^\nu_\rho\delta^\mu_\sigma. The commutation relations are : ^,M^= M^\eta^ - M^\eta^ + M^\eta^ - M^\eta^. There are different possible choices of convention in use. In physics, it is common to include a factor of i with the basis elements, which gives a factor of i in the commutation relations. Then M^ generate boosts and M^ generate rotations. The structure constants for the Lorentz algebra can be read off from the commutation relations. Any set of basis elements which satisfy these relations form a representation of the Lorentz algebra.


Generators of boosts and rotations

The Lorentz group can be thought of as a subgroup of the diffeomorphism group of R4 and therefore its Lie algebra can be identified with vector fields on R4. In particular, the vectors that generate isometries on a space are its
Killing vector In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal g ...
s, which provides a convenient alternative to the left-invariant vector field for calculating the Lie algebra. We can write down a set of six generators: * Vector fields on R4 generating three rotations ''i'' ''J'', *: -y \partial_x + x \partial_y \equiv i J_z ~, \qquad -z \partial_y + y \partial_z \equiv iJ_x~, \qquad -x \partial_z + z \partial_x \equiv iJ_y ~; * Vector fields on R4 generating three boosts ''i'' ''K'', *: x \partial_t + t \partial_x\equiv iK_x ~, \qquad y \partial_t + t \partial_y \equiv iK_y ~, \qquad z \partial_t + t \partial_z\equiv iK_z. The factor of i appears to ensure that the generators of rotations are Hermitian. It may be helpful to briefly recall here how to obtain a one-parameter group from a vector field, written in the form of a first order
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
partial differential operator such as : \mathcal = -y \partial_x + x \partial_y. The corresponding initial value problem (consider r = (x, y) a function of a scalar \lambda and solve \partial_ r = \mathcal r with some initial conditions) is : \frac = -y, \; \frac = x, \; x(0) = x_0, \; y(0) = y_0. The solution can be written : x(\lambda) = x_0 \cos(\lambda) - y_0 \sin(\lambda), \; y(\lambda) = x_0 \sin(\lambda) + y_0 \cos(\lambda) or : \begin t \\ x \\ y \\ z \end = \begin 1 & 0 & 0 & 0 \\ 0 & \cos(\lambda) & -\sin(\lambda) & 0 \\ 0 & \sin(\lambda) & \cos(\lambda) & 0 \\ 0 & 0 & 0 & 1 \end \begin t_0 \\ x_0 \\ y_0 \\ z_0 \end where we easily recognize the one-parameter matrix group of rotations exp(''i λ Jz'') about the z axis. Differentiating with respect to the group parameter and setting it ''λ''=0 in that result, we recover the standard matrix, : iJ_z = \begin 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end ~, which corresponds to the vector field we started with. This illustrates how to pass between matrix and vector field representations of elements of the Lie algebra. The exponential map plays this special role not only for the Lorentz group but for Lie groups in general. Reversing the procedure in the previous section, we see that the Möbius transformations that correspond to our six generators arise from exponentiating respectively ''η''/2 (for the three boosts) or ''iθ''/2 (for the three rotations) times the three
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 ...
: \sigma_1 = \begin 0 & 1 \\ 1 & 0 \end, \;\; \sigma_2 = \begin 0 & -i \\ i & 0 \end, \;\; \sigma_3 = \begin 1 & 0 \\ 0 & -1 \end.


Generators of the Möbius group

Another generating set arises via the isomorphism to the Möbius group. The following table lists the six generators, in which * The first column gives a generator of the flow under the Möbius action (after stereographic projection from the Riemann sphere) as a ''real'' vector field on the Euclidean plane. * The second column gives the corresponding one-parameter subgroup of Möbius transformations. * The third column gives the corresponding one-parameter subgroup of Lorentz transformations (the image under our homomorphism of preceding one-parameter subgroup). * The fourth column gives the corresponding generator of the flow under the Lorentz action as a real vector field on Minkowski spacetime. Notice that the generators consist of * Two parabolics (null rotations) * One hyperbolic (boost in the \partial_z direction) * Three elliptics (rotations about the ''x'', ''y'', ''z'' axes, respectively)


Worked example: rotation about the y-axis

Start with : \sigma_2 = \begin 0 & i \\ -i & 0 \end. Exponentiate: : \exp \left( \frac \, \sigma_2 \right) = \begin \cos\left(\frac\right) & -\sin\left(\frac\right) \\ \sin\left(\frac\right) & \cos\left(\frac\right) \end. This element of \text(2,\mathbb) represents the one-parameter subgroup of (elliptic) Möbius transformations: : \xi \mapsto \xi' = \frac. Next, : \left.\frac\_ = -\frac. The corresponding vector field on \mathbb (thought of as the image of S^2 under stereographic projection) is : -\frac \, \partial_\xi. Writing \xi = u + i v, this becomes the vector field on \mathbb^2 : -\frac \, \partial_u - u v \, \partial_v. Returning to our element of \text(2,\mathbb), writing out the action X \mapsto P X P^\dagger and collecting terms, we find that the image under the spinor map is the element of \text^+(1,3) : \begin 1 & 0 & 0 & 0 \\ 0 & \cos(\theta) & 0 & \sin(\theta) \\ 0 & 0 & 1 & 0 \\ 0 & -\sin(\theta) & 0 & \cos(\theta) \end. Differentiating with respect to \theta at \theta = 0, yields the corresponding vector field on \mathbb^, : z \partial_x - x \partial_z. \,\! This is evidently the generator of counterclockwise rotation about the y-axis.


Subgroups of the Lorentz group

The subalgebras of the Lie algebra of the Lorentz group can be enumerated, up to conjugacy, from which the
closed subgroup In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
s of the restricted Lorentz group can be listed, up to conjugacy. (See the book by Hall cited below for the details.) These can be readily expressed in terms of the generators X_n given in the table above. The one-dimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group: * X_1 generates a one-parameter subalgebra of parabolics SO(0, 1), * X_3 generates a one-parameter subalgebra of boosts SO(1, 1), * X_4 generates a one-parameter of rotations SO(2), * X_3 + a X_4 (for any a \neq 0) generates a one-parameter subalgebra of loxodromic transformations. (Strictly speaking the last corresponds to infinitely many classes, since distinct a give different classes.) The two-dimensional subalgebras are: * X_1, X_2 generate an abelian subalgebra consisting entirely of parabolics, * X_1, X_3 generate a nonabelian subalgebra isomorphic to the Lie algebra of the affine group Aff(1), * X_3, X_4 generate an abelian subalgebra consisting of boosts, rotations, and loxodromics all sharing the same pair of fixed points. The three-dimensional subalgebras use the
Bianchi classification In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras (up to isomorphism). The classification contains 11 classes, 9 of which contain a single Lie algebra and two of which contain a continuum-sized fam ...
scheme: * X_1, X_2, X_3 generate a Bianchi V subalgebra, isomorphic to the Lie algebra of Hom(2), the group of ''euclidean homotheties'', * X_1, X_2, X_4 generate a Bianchi VII subalgebra, isomorphic to the Lie algebra of E(2), the
euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations ...
, * X_1, X_2, X_3 + a X_4, where a \neq 0, generate a Bianchi VII subalgebra, * X_1, X_3, X_5 generate a Bianchi VIII subalgebra, isomorphic to the Lie algebra of SL(2, R), the group of isometries of the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...
, * X_4, X_5, X_6 generate a Bianchi IX subalgebra, isomorphic to the Lie algebra of SO(3), the rotation group. The Bianchi types refer to the classification of three-dimensional Lie algebras by the Italian mathematician
Luigi Bianchi Luigi Bianchi (18 January 1856 – 6 June 1928) was an Italian mathematician. He was born in Parma, Emilia-Romagna, and died in Pisa. He was a leading member of the vigorous geometric school which flourished in Italy during the later years of ...
. The four-dimensional subalgebras are all conjugate to * X_1, X_2, X_3, X_4 generate a subalgebra isomorphic to the Lie algebra of Sim(2), the group of Euclidean similitudes. The subalgebras form a lattice (see the figure), and each subalgebra generates by exponentiation a
closed subgroup In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
of the restricted Lie group. From these, all subgroups of the Lorentz group can be constructed, up to conjugation, by multiplying by one of the elements of the Klein four-group. As with any connected Lie group, the coset spaces of the closed subgroups of the restricted Lorentz group, or
homogeneous spaces In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ''G ...
, have considerable mathematical interest. A few, brief descriptions: * The group Sim(2) is the stabilizer of a ''null line''; i.e., of a point on the Riemann sphere—so the homogeneous space SO+(1, 3)/Sim(2) is the Kleinian geometry that represents
conformal geometry In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
on the sphere ''S''2. * The (identity component of the) Euclidean group SE(2) is the stabilizer of a
null vector In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms an ...
, so the homogeneous space SO+(1, 3)/SE(2) is the
momentum space In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
of a massless particle; geometrically, this Kleinian geometry represents the ''degenerate'' geometry of the light cone in Minkowski spacetime. * The rotation group SO(3) is the stabilizer of a timelike vector, so the homogeneous space SO+(1, 3)/SO(3) is the
momentum space In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
of a massive particle; geometrically, this space is none other than three-dimensional hyperbolic space H3.


Generalization to higher dimensions

The concept of the Lorentz group has a natural generalization to spacetime of any number of dimensions. Mathematically, the Lorentz group of (''n'' + 1)-dimensional Minkowski space is the indefinite orthogonal group O(''n'', 1) of linear transformations of R''n''+1 that preserves the quadratic form :(x_1, x_2, \ldots, x_n, x_) \mapsto x_1^2 + x_2^2 + \cdots + x_n^2 - x_^2. The group O(1, ''n'') preserves the quadratic form : (x_1, x_2, \ldots, x_n, x_) \mapsto x_1^2 - x_2^2 - \cdots - x_^2 It is isomorphic to O(''n'', 1) but enjoys greater popularity in mathematical physics, primarily because the algebra of 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 par ...
and, more generally, spinor and Clifford algebras, are "more natural" with this signature. A common notation for the vector space \mathbb^, equipped with this choice of quadratic form, is \mathbb^. Many of the properties of the Lorentz group in four dimensions (where ) generalize straightforwardly to arbitrary ''n''. For instance, the Lorentz group O(''n'', 1) has four connected components, and it acts by conformal transformations on the celestial (''n''−1)-sphere in (''n''+1)-dimensional Minkowski space. The identity component SO+(''n'', 1) is an SO(''n'')-bundle over hyperbolic ''n''-space H''n''. The low-dimensional cases and are often useful as "toy models" for the physical case , while higher-dimensional Lorentz groups are used in physical theories such as
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
that posit the existence of hidden dimensions. The Lorentz group O(''n'', 1) is also the isometry group of ''n''-dimensional
de Sitter space In mathematical physics, ''n''-dimensional de Sitter space (often abbreviated to dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canoni ...
dS''n'', which may be realized as the homogeneous space O(''n'', 1)/O(''n'' − 1, 1). In particular O(4, 1) is the isometry group of the
de Sitter universe A de Sitter universe is a cosmological solution to the Einstein field equations of general relativity, named after Willem de Sitter. It models the universe as spatially flat and neglects ordinary matter, so the dynamics of the universe are domin ...
dS4, a cosmological model.


See also


Notes


References


Reading List

*
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing l ...
(1957
''Geometric Algebra'', chapter III: Symplectic and Orthogonal Geometry
via
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
, covers orthogonal groups O(p,q) * A canonical reference; ''see chapters 1–6'' for representations of the Lorentz group. * An excellent resource for Lie theory, fiber bundles, spinorial coverings, and many other topics. * ''See Lecture 11'' for the irreducible representations of SL(2,C). * * . * ''See Chapter 6'' for the subalgebras of the Lie algebra of the Lorentz group. * ''See also'' the ''See Section 1.3'' for a beautifully illustrated discussion of covering spaces. ''See Section 3D'' for the topology of rotation groups. * §41.3 * (Dover reprint edition.) An excellent reference on Minkowski spacetime and the Lorentz group. * ''See Chapter 3'' for a superbly illustrated discussion of Möbius transformations. * *. {{DEFAULTSORT:Lorentz Group Lie groups Special relativity Group theory Hendrik Lorentz