Groups

Group theory

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In physics (and mathematics), the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all (nongravitational) physical phenomena. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity A classical field theory is a Physical theory that describes the study of how one or more physical fields interact with matter Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. The mathematical form of

• the kinematical laws of special relativity,
• Maxwell's field equations in the theory of electromagnetism,
• Dirac's equation in the theory of the electron,

are each invariant under Lorentz transformations. Kinematics ( Greek κινειν, kinein, to move is a branch of Classical mechanics which describes the motion of objects without Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J Therefore the Lorentz group can be said to express a fundamental symmetry of many of the known fundamental laws of nature. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or

Contents 1 Basic properties 2 Connected components 3 The restricted Lorentz group 4 Relation to the Möbius group 5 Appearance of the night sky 6 Conjugacy classes 7 The Lie algebra of the Lorentz group 8 Subgroups of the Lorentz group 9 Covering groups 10 Topology 11 General dimensions 12 See also 13 References

Basic properties

The Lorentz group is a subgroup of the Poincaré group, the group of all isometries of Minkowski spacetime. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime For the Mechanical engineering and Architecture usage see Isometric projection. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity The Lorentz transformations are precisely the isometries which leave the origin fixed. Thus, the Lorentz group is an isotropy subgroup of the isometry group of Minkowski spacetime. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. 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 transformations; general isometries of Minkowski spacetime are affine transformations. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector

Mathematically, the Lorentz group may be described as the generalized orthogonal group O(1,3), the matrix Lie group which preserves the quadratic form

on R⁴. In Mathematics, the indefinite orthogonal group, O( p, q) is the Lie group of all Linear transformations of a n = p In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables This quadratic form is interpreted in physics as the metric tensor of Minkowski spacetime, so this definition is simply a restatement of the fact that Lorentz transformations are precisely the linear transformations which are also isometries of Minkowski spacetime. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space

The Lorentz group is a 6-dimensional noncompact Lie group which is not connected, and whose connected components are not simply connected. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be The identity component (i. In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity e. the component containing the identity element) of the Lorentz group is often called the restricted Lorentz group and is denoted SO⁺(1,3).

In pure mathematics, the restricted Lorentz group arises in another guise as the Möbius group, which is the symmetry group of conformal geometry on the Riemann sphere. Möbius transformations should not be confused with the Möbius transform or the Möbius function. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a Riemannian manifold or Pseudo-Riemannian In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that This observation was taken by Roger Penrose as the starting point of twistor theory. Sir Roger Penrose, PhD, OM, FRS (born 8 August 1931) is an English Mathematical physicist and Emeritus The twistor theory, originally developed by Roger Penrose in 1967, is the mathematical theory which maps the Geometric objects of the four dimensional space-time It has a fascinating physical consequence for the appearance of the night sky as seen by an observer who is maneuvering at relativistic velocities relative to the "fixed stars", which is discussed below.

The restricted Lorentz group arises in other ways in pure mathematics. For example, it arises as the point symmetry group of a certain ordinary differential equation. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its This fact also has physical significance.

Note: the Lorentz group also preserves the quadratic form and is therefore sometimes denoted O(3,1). A similar remark applies to its identity component and the subgroups introduced below.

Connected components

Because it is a Lie group, the Lorentz group O(1,3) is both a group and a smooth manifold. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. As a manifold, it has four connected components. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of Intuitively, this means that it consists of four topologically separated pieces.

To see why, notice that a Lorentz transformation may or may not

• reverse the direction of time (or more precisely, transform a future-pointing timelike vector into a past-pointing one),
• reverse the orientation of a vierbein. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which This page covers applications of the Cartan formalism. For the general concept see Cartan connection.

Lorentz transformations which preserve the direction of time are called orthochronous. Those which 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 orthochronous transformations is often denoted O⁺(1,3).

The identity component of the Lorentz group is the set of all Lorentz transformations preserving both orientation and the direction of time. In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity It is called the proper, orthochronous Lorentz group, or restricted Lorentz group, and it is denoted by SO⁺(1, 3). It is a normal subgroup of the Lorentz group which is also six dimensional. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup.

Note: Some authors refer to SO(1,3) or even O(1,3) when they actually mean SO⁺(1, 3).

The quotient group O(1,3)/SO⁺(1,3) is isomorphic to the Klein four-group. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 Every element in O(1,3) can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group

{1, P, T, PT}

where P and T are the space inversion and time reversal operators:

P = diag(1, −1, −1, −1)

T = diag(−1, 1, 1, 1)

The four elements of this isomorphic copy of the Klein four-group label the four connected components of the Lorentz group. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate. T Symmetry is the symmetry of physical laws under a Time reversal transformation &mdash T t \mapsto -t

The restricted Lorentz group

As stated above, the restricted Lorentz group is the identity component of the Lorentz group. In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity This means that it consists of all Lorentz transformations which can be connected to the identity by a continuous curve lying in the group. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function The restricted Lorentz group is a connected normal subgroup of the full Lorentz group with the same dimension (in this case, 6 dimensions). In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup.

The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts (which can be thought of as hyperbolic rotations in a plane that includes a time-like direction). In Geometry and Linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a Rigid body around a fixed In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to The set of all rotations forms a Lie subgroup isomorphic to the ordinary rotation group SO(3). In Mathematics, a Subgroup H of a Lie group G is a Lie subgroup if the inclusion map from H to G is smooth This article is about rotations in three-dimensional Euclidean space The set of all boosts, however, does not form a subgroup, since composing two boosts does not, in general, result in another boost.

A boost in some direction, or a rotation about some axis, each generate a one-parameter subgroup. In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R An arbitrary rotation is specified by 3 real parameters, as is an arbitrary boost. In Mathematics, the Special orthogonal group in three dimensions otherwise known as the Rotation group SO(3 is a naturally occurring example of a Manifold Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation and a boost, it takes 6 real numbers (parameters) to specify an arbitrary proper orthochronous Lorentz transformation. This is one way to understand why the restricted Lorentz group is six dimensional. (We'll study this in more detail in a later section on the Lie algebra of the Lorentz group. ) To specify an arbitrary Lorentz transformation requires a further two bits of information, which pick out one of the four connected components. This pattern is typical of finite dimensional Lie groups.

Relation to the Möbius group

The restricted Lorentz group SO⁺(1, 3) is isomorphic to the Möbius group, which is, in turn, isomorphic to the projective special linear group PSL(2,C). In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Möbius transformations should not be confused with the Möbius transform or the Möbius function. In Mathematics, especially in area of Algebra called Group theory, the projective linear group (also known as the projective general linear group It will be convenient to work at first with SL(2,C). This group consists of all two by two complex matrices with determinant one

We can write two by two Hermitian matrices in the form

This trick has the pleasant feature that

Therefore, we have identified the space of Hermitian matrices (which is four dimensional, as real vector space) with Minkowski spacetime in such a way that the determinant of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime. A number of Mathematical entities are named Hermitian, after the Mathematician Charles Hermite: Hermitian adjoint But now SL(2,C) acts on the space of Hermitian matrices via

where P ^* is the Hermitian transpose of P, and this action preserves the determinant. In Mathematics, specifically in Functional analysis, each Linear operator on a Hilbert space has a corresponding adjoint operator. Therefore, SL(2,C) acts on Minkowski spacetime by (linear) isometries. We have now constructed a homomorphism of Lie groups from SL(2,C) onto SO⁺(1,3), which we will call the spinor map. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every The kernel of the spinor map is the two element subgroup ±I. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism Therefore, the quotient group PSL(2,C) is isomorphic to SO⁺(1,3).

Appearance of the night sky

This isomorphism has a very interesting physical interpretation. We can identify the complex number

ξ = u + iv

with a null vector in Minkowski space

or the Hermitian matrix

The set of real scalar multiples of this null vector, which we can call a null line through the origin, represents a line of sight from an observer at a particular place and time (an arbitrary event which we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars. In Linear algebra, the null vector or zero vector is the vector (0 0 &hellip 0 in Euclidean space, all of whose components are zero

But by stereographic projection, we can identify ξ with a point on the Riemann sphere. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that Putting it all together, we have identified the points of the celestial sphere with certain Hermitian matrices, and also with lines of sight. In Astronomy and Navigation, the celestial sphere is an imaginary rotating Sphere of "gigantic Radius " This implies that the Möbius transformations of the Riemann sphere precisely represent the way that Lorentz transformations change the appearance of the celestial sphere.

For our purposes here, we can pretend that the "fixed stars" live in Minkowski spacetime. Then, the Earth is moving at a nonrelativistic velocity with respect to a typical astronomical object which might be visible at night. But, an observer who is moving at relativistic velocity with respect to the Earth would see the appearance of the night sky (as modeled by points on the celestial sphere) transformed by a Möbius transformation. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial

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 four classes:

• elliptic transformations
• hyperbolic transformations
• loxodromic transformations
• parabolic transformations

(To be utterly pedantic, the identity element is in a fifth class, all by itself. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class )

In the article on Möbius transformations, 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 eigenspaces of restricted Lorentz transformations in their action on Minkowski spacetime. Möbius transformations should not be confused with the Möbius transform or the Möbius function. In Linear algebra, the null vector or zero vector is the vector (0 0 &hellip 0 in Euclidean space, all of whose components are zero In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

We will discuss a particularly simple example of each type, and in particular, the effect on the appearance of the night sky of the one-parameter subgroup which it generates. In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R At the end of the section we will briefly indicate how we can understand the effect of general Lorentz transformations on the appearance of the night sky in terms of these examples.

A typical elliptic element of SL(2,C) is

which has fixed points . Writing out the action and collecting terms, we find that our spinor map takes this to the (restricted) Lorentz transformation

This transformation represents a rotation about the z axis. The one-parameter subgroup it generates is obtained by simply taking θ 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 two fixed points, the North and South pole. They move all other points around latitude circles. In other words, this group yields a continuous counterclockwise rotation about the z axis as θ increases.

Notice the angle doubling; this phenomenon is a characteristic feature of spinorial double coverings.

A typical hyperbolic element of SL(2,C) is

which also has fixed points . Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this Möbius transformation is a dilation from the origin. Our homomorphism maps this to the Lorentz transformation

This transformation represents a boost along the z axis. The one-parameter subgroup it generates is obtained by simply taking β 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 longitudes away from the South pole and toward the North pole. Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement

A typical loxodromic element of SL(2,C) is

which also has fixed points . Our homomorphism maps this to the Lorentz transformation

Q₃ = Q₂Q₁ = Q₁Q₂

The one-parameter subgroup this generates is obtained by replacing β + iθ with any real multiple of this complex constant. (If we let β,θ vary independently, we obtain a two-dimensional abelian subgroup, consisting of simultaneous rotations about the z axis and boosts along the z axis; in contrast, the one-dimensional subgroup we are discussing 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 (always 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.

A typical parabolic element of SL(2,C) is

which has the single fixed point on the Riemann sphere. Under stereographic projection, it appears as ordinary translation along the real axis. Translation is the interpreting of the meaning of a text and the subsequent production of an equivalent text likewise called a " translation In Mathematics, the real numbers may be described informally in several different ways Our homomorphism maps this to the matrix (representing a Lorentz transformation)

This generates a one-parameter subgroup which is obtained by considering α to be a real variable rather than a constant. The corresponding continuous transformations of the celestial sphere move points along a family of circles which are all tangent at the North pole to a certain great circle. A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. All points other than the North pole itself move along these circles. (Except, of course, for the identity transformation. )

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), we will show how to determine the effect of our example of a parabolic Lorentz transformation on Minkowski spacetime, leaving the other examples as exercises for the reader. From the matrix given above we can read off the transformation

Differentiating this transformation with respect to the group parameter α and evaluate at α = 0, we read off the corresponding vector field (first order linear partial differential operator)

Apply this to an undetermined function f(t,x,y,z). The solution of the resulting first order linear partial differential equation can be expressed in the form

where F is an arbitrary smooth function. The arguments on the right hand side now give three rational invariants describing how points (events) move under our parabolic transformation:

(The reader can verify that these quantities standing on the left hand sides are invariant under our transformation. ) Choosing real values for the constants standing on the right hand sides gives three conditions, and thus defines a curve in Minkowski spacetime. This curve is one of the flowlines of our transformation. We see from the form of the rational invariants that these flowlines (or orbits) have a very simple description: suppressing the inessential coordinate y, we see that each orbit is the intersection of a null plane t = z + c₂ with a hyperboloid t² − x² − z² = c₃. In particular, the reader may wish to sketch the case c₃ = 0, in which the hyperboloid degenerates to a light cone; then orbits are parabolas lying in null planes just mentioned.

Parabolic transformations lead to the gauge symmetry of massless particles with helicity . In Particle physics, helicity is the projection of the spin \vec S onto the direction of momentum \hat p: h = \vec

Notice that a particular null line lying in the light cone is left invariant; this corresponds to the unique (double) fixed point on the Riemann sphere which was 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 α increases corresponds to following the motion of a point along one of the circular flow lines on the celestial sphere, as described above.

The Möbius transformations are precisely the conformal transformations of the Riemann sphere (or celestial sphere). In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane It follows that by conjugating with an arbitrary element of SL(2,C), we can obtain from the above examples 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 our examples by some conformal transformation. Thus, an arbitrary elliptic Lorentz transformation can have any two distinct fixed points on the celestial sphere, but points will still flow along circular arcs from one fixed point toward the other. Similarly for the other cases.

Finally, arbitrary Lorentz transformations can be obtained from the restricted ones by multiplying by a matrix which reflects across t = 0, or by an appropriate orientation reversing diagonal matrix.

The Lie algebra of the Lorentz group

As with any Lie group, the best way to study many aspects of the Lorentz group is via its Lie algebra. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Since the Lorentz group is SO ⁺ (1,3), its Lie algebra is reducible and can be decomposed to two copies of the Lie algebra of SL(2,R), as will be shown explicitly below (this is the Minkowski space analog of the SO(4) SU(2)SU(2) decomposition in a Euclidean space). In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, the Special linear group SL2( R) is the group of all real 2 × 2 matrices with Determinant one In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n In particle physics, a state that is invariant under one of these copies of SL(2,R) is said to have chirality, and is either left-handed or right-handed, according to which copy of SL(2,R) it is invariant under. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In Mathematics, the Special linear group SL2( R) is the group of all real 2 × 2 matrices with Determinant one A phenomenon is said to be chiral if it is not identical to its Mirror image (see Chirality) In Mathematics, the Special linear group SL2( R) is the group of all real 2 × 2 matrices with Determinant one

The Lorentz group is a subgroup of the diffeomorphism group of R⁴ and therefore its Lie algebra can identified with vector fields on R⁴. In particular, the vectors which generate isometries on a space are its Killing vectors, which provides a convenient alternative to the left-invariant vector field for calculating the Lie algebra. In Mathematics, a Killing vector field, named after Wilhelm Killing, is a Vector field on a Riemannian manifold (or Pseudo-Riemannian manifold We can immediately write down an obvious set of six generators:

• vector fields on R⁴ generating three rotations

• vector fields on R⁴ generating three boosts

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 partial differential operator such as

The corresponding initial value problem is

The solution can be written

or

where we easily recognize the one-parameter matrix group of rotations about the z axis. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. The word linear comes from the Latin word linearis, which means created by lines. In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator Differentiating with respect to the group parameter and setting λ = 0 in the result, we recover the matrix

which corresponds to the vector field we started with. This shows how to pass between matrix and vector field representations of elements of the Lie algebra.

Reversing the procedure in the previous section, we see that the Möbius transformations which correspond to our six generators arise from exponentiating respectively (for the three boosts) or (for the three rotations) times the three Pauli matrices

For our purposes, another generating set is more convenient. The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. We list the six generators in the following table, 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 direction),
• three elliptics (rotations about x,y,z axes respectively).

Vector field on R2	One-parameter subgroup of SL(2,C), representing Möbius transformations	One-parameter subgroup of SO+(1,3), representing Lorentz transformations	Vector field on R4
Parabolic
Hyperbolic
Elliptic

Let's verify one line in this table. Start with

Exponentiate:

This element of SL(2,C) represents the one-parameter subgroup of (elliptic) Möbius transformations:

Next,

The corresponding vector field on C (thought of as the image of S² under stereographic projection) is

Writing ξ = u + iv, this becomes the vector field on R²

Returning to our element of SL(2,C), writing out the action and collecting terms, we find that the image under the spinor map is the element of SO⁺(1,3)

Differentiating with respect the θ at θ = 0, we find that the corresponding vector field on R⁴ is

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 we can list the closed subgroups of the restricted Lorentz group, up to conjugacy. (See the book by Hall cited below for the details. ) We can readily express the result in terms of the generating set given in the table above.

The one-dimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group:

• X₁ generates a one-parameter subalgebra of parabolics SO(0,1),
• X₃ generates a one-parameter subalgebra of boosts SO(1,1),
• X₄ generates a one-parameter of rotations SO(2),
• X₃ + aX₄ (for any ) 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:

• X₁,X₂ generate an abelian subalgebra consisting entirely of parabolics,
• X₁,X₃ generate a nonabelian subalgebra isomorphic to the Lie algebra of the affine group A(1),
• X₃,X₄ generate an abelian subalgebra consisting of boosts, rotations, and loxodromics all sharing the same pair of fixed points.

The three dimensional subalgebras are:

• X₁,X₂,X₃ generate a Bianchi V subalgebra, isomorphic to the Lie algebra of Hom(2), the group of euclidean homotheties,
• X₁,X₂,X₄ generate a Bianchi VII_0 subalgebra, isomorphic to the Lie algebra of E(2), the euclidean group,
• X₂,X₂,X₃ + aX₄, where , generate a Bianchi VII_a subalgebra,
• X₁,X₃,X₅ generate a Bianchi VIII subalgebra, isomorphic to the Lie algebra of SL(2,R), the group of isometries of the hyperbolic plane,
• X₄,X₅,X₆ generate a Bianchi IX subalgebra, isomorphic to the Lie algebra of SO(3), the rotation group. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional In

(Here, the Bianchi types refer to the classification of three dimensional Lie algebras by the Italian mathematician Luigi Bianchi. In mathematics the Bianchi classification, named for Luigi Bianchi, is a classification of the 3-dimensional real Lie algebras into 11 classes 9 of which are single Luigi Bianchi ( January 18 1856 - June 6 1928) was an Italian mathematician ) The four dimensional subalgebras are all conjugate to

• X₁,X₂,X₃,X₄ 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 of the restricted Lie group. Closed may refer to Math Closure (mathematics Closed manifold Closed orbits Closed 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, have considerable mathematical interest. In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group We briefly describe some of them here:

• 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 which represents conformal geometry on the sphere S²,
• the (identity component of the) Euclidean group SE(2) is the stabilizer of a null vector, so the homogeneous space SO⁺(1,3)/SE(2) is the momentum space 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 of a massive particle; geometrically, this space is none other than three-dimensional hyperbolic space H³. In Mathematics, a Klein geometry is a type of Geometry motivated by Felix Klein in his influential Erlangen program. In Mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a Riemannian manifold or Pseudo-Riemannian In Linear algebra, the null vector or zero vector is the vector (0 0 &hellip 0 in Euclidean space, all of whose components are zero The Momentum space associated with a particle is a vector space in which every point {k_x k_y k_z} corresponds to a possible value of the Momentum vector \vec{k} In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity The Momentum space associated with a particle is a vector space in which every point {k_x k_y k_z} corresponds to a possible value of the Momentum vector \vec{k} In Mathematics, hyperbolic n -space, denoted H n, is the maximally symmetric Simply connected, n -dimensional

Covering groups

In a previous section we constructed a homomorphism SL(2,C) SO⁺(1,3), which we called the spinor map. Since SL(2,C) is simply connected, it is the covering group of the restricted Lorentz group SO⁺(1,3). In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism By restriction we obtain a homomorphism SU(2) SO(3). Here, the special unitary group SU(2), which is isomorphic to the group of unit norm quaternions, is also simply connected, so it is the covering group of the rotation group SO(3). Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician 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. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism One often says that the restricted Lorentz group and the rotation group are doubly connected. This means that the fundamental group of the each group is isomorphic to the two element cyclic group Z₂. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

Warning: in applications to quantum mechanics the special linear group SL(2, C) is sometimes called the Lorentz group. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Mathematics, the special linear group of degree n over a field F is the set of n × n matrices with

Twofold coverings are characteristic of spin groups. In Mathematics the spin group Spin( n) is the double cover of the Special orthogonal group SO( n) such that there exists a Short 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 all closely related to Clifford algebras. In Mathematics, Clifford algebras are a type of Associative algebra.

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 to hyperbolic 3-space H³, so we have exhibited the restricted Lorentz group as a principal fiber bundle with fibers SO(3) and base H³. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, hyperbolic n -space, denoted H n, is the maximally symmetric Simply connected, n -dimensional In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. 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. In Mathematics, real projective space, or RP n is the Projective space of lines in R n +1 Since the base space is contractible, this can be extended to a global homeomorphism.

General dimensions

The concept of the Lorentz group has a natural generalization to any spacetime dimension. Mathematically, the Lorentz group of n+1 dimensional Minkowski space is the group O(n,1) (or O(1,n)) of linear transformations of Rⁿ⁺¹ which preserve the quadratic form

Many of the properties of the Lorentz group in four dimensions (n=3) 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 Hⁿ.

The low dimensional cases n=1 and n=2 are often useful as "toy models" for the physical case n=3, while higher dimensional Lorentz groups are used in physical theories such as string theory which posit the existence of hidden dimensions. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings The Lorentz group O(n,1) is also the isometry group of n-dimensional de Sitter space dS_n, which may be realized as the homogeneous space O(n,1)/O(n-1,1). In Mathematics and Physics, n -dimensional De Sitter space, denoted dS_n is the Lorentzian analog of an ''n''-sphere (with its In particular O(4,1) is the isometry group of the de Sitter universe dS₄, a cosmological model. A de Sitter universe is a solution to Einstein 's field equations of General Relativity which is named after Willem de Sitter.

See also

References

• Artin, Emil (1957). In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to In Mathematics, inversive ring geometry is the extension to the context of Associative rings of the concepts of Projective line, Homogeneous In Mathematics, the indefinite orthogonal group, O( p, q) is the Lie group of all Linear transformations of a n = p This article is about rotations in three-dimensional Euclidean space Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime Möbius transformations should not be confused with the Möbius transform or the Möbius function. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial Geometric Algebra. New York: Wiley. ISBN 0-471-60839-4.   See Chapter III for the orthogonal groups O(p,q).
• Carmeli, Moshe (1977). Group Theory and General Relativity, Representations of the Lorentz Group and Their Applications to the Gravitational Field. McGraw-Hill, New York. ISBN 0-07-009986-3.   A canonical reference; see chapters 1-6 for representations of the Lorentz group.
• Frankel, Theodore (2004). The Geometry of Physics (2nd Ed. ). Cambridge: Cambridge University Press. ISBN 0-521-53927-7.   An excellent resource for Lie theory, fiber bundles, spinorial coverings, and many other topics.
• Fulton, William; & Harris, Joe (1991). Representation Theory: a First Course. New York: Springer-Verlag. ISBN 0-387-97495-4.   See Lecture 11 for the irreducible representations of SL(2,C).
• Hall, G. S. (2004). Symmetries and Curvature Structure in General Relativity. Singapore: World Scientific. ISBN 981-02-1051-5.   See Chapter 6 for the subalgebras of the Lie algebra of the Lorentz group.
• Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN 0-521-79540-0.   See also the online version. Retrieved on July 3, 2005. Events 324 - Battle of Adrianople Constantine I defeats Licinius, who flees to Byzantium. Year 2005 ( MMV) was a Common year starting on Saturday (link displays full calendar of the Gregorian calendar. See Section 1. 3 for a beautifully illustrated discussion of covering spaces. See Section 3D for the topology of rotation groups.
• Naber, Gregory (1992). The Geometry of Minkowski Spacetime. New York: Springer-Verlag. ISBN 0-486-43235-1 (Dover reprint edition).   An excellent reference on Minkowski spacetime and the Lorentz group.
• Needham, Tristam (1997). Visual Complex Analysis. Oxford: Oxford University Press. ISBN 0-19-853446-9.   See Chapter 3 for a superbly illustrated discussion of Möbius transformations.

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