Group theory  
Group theory


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. The mathematical form of
are each invariant under Lorentz transformations. Therefore the Lorentz group can be said to express a fundamental symmetry of many of the known fundamental laws of nature.
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The Lorentz group is a subgroup of the Poincaré group, the group of all isometries of Minkowski spacetime. 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. 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.
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 R^{4}. 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.
The Lorentz group is a 6dimensional noncompact real Lie group which is not connected. All four of its connected components are not simply connected. The identity component (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). It consists of those Lorentz transformations that preserve the orientation of space and direction of time. The restricted Lorentz group has often been presented through a facility of biquaternion algebra.
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. This observation was taken by Roger Penrose as the starting point of twistor theory. 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. 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.
Because it is a Lie group, the Lorentz group O(1,3) is both a group and a smooth manifold. As a manifold, it has four connected components. Intuitively, this means that it consists of four topologically separated pieces.
To see why, notice that a Lorentz transformation may or may not
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. 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.
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 fourgroup. 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
where P and T are the space inversion and time reversal operators:
The four elements of this isomorphic copy of the Klein fourgroup label the four connected components of the Lorentz group.
As stated above, the restricted Lorentz group is the identity component of the Lorentz group. This means that it consists of all Lorentz transformations which can be connected to the identity by a continuous curve lying in the group. The restricted Lorentz group is a connected normal subgroup of the full Lorentz group with the same dimension (in this case, 6 dimensions).
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 timelike direction). The set of all rotations forms a Lie subgroup isomorphic to the ordinary rotation group SO(3). 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 oneparameter subgroup. An arbitrary rotation is specified by 3 real parameters, as is an arbitrary boost. 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.
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). 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. 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. 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. The kernel of the spinor map is the two element subgroup ±I. Therefore, the quotient group PSL(2,C) is isomorphic to SO^{+}(1,3).
This isomorphism has a very interesting physical interpretation. We can identify the complex number
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.
But by stereographic projection, we can identify ξ with a point on the Riemann sphere. Putting it all together, we have identified the points of the celestial sphere with certain Hermitian matrices, and also with lines of sight. 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.
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:
(To be utterly pedantic, the identity element is in a fifth class, all by itself.)
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.
We will discuss a particularly simple example of each type, and in particular, the effect on the appearance of the night sky of the oneparameter subgroup which it generates. 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 oneparameter 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 oneparameter 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.
A typical loxodromic element of SL(2,C) is
which also has fixed points . Our homomorphism maps this to the Lorentz transformation
The oneparameter 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 twodimensional abelian subgroup, consisting of simultaneous rotations about the z axis and boosts along the z axis; in contrast, the onedimensional subgroup we are discussing here consists of those elements of this twodimensional 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. Our homomorphism maps this to the matrix (representing a Lorentz transformation)
This generates a oneparameter 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. 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 + c_{2} with a hyperboloid t^{2} − x^{2} − z^{2} = c_{3}. In particular, the reader may wish to sketch the case c_{3} = 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 .
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). 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 oneparameter 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.
As with any Lie group, the best way to study many aspects of the Lorentz group is via its Lie algebra. 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 particle physics, a state that is invariant under one of these copies of SL(2,R) is said to have chirality, and is either lefthanded or righthanded, according to which copy of SL(2,R) it is invariant under.
The Lorentz group is a subgroup of the diffeomorphism group of R^{4} and therefore its Lie algebra can be identified with vector fields on R^{4}. In particular, the vectors which generate isometries on a space are its Killing vectors, which provides a convenient alternative to the leftinvariant vector field for calculating the Lie algebra. We can immediately write down an obvious set of six generators:
It may be helpful to briefly recall here how to obtain a oneparameter 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 oneparameter matrix group of rotations about the z axis. 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. We list the six generators in the following table, in which
Notice that the generators consist of
Vector field on R^{2}  Oneparameter subgroup of SL(2,C), representing Möbius transformations 
Oneparameter subgroup of SO^{+}(1,3), representing Lorentz transformations 
Vector field on R^{4} 

Parabolic  
Hyperbolic  
Elliptic  
Let's verify one line in this table. Start with
Exponentiate:
This element of SL(2,C) represents the oneparameter subgroup of (elliptic) Möbius transformations:
Next,
The corresponding vector field on C (thought of as the image of S^{2} under stereographic projection) is
Writing ξ = u + iv, this becomes the vector field on R^{2}
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 R^{4} is
This is evidently the generator of counterclockwise rotation about the y axis.
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 onedimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group:
(Strictly speaking the last corresponds to infinitely many classes, since distinct a give different classes.) The twodimensional subalgebras are:
The three dimensional subalgebras are:
(Here, the Bianchi types refer to the classification of three dimensional Lie algebras by the Italian mathematician Luigi Bianchi.) The four dimensional subalgebras are all conjugate to
The subalgebras form a lattice (see the figure), and each subalgebra generates by exponentiation a closed subgroup 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 fourgroup.
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. We briefly describe some of them here:
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). 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). 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 of the each group is isomorphic to the two element cyclic group Z_{2}.
Warning: in applications to quantum mechanics the special linear group SL(2, C) is sometimes called the Lorentz group.
Twofold coverings are characteristic of spin groups. Indeed, in addition to the double coverings
we have the double coverings
These spinorial double coverings are all closely related to Clifford algebras.
The left and right groups in the double covering
are deformation retracts of the left and right groups, respectively, in the double covering
But the homogeneous space SO^{+}(1,3)/SO(3) is homeomorphic to hyperbolic 3space H^{3}, so we have exhibited the restricted Lorentz group as a principal fiber bundle with fibers SO(3) and base H^{3}. Since the latter is homeomorphic to R^{3}, while SO(3) is homeomorphic to threedimensional real projective space RP^{3}, we see that the restricted Lorentz group is locally homeomorphic to the product of RP^{3} with R^{3}. Since the base space is contractible, this can be extended to a global homeomorphism.
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 R^{n+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 (n1)sphere in n+1 dimensional Minkowski space. The identity component SO^{+}(n,1) is an SO(n)bundle over hyperbolic nspace H^{n}.
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. The Lorentz group O(n,1) is also the isometry group of ndimensional de Sitter space dS_{n}, which may be realized as the homogeneous space O(n,1)/O(n1,1). In particular O(4,1) is the isometry group of the de Sitter universe dS_{4}, a cosmological model.
