In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.
For example, the following laws, equations, and theories respect Lorentz symmetry:
The Lorentz group expresses the fundamental symmetry 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.
The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is the isotropy subgroup with respect to the origin 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.
Assume two inertial reference frames and, and two points,, the Lorentz group is the set of all the transformations between the two reference frames that preserve the speed of light propagating between the two points:
c2(\Deltat')2-(\Deltax')2-(\Deltay')2-(\Deltaz')2=c2(\Deltat)2-(\Deltax)2-(\Deltay)2-(\Deltaz)2
Λsf{T}ηΛ=η η=\operatorname{diag}(1,-1,-1,-1)
Mathematically, the Lorentz group may be described as the indefinite orthogonal group, the matrix Lie group that preserves the quadratic form
(t,x,y,z)\mapstot2-x2-y2-z2
on (the vector space equipped with this quadratic form is sometimes written). This quadratic form is, when put on matrix form (see Classical orthogonal group), interpreted in physics as the metric tensor of Minkowski spacetime.
The Lorentz group is a six-dimensional noncompact non-abelian real Lie group that is not connected. The four 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 . The restricted Lorentz group consists of those Lorentz transformations that preserve both the orientation of space and the direction of time. Its fundamental group has order 2, and its universal cover, the indefinite spin group, is isomorphic to both the special linear group and to the symplectic group . These isomorphisms allow the Lorentz group to act on a large number of mathematical structures important to physics, most notably spinors. Thus, in relativistic quantum mechanics and in quantum field theory, it is very common to call the Lorentz group, with the understanding that is a specific representation (the vector representation) of it.
A recurrent representation of the action of the Lorentz group on Minkowski space uses biquaternions, which form a composition algebra. The isometry property of Lorentz transformations holds according to the composition property .
Another property of the Lorentz group is conformality or preservation of angles. Lorentz boosts act by hyperbolic rotation of a spacetime plane, and such "rotations" preserve hyperbolic angle, the measure of rapidity used in relativity. Therefore the Lorentz group is a subgroup of the conformal group of spacetime.
Note that this article refers to as the "Lorentz group", as the "proper Lorentz group", and as the "restricted Lorentz group". Many authors (especially in physics) use the name "Lorentz group" for (or sometimes even) rather than . When reading such authors it is important to keep clear exactly which they are referring to.
Because it is a Lie group, the Lorentz group is a group and also has a topological description as a smooth manifold. 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:
Lorentz transformations that preserve the direction of time are called . The subgroup of orthochronous transformations is often denoted . Those that preserve orientation are called proper, and as linear transformations they have determinant . (The improper Lorentz transformations have determinant .) The subgroup of proper Lorentz transformations is denoted .
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 .
The set of the four connected components can be given a group structure as the quotient group, which is isomorphic to the Klein four-group. Every element in 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 parity and time reversal operators:
.
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.
The restricted Lorentz group is the identity component of the Lorentz group, which means that it consists of all Lorentz transformations that 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 with dimension six.
The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts (which are rotations in a hyperbolic space that includes a time-like direction[1]). 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 isomorphic to the ordinary rotation group . 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.
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)=
| 2 | |
| x | |
| 0 |
-
| 2 | |
| x | |
| 1 |
-
| 2 | |
| x | |
| 2 |
-
| 2. | |
| x | |
| 3 |
The surfaces of transitivity of the orthochronous Lorentz group, acting on flat spacetime are the following:
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.
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, due to the orbit-stabilizer theorem. Furthermore, this upper sheet also provides a model for three-dimensional hyperbolic space.
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 representations. 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 tachyons. Particles on the light cone are photons, and more hypothetically, gravitons. The "particle" corresponding to the origin is the vacuum.
Several other groups are either homomorphic or isomorphic to the restricted Lorentz group . These homomorphisms play a key role in explaining various phenomena in physics.
The Weyl representation or spinor map is a pair of surjective homomorphisms from to . They form a matched pair under parity transformations, corresponding to left and right chiral spinors.
One may define an action of on Minkowski spacetime by writing a point of spacetime as a two-by-two Hermitian matrix in the form
\overline{X}=\begin{bmatrix}ct+z&x-iy\ x+iy&ct-z\end{bmatrix} =ct11+x\sigmax+y\sigmay+z\sigmaz =ct11+\vec{x} ⋅ \vec{\sigma}
This presentation, the Weyl presentation, satisfies
\det\overline{X}=(ct)2-x2-y2-z2.
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 of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime. An element acts on the space of Hermitian matrices via
\overline{X}\mapstoS\overline{X}S\dagger~,
S\dagger
X=ct11-\vec{x} ⋅ \vec{\sigma}
X\mapsto\left(S-1\right)\daggerXS-1
\overline{X}X=\left(c2t2-\vec{x} ⋅ \vec{x}\right)11=\left(c2t2-x2-y2-z2\right)11
These maps are surjective, and kernel of either map is the two element subgroup . By the first isomorphism theorem, the quotient group is isomorphic to .
The parity map swaps these two coverings. It corresponds to Hermitian conjugation being an automorphism of . These two distinct coverings corresponds to the two distinct chiral actions of the Lorentz group on spinors. The non-overlined form corresponds to right-handed spinors transforming as, while the overline form corresponds to left-handed spinors transforming as .
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 physics, it is conventional to denote a Lorentz transformation as, thus showing the matrix with spacetime indexes . A four-vector can be created from the Pauli matrices in two different ways: as
\sigma\mu=(I,\vec\sigma)
Given a Lorentz transformation, the double-covering of the orthochronous Lorentz group by given above can be written as
x\prime\mu{\overline\sigma}\mu={\overline\sigma}\mu
| \mu} | |
| {Λ | |
| \nu |
x\nu=Sx\nu{\overline\sigma}\nuS\dagger
Dropping the
x\mu
{\overline\sigma}\mu
| \mu} | |
| {Λ | |
| \nu |
=S{\overline\sigma}\nuS\dagger
The parity conjugate form is
\sigma\mu
| \mu} | |
| {Λ | |
| \nu |
=\left(S-1\right)\dagger\sigma\nuS-1
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
ηΛsf{T}η=Λ-1
\omega\sigmak\omega-1=-\left(\sigmak\right)sf{T}=-\left(\sigmak\right)*
k=1,2,3
( ⋅ )sf{T}
( ⋅ )*
\omega
\omega=i\sigma2=\begin{bmatrix}0&1\ -1&0\end{bmatrix}
Written as the four-vector, the relationship is
| sf{T} | |
| \sigma | |
| \mu |
=
| * | |
| \sigma | |
| \mu |
=\omega\overline{\sigma}\mu\omega-1
This transforms as
\begin{align}
| sf{T} | |
| \sigma | |
| \mu |
| \mu} | |
| {Λ | |
| \nu |
&=\omega\overline{\sigma}\mu\omega-1
| \mu} | |
| {Λ | |
| \nu |
\\ &=\omegaS \overline{\sigma}\nuS\dagger\omega-1\\ &=\left(\omegaS\omega-1\right)\left(\omega\overline{\sigma}\nu\omega-1\right)\left(\omegaS\dagger\omega-1\right)\\ &=\left(S-1\right)sf{T}\sigma
| sf{T} | |
| \nu |
\left(S-1\right)* \end{align}
\sigma\mu
| \mu} | |
| {Λ | |
| \nu |
=\left(S-1\right)\dagger\sigma\nuS-1
The symplectic group is isomorphic to . This isomorphism is constructed so as to preserve a symplectic bilinear form on, that is, to leave the form invariant under Lorentz transformations. This may be articulated as follows. The symplectic group is defined as
\operatorname{Sp}(2,C)=\left\{S\in\operatorname{GL}(2,C):Ssf{T}\omegaS=\omega\right\}
\omega=i\sigma2=\begin{bmatrix}0&1\ -1&0\end{bmatrix}
Other common notations are
\omega=\epsilon
Given a pair of Weyl spinors (two-component spinors)
u=\begin{bmatrix}u1\ u2\end{bmatrix}~, v=\begin{bmatrix}v1\ v2\end{bmatrix}
\langleu,v\rangle=-\langlev,u\rangle=u1v2-u2v1=usf{T}\omegav
This form is invariant under the Lorentz group, so that for one has
\langleSu,Sv\rangle=\langleu,v\rangle
This defines a kind of "scalar product" of spinors, and is commonly used to defined a Lorentz-invariant mass term in Lagrangians. There are several notable properties to be called out that are important to physics. One is that
\omega2=-1
\omega-1=\omegasf{T}=\omega\dagger=-\omega
The defining relation can be written as
\omegaSsf{T}\omega-1=S-1
ηΛsf{T}η-1=Λ-1
η=\operatorname{diag}(+1,-1,-1,-1)
Λ\in\operatorname{SO}(1,3)
Since is simply connected, it is the universal covering group 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 quaternions, is also simply connected, so it is the covering group of the rotation group . 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 .
Twofold coverings are characteristic of spin groups. Indeed, in addition to the double coverings
we have the double coverings
These spinorial double coverings are constructed from 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 is homeomorphic to hyperbolic 3-space, so we have exhibited the restricted Lorentz group as a principal fiber bundle with fibers and base . Since the latter is homeomorphic to, while is homeomorphic to three-dimensional real projective space, we see that the restricted Lorentz group is locally homeomorphic to the product of with . Since the base space is contractible, this can be extended to a global homeomorphism.
Because the restricted Lorentz group is isomorphic to the Möbius group, its conjugacy classes also fall into five classes:
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.
An example of each type is given in the subsections below, along with the effect of the one-parameter subgroup 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 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.
An elliptic element of is
P1=\begin{bmatrix} \exp\left(
| i | |
| 2 |
\theta\right)&0\\ 0&\exp\left(-
| i | |
| 2 |
\theta\right) \end{bmatrix}
Q1=\begin{bmatrix} 1&0&0&0\\ 0&\cos(\theta)&\sin(\theta)&0\\ 0&-\sin(\theta)&\cos(\theta)&0\\ 0&0&0&1 \end{bmatrix}= \exp\left(\theta\begin{bmatrix} 0&0&0&0\\ 0&0&1&0\\ 0&-1&0&0\\ 0&0&0&0 \end{bmatrix}\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.
A hyperbolic element of is
P2=\begin{bmatrix} \exp\left(
| η | |
| 2 |
\right)&0\\ 0&\exp\left(-
| η | |
| 2 |
\right) \end{bmatrix}
The spinor map converts this to the Lorentz transformation
Q2=\begin{bmatrix} \cosh(η)&0&0&\sinh(η)\\ 0&1&0&0\\ 0&0&1&0\\ \sinh(η)&0&0&\cosh(η) \end{bmatrix}= \exp\left(η\begin{bmatrix} 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0 \end{bmatrix}\right)~.
This transformation represents a boost along the axis with rapidity . 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 longitudes away from the South pole and toward the North pole.
A loxodromic element of is
P3=P2P1=P1P2=\begin{bmatrix} \exp\left(
| 1 | |
| 2 |
(η+i\theta)\right)&0\\ 0&\exp\left(-
| 1 | |
| 2 |
(η+i\theta)\right) \end{bmatrix}
and has fixed points = 0, ∞. The spinor map converts this to the Lorentz transformation
Q3=Q2Q1=Q1Q2=\begin{bmatrix} \cosh(η)&0&0&\sinh(η)\\ 0&\cos(\theta)&\sin(\theta)&0\\ 0&-\sin(\theta)&\cos(\theta)&0\\ \sinh(η)&0&0&\cosh(η) \end{bmatrix}= \exp\begin{bmatrix} 0&0&0&η\\ 0&0&\theta&0\\ 0&-\theta&0&0\\ η&0&0&0 \end{bmatrix}~.
The one-parameter subgroup this generates is obtained by replacing with any real multiple of this complex constant. (If, 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.
A parabolic element of is
P4=\begin{bmatrix}1&\alpha\ 0&1\end{bmatrix}
and has the single fixed point = ∞ on the Riemann sphere. Under stereographic projection, it appears as an ordinary translation along the real axis.
The spinor map converts this to the matrix (representing a Lorentz transformation)
\begin{align} Q4&=\begin{bmatrix} 1+
| 1 | |
| 2 |
\vert\alpha\vert2&\operatorname{Re}(\alpha)&-\operatorname{Im}(\alpha)&-
| 1 | |
| 2 |
\vert\alpha\vert2\\ \operatorname{Re}(\alpha)&1&0&-\operatorname{Re}(\alpha)\\ -\operatorname{Im}(\alpha)&0&1&\operatorname{Im}(\alpha)\\
| 1 | |
| 2 |
\vert\alpha\vert2&\operatorname{Re}(\alpha)&-\operatorname{Im}(\alpha)&1-
| 1 | |
| 2 |
\vert\alpha\vert2 \end{bmatrix}\\[6pt] &=\exp\begin{bmatrix} 0&\operatorname{Re}(\alpha)&-\operatorname{Im}(\alpha)&0\\ \operatorname{Re}(\alpha)&0&0&-\operatorname{Re}(\alpha)\\ -\operatorname{Im}(\alpha)&0&0&\operatorname{Im}(\alpha)\\ 0&\operatorname{Re}(\alpha)&-\operatorname{Im}(\alpha)&0 \end{bmatrix}~. \end{align}
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. 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{bmatrix}t\ x\ y\ z\end{bmatrix} → \begin{bmatrix}t\ x\ y\ z\end{bmatrix}+\operatorname{Re}(\alpha) \begin{bmatrix}x\ t-z\ 0\ x\end{bmatrix}-\operatorname{Im}(\alpha) \begin{bmatrix}y\ 0\ z-t\ y\end{bmatrix}+
| \vert\alpha\vert2 | |
| 2 |
\begin{bmatrix}t-z\ 0\ 0\ t-z\end{bmatrix}.
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(\partialt+\partialz\right)+(t-z)\partialx.
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,t2-x2-z2\right),
y=c1,~~~~t-z=c2,~~~~t2-x2-z2=c3.
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 (such as photons) 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., in the special case discussed. (The other similar generator, as well as it and comprise altogether the little group of the light-like vector, isomorphic to .)
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, and can be identified with a null vector (a light-like vector) in Minkowski space
\begin{bmatrix}u2+v2+1\ 2u\ -2v\ u2+v2-1\end{bmatrix}
N=2\begin{bmatrix}u2+v2&u+iv\ u-iv&1\end{bmatrix}.
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
(t,x,y,z)\sim(t',x',y',z')
(t',x',y',z')=(λt,λx,λy,λz)
λ>0
t
From the Möbius side, acts on complex projective space, which can be shown to be diffeomorphic to the 2-sphere – this is sometimes referred to as the Riemann sphere. The quotient on projective space leads to a quotient on the group .
Finally, these two can be linked together by using the complex projective vector to construct a null-vector. If
\xi
2 x 2
As with any Lie group, a useful way to study many aspects of the Lorentz group is via its Lie algebra. Since the Lorentz group is a matrix Lie group, its corresponding Lie algebra
ak{so}(1,3)
ak{so}(1,3)=\left\{4 x 4R-valuedmatricesX\midetX\inSO(1,3)forallt\right\}
If
η
ak{o}(1,3)
4 x 4
X
ηXη=-Xsf{T}
Explicitly,
ak{so}(1,3)
4 x 4
\begin{pmatrix} 0&a&b&c\\ a&0&d&e\\ b&-d&0&f\\ c&-e&-f&0 \end{pmatrix}
a,b,c,d,e,f
ak{so}(1,3)
a
b
c
ak{so}(3)
The full Lorentz group, the proper Lorentz group and the proper orthochronous Lorentz group (the component connected to the identity) all have the same Lie algebra, which is typically denoted .
Since the identity component of the Lorentz group is isomorphic to a finite quotient of (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 . As a complex Lie algebra
ak{sl}(2,C)
The standard basis matrices can be indexed as
M\mu\nu
\mu,\nu
a,b, … ,f
(M\mu\nu)\rho\sigma=
| \nu{} | |
| \delta | |
| \sigma |
-
| \mu{} | |
| \delta | |
| \sigma |
[M\mu\nu,M\rho\sigma]=M\mu\sigmaη\nu\rho-M\nu\sigmaη\mu\rho+M\nu\rhoη\mu\sigma-M\mu\rhoη\nu\sigma.
i
i
Then
M0i
Mij
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.
The Lorentz group can be thought of as a subgroup of the diffeomorphism group of and therefore its Lie algebra can be identified with vector fields on . In particular, the vectors that 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. We can write down a set of six generators:
-y\partialx+x\partialy\equiviJz~, -z\partialy+y\partialz\equiviJx~, -x\partialz+z\partialx\equiviJy~;
x\partialt+t\partialx\equiviKx~, y\partialt+t\partialy\equiviKy~, z\partialt+t\partialz\equiviKz.
The factor of 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 partial differential operator such as
l{L}=-y\partialx+x\partialy.
The corresponding initial value problem (consider
r=(x,y)
λ
\partialλr=l{L}r
| \partialx | |
| \partialλ |
=-y,
| \partialy | |
| \partialλ |
=x, x(0)=x0, y(0)=y0.
The solution can be written
x(λ)=x0\cos(λ)-y0\sin(λ), y(λ)=x0\sin(λ)+y0\cos(λ)
\begin{bmatrix}t\ x\ y\ z\end{bmatrix}= \begin{bmatrix} 1&0&0&0\\ 0&\cos(λ)&-\sin(λ)&0\ 0&\sin(λ)&\cos(λ)&0\\ 0&0&0&1 \end{bmatrix} \begin{bmatrix}t0\ x0\ y0\ z0\end{bmatrix}
Differentiating with respect to the group parameter and setting it in that result, we recover the standard matrix,
iJz=\begin{bmatrix}0&0&0&0\ 0&0&-1&0\ 0&1&0&0\ 0&0&0&0\end{bmatrix}~,
Reversing the procedure in the previous section, we see that the Möbius transformations that correspond to our six generators arise from exponentiating respectively (for the three boosts) or (for the three rotations) times the three Pauli matrices
\sigma1=\begin{bmatrix}0&1\ 1&0\end{bmatrix}, \sigma2=\begin{bmatrix}0&-i\ i&0\end{bmatrix}, \sigma3=\begin{bmatrix}1&0\ 0&-1\end{bmatrix}.
Another generating set arises via the isomorphism to the Möbius group. The following table lists the six generators, in which
Notice that the generators consist of
\partialz
| Vector field on | One-parameter subgroup of, representing Möbius transformations | One-parameter subgroup of, representing Lorentz transformations | Vector field on | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Parabolic | |||||||||||||||||||
\partialu | \begin{bmatrix}1&\alpha\ 0&1\end{bmatrix} | \begin{bmatrix} 1+
\alpha2&\alpha&0&-
\alpha2\\ \alpha&1&0&-\alpha\\ 0&0&1&0\\
\alpha2&\alpha&0&1-
\alpha2 \end{bmatrix} | \begin{align} X1=x&(\partialt+\partialz)+{}\\ &(t-z)\partialx \end{align} | ||||||||||||||||
\partialv | \begin{bmatrix}1&i\alpha\ 0&1\end{bmatrix} | \begin{bmatrix} 1+
\alpha2&0&\alpha&-
\alpha2\\ 0&1&0&0\\ \alpha&0&1&-\alpha\\
\alpha2&0&\alpha&1-
\alpha2 \end{bmatrix} | \begin{align} X2=y&(\partialt+\partialz)+{}\\ &(t-z)\partialy \end{align} | ||||||||||||||||
| Hyperbolic | |||||||||||||||||||
\left(u\partialu+v\partialv\right) | \begin{bmatrix} \exp\left(
\right)&0\ 0&\exp\left(-
\right) \end{bmatrix} | \begin{bmatrix} \cosh(η)&0&0&\sinh(η)\\ 0&1&0&0\\ 0&0&1&0\\ \sinh(η)&0&0&\cosh(η) \end{bmatrix} | X3=z\partialt+t\partialz | ||||||||||||||||
| Elliptic | |||||||||||||||||||
\left(-v\partialu+u\partialv\right) | \begin{bmatrix} \exp\left(
\right)&0\ 0&\exp\left(
\right) \end{bmatrix} | \begin{bmatrix} 1&0&0&0\\ 0&\cos(\theta)&-\sin(\theta)&0\\ 0&\sin(\theta)&\cos(\theta)&0\\ 0&0&0&1 \end{bmatrix} | X4=-y\partialx+x\partialy | ||||||||||||||||
\partialu-uv\partialv | \begin{bmatrix} \cos\left(
\right)&-\sin\left(
\right)\ \sin\left(
\right)&\cos\left(
\right) \end{bmatrix} | \begin{bmatrix} 1&0&0&0\\ 0&\cos(\theta)&0&\sin(\theta)\\ 0&0&1&0\\ 0&-\sin(\theta)&0&\cos(\theta) \end{bmatrix} | X5=-x\partialz+z\partialx | ||||||||||||||||
uv\partialu+
\partialv | \begin{bmatrix} \cos\left(
\right)&i\sin\left(
\right)\ i\sin\left(
\right)&\cos\left(
\right) \end{bmatrix} | \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&\cos(\theta)&-\sin(\theta)\\ 0&0&\sin(\theta)&\cos(\theta) \end{bmatrix} | X6=-z\partialy+y\partialz | ||||||||||||||||
Start with
\sigma2=\begin{bmatrix}0&i\ -i&0\end{bmatrix}.
Exponentiate:
\exp\left(
| i\theta | |
| 2 |
\sigma2\right)= \begin{bmatrix} \cos\left(
| \theta | |
| 2 |
\right)&-\sin\left(
| \theta | |
| 2 |
\right)\\ \sin\left(
| \theta | |
| 2 |
\right)&\cos\left(
| \theta | |
| 2 |
\right) \end{bmatrix}.
This element of represents the one-parameter subgroup of (elliptic) Möbius transformations:
\xi\mapsto\xi'=
| |||||||||
|
.
Next,
| \left. | d\xi' |
| d\theta |
\right|\theta=0=-
| 1+\xi2 | |
| 2 |
.
The corresponding vector field on (thought of as the image of under stereographic projection) is
| - | 1+\xi2 |
| 2 |
\partial\xi.
Writing
\xi=u+iv
| - | 1+u2-v2 |
| 2 |
\partialu-uv\partialv.
Returning to our element of, writing out the action
X\mapstoPXP\dagger
\begin{bmatrix} 1&0&0&0\\ 0&\cos(\theta)&0&\sin(\theta)\\ 0&0&1&0\\ 0&-\sin(\theta)&0&\cos(\theta) \end{bmatrix}.
Differentiating with respect to at, yields the corresponding vector field on,
z\partialx-x\partialz.
This is evidently the generator of counterclockwise rotation about the -axis.
The subalgebras of the Lie algebra of the Lorentz group can be enumerated, up to conjugacy, from which the closed subgroups 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
Xn
The one-dimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group:
X1
X3
X4
X3+aX4
a ≠ 0
a
X1,X2
X1,X3
X3,X4
X1,X2,X3
X1,X2,X4
X1,X2,X3+aX4
a ≠ 0
X1,X3,X5
X4,X5,X6
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
X1,X2,X3,X4
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 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. A few, brief descriptions:
See main article: Indefinite orthogonal group. 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 of linear transformations of that preserves the quadratic form
(x1,x2,\ldots,xn,xn+1)\mapsto
| 2 | |
| x | |
| 1 |
+
| 2 | |
| x | |
| 2 |
+ … +
| 2 | |
| x | |
| n |
-
| 2. | |
| x | |
| n+1 |
The group preserves the quadratic form
(x1,x2,\ldots,xn,xn+1)\mapsto
| 2 | |
| x | |
| 1 |
-
| 2 | |
| x | |
| 2 |
- … -
| 2 | |
| x | |
| n+1 |
is isomorphic to, and both presentations of the Lorentz group are in use in the theoretical physics community. The former is more common in literature related to gravity, while the latter is more common in particle physics literature.
A common notation for the vector space, equipped with this choice of quadratic form, is .
Many of the properties of the Lorentz group in four dimensions (where) generalize straightforwardly to arbitrary . For instance, the Lorentz group has four connected components, and it acts by conformal transformations on the celestial -sphere in -dimensional Minkowski space. The identity component is an -bundle over hyperbolic -space .
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 that posit the existence of hidden dimensions. The Lorentz group is also the isometry group of -dimensional de Sitter space, which may be realized as the homogeneous space . In particular is the isometry group of the de Sitter universe, a cosmological model.