Lorentz group

Light cone in 2D space plus a time dimension.

Because it is a Lie group, the Lorentz group O(1,3) is both a group and admits 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:

• Some elements are reversed under time-inverting Lorentz transformations, for example, a future-pointing timelike vector would be inverted to a past-pointing vector
• Some elements have orientation reversed by improper Lorentz transformations, for example, certain vierbein (tetrads)

Lorentz transformations that preserve the direction of time are called orthochronous. The subgroup of orthochronous transformations is often denoted O⁺(1,3). Those that preserve orientation are called proper, and as linear transformations they have determinant +1. (The improper Lorentz transformations have determinant −1.) The subgroup of proper Lorentz transformations is denoted SO(1,3).

The subgroup of all Lorentz transformations preserving both orientation and direction of time is called the proper, orthochronous Lorentz group or restricted Lorentz group, and is denoted by SO⁺(1, 3). (Note that some authors refer to SO(1,3) or even O(1,3) when they actually mean SO⁺(1, 3).)

The set of the four connected components can be given a group structure as the quotient group O(1,3)/SO⁺(1,3), which is isomorphic to the Klein four-group. 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 parity and time reversal operators:

P = diag(1, −1, −1, −1)

T = diag(−1, 1, 1, 1).

Thus an arbitrary Lorentz transformation can be specified as a proper, orthochronous Lorentz transformation along with a further two bits of information, which pick out one of the four connected components. This pattern is typical of finite-dimensional Lie groups.

Hyperboloid of one sheet

Common conical surface

Hyperboloid of two sheets

If a group G acts on a space V, then a surface S ⊂ V is a surface of transitivity if S is invariant under G, i.e., ∀g ∈ G, ∀s ∈ S: gs ∈ S , and for any two points s₁, s₂ ∈ S there is a g ∈ G such that gs₁ = s₂. By definition of the Lorentz group, it preserves the quadratic form

${\displaystyle Q(x)=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}.}$

The surfaces of transitivity of the orthochronous Lorentz group O⁺(1, 3), Q(x) = const. of spacetime are the following:[3]

• Q(x) > 0, x₀ > 0 is the upper branch of a hyperboloid of two sheets. Points on this sheet are separated from the origin by a future time-like vector.
• Q(x) > 0, x₀ < 0 is the lower branch of this hyperboloid. Points on this sheet are the past time-like vectors.
• Q(x) = 0, x₀ > 0 is the upper branch of the light cone, the future light cone.
• Q(x) = 0, x₀ < 0 is the lower branch of the light cone, the past light cone.
• Q(x) < 0 is a hyperboloid of one sheet. Points on this sheet are space-like separated from the origin.
• The origin x₀ = x₁ = x₂ = x₃ = 0.

These surfaces are 3-dimensional, so the images are not faithful, but they are faithful for the corresponding facts about O⁺(1, 2). For the full Lorentz group, the surfaces of transitivity are only four since the transformation T takes an upper branch of a hyperboloid (cone) to a lower one and vice versa.

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.[4] 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 (m, 0, 0, 0). For each m ≠ 0, the vector pierces exactly one sheet. In this case the little group is SO(3), 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.

The Weyl representation or spinor map is a pair of surjective homomorphisms from SL(2,C) to SO⁺(1,3). They form a matched pair under parity transformations, corresponding to left and right chiral spinors.

One may define an action of SL(2,C) on Minkowski spacetime by writing a point of spacetime as a two-by-two Hermitian matrix in the form

${\displaystyle {\overline {X}}={\begin{bmatrix}ct+z&x-iy\\x+iy&ct-z\end{bmatrix}}=ct1\!\!1+x\sigma _{x}+y\sigma _{y}+z\sigma _{z}=ct1\!\!1+{\vec {x}}\cdot {\vec {\sigma }}}$

in terms of Pauli matrices. This presentation, the Weyl presentation, satisfies

${\displaystyle \det \,{\overline {X}}=(ct)^{2}-x^{2}-y^{2}-z^{2}.}$

Therefore, one has identified the space of Hermitian matrices (which is four-dimensional, as a real vector space) with Minkowski spacetime, in such a way that the determinant of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime. An element ${\displaystyle S\in SL(2,\mathbb {C} )}$ acts on the space of Hermitian matrices via

${\displaystyle {\overline {X}}\mapsto S{\overline {X}}S^{\dagger }~,}$

where ${\displaystyle S^{\dagger }}$ is the Hermitian transpose of ${\displaystyle S}$. This action preserves the determinant and so SL(2,C) acts on Minkowski spacetime by (linear) isometries. The parity-inverted form of the above is

${\displaystyle X=ct1\!\!1-{\vec {x}}\cdot {\vec {\sigma }}}$

which transforms as

${\displaystyle X\mapsto (S^{-1})^{\dagger }XS^{-1}}$

That this is the correct transformation follows by noting that

${\displaystyle {\overline {X}}X=(c^{2}t^{2}-{\vec {x}}\cdot {\vec {x}})1\!\!1=(c^{2}t^{2}-x^{2}-y^{2}-z^{2})1\!\!1}$

remains invariant under the above pair of transformations.

These maps are surjective, and kernel of either map is the two element subgroup ±I. By the first isomorphism theorem, the quotient group PSL(2,C) = SL(2,C) / {±I} is isomorphic to SO⁺(1,3).

The parity map swaps these two coverings. It corresponds to Hermitian conjugation being an automorphism of ${\displaystyle SL(2,\mathbb {C} ).}$ 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 ${\displaystyle \psi _{R}\mapsto S\psi _{R}~,}$ while the overline form corresponds to left-handed spinors transforming as ${\displaystyle \psi _{L}\mapsto (S^{\dagger })^{-1}\psi _{L}~.}$[lower-alpha 1]

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 ${\displaystyle \Lambda \in SO^{+}(1,3)}$ as ${\displaystyle {\Lambda ^{\mu }}_{\nu }~,}$ thus showing the matrix with spacetime indexes ${\displaystyle \mu ,\nu =0,1,2,3.}$ A four-vector can be created from the Pauli matrices in two different ways: as ${\displaystyle \sigma ^{\mu }=(I,{\vec {\sigma }})}$ and as ${\displaystyle {\overline {\sigma }}^{\mu }=(I,-{\vec {\sigma }})~.}$ The two forms are related by a parity transformation. Note that ${\displaystyle {\overline {\sigma }}_{\mu }=\sigma ^{\mu }~.}$

Given a Lorentz transformation ${\displaystyle x^{\mu }\mapsto x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu }~,}$ the double-covering of the orthochronous Lorentz group by ${\displaystyle S\in SL(2,\mathbb {C} )}$ given above can be written as

${\displaystyle x^{\prime \mu }{\overline {\sigma }}_{\mu }={\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }x^{\nu }=Sx^{\nu }{\overline {\sigma }}_{\nu }S^{\dagger }}$

Dropping the ${\displaystyle x^{\mu }}$ this takes the form

${\displaystyle {\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }=S{\overline {\sigma }}_{\nu }S^{\dagger }}$

The parity conjugate form is

${\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=(S^{-1})^{\dagger }\sigma _{\nu }S^{-1}}$

The symplectic group Sp(2,C) is isomorphic to SL(2,C). This isomorphism is constructed so as to preserve a symplectic bilinear form on ${\displaystyle \mathbb {C} ^{2},}$ that is, to leave the form invariant under Lorentz transformations. This may be articulated as follows. The symplectic group is defined as

${\displaystyle Sp(2,\mathbb {C} )=\{S\in GL(2,\mathbb {C} ):S^{T}\omega S=\omega \}}$

where

${\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}$

Other common notations are ${\displaystyle \omega =\epsilon }$ for this element; sometimes ${\displaystyle J}$ is used, but this invites confusion with the idea of almost complex structures, which are not the same, as they transform differently.

Given a pair of Weyl spinors (two-component spinors)

${\displaystyle u={\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}~,\quad v={\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}}$

the invariant bilinear form is conventionally written as

${\displaystyle \langle u,v\rangle =-\langle v,u\rangle =u_{1}v_{2}-u_{2}v_{1}=u^{T}\omega v}$

This form is invariant under the Lorentz group, so that for ${\displaystyle S\in SL(2,\mathbb {C} )}$ one has

${\displaystyle \langle Su,Sv\rangle =\langle u,v\rangle }$

This defines a kind of "scalar product" of spinors, and is commonly used to defined a Lorentz-invariant mass term in Lagrangians. There are several notable properties to be called out that are important to physics. One is that ${\displaystyle \omega ^{2}=-1}$ and so ${\displaystyle \omega ^{-1}=\omega ^{T}=\omega ^{\dagger }=-\omega }$

The defining relation can be written as

${\displaystyle \omega S^{T}\omega ^{-1}=S^{-1}}$

which closely resembles the defining relation for the Lorentz group

${\displaystyle \eta \Lambda ^{T}\eta ^{-1}=\Lambda ^{-1}}$

where ${\displaystyle \eta =\mathrm {diag} (+1,-1,-1,-1)}$ is the metric tensor for Minkowski space and of course, ${\displaystyle \Lambda \in SO(1,3)}$ as before.

An elliptic element of SL(2,C) is

${\displaystyle P_{1}={\begin{bmatrix}\exp \left({\frac {i}{2}}\theta \right)&0\\0&\exp \left(-{\frac {i}{2}}\theta \right)\end{bmatrix}}}$

and has fixed points ξ = 0, ∞. Writing the action as X ↦ P₁ X P₁^† and collecting terms, the spinor map converts this to the (restricted) Lorentz transformation

${\displaystyle Q_{1}={\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&-\sin(\theta )&0\\0&\sin(\theta )&\cos(\theta )&0\\0&0&0&1\end{bmatrix}}=\exp \left(\theta {\begin{bmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{bmatrix}}\right)~.}$

This transformation then represents a rotation about the z axis, exp(iθJ_z). The one-parameter subgroup it generates is obtained by taking θ 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 counterclockwise rotation about the z axis as θ increases. The angle doubling evident in the spinor map is a characteristic feature of spinorial double coverings.

A hyperbolic element of SL(2,C) is

${\displaystyle P_{2}={\begin{bmatrix}\exp \left({\frac {\eta }{2}}\right)&0\\0&\exp \left(-{\frac {\eta }{2}}\right)\end{bmatrix}}}$

and has fixed points ξ = 0, ∞. Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this Möbius transformation is a dilation from the origin.

The spinor map converts this to the Lorentz transformation

${\displaystyle Q_{2}={\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&1&0&0\\0&0&1&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}=\exp \left(\eta {\begin{bmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{bmatrix}}\right)~.}$

This transformation represents a boost along the z axis with rapidity η. The one-parameter subgroup it generates is obtained by taking η 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 SL(2,C) is

${\displaystyle P_{3}=P_{2}P_{1}=P_{1}P_{2}={\begin{bmatrix}\exp \left({\frac {1}{2}}(\eta +i\theta )\right)&0\\0&\exp \left(-{\frac {1}{2}}(\eta +i\theta )\right)\end{bmatrix}}}$

and has fixed points ξ = 0, ∞. The spinor map converts this to the Lorentz transformation

${\displaystyle Q_{3}=Q_{2}Q_{1}=Q_{1}Q_{2}.}$

The one-parameter subgroup this generates is obtained by replacing η+iθ with any real multiple of this complex constant. (If η, θ vary independently, then a two-dimensional abelian subgroup is obtained, consisting of simultaneous rotations about the z axis and boosts along the z-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 SL(2,C) is

${\displaystyle P_{4}={\begin{bmatrix}1&\alpha \\0&1\end{bmatrix}}}$

and has the single fixed point ξ = ∞ 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)

${\displaystyle {\begin{aligned}Q_{4}&={\begin{bmatrix}1+{\frac {1}{2}}\vert \alpha \vert ^{2}&\operatorname {Re} (\alpha )&\operatorname {Im} (\alpha )&-{\frac {1}{2}}\vert \alpha \vert ^{2}\\\operatorname {Re} (\alpha )&1&0&-\operatorname {Re} (\alpha )\\-\operatorname {Im} (\alpha )&0&1&\operatorname {Im} (\alpha )\\{\frac {1}{2}}\vert \alpha \vert ^{2}&\operatorname {Re} (\alpha )&\operatorname {Im} (\alpha )&1-{\frac {1}{2}}\vert \alpha \vert ^{2}\end{bmatrix}}\\[6pt]&=\exp {\begin{bmatrix}0&\operatorname {Re} (\alpha )&\operatorname {Im} (\alpha )&0\\\operatorname {Re} (\alpha )&0&0&-\operatorname {Re} (\alpha )\\-\operatorname {Im} (\alpha )&0&0&\operatorname {Im} (\alpha )\\0&\operatorname {Re} (\alpha )&\operatorname {Im} (\alpha )&0\end{bmatrix}}~.\end{aligned}}}$

This generates a two-parameter abelian subgroup, which is obtained by considering α 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

${\displaystyle {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}\rightarrow {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}+\operatorname {Re} (\alpha )\;{\begin{bmatrix}x\\t-z\\0\\x\end{bmatrix}}+\operatorname {Im} (\alpha )\;{\begin{bmatrix}y\\0\\z-t\\y\end{bmatrix}}+{\frac {\vert \alpha \vert ^{2}}{2}}\;{\begin{bmatrix}t-z\\0\\0\\t-z\end{bmatrix}}.}$

Now, without loss of generality, pick Im(α) = 0. Differentiating this transformation with respect to the now real group parameter α and evaluating at α=0 produces the corresponding vector field (first order linear partial differential operator),

${\displaystyle x\,\left(\partial _{t}+\partial _{z}\right)+(t-z)\,\partial _{x}.}$

Apply this to a function f(t, x, y, z), 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

${\displaystyle f(t,x,y,z)=F\left(y,\,t-z,\,t^{2}-x^{2}-z^{2}\right),}$

where F is an arbitrary smooth function. The arguments of F give three rational invariants describing how points (events) move under this parabolic transformation, as they themselves do not move,

${\displaystyle y=c_{1},~~~~t-z=c_{2},~~~~t^{2}-x^{2}-z^{2}=c_{3}.}$

Choosing real values for the constants on the right hand sides yields three conditions, and thus specifies a curve in Minkowski spacetime. This curve is an orbit of the transformation.

The form of the rational invariants shows that these flowlines (orbits) have a simple description: suppressing the inessential coordinate y, each orbit is the intersection of a null plane, t = z + c₂, with a hyperboloid, t² − x² − z² = c₃. The case c₃ = 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 α 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 Re(α) = 0 instead, produces similar orbits, now with the roles of x and y interchanged.

Parabolic transformations lead to the gauge symmetry of massless particles (like photons) with helicity |h| ≥ 1. In the above explicit example, a massless particle moving in the z direction, so with 4-momentum P=(p, 0, 0, p), is not affected at all by the x-boost and y-rotation combination K_x − J_y 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 α have disappeared. c₁ = c₂ = c₃ = 0, in the special case discussed. (The other similar generator, K_y+J_x as well as it and J_z comprise altogether the little group of the lightlike vector, isomorphic to E(2).)

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 ξ = u + iv, 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

${\displaystyle {\begin{bmatrix}u^{2}+v^{2}+1\\2u\\-2v\\u^{2}+v^{2}-1\end{bmatrix}}}$

or, in the Weyl representation (the spinor map), the Hermitian matrix

${\displaystyle N=2{\begin{bmatrix}u^{2}+v^{2}&u+iv\\u-iv&1\end{bmatrix}}.}$

The set of real scalar multiples of this null vector, called a null line through the origin, represents a line of sight from an observer at a particular place and time (an arbitrary event we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars. Then the points of the celestial sphere (equivalently, lines of sight) are identified with certain Hermitian matrices.

Another generating set arises via the isomorphism to the Möbius group. The following table lists the six generators, in which

• The first column gives a generator of the flow under the Möbius action (after stereographic projection from the Riemann sphere) as a real vector field on the Euclidean plane.
• The second column gives the corresponding one-parameter subgroup of Möbius transformations.
• The third column gives the corresponding one-parameter subgroup of Lorentz transformations (the image under our homomorphism of preceding one-parameter subgroup).
• The fourth column gives the corresponding generator of the flow under the Lorentz action as a real vector field on Minkowski spacetime.

Notice that the generators consist of

• Two parabolics (null rotations)
• One hyperbolic (boost in the ∂_z direction)
• Three elliptics (rotations about the 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
${\displaystyle \partial _{u}\,\!}$	${\displaystyle {\begin{bmatrix}1&\alpha \\0&1\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}1+{\frac {1}{2}}\alpha ^{2}&\alpha &0&-{\frac {1}{2}}\alpha ^{2}\\\alpha &1&0&-\alpha \\0&0&1&0\\{\frac {1}{2}}\alpha ^{2}&\alpha &0&1-{\frac {1}{2}}\alpha ^{2}\end{bmatrix}}}$	${\displaystyle {\begin{aligned}X_{1}={}&x(\partial _{t}+\partial _{z})+\\&(t-z)\partial _{x}\end{aligned}}}$
${\displaystyle \partial _{v}\,\!}$	${\displaystyle {\begin{bmatrix}1&i\alpha \\0&1\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}1+{\frac {1}{2}}\alpha ^{2}&0&\alpha &-{\frac {1}{2}}\alpha ^{2}\\0&1&0&0\\\alpha &0&1&-\alpha \\{\frac {1}{2}}\alpha ^{2}&0&\alpha &1-{\frac {1}{2}}\alpha ^{2}\end{bmatrix}}}$	${\displaystyle {\begin{aligned}X_{2}={}&y(\partial _{t}+\partial _{z})+\\&(t-z)\partial _{y}\end{aligned}}}$
Hyperbolic
${\displaystyle {\frac {1}{2}}\left(u\partial _{u}+v\partial _{v}\right)}$	${\displaystyle {\begin{bmatrix}\exp \left({\frac {\eta }{2}}\right)&0\\0&\exp \left(-{\frac {\eta }{2}}\right)\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&1&0&0\\0&0&1&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}}$	${\displaystyle X_{3}=z\partial _{t}+t\partial _{z}\,\!}$
Elliptic
${\displaystyle {\frac {1}{2}}\left(-v\partial _{u}+u\partial _{v}\right)}$	${\displaystyle {\begin{bmatrix}\exp \left({\frac {i\theta }{2}}\right)&0\\0&\exp \left({\frac {-i\theta }{2}}\right)\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&-\sin(\theta )&0\\0&\sin(\theta )&\cos(\theta )&0\\0&0&0&1\end{bmatrix}}}$	${\displaystyle X_{4}=-y\partial _{x}+x\partial _{y}}$
${\displaystyle {\frac {v^{2}-u^{2}-1}{2}}\partial _{u}-uv\,\partial _{v}}$	${\displaystyle {\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&-\sin \left({\frac {\theta }{2}}\right)\\\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&0&\sin(\theta )\\0&0&1&0\\0&-\sin(\theta )&0&\cos(\theta )\end{bmatrix}}}$	${\displaystyle X_{5}=-x\partial _{z}+z\partial _{x}}$
${\displaystyle uv\,\partial _{u}+{\frac {1-u^{2}+v^{2}}{2}}\partial _{v}}$	${\displaystyle {\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&i\sin \left({\frac {\theta }{2}}\right)\\i\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos(\theta )&-\sin(\theta )\\0&0&\sin(\theta )&\cos(\theta )\end{bmatrix}}}$	${\displaystyle X_{6}=-z\partial _{y}+y\partial _{z}}$

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

${\displaystyle \sigma _{2}={\begin{bmatrix}0&i\\-i&0\end{bmatrix}}.}$

Exponentiate:

${\displaystyle \exp \left({\frac {i\theta }{2}}\,\sigma _{2}\right)={\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&-\sin \left({\frac {\theta }{2}}\right)\\\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}.}$

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

${\displaystyle \xi \mapsto \xi '={\frac {\cos \left({\frac {\theta }{2}}\right)\,\xi -\sin \left({\frac {\theta }{2}}\right)}{\sin \left({\frac {\theta }{2}}\right)\,\xi +\cos \left({\frac {\theta }{2}}\right)}}.}$

Next,

${\displaystyle \left.{\frac {d\xi '}{d\theta }}\right|_{\theta =0}=-{\frac {1+\xi ^{2}}{2}}.}$

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

${\displaystyle -{\frac {1+\xi ^{2}}{2}}\,\partial _{\xi }.}$

Writing ${\displaystyle \xi =u+iv}$, this becomes the vector field on R²

${\displaystyle -{\frac {1+u^{2}-v^{2}}{2}}\,\partial _{u}-uv\,\partial _{v}.}$

Returning to our element of SL(2,C), writing out the action ${\displaystyle X\mapsto PXP^{\dagger }}$ and collecting terms, we find that the image under the spinor map is the element of SO⁺(1,3)

${\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&0&\sin(\theta )\\0&0&1&0\\0&-\sin(\theta )&0&\cos(\theta )\end{bmatrix}}.}$

Differentiating with respect to θ at θ=0, yields the corresponding vector field on R⁴,

${\displaystyle z\partial _{x}-x\partial _{z}.\,\!}$

This is evidently the generator of counterclockwise rotation about the y axis.