Lorentz group ($O(1,3)$)

The Lorentz group consists of real matrices preserving the Minkowski metric: ${\Lambda }^T\eta \Lambda =\eta$.

For $J=\mathrm{diag}(1,-1)$, the matrices satisfying $A^{T}JA=J$ and $A_{11}>0$ form the time-orientation-preserving Lorentz group in one spatial dimension. They are exactly $A=H(t)D_{\epsilon }$, where $H(t)=\left(\begin{array}{cc}\cosh t& \sinh t\\ \sinh t& \cosh t\end{array}\right)$, $D_{\epsilon }=\mathrm{diag}(1,\epsilon )$ and $\epsilon =\pm 1$. Preservation of the Minkowski metric makes the first column a future-pointing unit timelike vector; Minkowski inner product orthogonality determines the second up to sign. The identity $D_{\epsilon }H(s)=H(\epsilon s)D_{\epsilon }$ gives the multiplication law $(t,\epsilon )(s,\eta )=(t+\epsilon s,\epsilon \eta )$ and inverse $(-\epsilon t,\epsilon )$.

For $M=\mathrm{diag}(-1,1,1,1)$,Every such matrix has the block formso the Lie algebra and the Lorentz group have dimension six. At $A\in O(1,3)$, the tangent space is $A\mathfrak{o}(1,3)$.

The orthochronous Lorentz group consists of Lorentz transformations preserving the future sheet of the unit timelike hyperboloid. Equivalently, $A_{00}>0$. This is an open subgroup of $O(1,3)$ and hence a full-dimensional submanifold.

The proper orthochronous Lorentz group is the identity component of the Lorentz group. Its elements have determinant one and preserve the future light cone.

Use the Lorentz spinor double cover $SL(2,\mathbb{C})\to S{O}^{+}(1,3)$. For a lift with distinct eigenvalues $\lambda ,{\lambda }^{-1}$, diagonalize and choose logarithms $z,-z$; this gives a traceless matrix logarithm. For a repeated eigenvalue, choose the sign of the lift so it is $I+N$, where $N^2=0$, and use $\log (I+N)=N$. The two lifts have the same Lorentz image. Exponentiating the differential of the covering map therefore expresses every proper orthochronous transformation as the exponential of a real element of the Lie algebra of the Lorentz group. This argument does not assert that every element of $SL(2,\mathbb{C})$ itself has a traceless logarithm.

Identify Minkowski vectors with Hermitian matrices $X=x^{0}I+x^a{\sigma }_a$. The action $X\mapsto SX{S}^{†}$ preserves $\det X=(x^0{)}^2-|\mathbf{x}{|}^2$. It gives a surjective homomorphism from $SL(2,\mathbb{C})$ to the proper orthochronous Lorentz group with kernel $\{I,-I\}$. Polar decomposition retracts the covering group onto SU(2), making it simply connected. This cover permits half-integer spin group representations that do not descend to the Lorentz group itself.

A real four-vector corresponds to a two-by-two Hermitian matrix with determinant equal to its Minkowski squared norm. The action $X\mapsto NX{N}^{†}$ for $N\in SL(2,\mathbb{C})$ preserves that determinant and the future cone. Expanding with $\frac{1}{2}\mathrm{Tr}({\bar{\sigma }}^{\mu }{\sigma }_{\nu })={\delta }^{\mu }{}_{\nu }$ yields ${\Lambda }^{\mu }{}_{\nu }(N)=\frac{1}{2}\mathrm{Tr}({\bar{\sigma }}^{\mu }N{\sigma }_{\nu }{N}^{†})$. The map has kernel $\{\pm I\}$ and image $S{O}^{+}(1,3)$.

The real matrix space defined by $C+C^{\ast }=\mathrm{tr}(C)I$ consists exactly of $C=aI+i\mathbf{v}\cdot \mathbf{\sigma }$, with $a$ and the three components of $\mathbf{v}$ real. The Pauli matrices have $\mathrm{tr}({\sigma }_i{\sigma }_j)=2{\delta }_{ij}$, giving the displayed real symmetric bilinear form of signature $(3,1)$. The real isomorphism $C\mapsto aI-\mathbf{v}\cdot \mathbf{\sigma }$ to Hermitian matrices preserves this form when the latter have the polarized negative-determinant form $[\mathrm{tr}(AB)-\mathrm{tr}A\mathrm{tr}B]/2$.

The Poincare group is the semidirect product of spacetime translations by the Lorentz group. It is the isometry group of Minkowski spacetime.

The Poincare algebra is generated by translations $P^{\mu }$ and Lorentz transformations $M^{\mu \nu }$. Its mixed commutator is $[M^{\mu \nu },P^{\rho }]=i({\eta }^{\nu \rho }{P}^{\mu }-{\eta }^{\mu \rho }{P}^{\nu })$.

The Pauli-Lubanski pseudovector is $W_{\mu }={ϵ}_{\mu \nu \rho \sigma }{M}^{\nu \rho }{P}^{\sigma }/2$, up to an overall sign convention. Its square is a Casimir of the Poincare algebra and measures spin in a massive irreducible representation.

In $W^{\mu }=-{ϵ}^{\mu \nu \rho \sigma }{P}_{\nu }{M}_{\rho \sigma }/2$, contracting with $P_{\mu }$ gives zero because the commuting momentum products are symmetric in $\mu ,\nu$ while the alternating tensor is antisymmetric. Reordering $P$ and $M$ makes no difference because the resulting metric contractions with the alternating tensor vanish.

// Target: special-relativity.bigb

For a massive positive-energy unitary irreducible representation of the Poincaré group with signature $(+---)$, choose a rest frame. The Pauli-Lubanski pseudovector has zero time component and spatial components $m{J}_i$, up to a common sign convention. Its square is therefore $-m^2{\mathbf{J}}^2$. The SU(2) little-group representation has spin $s$ and ${\mathbf{J}}^2=s(s+1)$, proving the invariant value. The mass and spin label the massive induced representation of the Poincare double cover.

With signature $(+---)$ and ${ϵ}_{0123}=+1$, a momentum $(k,0,0,k)$ and helicity $j_3$ give lower components $W_0=k{j}_3$ and $W_3=-k{j}_3$. They obey $W_{\mu }{P}^{\mu }=k(W_0+W_3)=0$. These longitudinal identities require only the momentum and rotation-projection eigenvalues, not a statement about transverse little-group translations.

With signature $(+---)$, ${ϵ}_{0123}=+1$, $J_3=M^{12}$ and the lower-index definition of $W_{\mu }$, a rest momentum $(m,0,0,0)$ gives $W_0=0$ and $W_3=-m{J}_3$ on that momentum eigenspace. A Hermitian spin projection $j_3$ consequently gives eigenvalues $0$ and $-m{j}_3$. Raising the third index changes its sign.

The little group of a momentum $p$ is its stabilizer in the Lorentz group. Its unitary irreducible representations label the internal degrees of freedom of one-particle states on the orbit of $p$.

These stabilizer representations are used in Wigner's classification of relativistic particle states.

A particle polarization specifies a physical spin state at fixed four-momentum. Massive spin states transform under the rotation little group $SO(D-1)$; finite-spin massless states transform under its transverse rotation part $SO(D-2)$. A massive two-form field has $\left(\genfrac{}{}{0ex}{}{D-1}{2}\right)$ such states, while a two-form gauge field has $\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)$. Polarization components must be counted after field equations and any gauge equivalence are imposed.

The stabilizer of a nonzero null four-momentum has connected group $ISO(D-2)$. For ordinary finite-polarization particles its translation part acts trivially, leaving a representation of $SO(D-2)$.

For a future null standard momentum $k$, choose Lorentz transformations $L(p)k=p$ and a unitary irreducible representation $d$ of its little group. The Wigner rotation $R(\Lambda ,p)=L(\Lambda p{)}^{-1}\Lambda L(p)$ acts on the internal state. The resulting induced representation acts on wavefunctions with invariant measure $d^{3}p/(2p^0)$. Trivial little-group translations give fixed helicity; nontrivial translations give a continuous-spin representation with infinitely many polarization states.

In a finite-dimensional unitary representation of $ISO(2)$, its two commuting Hermitian translation generators have a finite joint spectrum. Conjugating by every planar rotation rotates that spectrum. A finite set invariant under all rotations contains only the origin, so both generators vanish. An irreducible representation is then a one-dimensional rotation character, giving a helicity state. Nontrivial translations instead require the infinite-dimensional continuous-spin representation. The same argument applies to the physical double cover of this little group.

For a standard null momentum $(E,0,0,E)$ in signature $(+---)$, define $K_i=M^{0i}$ and $J_i={ϵ}_{ijk}{M}^{jk}/2$. The stabilizing generators are $J_3,N_1,N_2$. The Poincare algebra gives $[J_3,N_1]=i{N}_2$, $[J_3,N_2]=-i{N}_1$ and $[N_1,N_2]=0$, the Lie algebra of planar Euclidean motions. Evaluating the Pauli-Lubanski pseudovector gives $W^2=-E^2(N_1^2+N_2^2)$.

An ordinary massless spin-one particle has transverse vector polarizations under $SO(D-2)$. A covariant polarization satisfies $p\cdot ϵ=0$ and is identified under $ϵ\mapsto ϵ+\lambda p$. The quotient has $D-2$ positive polarizations.

The stabilizer of a timelike four-momentum in $D$-dimensional Minkowski spacetime has connected group $SO(D-1)$. Massive particle polarizations form representations of this group.

A massive spin-one particle transforms as a vector under its massive particle little group $SO(D-1)$. Its $D-1$ polarizations are orthogonal to its timelike four-momentum. Under transverse $SO(D-2)$ rotations they split into a vector and a scalar.

The symmetry used to classify an open string must preserve its endpoint boundary background as well as its momentum. A Neumann-Dirichlet open-string boundary condition breaks target Lorentz transformations mixing ND and common directions with Neumann boundary conditions. For a D1–D25 pair with 24 ND directions, the common worldvolume has Lorentz group $SO(1,1)$, whose connected massive little group is trivial. The transverse $SO(24)$ remains as an internal rotation group. Thus a massive 24-component oscillator multiplet is not an $SO(25)$ vector of the full 26-dimensional massive little group; a vector of that full group would have 25 components.

A massive vector in $D$ spacetime dimensions has $D-1$ polarizations perpendicular to its timelike momentum. A massless vector has transverse polarizations modulo its null longitudinal polarization, leaving $D-2$ physical degrees of freedom.

Helicity is the projection of angular momentum onto the momentum direction. For a finite-helicity massless particle, the Pauli-Lubanski vector satisfies $W^{\mu }=\lambda P^{\mu }$.

For $\mathbf{k}=k\hat{z}$, these complex vectors form an orthonormal basis of the transverse plane using the Hermitian inner product. They have opposite phases under rotations about $\hat{z}$, so they describe the two helicity components. Since $\mathbf{e}\cdot m^{(\pm )}=\sin \theta e^{\pm i\phi }/\sqrt{2}$, transverse-vector sources carry only the angular sectors $m=\pm 1$ of spherical harmonics. Streaming by $ik\cos \theta$ preserves $m$ and couples neighboring angular degrees.

A continuous-spin representation is a massless unitary irreducible representation of the Poincare group in which the translation subgroup of the $ISO(2)$ little group acts nontrivially. It contains infinitely many helicities.

Wigner's classification obtains irreducible positive-energy representations of the Poincaré group by inducing from unitary irreducible representations of the little group of a standard momentum.

Fix $m>0$, $k=(m,0)$, and a rotationless Lorentz boost $L(p)$ taking $k$ to $p$ on the positive-energy mass shell. The little group on the Lorentz spinor double cover is SU(2). Its spin-$s$ irreducible representation induces a unitary representation on $L^2(d^{3}p/(2p^0))\otimes {\mathbb{C}}^{2s+1}$. Here $W(\Lambda ,p)=L(\Lambda p{)}^{-1}\Lambda L(p)$ is the Wigner rotation, lifted to SU(2) when necessary, and the translation convention is $U(I,a)|p⟩=e^{ip\cdot a}|p⟩$. The little-group cocycle proves the group law, and Wigner's classification gives irreducibility. Every nonnegative integer or half-integer spin is allowed on the double cover; descent to the ordinary proper orthochronous Poincaré group requires integer spin, because its $2\pi$ rotation must act as the identity.

The determinant-one Lorentz group in one space and one time dimension has two connected components. Its identity component consists of boostsand the other component is its negative.

A boost of rapidity $\phi$ acts on null coordinates by $u\mapsto e^{\phi }u$ and $v\mapsto e^{-\phi }v$, preserving $uv=x^2-y^2$.

In the $(t,z)$ plane, $u=t-z$ is retarded time and $v=t+z$ is advanced time. The radial part of the Minkowski metric is $-d{t}^2+d{z}^2=-dudv$.

For $u=t-z$ and $v=t+z$, the four-dimensional Minkowski wave operator isEvery profile depending only on one null coordinate solves the wave equation when it is uniform in the transverse coordinates.