Lorentz transformation ($\Lambda$)

A Lorentz transformation preserves the Minkowski metric. A boost of speed $u$ along the $x$-axis uses $\beta =u/c$ and

Spatially separated simultaneous events in one inertial frame are generally not simultaneous in a relatively moving frame. The Lorentz transformation explicitly mixes position into time. This is essential when comparing distances between moving light pulses: one must compare their positions at the same time in the measuring frame.

For two photons emitted simultaneously a distance $d$ apart in one frame and moving in the same positive direction, the world-line intercept $x-ct$ transforms to $\gamma (1+v/c)(x-ct)$. Their equal-time separation in the boosted frame is therefore the displayed constant $d^{\prime }$. It is different from length contraction of the emitters, because the emission events are not simultaneous after the boost.

A physical quantity is Lorentz invariant when its value is unchanged under Lorentz transformations. Tensor equations can be Lorentz covariant while their components transform. Fixed-vector gauge choices need not make covariance manifest, although physical gauge-invariant predictions retain the symmetry.

A Lorentz boost relates inertial frames moving at a constant relative velocity without an additional spatial rotation. In units with light speed one, the displayed transformation preserves $t^2-x^2$, with Lorentz factor $\gamma =(1-v^2{)}^{-1/2}$. Applying it to a static classical field-theory soliton produces a uniformly moving solution in a Lorentz-invariant theory.

Two noncollinear Lorentz boosts produce a Lorentz transformation with both a boost and a rotation. In the SL(2,C) lift, the product of two positive Hermitian boost matrices generally ceases to be Hermitian. Its polar decomposition of an invertible complex matrix separates a positive Hermitian boost factor and a unitary rotation factor. The rotation depends on both boosts and their order, and is called the Wigner rotation.

For boost matrices $A=cI+s\mathbf{n}\cdot \mathbf{\sigma }$ and $A^{\prime }=c^{\prime }I+s^{\prime }{\mathbf{n}}^{\prime }\cdot \mathbf{\sigma }$, the Pauli matrix multiplication law gives the displayed anti-Hermitian part. If both rapidities are nonzero and the axes are noncollinear, it cannot vanish, while either lift of a pure boost is Hermitian. Thus an additional Wigner rotation is necessary. A zero boost is a degenerate exception.

For one spatial dimension and $c=1$, a massive particle has $v=\tanh \theta$, $E=M\cosh \theta$ and $p=M\sinh \theta$. A Lorentz boost adds a constant to all rapidities, so a rapidity difference is invariant. Rapidity is useful for factorized scattering, where two-body S-matrices depend only on that difference.

The relativistic velocity-addition formula becomes addition of rapidities for collinear Lorentz boosts. Starting from rest, applying the same positive rest-frame velocity increment $c\tanh \alpha$ repeatedly gives $u_n=c\tanh (n\alpha )$. For increment $c/2$, $\alpha =\frac{1}{2}\log 3$ and $u_n=c(3^n-1)/(3^n+1)$. Every finite sequence remains below the speed of light.

An active transformation keeps the coordinate argument fixed and changes a contravariant vector field by $A^{\prime \mu }(x)={\Lambda }^{\mu }{}_{\nu }{A}^{\nu }({\Lambda }^{-1}x)$. Infinitesimally, $\delta A^{\mu }=\alpha {\omega }^{\mu }{}_{\nu }{A}^{\nu }-\alpha {\omega }^{\rho }{}_{\sigma }{x}^{\sigma }{\partial }_{\rho }{A}^{\mu }$. For a Lorentz scalar density, the spin-index term is absent, giving $\delta \mathcal{L}=-\alpha {\omega }^{\rho }{}_{\sigma }{x}^{\sigma }{\partial }_{\rho }\mathcal{L}$.

A Lorentz scalar is unchanged under a Lorentz transformation. Contracting every Lorentz index in a tensor expression produces a scalar when the constituent fields transform covariantly.

A Lorentz tensor transforms by one Lorentz matrix for each contravariant index and one inverse matrix for each covariant index. Complete contraction of all indices produces a Lorentz scalar.

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

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.

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 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.

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 $p$.

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 $SO(D-2)$.

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 Poincare group by inducing from unitary irreducible representations of the little group of a standard momentum.

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.

The nonrelativistic limit takes characteristic speeds much smaller than $c$. Lorentz transformations then reduce to Galilean transformations at leading order.

For a boost of speed $V$ along $x$,