Minkowski Space and Its Symmetries

Contents

Conserved quantities
The Lorentz group
The Poincaré group

The Lorentz transformations of Lorentz Transformations are the symmetries of Minkowski space, and this chapter assembles them into groups: the Lorentz group of linear transformations preserving the Minkowski metric, its distinguished subsets, and the Poincaré group obtained by adjoining space-time translations, together with the Lie algebras of both. Standard accounts are found in [Maggiore:2008] [Wald:1984]. The source also reserves a chapter for the conserved quantities associated with these symmetries; that heading is retained below as pending.

Conserved quantities

The reason to assemble the Lorentz transformations into a group is that symmetries and conservation laws are the same thing seen twice. Noether's theorem (Calculus of Variations) attaches to every one-parameter group of symmetries of an action a quantity conserved along the solutions, and the ten parameters of the Poincaré group of Section 39.3 deliver the ten conservation laws that organize all of mechanics:

the four translations of spacetime give conservation of the four-momentum $p^{\mu}$ of Definition 40.4 — energy from the time translation, the three momenta from the spatial ones;
the three rotations give conservation of angular momentum;
the three boosts give conservation of $\vect{K}=t\vect{p}-E\vect{x}/c^{2}$, which is the statement that the centre of energy moves uniformly.

The last is unfamiliar only because it is rarely named; it is the relativistic form of the elementary fact that the centre of mass of an isolated system does not accelerate. All ten are components of two tensors, $p^{\mu}$ and the antisymmetric $J^{\mu\nu}$, whose conservation is the content of Equations (39.18), (39.19) and (39.20). Which quantities are conserved is therefore fixed by the symmetry group of spacetime, and that is the subject of this chapter.

The Lorentz group

Foundations

Definition 39.1 (Lorentz group).

The set of all Lorentz transformations (those that leave the Minkowski metric invariant) is given by

\begin{equation}\tag{39.1} L=\set{\Lambda\in\GL(4,\R)\mid\Lambda\eta\Lambda\transpose=\eta}\ep \end{equation}

Rests on Definition 38.6.

Since $\eta$ is a symmetric bilinear form of signature $(1,3)$, the set Equation (39.1) is by definition the orthogonal group of that form, $L=\Ogrp(3,1)$.

Proposition 39.2 ($L$ is a Lie group).

$L$ is a group under matrix multiplication and an embedded submanifold of $\GL(4,\R)$ of dimension six. Rests on Definition 39.1 and Equation (39.2).

Proof.

Derives Proposition 39.2. Group. If $\Lambda_{1},\Lambda_{2}\in L$ then $\left(\Lambda_{1}\Lambda_{2}\right)\transpose\eta \Lambda_{1}\Lambda_{2} =\Lambda_{2}\transpose\left(\Lambda_{1}\transpose\eta\Lambda_{1}\right) \Lambda_{2}=\Lambda_{2}\transpose\eta\Lambda_{2}=\eta$, so $L$ is closed; $\identity\in L$; and taking determinants in Equation (39.2) gives $\left(\det\Lambda\right)^{2}=1$, so every $\Lambda\in L$ is invertible, and inverting the defining relation shows $\Lambda^{-1}\in L$.

Manifold. Consider $F:\GL(4,\R)\longrightarrow\mathrm{Sym}(4)$ defined by $F(\Lambda)=\Lambda\transpose\eta\Lambda-\eta$, whose target is the ten-dimensional space of symmetric matrices. Then $L=F^{-1}(0)$, and the derivative of $F$ at $\Lambda\in L$ sends $X$ to $X\transpose\eta\Lambda+\Lambda\transpose\eta X$, which is surjective onto the symmetric matrices — given symmetric $S$, take $X=\tfrac{1}{2}\eta^{-1}\left(\Lambda\transpose\right)^{-1}S$. Since $0$ is a regular value, the preimage theorem makes $L$ an embedded submanifold of dimension $16-10=6$, matching Proposition 39.9. Multiplication and inversion are rational in the entries, hence smooth, so $L$ is a Lie group (Lie Groups, Lie Algebras, and Fibre Bundles).

∎

Non-compactness of the Lorentz group

The condition satisfied by the elements of the Lorentz group is

\begin{equation}\tag{39.2} \Lambda\eta\Lambda\transpose=\eta \iff \eta_{\lambda\rho}\Lambda^{\lambda}_{\ \mu}\Lambda^{\rho}_{\ \nu} =\eta_{\mu\nu}\ec \end{equation}

whence, taking the $\mu=\nu=0$ component and using $\eta_{00}=1$ and $\eta_{ij}=-\delta_{ij}$,

\begin{align*} \eta_{\lambda\rho}\Lambda^{\lambda}_{\ 0}\Lambda^{\rho}_{\ 0} &=\eta_{00}\ec\\ \eta_{00}\Lambda^{0}_{\ 0}\Lambda^{0}_{\ 0} +\eta_{ij}\Lambda^{i}_{\ 0}\Lambda^{j}_{\ 0}&=1\ec\\ \left(\Lambda^0_{\ 0}\right)^2 -\delta_{ij}\Lambda^{i}_{\ 0}\Lambda^{j}_{\ 0}&=1\ec\\ \left(\Lambda^0_{\ 0}\right)^2 &=1+\Lambda^{i}_{\ 0}\Lambda^{i}_{\ 0}\ec \end{align*}

from which we see that $\left(\Lambda^0_{\ 0}\right)^2\geq1$ and therefore

\begin{equation}\tag{39.3} \abs{\Lambda^0_{\ 0}(\beta,\theta)}\geq1\ep \end{equation}

From Equation (39.2) (the $\mu=\nu=0$ component of the defining relation).

The result obtained in Equation (39.3) tells us that, as a manifold, the Lorentz group is not compact.

Derivation. A compact subset of $\R^{16}$ is bounded. The boosts Equation (38.32) form a curve in $L$ parametrized by the rapidity $\varphi\in\R$, along which $\Lambda^{0}{}_{0}=\cosh\varphi\to\infty$ as $\varphi\to\infty$. The entries of $\Lambda$ are therefore unbounded on $L$, so $L$ is not compact — and neither is $L^{\uparrow}_{+}$, which contains that whole curve.

The contrast with the rotation group is the point. $\SO(3)$ is compact: its entries are sines and cosines, bounded by $1$. Replacing the angle by a rapidity replaces the bounded circular functions by unbounded hyperbolic ones, which is the same statement as the sign difference in the last bracket of Equation (39.15). The consequence matters in Particles as Poincaré Representations: a non-compact group has no finite-dimensional unitary representations except the trivial one, which is why the unitary representations describing particles are infinite dimensional, and why the finite-dimensional representations that classify fields — scalars, spinors, vectors — are not unitary.

∎

The proper Lorentz group

Definition 39.3 (Proper Lorentz group).

The proper Lorentz group is

\begin{equation}\tag{39.4} L_+=\set{\Lambda\in L\mid\det\left[\Lambda\right]=1}\ep \end{equation}

Rests on Definition 39.1.

Evidently we have $L_+=\SO(3,1)$.

The improper Lorentz group

Taking the determinant of Equation (39.2) gives $\left(\det\Lambda\right)^{2}\det\eta=\det\eta$, so $\det\Lambda=\pm1$ and the transformations with $\det\Lambda=-1$ form the complement of $L_{+}$ in $L$.

Definition 39.4 (Improper Lorentz transformations).

$L_{-}:=\set{\Lambda\in L\mid\det\left[\Lambda\right]=-1}$. Rests on Definition 39.1 and Equation (39.2).

$L_{-}$ is not a subgroup — it does not contain the identity, and the product of two of its elements has determinant $+1$ — but it is a coset of $L_{+}$, and every element of it is a proper transformation composed with one fixed reflection. The canonical choice is parity, the spatial reflection

\begin{equation}\tag{39.5} \left[P\right]=\diag\left(1,-1,-1,-1\right)\ec \qquad\det P=-1\ec\qquad P\in L^{\uparrow}_{-}\ec \end{equation}

which reverses the three spatial axes and leaves time alone. That $P$ satisfies Equation (39.2) is immediate, since $\left[P\right]=\left[\eta\right]$ and $\eta\eta\eta=\eta$.

Parity is a symmetry of gravitation, electromagnetism and the strong interaction, and is not a symmetry of the weak interaction: that discovery (Experiment: Parity Violation) is the reason $L^{\uparrow}_{+}$ rather than $L$ is the group of physical relevance in particle physics.

The orthochronous Lorentz group

By Equation (39.3) every Lorentz transformation has $\abs{\Lambda^{0}{}_{0}} \ge 1$, so the sign of $\Lambda^{0}{}_{0}$ splits $L$ in two.

Definition 39.5 (Orthochronous Lorentz group).

\begin{equation}\tag{39.6} L^{\uparrow} = \set{\Lambda \in L \mid \Lambda^{0}{}_{0} \ge 1}\ec \qquad L^{\uparrow}_{+} = L^{\uparrow} \cap L_{+}\ep \end{equation}

$L^{\uparrow}_{+}$ is the proper orthochronous (or restricted) Lorentz group. Rests on Definition 39.1 and Equation (39.3).

Proposition 39.6 ($L^{\uparrow}$ and $L^{\uparrow}_{+}$ are subgroups).

$L^{\uparrow}$ is closed under products and inverses, hence so is $L^{\uparrow}_{+}$. Moreover the two defining conditions, $\det\Lambda = 1$ and $\Lambda^{0}{}_{0} \ge 1$, are open and closed on $L$, so $L^{\uparrow}_{+}$ contains every connected subset of $L$ through the identity. Rests on Definition 39.5, Equation (39.2) and Equation (39.3).

Proof.

Derives Proposition 39.6. The $\mu = \nu = 0$ component of Equation (39.2) is the column identity $\left(\Lambda^{0}{}_{0}\right)^{2} = 1 + \sum_i \Lambda^{i}{}_{0}\Lambda^{i}{}_{0}$ (the computation leading to Equation (39.3)); applied to $\Lambda^{-1} = \eta\Lambda\transpose\eta \in L$ it becomes the row identity $\left(\Lambda^{0}{}_{0}\right)^{2} = 1 + \sum_i \Lambda^{0}{}_{i}\Lambda^{0}{}_{i}$. For $\Lambda, \Lambda' \in L^{\uparrow}$,

\begin{equation*} (\Lambda\Lambda')^{0}{}_{0} = \Lambda^{0}{}_{0}\,\Lambda'^{0}{}_{0} + \sum_i \Lambda^{0}{}_{i}\,\Lambda'^{i}{}_{0}\ec \end{equation*}

and the Cauchy–Schwarz inequality together with the two identities bounds the cross term strictly:

\begin{equation*} \abs{\textstyle\sum_i \Lambda^{0}{}_{i}\,\Lambda'^{i}{}_{0}} \le \sqrt{\left(\Lambda^{0}{}_{0}\right)^{2}-1}\; \sqrt{\left(\Lambda'^{0}{}_{0}\right)^{2}-1} < \Lambda^{0}{}_{0}\,\Lambda'^{0}{}_{0}\ec \end{equation*}

so $(\Lambda\Lambda')^{0}{}_{0} > 0$, hence $\ge 1$ by Equation (39.3). For the inverse, $(\Lambda^{-1})^{0}{}_{0} = (\eta\Lambda\transpose\eta)^{0}{}_{0} = \Lambda^{0}{}_{0} \ge 1$. Clopenness: $\det$ and $\Lambda \mapsto \Lambda^{0}{}_{0}$ are continuous, $\det = \pm1$ on $L$ and $\abs{\Lambda^{0}{}_{0}} \ge 1$, so each condition singles out a union of connected components; a connected set containing the identity (which satisfies both) therefore stays in $L^{\uparrow}_{+}$.

∎

That $L^{\uparrow}_{+}$ is itself connected — and is therefore precisely the identity component of $L$ — is proved in Particles as Poincaré Representations, as the continuous image of the connected covering group $\SL(2,\C)$.

The antichronous Lorentz group

Definition 39.7 (Antichronous transformations).

\begin{equation}\tag{39.7} L^{\downarrow}:=\set{\Lambda\in L\mid\Lambda^{0}{}_{0}\leq-1}\ec \end{equation}

the complement of $L^{\uparrow}$ in $L$ by Equation (39.3). Rests on Definition 39.5 and Equation (39.3).

An antichronous transformation reverses the direction of time: it maps the future light cone of Figure 38.1 onto the past one. Like $L_{-}$ it is a coset rather than a subgroup, and its canonical representative is time reversal

\begin{equation}\tag{39.8} \left[T\right]=\diag\left(-1,1,1,1\right)\ec \qquad\det T=-1\ec\qquad T\in L^{\downarrow}_{-}\ep \end{equation}

The four components

The two independent signs — $\det\Lambda=\pm1$ from Definition 39.4 and $\sgn\Lambda^{0}{}_{0}=\pm1$ from Equation (39.3) — are continuous functions taking discrete values, so neither can change along a path inside $L$. They therefore partition the group into four pieces that no continuous motion connects.

Proposition 39.8 (Component structure).

$L$ has exactly four connected components,

\begin{equation}\tag{39.9} L^{\uparrow}_{+}\ec\qquad L^{\uparrow}_{-}=P\,L^{\uparrow}_{+}\ec\qquad L^{\downarrow}_{-}=T\,L^{\uparrow}_{+}\ec\qquad L^{\downarrow}_{+}=PT\,L^{\uparrow}_{+}\ec \end{equation}

and the quotient of $L$ by the identity component is the discrete group

\begin{equation}\tag{39.10} \frac{L}{L^{\uparrow}_{+}}\cong\set{\identity,P,T,PT} \cong\Z_{2}\times\Z_{2}\ec \end{equation}

the Klein four-group. Rests on Definition 39.5, Definition 39.4, Equation (39.5), Equation (39.8) and Proposition 94.10.

Proof.

Derives Proposition 39.8. That the four sets are disjoint and exhaust $L$ is the case analysis on the two signs. That each is a coset of $L^{\uparrow}_{+}$ follows because $P$, $T$ and $PT=-\identity$ realize every combination of the two signs, and multiplying by a fixed element is a homeomorphism of $L$ onto itself, so each coset is connected if $L^{\uparrow}_{+}$ is. Connectedness of $L^{\uparrow}_{+}$ is proved in Particles as Poincaré Representations as the continuous image of $\SL(2,\C)$, which is connected. Finally $P^{2}=T^{2}=\identity$ and $PT=TP$, which is the multiplication table of $\Z_{2}\times\Z_{2}$.

∎

Only $L^{\uparrow}_{+}$ can be reached from the identity by a continuous sequence of physical motions, which is why it alone is generated by the Lie algebra of Section 39.2.7. The other three components are reached only by adjoining a discrete operation, and whether Nature respects those operations is an experimental question with a surprising answer: $P$ is violated by the weak interaction (Experiment: Parity Violation), the combination $CP$ is violated in neutral-meson decays (Experiment: CP Violation), and only $CPT$ — with $C$ the charge conjugation of Discrete Symmetries and CPT — is protected by a theorem.

The algebra of the proper Lorentz group

In the context of Minkowski space we must find the generators of the rotations $\Lambda(0,\theta)$ and of the boosts $\Lambda(\beta,0)$, and then the algebra they satisfy.

Proposition 39.9 (The generators).

Write an infinitesimal Lorentz transformation as $\Lambda^{\mu}{}_{\nu}=\delta^{\mu}{}_{\nu} +\omega^{\mu}{}_{\nu}$ with $\abs{\omega}\ll1$. Then Equation (39.2) holds to first order if and only if $\omega_{\mu\nu}:=\eta_{\mu\lambda}\omega^{\lambda}{}_{\nu}$ is antisymmetric,

\begin{equation}\tag{39.11} \omega_{\mu\nu}=-\omega_{\nu\mu}\ec \end{equation}

so the algebra has $\binom{4}{2}=6$ parameters: three rotations and three boosts. Rests on Definition 39.1 and Equation (39.2).

Proof.

Derives Proposition 39.9. Substituting into $\Lambda\transpose\eta\Lambda=\eta$ and keeping first-order terms,

\[ \eta_{\mu\nu} =\left(\delta^{\lambda}{}_{\mu}+\omega^{\lambda}{}_{\mu}\right) \eta_{\lambda\rho} \left(\delta^{\rho}{}_{\nu}+\omega^{\rho}{}_{\nu}\right) =\eta_{\mu\nu}+\omega_{\nu\mu}+\omega_{\mu\nu} +O\!\left(\omega^{2}\right)\ec \]

so $\omega_{\mu\nu}+\omega_{\nu\mu}=0$. An antisymmetric $4\times4$ matrix has six independent entries; the three with both indices spatial generate rotations, and the three with one index equal to $0$ generate boosts, as Equation (38.32) shows for a single boost.

∎

The six parameters are collected into an antisymmetric $\omega^{\mu\nu}$ and the six generators into an antisymmetric $J_{\mu\nu}=-J_{\nu\mu}$, so that a finite transformation is

\begin{equation}\tag{39.12} \Lambda=\exp\left(\tfrac{1}{2}\omega^{\mu\nu}J_{\mu\nu}\right)\ep \end{equation}

Realizing the generators as the differential operators that implement these transformations on functions of $x$,

\begin{equation}\tag{39.13} J_{\mu\nu}=x_{\mu}\pp_{\nu}-x_{\nu}\pp_{\mu}\ec \end{equation}

one obtains by direct computation the algebra $\mathfrak{so}(3,1)$:

\begin{equation}\tag{39.14} \comm{J_{\mu\nu}}{J_{\lambda\rho}} =\eta_{\nu\lambda}J_{\mu\rho}+\eta_{\mu\rho}J_{\nu\lambda} -\eta_{\mu\lambda}J_{\nu\rho}-\eta_{\nu\rho}J_{\mu\lambda}\ec \end{equation}

in agreement with the general $\mathfrak{so}(p,q)$ relations derived in Section 14.3.2 (Equation (14.90)).

Derivation. Derives Equation (39.14). Insert Equation (39.13) into the commutator and expand, using $\pp_{\nu}x_{\lambda}=\eta_{\nu\lambda}$ and the fact that the second-derivative terms cancel in pairs:

\begin{align*} \comm{J_{\mu\nu}}{J_{\lambda\rho}} &=\comm{x_{\mu}\pp_{\nu}-x_{\nu}\pp_{\mu}} {x_{\lambda}\pp_{\rho}-x_{\rho}\pp_{\lambda}}\\ &=x_{\mu}\comm{\pp_{\nu}}{x_{\lambda}}\pp_{\rho} -x_{\mu}\comm{\pp_{\nu}}{x_{\rho}}\pp_{\lambda} -x_{\nu}\comm{\pp_{\mu}}{x_{\lambda}}\pp_{\rho} +x_{\nu}\comm{\pp_{\mu}}{x_{\rho}}\pp_{\lambda}\\ &\quad +x_{\lambda}\comm{x_{\mu}\pp_{\nu}-x_{\nu}\pp_{\mu}}{\pp_{\rho}} -x_{\rho}\comm{x_{\mu}\pp_{\nu}-x_{\nu}\pp_{\mu}}{\pp_{\lambda}}\ep \end{align*}

Each commutator of a derivative with a coordinate is a metric component, $\comm{\pp_{\nu}}{x_{\lambda}}=\eta_{\nu\lambda}$, and collecting the eight resulting terms in pairs gives

\[ \comm{J_{\mu\nu}}{J_{\lambda\rho}} =\eta_{\nu\lambda}\left(x_{\mu}\pp_{\rho}-x_{\rho}\pp_{\mu}\right) +\eta_{\mu\rho}\left(x_{\nu}\pp_{\lambda}-x_{\lambda}\pp_{\nu}\right) -\eta_{\mu\lambda}\left(x_{\nu}\pp_{\rho}-x_{\rho}\pp_{\nu}\right) -\eta_{\nu\rho}\left(x_{\mu}\pp_{\lambda}-x_{\lambda}\pp_{\mu}\right)\ec \]

each bracket being again a generator Equation (39.13). This is Equation (39.14). The computation nowhere used the signature, only the symmetry of $\eta$, which is why the same relations hold for $\mathfrak{so}(p,q)$ generally.

∎

Splitting the six generators into rotations $J_{i}=\tfrac{1}{2} \epsilon_{ijk}J^{jk}$ and boosts $K_{i}=J^{0i}$ makes the structure familiar: the $J_{i}$ close among themselves — rotations form a subgroup — while

\begin{equation}\tag{39.15} \comm{J_{i}}{J_{j}}=\epsilon_{ijk}J_{k}\ec\qquad \comm{J_{i}}{K_{j}}=\epsilon_{ijk}K_{k}\ec\qquad \comm{K_{i}}{K_{j}}=-\epsilon_{ijk}J_{k}\ep \end{equation}

From Equation (39.14) (the rotation–boost split just made). The last bracket is the algebraic statement of Remark 38.24: two boosts do not commute, and their commutator is a rotation — the Thomas rotation. The minus sign distinguishes $\mathfrak{so}(3,1)$ from $\mathfrak{so}(4)$ and is the source of the non-compactness of Equation (39.3).

The Poincaré group

Foundations

The Lorentz group holds one event fixed: every $\Lambda$ maps the origin to itself. But no event in Minkowski space is distinguished, so the full symmetry group must include the translations that move the origin.

Definition 39.10 (Poincaré group).

The Poincaré group, or inhomogeneous Lorentz group $\ISO(3,1)$, is the set of transformations

\begin{equation}\tag{39.16} x^{\mu}\longmapsto\bar{x}^{\mu} =\Lambda^{\mu}{}_{\nu}x^{\nu}+a^{\mu}\ec \qquad\Lambda\in L\ec\quad a^{\mu}\in\R^{4}\ec \end{equation}

with the composition law

\begin{equation}\tag{39.17} \left(\Lambda_{2},a_{2}\right)\left(\Lambda_{1},a_{1}\right) =\left(\Lambda_{2}\Lambda_{1},\ \Lambda_{2}a_{1}+a_{2}\right)\ep \end{equation}

Rests on Definition 39.1.

Equation (39.17) is read off by applying the two transformations in succession, and it shows that the group is not a direct product: the translation part of the composite depends on the Lorentz part of the second factor. It is a semidirect product $\R^{4}\rtimes L$, with the translations forming a normal subgroup and the Lorentz group acting on them.

Proposition 39.11 (Invariance of the interval).

The Poincaré transformations are exactly the maps of Minkowski space preserving the interval Equation (38.13) between every pair of events. Rests on Definition 39.10, Equation (38.13) and Equation (39.2).

Proof.

Derives Proposition 39.11. The separation of two events transforms as $\Delta\bar{x}^{\mu}=\Lambda^{\mu}{}_{\nu}\Delta x^{\nu}$ — the constant $a^{\mu}$ cancels in the difference — so Equation (39.2) gives invariance of $\eta_{\mu\nu}\Delta x^{\mu}\Delta x^{\nu}$. Conversely, a map preserving all intervals preserves in particular those from a fixed event, and the polarization identity then forces it to be affine with linear part satisfying Equation (39.2).

∎

The group has ten parameters: six of $\Lambda$ from Proposition 39.9 and four of $a^{\mu}$. These are the ten symmetries whose conserved charges were listed in Section 39.1, and the classification of the irreducible unitary representations of this group is the classification of the possible kinds of elementary particle — Wigner's theorem, carried out in Particles as Poincaré Representations.

The algebra of the Poincaré group

The commutation relations of the Poincaré algebra are

\begin{align} \comm{J_{\mu\nu}}{J_{\lambda\rho}} &=\eta_{\nu\lambda}J_{\mu\rho}+\eta_{\mu\rho}J_{\nu\lambda} -\eta_{\mu\lambda}J_{\nu\rho}-\eta_{\nu\rho}J_{\mu\lambda}\ec \tag{39.18}\\ \comm{J_{\mu\nu}}{P_{\lambda}} &=\eta_{\nu\lambda}P_{\mu}-\eta_{\mu\lambda}P_{\nu}\ec \tag{39.19}\\ \comm{P_{\mu}}{P_{\nu}}&=0\ep \tag{39.20} \end{align}

Derivation. Derives Equation (39.19). The Lorentz generators are Equation (39.13) and the translation generators are

\begin{equation}\tag{39.21} P_{\mu}=\pp_{\mu}\ec \end{equation}

since $\exp\left(a^{\mu}\pp_{\mu}\right)$ shifts the argument of a function by $a^{\mu}$. Equation (39.18) was derived in Section 39.2.7. For Equation (39.19),

\begin{align*} \comm{J_{\mu\nu}}{P_{\lambda}} &=\comm{x_{\mu}\pp_{\nu}-x_{\nu}\pp_{\mu}}{\pp_{\lambda}}\\ &=-\left(\pp_{\lambda}x_{\mu}\right)\pp_{\nu} +\left(\pp_{\lambda}x_{\nu}\right)\pp_{\mu} =-\eta_{\lambda\mu}\pp_{\nu}+\eta_{\lambda\nu}\pp_{\mu} =\eta_{\nu\lambda}P_{\mu}-\eta_{\mu\lambda}P_{\nu}\ec \end{align*}

the derivative acting only on the explicit coordinate because the second-derivative terms cancel. Equation (39.20) is the commutativity of partial derivatives, $\comm{\pp_{\mu}}{\pp_{\nu}}=0$: translations in different directions commute, so the translation subgroup is abelian.

The three relations say exactly what Definition 39.10 already said in words: Equation (39.20) that the translations are abelian, Equation (39.19) that they form a normal subgroup transforming as a four-vector under Lorentz transformations, and Equation (39.18) that the Lorentz generators close among themselves. The semidirect structure is visible in the fact that Equation (39.19) does not vanish, while it would for a direct product.

∎

Remark 39.12 (The Casimir invariants).

Two combinations commute with every generator of the algebra:

\begin{equation}\tag{39.22} P^{2}=\eta^{\mu\nu}P_{\mu}P_{\nu}\ec\qquad W^{2}=\eta^{\mu\nu}W_{\mu}W_{\nu}\ec\qquad W^{\mu}:=-\tfrac{1}{2}\epsilon^{\mu\nu\lambda\rho} J_{\nu\lambda}P_{\rho}\ec \end{equation}

the second built from the Pauli–Lubański vector. On an irreducible representation each takes a fixed value, $P^{2}=m^{2}c^{2}$ and $W^{2}=-m^{2}c^{2}s\left(s+1\right)\hbar^{2}$ for $m>0$. Those two numbers are the mass and the spin: a particle is an irreducible representation of the Poincaré group, labelled by the two invariants of its algebra. This is the point at which the kinematics of this part becomes the particle classification of Particles as Poincaré Representations. Rests on Equations (39.18), (39.19) and (39.20).