Galilei group and Bargmann central extension

Intuition [Beginner]

The Galilean group encodes the symmetries of Newtonian spacetime: shifts in space and time, rotations, and boosts between uniformly-moving frames. But something is missing. Two particles with the same position and velocity but different masses respond differently to the same forces, and mass does not appear in the Galilean group itself.

The fix is to enlarge the group by one extra dimension. This extra piece sits in the centre — it commutes with everything — and the parameter it carries is the mass. The enlarged group is the Bargmann group, and the construction is the Bargmann central extension.

Why does this matter? In quantum mechanics, wave functions do not transform under the Galilean group the way you might expect. Under a boost, they pick up an extra phase that depends on the mass. This phase is invisible in classical mechanics but is the difference between correct and incorrect quantum predictions. The central extension is the bookkeeping device that captures it.

Visual [Beginner]

A diagram showing the Galilean group as a ten-parameter family of transformations of spacetime, with an extra vertical axis representing the central extension parameter (mass). The projection from the Bargmann group down to the Galilean group collapses this vertical axis to a point. An inset shows a quantum wave function picking up a mass-dependent phase under a boost.

The picture to keep in mind: the Bargmann group is one dimension taller than the Galilean group, the extra direction is the mass parameter, and the extension is classified by a single cohomology class.

Worked example [Beginner]

Take a free quantum particle of mass $m=1$ moving in one dimension. Its wave function in frame $A$ is a plane wave with momentum $p_A=0$ (the particle is at rest). Now switch to frame $B$ moving at velocity $v=2$ relative to $A$. In classical mechanics, the particle is now moving at velocity $-2$ and that is the end of the story.

In quantum mechanics, the boost also shifts the phase of the wave function. The phase change depends on the mass, the velocity, the position, and the time. With $m=1$ and $v=2$, the extra phase at position $x$ and time $t$ is proportional to $2x+2t$ (in suitable units). A particle of mass $m=3$ boosted by the same velocity would pick up three times the phase: $6x+6t$.

What this tells us: the mass enters the transformation law of quantum states under boosts, so the symmetry group of non-relativistic quantum mechanics must carry mass as a parameter. The Galilean group alone does not have a mass parameter; the Bargmann central extension supplies one.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $\mathfrak{g}=\mathfrak{gal}(4)$ be the 10-dimensional Lie algebra of the Galilean group with generators $H$ (time translation), $P_i$ (spatial translations, $i=1,2,3$), $J_i$ (rotations), $K_i$ (boosts). The non-vanishing brackets are

$$[J_i,J_j]={\epsilon }_{ijk}{J}_k,[J_i,P_j]={\epsilon }_{ijk}{P}_k,[J_i,K_j]={\epsilon }_{ijk}{K}_k,[H,K_i]=-P_i,$$

with all other brackets among generators vanishing. In particular, $[K_i,P_j]=0$ in the centreless algebra.

A Lie-algebra 2-cocycle on $\mathfrak{g}$ with values in $\mathbb{R}$ is a bilinear, antisymmetric map $\omega :\mathfrak{g}\times \mathfrak{g}\to \mathbb{R}$ satisfying the cocycle identity

$$\omega ([X,Y],Z)+\omega ([Y,Z],X)+\omega ([Z,X],Y)=0$$

for all $X,Y,Z\in \mathfrak{g}$. A 2-cocycle is a coboundary if there exists a linear map $\varphi :\mathfrak{g}\to \mathbb{R}$ such that $\omega (X,Y)=\varphi ([X,Y])$ for all $X,Y$. The second Lie-algebra cohomology $H^2(\mathfrak{g};\mathbb{R})$ is the quotient of cocycles modulo coboundaries.

The Bargmann cocycle. Define ${\omega }_B:\mathfrak{g}\times \mathfrak{g}\to \mathbb{R}$ on generators by

$${\omega }_B(K_i,P_j)={\delta }_{ij},{\omega }_B=0\text{ on all other pairs},$$

extended bilinearly and antisymmetrically. The associated central extension is the Lie algebra $\tilde{\mathfrak{g}}=\mathfrak{g}\oplus \mathbb{R}M$ with the bracket of $\mathfrak{g}$ modified by $[K_i,P_j{]}_{\tilde{\mathfrak{g}}}={\delta }_{ij}M$ and $M$ central (commuting with everything).

Counterexamples to common slips

• The cocycle is not identically zero on all pairs involving $K$ and $P$. The cocycle ${\omega }_B$ pairs $K_i$ with $P_j$ specifically; it vanishes on $(K_i,H)$, $(K_i,J_j)$, and $(K_i,K_j)$. The only non-vanishing pairs are the mixed boost-translation pairs.
• A central extension is not a direct product. The Bargmann algebra $\tilde{\mathfrak{g}}$ is not $\mathfrak{g}\times \mathbb{R}$ as a Lie algebra; the bracket is modified by the cocycle. The extension is split if and only if the cocycle is a coboundary.
• Not every Lie algebra has a substantive central extension. Semi-simple Lie algebras satisfy $H^2(\mathfrak{g};\mathbb{R})=0$ (Whitehead's lemma); the Galilean algebra is not semi-simple, which is what allows a non-zero cohomology class.

Key theorem with proof [Intermediate+]

Theorem (Bargmann 1954). The second Lie-algebra cohomology of the Galilean algebra is one-dimensional: $H^2(\mathfrak{gal}(4);\mathbb{R})\cong \mathbb{R}$, generated by the class $[{\omega }_B]$ of the Bargmann cocycle. The central extension $\widetilde{\mathfrak{gal}}(4)=\mathfrak{gal}(4)\oplus \mathbb{R}M$ with $[K_i,P_j]={\delta }_{ij}M$ is a genuine (non-split) extension.

Proof. The proof has three parts: verify ${\omega }_B$ is a cocycle, show it is not a coboundary, and show the cohomology is one-dimensional.

Step 1: ${\omega }_B$ is a cocycle. Bilinearity and antisymmetry are immediate. For the cocycle identity, consider all triples of generators. The only triples producing potentially non-vanishing terms are those containing at least one $K$ and at least one $P$. Take $(K_i,P_j,J_k)$: the brackets in $\mathfrak{g}$ are $[K_i,P_j]=0$, $[P_j,J_k]=-{\epsilon }_{jk\ell }{P}_{\ell }$, $[J_k,K_i]=-{\epsilon }_{ki\ell }{K}_{\ell }$. The cyclic sum is

$${\omega }_B(0,J_k)+{\omega }_B(-{\epsilon }_{jk\ell }{P}_{\ell },K_i)+{\omega }_B(-{\epsilon }_{ki\ell }{K}_{\ell },P_j)$$ $$=0+{\epsilon }_{jk\ell }{\omega }_B(P_{\ell },K_i)-{\epsilon }_{ki\ell }{\omega }_B(K_{\ell },P_j)=-{\epsilon }_{jk\ell }{\delta }_{i\ell }+{\epsilon }_{ki\ell }{\delta }_{j\ell }=-{\epsilon }_{jki}+{\epsilon }_{kij}=0$$

by the cyclic symmetry ${\epsilon }_{jki}={\epsilon }_{kij}$. Triples involving $H$ with one $K$ and one $P$: $[H,K_i]=-P_i$ and ${\omega }_B(P_i,P_j)=0$, so the sum reduces to ${\omega }_B(-P_i,P_j)+0+0=0$. Triples with two $K$s and one $P$: $[K_i,K_j]=0$, so all terms vanish. Triples with two $P$s and one $K$: $[P_i,P_j]=0$, so all terms vanish. All other triples give zero by inspection. So ${\omega }_B$ is a cocycle.

Step 2: ${\omega }_B$ is not a coboundary. Suppose ${\omega }_B=d\varphi$ for some linear $\varphi :\mathfrak{g}\to \mathbb{R}$, meaning ${\omega }_B(X,Y)=\varphi ([X,Y])$. Then ${\omega }_B(K_i,P_j)=\varphi ([K_i,P_j])=\varphi (0)=0$. But ${\omega }_B(K_i,P_j)={\delta }_{ij}\ne 0$, a contradiction.

Step 3: dimension of $H^2$. A general 2-cocycle on $\mathfrak{g}$ is determined by its values on ordered pairs of generators. The cocycle identity constrains most of these to vanish. Any cocycle must vanish on $(J_i,J_j)$, $(J_i,P_j)$, $(J_i,K_j)$, $(H,P_i)$, $(H,J_i)$, $(P_i,P_j)$, $(K_i,K_j)$ by the same cyclic-identity argument. The only remaining freedom is $\omega (K_i,P_j)=c{\delta }_{ij}$ for some constant $c$, plus possible $\omega (H,K_i)=d_i$ and $\omega (H,P_i)=e_i$. The cocycle identity on $(H,K_i,P_j)$ forces $e_j{\delta }_{ij}=0$, so $e_i=0$. The identity on triples involving spatial rotations forces $d_i=0$ by $\mathrm{SO}(3)$-invariance. So $H^2\cong \mathbb{R}$, generated by ${\omega }_B$. $\square$

Bridge. The cohomology computation here builds toward the quantum-mechanical interpretation of the Bargmann extension: the central charge $M$ acts as multiplication by mass in any unitary representation, and this is exactly the content of the mass superselection rule. The bridge between the algebraic cocycle and the physics is the theorem that projective unitary representations of $\mathrm{Gal}(4)$ are in bijection with genuine unitary representations of $\widetilde{\mathrm{Gal}}(4)$ at fixed $M$, which appears again in the representation theory of the Poincare group where $H^2=0$ and the distinction vanishes. The foundational reason the Galilean group needs an extension but the Poincare group does not is that the Galilean algebra is the Inonu-Wigner contraction $c\to \infty$ of the Poincare algebra, and the contraction kills the bracket $[{\tilde{K}}_i,{\tilde{P}}_j]=-{\delta }_{ij}{\tilde{P}}_0/c^2$ whose leading piece survives as the central charge. Putting these together identifies the Bargmann mass with the non-relativistic shadow of the rest energy $E_0=M{c}^2$.

Exercises [Intermediate+]

Advanced results [Master]

The Bargmann group as a central extension. The Bargmann group $\widetilde{\mathrm{Gal}}(4)$ is the simply-connected Lie group with Lie algebra $\widetilde{\mathfrak{gal}}(4)$. It fits into an exact sequence

$$1⟶\mathbb{R}\stackrel{\iota }{⟶}\widetilde{\mathrm{Gal}}(4)\stackrel{\pi }{⟶}\mathrm{Gal}(4)⟶1$$

where $\iota$ embeds $\mathbb{R}$ as the central subgroup $\{(0,0,0,0,s):s\in \mathbb{R}\}$ and $\pi$ projects out the mass coordinate. The extension is classified by the cohomology class $[{\omega }_B]\in H_{\mathrm{Lie}}^2(\mathrm{Gal}(4);\mathbb{R})$. In coordinates adapted to a fixed inertial frame, an element of $\widetilde{\mathrm{Gal}}(4)$ acts on $(t,x)\in \mathbb{R}\times {\mathbb{R}}^3$ by the usual Galilean action $(t,x)\mapsto (t+\tau ,Rx+vt+a)$, and the central coordinate $s\in \mathbb{R}$ (the mass parameter) is carried along passively.

Projective representations and the Bargmann correspondence. A projective unitary representation of a group $G$ on a Hilbert space $\mathcal{H}$ is a map $\rho :G\to U(\mathcal{H})/U(1)$ such that $\rho (g_1)\rho (g_2)=c(g_1,g_2)\rho (g_1{g}_2)$ for a phase $c(g_1,g_2)\in U(1)$. The function $c:G\times G\to U(1)$ is the Schur multiplier. Bargmann's theorem (1954) states: for the Galilean group, the Schur multiplier is determined by the cohomology class $[{\omega }_B]$, and there is a bijection

$$\left\{\text{projective unitary reps of }\mathrm{Gal}(4)\right\}⟷\left\{\text{unitary reps of }\widetilde{\mathrm{Gal}}(4)\text{ at fixed }M\right\}.$$

In an irreducible representation with $M=m\ne 0$, the generators $P_i,J_i,K_i,H$ act as operators on $L^2({\mathbb{R}}^3)$ satisfying the commutation relations $[K_i,P_j]=i\hslash m{\delta }_{ij}$. The boost generator acts as $K_i=m{Q}_i-t{P}_i$ where $Q_i$ is the position operator and $P_i$ the momentum operator; the $m{Q}_i$ term is the quantum signature of the central extension.

Mass superselection rule. In non-relativistic quantum mechanics, the mass $M$ is fixed within any irreducible representation of $\widetilde{\mathrm{Gal}}(4)$. A superposition $\alpha |m_1⟩+\beta |m_2⟩$ of states with different masses transforms under a boost by the phase $e^{i{m}_j(vx+v^{2}t/2)/\hslash }$, which depends on $m_j$. The relative phase between $|m_1⟩$ and $|m_2⟩$ changes with $v$, so no Galilean-invariant probability amplitude can be formed from such a superposition. This is the Bargmann mass superselection rule: observables in non-relativistic QM commute with the central charge $M$, and physical states live in a single mass sector.

The Galilean particle and Souriau's classification. Souriau (1970) classified elementary classical particles by coadjoint orbits of the Galilean (equivalently, Bargmann) group. A coadjoint orbit of $\widetilde{\mathrm{Gal}}(4)$ with central charge $m>0$ and energy $E$ carries a canonical symplectic structure (the Kirillov-Kostant-Souriau form). The orbit through $(m,p,E,\ell )$ with $m>0$ has the symplectic manifold structure of the phase space of a single Newtonian particle with mass $m$, momentum $p$, energy $E$, and angular momentum $\ell$. The symplectic volume of the orbit reproduces the quantum partition function of the non-relativistic particle.

Inonu-Wigner contraction and the relativistic origin of the central charge. The Poincare algebra $\mathfrak{poin}(1,3)$ has brackets $[{\tilde{K}}_i,{\tilde{P}}_j]=-{\delta }_{ij}{\tilde{P}}_0/c^2$. Under the rescaling $K_i=c{\tilde{K}}_i$, $H=c{\tilde{P}}_0$, the bracket becomes $[K_i,P_j]=-{\delta }_{ij}H/c^2$. In the limit $c\to \infty$ this vanishes, recovering the centreless Galilean bracket $[K_i,P_j]=0$. The Bargmann central charge $M$ is the leading $1/c^2$ correction that survives projectively: setting $M=E_0/c^2$ where $E_0=m{c}^2$ is the rest energy, the bracket $[K_i,P_j]={\delta }_{ij}M$ is exactly the $c\to \infty$ remnant of the relativistic $[{\tilde{K}}_i,{\tilde{P}}_j]=-{\delta }_{ij}{E}_0/c^2$. Read in the other direction, the Bargmann mass is the non-relativistic shadow of the relativistic rest energy.

The Heisenberg algebra embedded in the Bargmann algebra. The subalgebra $\mathfrak{h}=\mathrm{span}\{K_1,P_1,M\}$ (one boost direction, one translation, the central element) is a 3-dimensional Heisenberg algebra with $[K_1,P_1]=M$. The Stone-von Neumann theorem classifies its irreducible unitary representations: each is determined by the value $\hslash m$ of the central element. The full Bargmann algebra contains three commuting copies of this Heisenberg algebra (one per spatial dimension), intertwined by the rotation generators.

Synthesis. The Bargmann central extension is the cohomological signature of the fact that the Galilean group is not semi-simple. The foundational reason the extension exists is that the boost-translation bracket $[K_i,P_j]$ vanishes in $\mathfrak{gal}(4)$ but is obstructed from being a coboundary — the obstruction is the single class $[{\omega }_B]\in H^2(\mathfrak{gal}(4);\mathbb{R})\cong \mathbb{R}$. The bridge between this algebraic fact and physics is the Bargmann correspondence: projective representations of $\mathrm{Gal}(4)$ are genuine representations of $\widetilde{\mathrm{Gal}}(4)$, and the central charge $M$ acts as the mass.

This is exactly the structure that distinguishes non-relativistic from relativistic quantum mechanics: the Poincare group has $H^2=0$, so relativistic QM has no mass superselection rule. The central insight is that the mass superselection rule of non-relativistic QM — the impossibility of coherently superposing states of different mass — is not a physical law but an algebraic consequence of $H^2(\mathfrak{gal}(4))\ne 0$. Putting these together with the Inonu-Wigner contraction identifies the Bargmann mass with the non-relativistic limit of the rest energy, and the pattern recurs in the classification of elementary systems by coadjoint orbits: the Galilean coadjoint orbits of fixed mass $m$ are the non-relativistic counterparts of the Poincare coadjoint orbits classified by Wigner (massive, massless, tachyonic), and the Bargmann extension is what makes the non-relativistic classification richer at the projective level.

Full proof set [Master]

Proposition (the Bargmann cocycle satisfies the cocycle identity). The bilinear antisymmetric form ${\omega }_B$ on $\mathfrak{gal}(4)$ defined by ${\omega }_B(K_i,P_j)={\delta }_{ij}$ and zero on all other ordered pairs of generators satisfies ${\omega }_B([X,Y],Z)+{\omega }_B([Y,Z],X)+{\omega }_B([Z,X],Y)=0$ for all generators $X,Y,Z$.

Proof. The only triples that could give non-zero contributions involve at least one $K$-generator and at least one $P$-generator, since ${\omega }_B$ is zero on all other pairs. Case by case:

$(K_i,P_j,J_k)$: $[K_i,P_j]=0$, $[P_j,J_k]=-{\epsilon }_{jk\ell }{P}_{\ell }$, $[J_k,K_i]=-{\epsilon }_{ki\ell }{K}_{\ell }$. The cyclic sum is $0+{\omega }_B(-{\epsilon }_{jk\ell }{P}_{\ell },K_i)+{\omega }_B(-{\epsilon }_{ki\ell }{K}_{\ell },P_j)={\epsilon }_{jk\ell }{\delta }_{i\ell }-{\epsilon }_{ki\ell }{\delta }_{j\ell }={\epsilon }_{jki}-{\epsilon }_{kij}=0$.

$(K_i,P_j,H)$: $[K_i,P_j]=0$, $[P_j,H]=0$, $[H,K_i]=-P_i$. The cyclic sum is $0+0+{\omega }_B(-P_i,P_j)=0$ since ${\omega }_B(P_i,P_j)=0$.

$(K_i,P_j,P_k)$: $[K_i,P_j]=0$, $[P_j,P_k]=0$, $[P_k,K_i]=0$. Sum is zero.

$(K_i,P_j,K_k)$: $[K_i,P_j]=0$, $[P_j,K_k]=0$, $[K_k,K_i]=0$. Sum is zero.

All other triples either have no $K$ or no $P$ among the three entries, making every ${\omega }_B$ term vanish. $\square$

Proposition (the Bargmann cocycle is not a coboundary). There is no linear map $\varphi :\mathfrak{gal}(4)\to \mathbb{R}$ such that ${\omega }_B(X,Y)=\varphi ([X,Y])$ for all $X,Y\in \mathfrak{gal}(4)$.

Proof. Suppose such a $\varphi$ exists. Then ${\omega }_B(K_i,P_j)=\varphi ([K_i,P_j])=\varphi (0)=0$ for all $i,j$. But ${\omega }_B(K_i,P_i)=1\ne 0$. Contradiction. $\square$

Proposition ($H^2(\mathfrak{gal}(4);\mathbb{R})\cong \mathbb{R}$). Every 2-cocycle on $\mathfrak{gal}(4)$ is cohomologous to a scalar multiple of ${\omega }_B$.

Proof. Let $\omega$ be a 2-cocycle. The cocycle identity on $(J_i,X,Y)$ for $X,Y\in \{P_j,K_j\}$ implies that $\omega$ transforms as an $\mathrm{SO}(3)$-invariant bilinear form on the 3-vector representations. The only invariant pairing between two 3-vectors is the Kronecker delta: $\omega (K_i,P_j)=c{\delta }_{ij}$ for some scalar $c$.

The identity on $(H,K_i,P_j)$: $\omega ([H,K_i],P_j)+\omega ([K_i,P_j],H)+\omega ([P_j,H],K_i)=\omega (-P_i,P_j)+0+0=0$, which is automatic since $\omega$ vanishes on $(P,P)$ pairs.

The identity on $(H,K_i,K_j)$: $\omega (-P_i,K_j)+0+0=-\omega (P_i,K_j)=c^{\prime }{\delta }_{ij}$. But $P_i$ and $K_j$ live in different representations (both 3-vectors but with different transformation properties under boosts). The cocycle identity on $(J_k,H,K_i)$ forces $\omega (H,K_i)$ to be an $\mathrm{SO}(3)$-invariant map from the 3-vector $K$ to $\mathbb{R}$, which is zero.

All remaining freedom is the single parameter $c$. The cocycle $\omega -c{\omega }_B$ vanishes on all generator pairs, hence is zero. So $\omega =c{\omega }_B$ in cohomology. $\square$

Connections [Master]

• Galilean group and Newtonian mechanics 05.00.06. The Galilean group and its Lie algebra are developed in 05.00.06; this unit specialises to the central extension and the cohomological classification. The ten conserved quantities of 05.00.06 are the Noether charges of the centreless group, and the Bargmann extension modifies the boost-momentum bracket to record the projective phase.

• Noether's theorem 05.00.04. The conserved quantities associated to the Bargmann group include the same energy, momentum, and angular momentum as the Galilean group, plus the central charge $M$ (mass) conserved by virtue of being central. The cocycle-modified boost charge is the algebraic origin of the centre-of-mass motion theorem in 05.00.04.

• Symplectic manifold 05.01.02. The coadjoint orbits of the Bargmann group carry canonical symplectic structures (Kirillov-Kostant-Souriau form). The orbit through $(m,p,E,\ell )$ is the phase space of a Newtonian particle, and the symplectic form on it reproduces the canonical symplectic form on $T^{\ast }{\mathbb{R}}^3$ restricted to a mass shell. The Bargmann extension enriches the coadjoint-orbit classification by adding the mass parameter.

• Coadjoint orbit 05.03.01. The Souriau classification of elementary Newtonian particles by coadjoint orbits of $\widetilde{\mathrm{Gal}}(4)$ is the direct analogue of the Wigner classification of relativistic particles by coadjoint orbits of the Poincare group. The central extension produces an additional orbit parameter (mass) that has no relativistic counterpart in $H^2$.

• Lagrangian on $TQ$ 05.00.01. The Lagrangian $L=T-V$ on the tangent bundle of configuration space is the classical-mechanical shadow of the Bargmann-group action: the mass parameter in $T=\frac{1}{2}m\Vert \dot{q}{\Vert }^2$ is the value of the central charge $M$ on the irreducible representation that describes the particle.

Historical & philosophical context [Master]

Valentine Bargmann's paper On unitary ray representations of continuous groups (1954) ^{[Bargmann 1954]} introduced the central extension of the Galilean group and established the correspondence between projective representations of $\mathrm{Gal}(4)$ and genuine representations of the extended group. Bargmann was working in the context of Wigner's classification of relativistic particles (1939) and the emerging understanding that non-relativistic quantum mechanics required its own representation-theoretic foundations.

Jean-Marc Levy-Leblond's Galilei group and Galilean invariance (1963) ^{[Levy-Leblond 1963]} made the consequences explicit: the mass superselection rule, the projective phase under boosts, and the physical meaning of the central charge. Levy-Leblond showed that the Schrodinger equation for a free particle is the statement that the wave function carries an irreducible projective representation of the Galilean group, with mass as the label.

Jean-Marie Souriau's Structure des systemes dynamiques (1970) ^{[Souriau 1970]} placed the Bargmann extension in the symplectic-geometric framework: the coadjoint orbits of the Bargmann group are the phase spaces of elementary Newtonian particles, and the Kirillov-Kostant-Souriau symplectic form on each orbit is the canonical symplectic structure of classical mechanics.

Bibliography [Master]

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  journal   = {Annals of Mathematics},
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  number    = {1},
  pages     = {1--46},
  year      = {1954}
}

@article{LevyLeblond1963Galilei,
  author    = {L{\'e}vy-Leblond, Jean-Marc},
  title     = {Galilei group and Galilean invariance},
  journal   = {Journal of Mathematical Physics},
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  pages     = {776--788},
  year      = {1963}
}

@book{Souriau1970Structure,
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}

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