Galilei Symmetry and the Mass Superselection Argument

The Galilei group’s quantum boost and translation operators commute only up to a phase proportional to mass. Different fixed masses therefore carry different projective multipliers. Combining them on the full ray space of a direct sum obstructs a projective action of the unextended Galilei group. Turning that obstruction into a mass superselection rule requires retaining the assumptions about the symmetry group, the available observables, and the physical meaning of a variable mass.

Required background. Combining Ray Representations supplies the direct-sum obstruction. Central Extensions explains how a phase becomes a group coordinate. Helpful background. Galilean Boosts develops the passive-frame transformation used here.

Fix the boost and translation convention

At time zero, on $L^2(\mathbb R^3)$ for a spinless particle of fixed mass $m>0$, use

$$(T(a)\psi)(x)=\psi(x-a), \qquad (B_m(v)\psi)(x)=e^{-imv\cdot x/\hbar}\psi(x).$$

This boost convention is passive: the momentum of a state is shifted by $-mv$. The full time-dependent boost law also transports the spatial argument and includes the kinetic phase; its derivation belongs to Galilean Boosts. The time-zero operators already expose the central obstruction.

These are bounded unitaries on the entire Hilbert space. Their exact composition is therefore safer than inferring a global group identity solely from an unbounded-generator commutator.

A classically trivial loop produces a quantum phase

Direct substitution gives

$$B_m(v)T(a) =e^{-imv\cdot a/\hbar}T(a)B_m(v).$$

Consequently,

$$B_m(v)T(a)B_m(v)^{-1}T(a)^{-1} =e^{-imv\cdot a/\hbar}I.$$

Spatial translations and boosts commute in the unextended Galilei group: the corresponding loop returns to the identity spacetime transformation. Quantum mechanically, the loop is identity on rays within a fixed-mass sector because its action is scalar.

Changing the phases assigned separately to boosts or translations cannot change this scalar commutator. Its nontrivial value therefore detects a multiplier that cannot be removed by rephasing.

On a common canonical test domain, the same structure appears infinitesimally as

$$[K_j,P_k]=i\hbar m\,\delta_{jk}I, \qquad K_j=mQ_j$$

at time zero. The bounded composition calculation above establishes the relevant global relation without assuming that the infinitesimal relation alone integrates.

Compare two mass sectors on their full ray space

Consider $\mathcal H_{m_1}\oplus\mathcal H_{m_2}$ with the standard sector actions. The loop acts as

$$e^{-im_1v\cdot a/\hbar}I_1 \oplus e^{-im_2v\cdot a/\hbar}I_2.$$

For $m_1\ne m_2$, choose $a,v$ so the phases differ. A vector with nonzero components in both sectors then acquires a nontrivial relative phase. The classically identity group element does not act as identity on that combined ray.

Thus these sector actions cannot define one projective representation of the unextended Galilei group on all rays of the direct sum. Independent sector rephasings do not help because the commutator phase is invariant under them.

One possible resolution is to restrict the physical pure-state description to individual mass sectors and the observables to operators that do not compare their phases. Under those assumptions, the familiar mass superselection conclusion follows. The obstruction itself does not specify that observable restriction.

The central extension changes what the loop means

In the centrally extended group, the boost-translation loop is a nontrivial central element rather than the identity. If mass is represented by a self-adjoint operator $M$, the central subgroup may act as

$$s\longmapsto e^{-isM/\hbar},$$

with $s$ having the units needed for $sM$ to be an action. A direct sum of different central characters is then an ordinary reducible representation of the extended group.

This removes the algebraic contradiction with an identity transformation: the loop is no longer identity in the chosen group. It does not by itself establish that arbitrary mass-changing preparations or measurements are physically available. Those depend on the model and its observable algebra.

Giulini’s analysis makes this limitation explicit by distinguishing a fixed mass parameter from an enlarged dynamics in which mass is a variable. A superselection assertion needs such a dynamical and operational context, not only a comparison of parameter-labeled Hamiltonians.

Keep the nonrelativistic assumptions attached

The argument concerns Galilei symmetry and its standard mass-dependent representations. It is not a general prohibition on coherent superpositions of different energies, internal levels, or relativistic mass observables. Relativistic symmetry groups and mass-energy relations change the representation problem.

Even within a nonrelativistic treatment, adding a reference or enlarging the central symmetry can change which phase comparisons are meaningful. The general superselection discussion separates conservation, restricted observables, and relational coherence. None should be silently substituted for another.

Exercise: detect the relative phase

Take normalized sector vectors $\psi_1,\psi_2$ and the full-space vector $(\psi_1\oplus\psi_2)/\sqrt2$. What relative phase does the loop add? Under what condition is the full ray unchanged for every $a,v$?

Solution

After removing the common phase of the first component, the second acquires $e^{-i(m_2-m_1)v\cdot a/\hbar}$. Since $v\cdot a$ ranges continuously over real values, this is one for every loop exactly when $m_1=m_2$. For distinct masses, a sector-restricted algebra cannot detect the change, whereas a full-space interference observable could. The availability of the latter is precisely the extra physical assumption at issue.

References

• V. Bargmann, “On Unitary Ray Representations of Continuous Groups”, Annals of Mathematics 59, 1–46, 1954 — mass-dependent projective representations.
• D. Giulini, “On Galilei Invariance in Quantum Mechanics and the Bargmann Superselection Rule”, Annals of Physics 249, 222–235, 1996, arXiv:quant-ph/9508002 — the dynamical assumptions and extended-group interpretation.
• D. Giulini, “Superselection Rules”, arXiv:0710.1516 [quant-ph], 2007, revised 2009 — the broader scope and limitations of superselection arguments.