Superselection Rules and Charge Invariance

A charge superselection rule restricts the observable algebra so that it cannot compare coherence between charge sectors. The precise criterion uses spectral projections or the charge’s unitary group, avoiding ambiguous commutators with an unbounded operator. The elementary block-mixture calculation is in Direct Sums and Superselection; this page identifies the hypotheses behind that model and explains how reference systems change the question.

Required background. Direct Sums and Superselection proves the basic indistinguishability statement. Projection-Valued Measures provides the spectral language for general charges.

Express charge invariance with bounded operators

Let $\mathcal A\subseteq\mathcal B(\mathcal H)$ be a unital observable algebra, and let $Q$ be self-adjoint. A precise charge-invariance condition is

$$A e^{itQ}=e^{itQ}A \quad\text{for all }A\in\mathcal A,\ t\in\mathbb R,$$

where the parameter has units making $tQ$ dimensionless. By spectral calculus, this is equivalent to

$$A E_Q(\Delta)=E_Q(\Delta)A \quad\text{for every Borel }\Delta.$$

The condition is stronger and clearer than a formal assertion $[A,Q]=0$ on an unspecified test space. For bounded $A$, strong commutation also implies preservation of $D(Q)$ and $QA\psi=AQ\psi$ there.

If $Q$ has a finite or countable pure-point decomposition, its eigenspace projections give the usual sector blocks. Continuous charge decompositions require direct integrals rather than a sum of normalizable charge eigenvectors.

A commutant projection is not automatically a central observable

The displayed criterion places $E_Q(\Delta)$ in the commutant $\mathcal A'$. To call it central in $\mathcal A$, it must also belong to $\mathcal A$ itself. This second condition matters: an invariant decomposition can arise from multiplicity of a representation without supplying an observable sector label.

For example, let

$$\mathcal H=\mathbb C^2\otimes\mathbb C^2, \qquad \mathcal A=\mathcal B(\mathbb C^2)\otimes I.$$

The projection $I\otimes|0\rangle\langle0|$ commutes with all of $\mathcal A$ but is not in $\mathcal A$. Its range is an invariant copy of the same observable representation. Calling the two copies different observable charges would add a measurement that the stated algebra does not contain.

For a von Neumann observable algebra, genuine central projections belong to $\mathcal Z(\mathcal A)=\mathcal A\cap\mathcal A'$ and separate distinguishable classical alternatives in that representation. More general superselection theory also compares inequivalent representations of an abstract algebra; it should not be reduced to an arbitrary choice of basis blocks.

Conservation constrains dynamics, not every measurement

If a Hamiltonian’s unitary group commutes with the spectral projections of $Q$, closed-system evolution preserves charge sectors. This is a conservation statement. A superselection restriction concerns the entire admitted observable or operation class.

A qubit with $H\propto Z$ conserves $Z$, but a detector for $X$ can reveal coherence between the $Z$ eigenstates. Excluding that detector requires an additional physical assumption. Likewise, environmental suppression of off-diagonal entries is an approximate dynamical claim; it does not alone prove an exact restriction on the observable algebra.

The symmetry introduction to sectors develops the physical distinction. Here the point is to keep the quantifier visible: “commutes with this Hamiltonian” and “commutes with every allowed observable” are different assertions.

A reference system can reveal relational coherence

Take two systems with charge eigenstates $|0\rangle,|1\rangle$ and total charge $Q_{\rm tot}=Q_A\otimes I+I\otimes Q_B$. The states

$$|\Psi_\theta\rangle =\frac{|0,1\rangle+e^{i\theta}|1,0\rangle}{\sqrt2}$$

lie entirely in the total-charge-one sector. The bounded observable

$$C=|0,1\rangle\langle1,0| +|1,0\rangle\langle0,1|$$

commutes with $Q_{\rm tot}$ and has expectation $\cos\theta$. Yet each reduced state is diagonal and independent of $\theta$.

The joint observable compares charge transferred between the two systems while preserving the total. It reveals a relative coherence invisible to charge-invariant observables on either subsystem alone. This does not violate total-charge superselection: the superposition never mixes different total charges.

Thus statements about unavailable local phases must name the system boundary and the admitted reference. Enlarging the system changes the allowed relational questions; it does not automatically remove every fundamental charge restriction.

Symmetry arguments need their physical assumptions

Different projective multipliers can obstruct a single ray action on a full direct sum. The ray-representation analysis proves that representation-theoretic statement. To infer a superselection rule, one must additionally specify the physical symmetry group, the state space on which it acts, and the observable algebra.

The Galilei mass example is especially sensitive to this qualification. A mass parameter in one fixed model is not automatically a dynamical observable whose superpositions have been forbidden by a theorem.

Exercise: distinguish multiplicity from a measured charge

In the algebra $\mathcal B(\mathbb C^2)\otimes I$, can the states $u\otimes|0\rangle$ and $u\otimes|1\rangle$ be distinguished by an allowed observable? What changes if the algebra is enlarged to include $I\otimes Z$?

Solution

Every allowed expectation is $\langle u,Au\rangle$ and is independent of the second factor, so the states are indistinguishable. Adding $I\otimes Z$ makes the two labels distinguishable. If the enlarged algebra is generated only by these operators, the second-factor spectral projections become central, while operators mixing the two labels remain absent. The chosen algebra determines which distinction is physically available.

References

• D. Giulini, “Superselection Rules”, arXiv:0710.1516 [quant-ph], 2007, revised 2009 — algebraic criteria, conservation, and the interpretation of superselection.
• K. Landsman, Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, Fundamental Theories of Physics 188, Springer, 2017 — observable algebras, representations, and quantum symmetries.