Direct Sums and Superselection

A direct sum separates mutually orthogonal alternatives inside a Hilbert space. It becomes a superselection decomposition only when the admitted observable algebra cannot connect the alternatives. Under that restriction, phases between sectors have no observable effect, and a vector with several nonzero sector components induces a mixed state on the restricted algebra. This page proves that elementary finite or countable sector statement; physical origins and symmetry-specific sector rules are treated separately in Superselection Sectors.

Required background. Physical States as Rays separates rays from representatives. Expectation Functionals explains why specifying the observable algebra is part of specifying a state.

Orthogonal sector sums

For Hilbert spaces $\mathcal H_\alpha$, with a finite or countable label set, the Hilbert direct sum consists of vectors

$$\mathcal H=\bigoplus_\alpha\mathcal H_\alpha, \qquad \psi=(\psi_\alpha)_\alpha, \qquad \sum_\alpha\|\psi_\alpha\|^2<\infty.$$

Its inner product is $\langle\psi,\phi\rangle=\sum_\alpha\langle\psi_\alpha,\phi_\alpha\rangle$. The orthogonal projection $P_\alpha$ extracts one component, and the sum of these projections converges strongly to the identity. A direct sum describes alternatives; a tensor product describes simultaneous subsystem degrees of freedom. A sector decomposition is not, by itself, a bipartite decomposition.

There are many possible orthogonal decompositions of the same space. A qubit can be written as the direct sum of its two computational basis lines, yet ordinary qubit experiments measure $\sigma_x$ and detect coherence between them. Direct-sum notation alone therefore establishes no superselection rule.

The block-diagonal observable assumption

Consider the bounded algebra

$$\mathcal A= \left\{\bigoplus_\alpha A_\alpha: A_\alpha\in\mathcal B(\mathcal H_\alpha), \sup_\alpha\|A_\alpha\|<\infty\right\}.$$

Equivalently, every $A\in\mathcal A$ commutes with every $P_\alpha$. The sector projections belong to the center of this algebra: they commute with all its elements. Thus the sector label is a classical alternative within this particular quantum description.

For a unit vector $\psi$, let $p_\alpha=\|\psi_\alpha\|^2$. When $p_\alpha>0$, define $\widehat\psi_\alpha=\psi_\alpha/\sqrt{p_\alpha}$. The block structure gives

$$\langle\psi,A\psi\rangle =\sum_\alpha p_\alpha \langle\widehat\psi_\alpha,A_\alpha\widehat\psi_\alpha\rangle.$$

Replacing $\psi_\alpha$ independently by $e^{i\theta_\alpha}\psi_\alpha$ changes none of these expectations. More strongly, the vector projection $P_\psi$ and the density operator

$$\rho_{\rm diag} =\sum_{\alpha:p_\alpha>0} p_\alpha|\widehat\psi_\alpha\rangle \langle\widehat\psi_\alpha|$$

define the same functional on $\mathcal A$. In a countable sum the positive series converges in trace norm, since its tail trace is the remaining sector probability.

The equality is relative to $\mathcal A$. On the full algebra $\mathcal B(\mathcal H)$, $P_\psi$ is still pure, and off-diagonal operators can distinguish it from $\rho_{\rm diag}$. If two or more sector weights are nonzero, the restriction to $\mathcal A$ is a nontrivial convex mixture of sector-supported functionals. It is consequently mixed on that algebra.

Two sectors expose the distinction

Take $\mathcal H=\mathbb C\oplus\mathbb C$ and restrict observables to diagonal matrices. The vectors

$$\psi_\theta=\sqrt p\,e_0 +e^{i\theta}\sqrt{1-p}\,e_1$$

all yield the same expectation $pa+(1-p)b$ for $A=\operatorname{diag}(a,b)$. Their restricted state equals the classical distribution $(p,1-p)$.

If $\sigma_x$ becomes an admitted observable, however,

$$\langle\psi_\theta,\sigma_x\psi_\theta\rangle =2\sqrt{p(1-p)}\cos\theta,$$

while the corresponding diagonal density operator gives zero. Enlarging the observable algebra changes which preparations can be distinguished. It does not merely change notation for the same restricted experiment.

This is why physical superselection discussions must identify available observables and reference systems. A statement that a phase is inaccessible within one model does not, without further argument, prove that no enlarged system or reference can reveal relational coherence. Giulini’s review discusses the conceptual status of such restrictions and distinguishes exact kinematical assumptions from dynamical explanations.

Sector-preserving dynamics and its domains

An autonomous dynamics preserves these sectors when its unitary group satisfies

$$U(t)P_\alpha=P_\alpha U(t) \quad\text{for all }t,\alpha.$$

For a self-adjoint generator, this means the sector subspaces reduce the Hamiltonian. In a diagonal sector representation $H=\bigoplus_\alpha H_\alpha$, its domain is

$$D(H)=\left\{\psi: \psi_\alpha\in D(H_\alpha), \sum_\alpha\|H_\alpha\psi_\alpha\|^2<\infty\right\}.$$

The square-summability condition is essential. Belonging componentwise to every sector domain does not alone make the full Hamiltonian action a Hilbert-space vector. Formal vanishing of a commutator on an unspecified test domain is likewise weaker than the displayed unitary commutation relation.

Finally, an approximate loss of off-diagonal entries through environmental decoherence does not establish an exact algebraic superselection rule. Nor does the elementary countable sum cover every sector construction: continuous decompositions require direct-integral theory, and general observable algebras can have more complicated representation structure.

Exercise: restriction can turn a pure state into a mixture

For $0<p<1$, write the restricted state of $\psi_\theta$ as a convex combination of the two evaluation functionals on diagonal matrices. Show that these functionals are distinct.

Solution

Define $\omega_0(\operatorname{diag}(a,b))=a$ and $\omega_1(\operatorname{diag}(a,b))=b$. Then $\omega_{\psi_\theta}|_{\mathcal A}=p\omega_0+(1-p)\omega_1$. They are distinct because $\omega_0(P_0)=1$ and $\omega_1(P_0)=0$. Both weights are positive, so the restricted functional is not extreme. The full-space density operator $P_{\psi_\theta}$ remains rank one.

References

• D. Giulini, “Superselection Rules”, arXiv:0710.1516 [quant-ph], 2007 — observable restrictions, sector interpretations, and their physical qualifications.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Sections 1.4 and 2.5 — Hilbert direct sums and direct sums of operators.