# Why doesn't superselection forbid almost every superposition?

Superselection makes sense not abstractly in an arbitrary Hilbert space but only in Hilbert spaces structured by the introduction of an algebra of distinguished observables of interest (i.e., that can be combined to prepare states), with prescribed commutation rules. These define the physics that is possible in the class of models considered, and superselection is a concept defined relative to these. (In particular, if the class of relevant observables changes, the concept of superselection changes with it. Enlarging the observable algebra may merge some sectors but typically creates others.)

Typical examples are the universal enveloping algebras of Lie algebras, or the $C^*$ algebras generated by the corresponding Lie groups. For example Heisenberg algebras and Heisenberg (or Weyl) groups correspond to canonical commutation rules, which are the basis of much of quantum physics.

The Hilbert spaces of interest are the (continuous unitary) irreducible representation spaces of these algebras, Lie algebras, or groups. These (or more precisely the classes of equivalent such representation spaces) are called the **(superselection) sectors of the theory**. Since they constitute different Hilbert spaces it makes no sense to superimpose vectors of the different sectors. One can define an inner product on the direct sum of these Hilbert spaces, but the algebra of operators still map each sector into itself, hence there is no way to create (in a physically relevant way) a superposition from pure states within the sectors.

For finite-dimensional Heisenberg algebras/groups, all continuous unitary irreducible representations are equivalent (Stone-von Neumann theorem); hence for nonrelativistic N-particle theories, there are no superselection rules (that would specify superselection sectors).

Once one also accounts for spin, the situation becomes more complicated: a mixture of a fermionic and a bosonic state makes no longer physical sense since the two state vectors behave differently under a rotation by 360 degree - though formally it is still defined. No amount of new physics will change that.

For infinite-dimensional Heisenberg algebras/groups as they occur in (relativistic or nonrelativistic) quantum field theory, the Stone-von Neumann theorem is no longer valid, and there are uncountably many inequivalent continuous unitary irreducible representations, hence there are uncountably many superselection sectors, distinguished by their essentially different behavior at spacelike infinity.

In more technical terms: The most interesting superselection rules, accounting for superconductivity, charge, baryon number, etc., arise due to unimplementable Bogoliubov transformations, involving limits that are so singular that they lead out of the Hilbert space representing the vacuum sector. In particular, charged states have a sufficiently different asymptotic structure from uncharged ones, since the Coulomb field is long range, and they belong to different superselection sectors. This is a general property of charges in gauge theories, see Strocchi, F., & Wightman, A. S. (1974). Proof of the charge superselection rule in local relativistic quantum field theory. Journal of Mathematical Physics, 15(12), 2198-2224. No amount of new physics will change that.

Under certain conditions, superselection sectors can be classified; see, e.g., the article *DHR superselection theory* from nLab. DHR stands for Doplicher, Haag, and Roberts; see, e.g.,

- Doplicher, S., & Roberts, J. E. (1990). Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Communications in Mathematical Physics, 131(1), 51-107.

Superselection rules have nothing to do with conservation laws. In spite of momentum conservation, states of different momentum can be superimposed since the Lorentz transformations that turn one momentum state into another are unitary and hence are defined on the same representation space.

I think the OP is getting close to answering his own question by suggesting that "superselection rules are just functions of our current technology." In my opinion, the answer is just a more precise statement of this suggestion: superselection rules capture beliefs we have about the **nature of possible physical observables** in the universe, limited by current experimental probes and theoretical models.

To explain that, we first give a mathematical definition of superselection sectors:

Definition 1: Considera Hilbert space$H$ and a collection of subspaces $\{H_i\}$ such that: $H = \oplus_{i} H_i$. We say that $\{H_i\}$ aresuperselection sectorsin $H$ if $\langle \psi_i| \mathcal{O}|\psi_j\rangle = 0$ $\forall$ $\psi_{i,j} \in H_{i,j}$, $i \neq j$, $\mathcal{O} \in O_{phy}$, where $O_{phy}$ is the set of allphysical observables$\mathcal{O}$ acting on the Hilbert space $H$.

There are two important notions that have to exist a priori in this definition:

We have to

**specify a Hilbert space**. To solve any quantum mechanical problem dogmatically, we would need a Hilbert space that contains states of the entire universe to ensure unitary evolution. However, practically, we often assume weak enough interactions between a test system $A$ and the rest of the universe $B$, so weak that we approximate $A$ as isolated (i.e. $H = H_A \otimes H_B$). So we can approximately obtain unitary evolution by restricting to the test system subspace $H_A$ for all interactions that act only on $H_A$. This freedom of choosing Hilbert spaces gives a bit of ambiguity to the meaning of superselection. Because it is possible that $\{H_i\}$ define superselection sectors in $H_A$, while $\{H_i \otimes H_B\}$ doesn't define superselection sectors in $H_A \otimes H_B = H$. We will see examples illustrating this later.We have to define

**physical observables**. Technically all self-adjoint operators on $H$ should be admissible candidates. But based on the current understanding of fundamental physics, we believe that certain self-adjoint operators are impossible to measure and certain transitions between states are impossible to engineer. Thus superselection rules today are practical tools that can be updated as we learn more about the world and find more physical observables that possibly break existing rules.

We now look at two examples to clarify why these two points above can cause confusion and to postulate a more complete definition of superselection rules.

**Example 1**: To illustrate the importance of the first bullet point, consider the OP's example of spins. Suppose we look at the Stern-Gerlach experiment and define $|\uparrow \rangle$ and $|\downarrow\rangle$ as eigenstates of the $\sigma_z$ operator. We take the Hilbert space $H_A$ spanned by these two states and postulate that the only physical observables on $H_A$ are $f(\sigma_z)$ where $f$ is some arbitrary analytic function. Now you may easily check that $|\uparrow \rangle$ and $|\downarrow\rangle$ define superselection sectors on $H_A$ alone!

But remember we have a freedom of choosing Hilbert space! Suppose now that we enlarge our Hilbert space to include a second particle $B$. Although we constrained physical observables on $H_A$ to be $f(\sigma_z)$, we don't have to constrain observables mixing $H_A$ and $H_B$. Even if we maintain conservation of total angular momentum $\sigma_z(A) + \sigma_z(B)$, we may well break conservation of $\sigma_z(A)$ or $\sigma_z(B)$ separately, thus mixing $\{|\uparrow\rangle \otimes H_B\}$ and $\{|\downarrow \rangle \otimes H_B\}$. This would imply $\{|\uparrow\rangle \otimes H_B\}$ and $\{|\downarrow \rangle \otimes H_B\}$ do not define superselection sectors on $H_A \otimes H_B$, as advertised in the first bullet point above. This is similar in spirit to OP's consideration of $p(particle) + p(recoil)$ on the Hilbert space $H_{\text{particle}} \otimes H_{\text{apparatus}}$.

**Example 2**: To illustrate the second bullet point, we consider OP's spin system without adding particle $B$. Instead we enlarge the set of physical observables by adding $\sigma_x$. In particular, we engineer a Hamiltonian $H= -\sigma_x$ by putting the spin in a unit of transverse magnetic field (as usual, we assume the interactions between the source of the magnetic field and the particle $A$ to be sufficiently weak that we can separate out $H_A$). Suppose we start the system in $|\downarrow \rangle$ at $t=0$, then at $t= \pi/2$, we flip the two sectors completely:
$$ e^{-i Ht} |\downarrow\rangle = e^{i \pi \sigma_x/2} |\downarrow\rangle = i \sigma_x |\downarrow\rangle = i|\uparrow \rangle $$
Again, as advertised, the introduction of more possible physical observables (in this case the introduction of $\sigma_x$ into the Hamiltonian) breaks the conservation of angular momentum and therefore the superselection sectors.

**Example 3**: Another example to illustrate the second bullet point. Let's escape from reality a bit more and imagine for fun that QM was discovered in 18th century, back when we believed in Galilean symmetry. To study states in QM, we found the projective representation of the Galilean algebra with $\{K_i\}$ generating Galilean boosts and $\{P_i\}$ generating translations. When we compose boosts and translations, we found that for a state with mass $M$:
$$ e^{-i \vec K \cdot \vec v} e^{-i \vec P \cdot \vec a} = e^{i M \vec a \cdot \vec v/2} e^{-i (\vec K \cdot \vec v + \vec P \cdot \vec a)} $$
This means a superposition of states with different masses would break Galilean symmetry, a sacred law of physics back then. In order to avoid that problem, we would have guessed a superselection rule forbidding transitions between states with different masses. And that rule would have stood the test of experiments and theoretical models up until the introduction of Einstein's relativity (which effectively modified the Hamiltonian, one of the physical observables, and its fundamental symmetries)! This is a rather dramatic illustration but I think it makes the point that **what we call physical is always a matter of contemporary belief.**

To summarize the discussion above, we come back to the classic example of electric charge. Why have physicists established the convention that charges define superselection sectors? It is because the superselection of charges is in some sense **universal**. While the superselection rule of angular momentum depends on the choice of Hilbert space, and the superselection rule of mass depends on choice of physical observables, electric charge is somehow different. As long as we believe in the Standard Model, no matter what Hilbert space $H \subset H_{\text{universe}}$ we choose (it can be the Hilbert space of the lab, of the earth, or of the whole universe), the subspaces $\{H_Q \subset H\}$ indexed by the restricted total charge operator $Q|_H$, define superselection sectors in $H$ in the sense of definition 1. Colloquially, this is to say: states with different charges don't talk to each other, for all choices of Hilbert space and physical observables within the Standard Model.

This last summary makes it clear that superselection rules we have today are intimately connected to contemporary beliefs about the character of physical laws. Maybe one day, there will be a new physical observable that mixes charge sectors, or a larger Hilbert space in which charge in one universe can be traded for charge in another. By that time, we would have to repeal the principle of charge superselection and hopefully find new superselection rules that will make our lives easier.