I thought it was about time I got around to finishing my comments on Rovelli’s “Relational QM” programme.

Relational QM (RQM) has a lot in common with the Everett/many-worlds interpretation, so it should be no surprise that it shares some of the same difficulties. In my opinion, the “basis problem” also applies to RQM, and one cannot appeal to decoherence in order to solve it as one does in many-worlds. Before discussing this, let me summarize the main differences between RQM and many-worlds:

- In Everett, the state-vector of the universe is the full description of reality. It always evolves unitarily, but different observers can have different subjective impressions of reality depending on how they are described within this state.
- In RQM there is no state-vector of the universe. State-vectors always describe the point of view that one subsystem has about another system. State vectors are therefore always subjective descriptions of reality.
- In Everett, the concept of measurement is an emergent phenomenon that applies when a macroscopic system interacts with a microscopic one. The theory of decoherence is used to explain why measurement results appear to be stable.
- In RQM, Rovelli states explicitly that he doesn’t want to treat microscopic systems any differently from microscopic ones. For example, if two electrons interact, then it is valid to think that one of the electrons acts as a measuring device on the other and vice versa. One description is valid from the point of view of one of the electrons and the other is valid from the point of view of the other electron.

The appeal to decoherence in Everett is designed to address the “basis problem”, which arises due to the ambiguity over which baisis the states are decomposed in. For example, suppose two qubits start in the (unnormalized) state

(|0> + |1>)|0>

and interact so that they end up in the state

|00> + |11> .

This is a typical example of a “measurement” interaction and we might want to say that the second qubit has measured whether the first qubit is in the state |0> or |1>. In Rovelli’s formulation, the state of the first qubit is definitely either |0> or |1>, relative to the second qubit, with 50/50 probabilities of each being the case.

However, we could equally well write the final state as

(|0> + |1>)(|0> + |1>) + (|0> – |1>)(|0> – |1>).

If the qubits are actually spin-1/2 particles and |0> and |1> are spin up and spin down in the Z-direction, then this is a decomposition of the state in the spin-X basis. Therefore, we might equally well say that the second qubit has measured whether the first qubit is in the state (|0> + |1>) or (|0> – |1>). In Rovelli’s formulation, we ought to be able to say that the first qubit is either in the state (|0> + |1>) or the state (|0> – |1>) with 50/50 probabilities.

Note that, this is not only an issue with the particular state |00> + |11>. Any bipartite state can be decomposed according to any basis for one subsystem, although the relative states of the other system will not generally be orthogonal.

I have seen no discussion of this issue from Rovelli. He seems to assume that there just is some natural basis in which to do the decomposition. I think the possible solutions are:

- Accept multiple descriptions: The state of one subsystem is not only relative to another subsystem, but it is also relative to an arbitrary basis choice. The problem with this is that it does not explain why our subjective experience is always according to one particular basis. I always feel like I am in one particular location, observing one particular thing, and I am incapable of regarding myself as being in a superposition of two locations, despite the fact that such a decomposition of the wavefunction almost certainly exists.
- Stipulate a basis: For example, the position basis might be a natural choice, since it generically corresponds to our everyday subjective experience. The question then arises as to why this basis is chosen rather than some other. What is there within the formalism of QM that compells us to make this choice?
- Appeal to decoherence: Decoherence theory usually supplies a “pointer basis” in which the results of measurement outcomes are almost exactly stable. This is the usual solution of the Everettians. However, if Rovelli takes this option then he has to back away from the position that microscopic systems are to be treated in exactly the same way as macroscopic one. It would no longer make sense to talk of a single electron acting as a measuring device.
- Use the biorthogonal decomposition: Most bipartite states have a unique decomposition of the form \sum_j a_j | phi_j> |psi_j>, where <phi_j | phi_k> = \delta_{jk} and <psi_j | psi_k > = \delta_{jk}. We could simply stipulate that this basis is the correct one to do the decomposition in. This is the solution advocated in some variants of the modal interpretation. Problems include the fact that there are special states like the one above (admittedly they form a set of measure zero) for which the decomposition is not unique. Also, the biorthogonal basis does not always correspond exactly to our subjective experience, e.g. it may be close to, but not exactly equal to, the position basis.

My impression is that none of these solutions would completely appeal to Rovelli, so it would be interesting to see what he says about the matter. However, if we combine this issue with the previous comments I made, then I have a hard time seeing how the Everettian/many-worlds ontology can really be avoided in this sort of approach.