Paul Merriam posted a paper called Quantum Relativity: Physical Laws Must be Invariant Over Quantum Systems in which he puts forth a conceptual strategy for understanding how a relational interpretation addresses the foundational issues of quantum mechanics. Please see this prior post for more background. What follows is a summary and attempted interpretation of what I found to be key aspects of the paper. The usual caveats are in place: my summaries may be not only incomplete (including omission of formalisms) but also misleading due to errors in interpretation. Please read the paper to judge.
The paper starts with a section which discusses why decoherence does not solve the foundational issues of QM. Since I believe this is generally acknowledged (see this recent blog post from Matt Leifer; an old blog post of mine is here), I’ll just focus on the most important part of this discussion. Recall that one of the perceived shortcomings of the relational interpretation of QM revolves around the question of how two or more interacting systems come to “choose” the same basis. Merriam says that decoherence has a “change of basis” problem of its own.
To see this, Merriam returns to the “Wigner’s friend” framework and replaces "Wigner" with the "environment" to create a decoherence version of the scenario. Relative to the environment E, the experimenter (called A) and the system he or she is measuring (S) are in superposition and evolve according the Schrödinger picture. Decoherence would lead to the selection of relatively stable “classical” appearances of the observable which is the basis of the measurement. But suppose A decides to measure a different observable of S (change of basis). Decoherence takes place over a period of time (decoherence time); this time depends on many factors, but the “change of basis” is a problem for the time between zero and the decoherence time. (Decoherence is not measurement).
Next Merriam discusses (repeating the arguments of his older paper) the issues highlighted by the Wigner’s friend setup, arguing again that the quantum state describes a system relative to another system. Quantum mechanics is an intransitive theory.
The next section is titled “Quantum Relativity”. So having acknowledged the perspectivist nature of QM, what’s the next step? When considering two quantum systems: “The essential point of this paper is that since both systems physically exist they are both valid coordinate frames from which the laws of physics must hold. Quantum mechanics is as valid in S as it is in A.” If A describes S in terms of a superposition across some measurement basis, then S will describe A as starting out in a corresponding superposition. When A observes (measures) S to be in some eigenstate, “S must also observe A to be in some corresponding eigenstate…”
The key point is brought out by the word “must” here and in the title of the paper. The conceptual hurdle we are jumping here is as follows: if QM is valid from the point of view of all “quantum systems” (including everything from electrons to physicists), then when they interact they necessarily select the consistent basis for interaction. The basis problem is solved by asserting that basis choices must match if QM is to be valid from all points of view.
Merriam believes this conceptual leap has consequences analogous to special relativity. The next passage (see p. 6) looks at the formalism of the Schrödinger equation from A’s and S’s perspective and wonders how they can be consistent if the mass is so different in the two cases. But he notes the values for length or distance between the two quantum observations do not have to have the same numeral values in both systems. If distance is scaled to the relationship of the masses, then it is possible to create a transformation from the superposition of S as described by A to that of A described by S. There can be a group of such transformations for any number of systems. Merriam derives a transformation constant in analogy to the role the speed of light c plays in relativistic transformation.
Merriam also speculates about that one could extend the idea to include gravity by taking the equivalence of gravitational force and acceleration to be relative to the local quantum reference system. He suggests the shape a quantum version of Einstein’s equation would take. I will skip for now further discussion of this idea and a section on how gauge invariance might be impacted, since I think the key concept is in place with the analogue to special relativity.
Key to special relativity is the postulate that physical laws valid from one reference frame should be form-invariant when translated to another frame. To review, we assume that QM gives a valid physical description from the point of view of a system, and each quantum system forms a physically valid coordinate frame. Note that systems only share a reference frame when they interact. We should be able to translate the state of a system S which is in superposition relative to system A to the state of A relative to S. Again, this only works if we stipulate that if an interaction takes place, the “basis choice” is necessarily consistent from both perspectives.