Consistent Histories (What's the catch?)

"The consistent histories approach combines wave functions and probabilities in a fully consistent way which does not rely upon the use of measurements"....."Histories can be used to describe how a particle interacts with a measuring apparatus, and how the outcome of a measurement (e.g., the position of a pointer) is related to some property of the particle before the measurement took place. However, they can also be employed for a single particle, or any number of particles, in the absence of any measurement. For example, by using consistent histories it is possible to assign a probability for the time at which an unstable particle, such as a radioactive atom, will decay, even if it is out in interstellar space far from any measuring device."

What are frameworks, and what is the single-framework rule?
A framework or consistent family is a set of mutually-exclusive possibilities to which one can assign probabilities according to the rules of quantum theory. It is the analog of a sample space in ordinary probability theory. For example, the two possibilities +1/2 and -1/2 for the z component S_z of spin angular momentum of a spin-half particle are mutually exclusive and form a quantum framework. On the other hand, +1/2 for the z component and +1/2 for the x component S_x are incompatible in that it is meaningles to combine S_z=+1/2 and S_x=+1/2 in a single quantum description. The reason is that there is nothing in the quantum Hilbert space which can correspond to this combination, and assuming that S_z=+1/2 AND S_x=+1/2 makes sense leads to logical difficulties [4]. Values for S_z and S_x are not mutually exclusive in the sense that if one is true the other is necessarily false. Instead, they are non-comparable: trying to construct logical relationships between them does not make sense. Since they are not mutually exclusive, S_z and S_x cannot appear in the same framework. While S_z and S_x can be described by means of separate frameworks, these two frameworks are incompatible . One cannot combine a quantum description based upon some framework with a description based upon an incompatible framework, because the result would be meaningless (i.e., quantum theory can assign it no meaning). This is known as the single-framework rule.

Frameworks are used for families of histories as well as for quantum states at a single time. In this case, not only must the different possibilities be mutually exclusive, but they must satisfy consistency conditions in order that quantum probabilities can be assigned in a consistent way. Frameworks of histories that cannot be combined in a way that satisfies the consistency conditions are by definition incompatible, and once again the single-framework rule states that descriptions using incompatible frameworks cannot be combined.

Given two or more frameworks, which one provides the correct description of a quantum system?
There is no fundamental principle of quantum mechanics, no law of nature, that singles out one framework as the only possibility for a "correct" description. Consider a classical spinning body and a description, X, which assigns some value to the x component of its angular momentum L_x. Let Z be a second description that assigns a value to the z component of angular momentum L_z. Can one say that X rather than Z is the correct description? Clearly this would be silly. Instead one thinks of X and Z as part of a total description of the angular momentum that includes both of them. The case of a quantum spin-half particle is similar: it makes no sense to say that a description assigning a value to S_x is the correct description rather that a description that assigns a value to S_z. However, unlike the classical case, there is no total description that includes both S_x and S_z, for these two frameworks are incompatible and cannot be combined, due to the mathematical properties of the quantum Hilbert space.

In practice, physicists choose among the various possible quantum descriptions depending upon the question they want to answer. In the case of Schrödinger's cat there is a framework which is useful for answering (at least in a probabilistic sense) the question of whether the cat is dead or alive, and this framework is incompatible with the one corresponding to unitary time evolution according to Schrödinger's equation, which leads to a superposition state. Either description is a valid one from the perspective of fundamental quantum theory. On the other hand, they are not at all the same in terms of the sorts of questions that they allow one to address. If one employs the framework based on unitary time evolution, the question of whether the cat is dead or alive is not meaningful, since it is like asking for the value of S_z when S_x=+1/2 for a spin-half particle. Indeed, it does not even make sense to talk about a cat, since those properties that we normally associate with a cat (small furry animal with four legs and a tail, etc.) are incompatible with the description based on the superposition state.

It is worth emphasizing that as long as one is considering a particular physical question, such as the probability that the cat will be dead or alive, the multiplicity of possible frameworks causes no problem, for they will all give the same answer (for a given experimental situation, initial conditions, etc.) to the same question.

There is a catch: it's indeterministic. Making it severily ugly

But that's true of most interpretations, other than things like Bohmian mechanics.

Bhobba, I disagree.
Indeterminism is immensly ugly philosophically.

Why would any sort of indeterminism follow a statistical law (Born rule) ? That needs explanation (cause) and then we are back at determinism. But no, people like that want to believe that the universe magically CHOOSE free willingly to follow a statistical law. It makes absolutely no sense whatsoever.

In the history of science, determinism has ALWAYS triumphed and I would be genuinely shocked if it does not happen in QM too. For me indeterminism is as likely as solipsism.