Quantum Logic for the Eurycosm?

I don't think quantum mechanics, in itself, is sufficient to explain the structure and dynamics of what I'm here calling the "Eurycosm."

However, I do think that QM has something to teach us about euryphysics -- possibly quite a lot.

One relevant issue is the need to apply quantum rather than classical logic in particular cases.

When To Apply Quantum versus Classical Logic?

The story (based on various experiments, e.g. delayed choice double slit, various quantum erasers and teleporters, etc. etc.) seems to be that: If a certain event cannot in principle be measured, then it needs to be modeled using quantum logic rather than classical logic.

However, it isn't entirely clear how this should be formulated.   If one accepts the "relational interpretation" of QM thoroughly, then one may want to say "If a certain event cannot in principle be measured by a certain observer O, then it needs to be modeled using quantum rather than classical logic, by observer O."    (But, you ask, could the same event then be modeled using classical logic, by another observer O1?  Well in the relational interpretation of QM, this possibility is not well-formed, because one deals only with (event, observer) pairs, i.e. all events are at foundation considered as observer-dependent.  Of course there may be various mappings between E and E1 with (E,O), (E1, O1), so that one may have E and E1 that are very structurally similar but with E quantum-modelable by E and E1 classical-modelable by O1....)

The question then arises whether it makes sense for a Turing machine to be "properly modelable only using quantum logic" by a certain observer.  If you buy the work of Dirk Aerts on quantum models of classical systems (search "Diederik Aerts" on arxiv for some of his relevant papers), then the answer is yes.   The "trick" is that one must assume certain constraints on the observer.

I suspect that, when all this is cashed out in detail, one will get the implication that:  For a system with a high degree of complexity and a limited amount of reflective capability, this system needs to model certain aspects of its own state in terms of quantum logic.   Regardless of the fact that from the perspective of a hypothetical observer with full knowledge of all bits in the system and the system's hardware underpinnings, the system should be modelled classically.

Remember that in the relational view, systems do not exist in themselves, only (system, observer) pairs.   In this view, there are no classical systems, only (system, observer) pairs in which the sensible model of the system by the observer is classical...

Note also the hypothesis in this paper  that an observer should be understood as a system identification algorithm, and quantified using algorithmic information (aka Kolmogorov information, aka the length of the shortest program for
producing the observer).   E.g. this paper posits

System S  is called quantum with respect to observer  X if K(S) <
K(X), meaning that X  will be able to maintain a complete list of all
its degrees of freedom. Otherwise X  is called classical with respect
to  X.

Quantum Logic for the Eurycosm?

Suppose we have a local time-bundle T, with which observer O intersects.   Then we can ask whether: within time-bundle T, there are any observations in which observer O is observing event E.

If yes, then we can say that O can "in principle" observe E, relative to T.

One implication of quantum theory, then, is that in cases where O cannot in principle observe E, relative to T, the right way for the probability of E to be quantified (relative to observation by O) is using complex-number probabilities or quantum amplitudes.

On the other hand, if O can in principle observe E, relative to T, then the right way for the probability of E to be quantified (relative to observation by O) is using plain old real-number probabilities.

Note that from a eurycosmic perspective, this logic may be posited to hold regardless of the degree of intersection of E, O or T with our spacetime continuum.

Is this really the case?  Does the logic of when to apply quantum vs. classical logic really apply in this out-there setting?   I'm not sure.   But it seems the best hypothesis to make at this point.

(As an aside, I have long wanted to find some excuse to introduce quaternionic or octonionic probabilities, extending the complex probabilities one finds in QM.  But I haven't found a good reason yet...)

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