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Sunday, July 14, 2019

"Deriving" Quantum Logic from Reason, Beauty and Freedom

One Basic Principle of this Corner of the Eurycosm: The Universe Maximizes Freedom Given Constraints of Reason and Beauty

I was musing a bit about the basic concept at the heart of quantum logic and quantum probability: That a particular observer, when reasoning about properties of a system that it cannot in principle ever observe, should use quantum logic / quantum probabilities instead of classical ones.

I kept wondering: Why should this be the case?

Then it hit me: It’s just the maximum-entropy principle on a higher level!

The universe tends to maximize entropy/uncertainty as best it can given the imposed constraints.   And quantum amplitude (complex probability) distributions are in a way “more uncertain” than classical ones.   So if the universe is maximizing entropy it should be using quantum probabilities wherever possible.

A little more formally, let’s assume that an observer should reason about their (observable or indirectly assessable) universe in a way that is:

  1. logically consistent: the observations made at one place or time should be logically consistent with those made at other places and times
  2. pleasantly symmetric: the ways uncertainty and information values are measured should obey natural-seeming symmetries, as laid out e.g. by Knuth and Skilling in their paper on Foundations of Inference [] , and followup work on quantum inference []
  3. maximally entropic: having maximum uncertainty given other imposed constraints.  Anything else is assuming more than necessary.  This is basically an Occam’s Razor type assumption.

(I note that logical consistency is closely tied with the potential for useful abstraction.  In an inconsistent perceived/modeled world, one can't generalize via the methodology of making a formal abstraction and then deriving implications of that formal abstraction for specific situations (because one can't trust the "deriving" part).... In procedural terms, if a process (specified in a certain language L) starting from a certain seed produces a certain result, then we need it still to be the case later and elsewhere that the same process from the same seed will generate the same result ... if that doesn't work then "pattern recognition" doesn't work so well....   So this sort of induction involving patterns expressed in a language L appears equivalent to logical consistency according to Curry-Howard type correspondences.)

To put the core point more philosophico-poetically, these three assumptions basically amount to declaring that an observer’s subjective universe should display the three properties of:

  1. Reason
  2. Beauty
  3. Freedom

Who could argue with that?

How do reason, beauty and freedom lead to quantum logic?

I’m short on time as usual so I’m going to run through this pretty fast and loose.   Obviously all this needs to be written out much more rigorously, and some hidden rocks may emerge.   Let’s just pretend we’re discussing this over a beer and a joint with some jazz in the background…

We know basic quantum mechanics can be derived from a principle of stationary quantropy (complex valued entropy) [], just as basic classical physics can be derived from a principle of stationary entropy …

Quantropy ties in naturally with  Youssef’s complex-valued truth values [], though one can also interpret/analyze it otherwise…

It seems clear that modeling a system using complex truth values in a sense reflects MORE uncertainty than modeling a system using real truth values.   What I mean is: The complex truth values allow system properties to have the same values they would if modeled w/ real truth values, but also additional values.

Think about the double-slit experiment: the quantum case allows the electrons to hit the same spots they would in the classical case, but also other spots.

On the whole, there will be greater logical entropy [] for the quantum case than the classical case, i.e. the percentage of pairs of {property-value assignments for the system} that are considered different will be greater.   Double-slit experiment is a clear example here as well.

So, suppose we had the meta-principle: When modeling any system’s properties, use an adequately symmetric information-theoretic formalism that A) maximizes uncertainty in one’s model of the system, B) will not, in any possible future reality, lead to logical contradictions with future observations.

By these principles — Reason, Beauty and Freedom — one finds that

  • for systems properties whose values cannot in principle be observed by you, you should use quantum logic, complex truth values, etc. in preference to regular probabilities (because these have greater uncertainty and there is no problematic contradiction here)
  • for system properties whose values CAN in principle be observed by you, you can’t use the complex truth values because in the possible realities where you observe the system state, you may come up with conclusions that would contradict some of the complex truth-value assignments

(E.g. in the double-slit experiment, in the cases where you can in principle observe the electron paths, the quantum assumptions can’t be used as they will lead to conclusions contradictory to observation…)

A pending question here is why not use quaternionic or octonionic truth values, which Youssef shows also display many of the pleasant symmetries needed to provide reasonable measure of probability and information.  The answer has to be that these lack some basic symmetry properties we need to have a workable universe….  This seems plausibly true but needs more detailed elaboration…

So from the three meta-principles

  1. logical consistency of our models of the world at various times
  2.  measurement of uncertainty according to a formalism obeying certain nice symmetry axioms
  3.  maximization of uncertainty in our models, subject to the constraints of our observation

we can derive the conclusion that quantum logic / complex probability should be used for those things an observer in principle can’t measure, whereas classical real probability should be used for those things they can…

That is, some key aspects our world seem to be derivable from the principle that: The Universe Maximizes Freedom Given Constraints of Reason and Beauty

What is the use of this train of thought?

I’m not sure yet.  But it seems interesting to ground the peculiarity of quantum mechanics in something more fundamental.

The weird uncertainty of quantum mechanics may seem a bit less weird if one sees it as coming from a principle of assuming the maximum uncertainty one can, consistent with principles of consistency and symmetry. 

Assuming the  maximum uncertainty one can, is simply a matter of not assuming more than is necessary.  Which seems extremely natural — even if some of its consequences, like quantum logic, can seem less than natural if (as evolution has primed us humans to do) you bring the wrong initial biases to thinking about them.


Anonymous said...

Hello sir. I have developed a theory that would apply on the cuantum realm but also on the universe that we see above us. I was able to get to the beggining and moving on to the where infinite=0,0=1 and so on. It would apply on the dimension of matter as we see it around us( it contradicts the theory of expansion and the relativity unfortunatly) but it applies on the solar systems, galaxies and so on. Please announce me if you have 5 minutes of your time to read my theory. Ty :)

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