tom.stoer said:
On the contrary, it hides them as it draws our attention to classical entities like the Lagrangian and paths. But we know that there may be physically relevant systems w/o classical Lagrangian!
...
with an emergent classical domain (including an emergent classical apparatus and classical states)
On your point about the classical references, and your desire to find a non-classical starting point, we fully agree.
But for me constructing and action as a sum of weighted transitions really does provide a good handle on this. There are indeed things in the PI that needs fixing. just like there is in the canonical approach.
I think we are probably seeking more or later to fix the same things? but maybe from different conjectures.
In my crazy picture I'm considering something loosely like this (which conceptually is close to PI but contains more)
The probability for a transition between two states, relative to a given observer, is intuituvely constructed as a rational expectaion by simply "adding all the information at hand" in a rational "averaging process". Generally the information we have seen are sometimes conflicting, and then we need to measure the "weight" of each evidence and let them "interfere" and the results is a rationally constructed subjective proability.
I am not at all picturing classical paths or anything like that, I agree that's not a good abstraction.
Instead the "paths" I talk think about are, consistent transition paths between two information states. There is nothing classical about this, it's a pure abstraction, because each observer sees a different "space of paths". Ultimaltey my vision is that these spaces are defined purely combinatorically. Ie they are observer dependent discrete spaces (there is no observer independent discreteness, so no issue with relativity).
Now, if we take a pure inferencial perspective like I suggest, then the acual "actions" rather than coming from classicla baggage, are reconstructed similar from combinatorical expressions.
One can look at a toy models, without non-commuting information where these transition probabilities takes the form of exponetials where interestingly enough the weight factor is a kull-back liebler information divergence - this follows simply from evaluating the multinomial distribution, so it's nothing fancy. From this picture, one can then classical define an observer like a finite history, which defines the prior, which further defines a "perturbation space" on which one can consider eovlution, and the evolution in this picture is just decay type entropic flows.
This very picture I'm working on refining, by defining a real probability of two non-commuting information sets. the trick to do this is to realize that in the actual inference, they are dependent, by lossy information transformations, and which representation that's chosen is balance in the evolutio npicture. So my vision is that if this works, a PI like picture weill reappear, where the action (s) is defined as a information theoretic abstraction completely without classical analoges.
But I don't want to sprinkle out any details until I've mature this picture. But in this problem, I face several open issues as they are entangle with this. In particular does it seem impossible to DEFINE the measures (corresponding to the classical acitons) WITHOUT considering it in hte context of evolution, because there is no LOGICAL reason direct reason why the action is the way it is, it's only selected during interactions with the environment. This means interacting observers. And in this picture each observer has a complexity measure (which is close enough analog to it's mass) that in a nontrivial way affects ALL interactions because it constrains the spaces where the permutations takess place.
So I have some reasonly concrete ideas, even though "crazy" and for me the closest fit with the standard pictures is hte PI. But what I'm picturing is that the SELECTION of the S-measure, is defined only in terms of what one may see as interacting PI's. Ie. if you FIX the background, and just write down a fixed PI, for a fixed path space, then the logic that explains the S is frozen, and you have no opton put to put it in mnualla.y But there is no reason why this picture can't be improved.
Edit: please see this as my "short" remark on your short remark it was meant to be
Edit: I'm sorry but to add one more thing. I see that the central thing is the transition probability, and how it's constructed from the inside and how this influences the action of the observer. And the most generic "action principle" is simply "maximize the transition probability". Of course the observer does not "maximize" anything, it just does a random walk, but on average it will then follow the peak according the action principle. But what's interesting is that when you write down the expression for the transition probability in the example of trying to predict a future sequence, from a past sequence there information about the OBSERVERs prior, factors out from other details and ends up in the information divergence measures. Which one can interpret as a a kind of action. (example
http://en.wikipedia.org/wiki/Kullback–Leibler_divergence). One can then see the action of a path, as a mesaure of the AMOUNT of information that deviates from the prior. So principles of least action is simply the principle of "minimal inconsistency" with the prior. but since the prior evolves, selection takes place when two such measures interact.
/Fredrik
Well I am silently working on new reconstruction of measurement theory as part of my own ideas. But not as a profession, so progress is slow. I've learned that in particular since time isn't abundant optimal use of time means I should not oversee the constructing principles in the calculational machinery.
I've seen enough of that I decided 10 years ago to not let myself fall into that trap.