I think the questions we now start to discuss are good ones, so I probably will complement my answers later when I've formulated the better answers, but here are some quick ones.
LuisPe said:
Now, by observer you mean an environment then?
Yes it could be, but more often it's just a part of the environment, not necessarily the entire environment.
Also, conceptually I think of physical law as a tool used by each observer, to interact with its environment. In this abstaction I flip the notions, the "environment" is "the system under observation", and the observer is what "interfaces" to the environment.
But the key in my view, is that one must account for the action of the observer on the system, as well as the backreaction from the system on the observer.
I don't have all answers here, but to flesh it out a bit as part my starting point I picture that each observer, at each moment in time to to speak, has what I call a system of microstructures which defines or constraints the state space in complexity, and a this structure also has a state, corresponding to the "information state". I picture a entropic action that induces a natural flow on this state space, thus the structure of the statespace encodes the hamiltonian. Once the statespace is set, the hamiltonian flow is basically an entropic flow.
Hilbert spaces and QM state vectors would be a special case, that I expect to be derived from this deeper picture. But I have not been ablt to do so yet, but I'm somehow convinced it's possible.
LuisPe said:
Just to be sure, are thinking in the Schrodinger picture?
Since I'm picturing a reconstruction, where QM structure eventually would be emergent or induced, it may be confusing to borrow all the usual notions from QM. In the writing I made abover, the schrodinger picture was probably closest, but otoh I don't think it makes any difference to my point if we have a space of evolving state vectors, or we have a space of evolving operators.
LuisPe said:
So you propose that one cannot fully know what is the hilbert space of some system, and that it might also change in time, ie evolve. I am having a hard time picturing it in some concrete example (probably because I am sort of used to thinking in the ortodox way, the one that is the basis of our work). For instance, say I consider some spin interacting with a bunch of other spins. There I believe there is no problem with assuming that the state space is known.
This is all subtle I guess. First, my own preferred phrasings would not be to say that the hilbert space "evolves IN TIME", it would be rather silly, it's rather that the evolution of the hilbert space, as quantified by a kind of information divergence, IS a kind of time. An incomplete analogy, think about how we used the expansion of our universe to somehow "define" cosmological time. I'm thinking similarly, but in a differeny way.
The example you take, is the type of example where indeed the timeless statespaces make the MOST sense! This is also what Smolin called the case of "subsystems" which is the typical scenario for most particle physics experiments, which after all is WHERE QM as we know it, is confirmed.
The opposite to the "subsystem" scenario, is the "cosmological scenario".
To elaborate: The subsystem scenariou really corresponds in my use of the word, with the entire environment of the observed system (say a system of interacting spin-systems) IS "the observer".
In a particle physics lab, with decectors etc, I think it's fair to say that loosely speaking the ENTIRE environment of the collision domain, is under our control, and here it's fair to think of the ENTIRE environment of the localized events as "the observer". Also the environment is MASSIVE in complexity(and energy) relative to the system. In this example, the usualy logic works very well (but not perfect, but I could expand on that another time since it's a more subtle thing which will only confuse things here). Here I picture that repetivity experiments, and storing large time histories of experimental sequences including preparations in the environment (laboratory) are actually possible, here one can also infer effectively stable "timeless" state spaces. So QM works fine.
But, if we now consider the flip situation, that we sciencetist are the ones stuck inside a some small detector and are making our "observations" not towards a subsytem but out towards our cosmological horizon, into effectively an "open system", then our possibilities to infer, hold and store, histories of experiments are limited simply by _complexity_ or an information bound. For this kind of situation, the inference of a timelss state space of the environemtn just doesn't make any sense to me. As smoling puts it a bit provocative in this talk bout evolving law, it's a "fallacy" to apply the logic of subsystems to the cosomological situation. And I agree with him.
So, I am basically consider that the physical law, and theory itself, LIVES or is encoded, in the microstructure of the observer - in the above examples it's either encoded in the lab-environment, which is relative to the microscopic particles "infinite", or in the other scenario, which we can also call the "inside perspective" corresponding to how the subatomic particles "see" physical law, they must encode the laws themselves! And of course, this is a different situation, but I also think thta this is the reason why the interactions are bound to be unified as we consider less and less (complex) "observers" (read subatomic fragments) which is what happens in High energy experiments and we try to break matter into smaller fragments, then the interactions between the fragments are bound to have simpler and simpler "logic".
This is something I think we need to systemize and take seriously, and it's a natural part in the intrinsic model to me.
About event spaces, I distinguish between expected, and unexpected events. To any observer, the expected evolution in the knowns event space is always "information preserving" (to use a more neutral notion) for obvious reasons, but the undecidability makes it impossible to predict the full evolution. Now my exploit is that I think this will have observable effects to a second observer. The system will have an action, that reveals that it can not predict everything! This is a key exploit, and how I envision that the hamiltonian or action eventually follows and evolves along with the state spaces.
anothre thing is that I've come to the conclusion that the state spaces is rather more like a system of state spaces, that corresponds to "memory compartments" having different compression codings, and that THIS eventually is the origin of non-commuative structures, and QM. But there is a lot of work left here. I have reasonable ideas on all this but progress is snail-speed.
More later, let me know if any of this makes sense to you.
/Fredrik
