Path integrals, action principles and transition probabilities

[fredrik]

Some other threads make me pose another question...

 

In the introductory courses of classical analytical mechanics, the classical action principles where to my experience argued by showing that they yield equivalent solutions to the ordinary newtons mechanics.

 

So they are in a sense nothing new, just a sort of reformulation, whose main justification is the agreement with the standard newtons equations of motion.

 

Still it is intuitively appealing, beatiful approach that appeals to variational principles and ideas that seem very intuitive such that "nature chooses the quickest path", giving the student the impression that there is after all something deeper hear. It is appealing in short.

 

Then feynmann's pathintegral formulation of QM which also makes use of similar looking formalisms to evaluate the transition probability, but there is now the concept of the complex amplitudes. A connection to stat mech can be done by relating time with complex temperatures. Interesting connections but I lack a rigorous and consistent connection.

 

Exactly how does the feynmann pathintegrals and complex action connect to probabilistic reasoning?

 

After all, set aside any exotic formalisms, what we want is to estimate the prior probability for a observing a new state, given the information at hand, right?

 

How can we _define_ a prior probability? Is there a physical meaning of such prior probability?

 

What forms does "information" at hand take, and how does one define "addition" of such information? We have learned that superposition of complex amplitudes are generally quite a different thing than classical union of events, but what's the connection?

 

Many QG papers use these concepts that are IMO still awaiting full understanding.

 

What is the general view on this? Do you think the problem of QG to be independent from the foundational consistency of QM? Or is the foundational issue of QM a non-issue?

 

Ideas? comments?

 

/Fredrik

 

What I feel is suspect and currently reviewing is the lack of detail in the constant of proportionality of [math]

e^{iS/ \hbar}

[/math] and this could possibly be the reason for why the expression diverges at times.

 

If we compare this to a simpler classical case of the estimated transition probability to one relative frequency given a prior frequency distribution that is proportional to [math]e^{-S_KL}[/math] where S_KL is the kull-back entropy. This can be found by basic combinatorics.

 

But the constant in front of the factor has a complex dependency on the scale! As the set population goes to infinity so that we get a continuum this goes to 1, but in the discrete domain there is a complex dependence on the factor. In simple combinatorical classical cases this factor can be derived explicitly.

 

I think there is a link to this, combined with transformations between probability spaces whose results will be very close to the path integral formulation. And most probably it will provide a much more general setting where feynmanns original ideas can be understood from a deeper analysis from probabilistic reasonings. And I think the connection to gravity will be understood much better since this relates to complexity scaling (discrete -> continuum)

 

Still it takes some time and I resumed this again not so long ago. But there must be tons of papers that has ripped this to pieces over and over again. But I haven't yet found them. Anyone knows what attempts that has been made already along these lines?

 

/Fredrik

[Fred56]

Are you thinking about behaviour of external objects (i.e. mechanics and dynamics) and what the relation is to QM, or about experience (how we perceive the world)?

[fredrik]

Fred, I'm not sure what you mean. I am not talking specifically about human perceptions, I am talking about trying to understand the connection to ontology and epistemology.

 

I adhere to some kind of measurement ideals and the philosophy of Bohr that the task of physics is not to explain what nature is, but to explain what we can say about nature. And then I extend that. With "we" I don't mean just me, or humanity. I am talking about an arbitrary observer, a system in general. Not only does the environment scale, the observer itself can scale.

 

I am still trying to find a information theoretic understand that to the limit of my understanding is consistent. Part of the problem is to define both ontological as well as epistemological things. And I think of them as complementary and inseparable, not something which you choose between.

 

We learn by interacting, observing. But one can not just talke about observations in free space. There has to be an observer or a system that relates to the observations, that retains the results of the measurements in some kind of structure. This observer is itself part of the dynamics, and it seems clear that the observers "strategy" of information processing is crucial for it's persistence and survival.

 

I think that the probability interpretations can be consistently restored, but it is not possible to make a fixed distinction between ontology and epistemology, observer and observations, matter and space.

 

Empirist ideals are excellent. But things aren't that easy, because the empirically collected experience need to be processed and stored efficiently, because we have limited resources. A selection process will clearly favour those structures that can do this.

 

The question is not what we can predict with certainty, the question is more what is our best expectation. I think this can be furthermore reformulated as what is most likely the more beneficial expectation. This can I think be attacked by a generalised probabilit theory, and I think the path integral can be consistently seen as as generalized diffusion. Of course this idea is old, but I have yet to see a proper formulation of this. I think a proper review should also fix some of the loose/unclear ends in this reasoning.

 

With an "action" is associate a prior probability for the relation transformation. This probability evolves by two mechanisms - self evolution, which occurs because the condition consists of mixed information (this is why it's no "simple diffusion", more interesting thingsh append - supposedly this is described by the dynamical equations. but there may also a feedback that is not predictable. This updates the "self-evolution".

 

The main philosophy behind this is that we know what we know, we don't know what the don't know and it's not possible to put bounds on that. The best we have is expectations. Continuous agreement with interactions means that a steady-state or equilibrium of is attained in a certain sense.

 

This relates closely I think to the path integral ideas, the action ideas and the fundaments of QM. But most probably also to gravity. I think gravity will be emergent from the complexity.

 

Maybe it is the very simplification in our description that shaves off gravity? I think we would benefit from a reconstruction here, and see if gravity pops up as another effect of higher order complexity.

 

/Fredrik

 

Now I'll mix everyday language with physics but it happens to be interesting so I'll do it...

 

What determines Your initiated actions, from Your point of view? Clearly it's your expectations of the world right? Wether your expectations are "correct" (whatever that means) is completely irrelevant because it's all you've got.

 

But what about things that are forced upon you by things beyond control? New things, new information, clearly causes you to revise your expectations, so this is still consistent.

 

The behaviour of a poker player is completely determined what he expects of the other players hands and strategies. If he is wrong, is irrelevant, his expectations still governs his behaviour until his expectations are updated.

 

If we insist on that we are never wrong, that our expectations are always met, then we risk failing to learn and that's when we get toasted.

 

/Fredrik