Path integrals and foundations of quantum mechanics

In summary, there is a debate over whether the path integral formulation of quantum mechanics is truly equivalent to the canonical quantization approach. While the path integral method is useful for calculating states at later times, it is not self-sufficient as it does not contain a notion of quantum states living in a Hilbert space. This has led to the borrowing of quantum states from the canonical approach in order to fully define the theory. The question remains whether there are calculations that cannot be done through the path integral method alone, such as deriving Bell's inequalities. Some argue that canonical quantization is also not self-sufficient as it borrows from classical mechanics, while others believe it is a self-sufficient theory. There are also alternative formulations, such as using
  • #36
I did not say that decoherence removes the observer but that it removes the classical apparatus.

But as far as I can see these questions are irrelevant for the discussion of PI vs. canonical approach as the two approaches both agree regarding the basic ideas.
 
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  • #37
One thing in favor of the PI formulation is that it can be expanded into perturbative terms that include every interaction imaginable everywhere. It incompasses everything, everywhere, in one formula. Whereas the canonical only seems to concentrate on one interaction.
 
  • #38
I don't think so. The PI formalism seems to more elegant in some cases, but in QM they are strictly equivalent; you can derive the perturbation series via canonical quantization as well.

Why do you think that the application of one approximation method is an argument for one formalism? There are other approximations for which the canonical formalism is better suited.
 
  • #39
tom.stoer said:
I did not say that decoherence removes the observer but that it removes the classical apparatus.

But as far as I can see these questions are irrelevant for the discussion of PI vs. canonical approach as the two approaches both agree regarding the basic ideas.
My train of thought tried to suggest connections, but I since I see you think that QM needs no change, this is why I think you lack the motivator for seeing the arguments I had. For me it's exactly my belieif that QM needs reconstruction, that is the startingpoint, and in the "PI-view" it's easier to see these issues, that appear more "hidden" in the other views.

But of course, if one does not think the issues are there, they are hidden anyhow.

Another question for you Tom, not sure if I explicitly asked before. Do you, or do you not agree with Rovelli that the problem of unifying interactions/forces and the problem of QG are disjoint? Ie that the problems don't couple?

/Fredrik
 
  • #40
Fra, to short remarks:

I don't agree that the PI formalism shows the problems more explicitly. 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! And usually we omit (or forget) to think about the measure which is not a classical entity. Therefore the PI formalism pretends a classical world which could very well be an illusion restricted or constructed by our perception but not by nature. That's why I think the canonical formalism with an emergent classical domain (including an emergent classical apparatus and classical states) as found in the decoherence approach is much more promising (for me the PI is just tool - btw.: this agrees with Feynman's view - and he should know rather well).

I agree with Rovelli (but to be honest that's of little relevance) that unification and the foundational problems of QM are not related. But I am not so sure whether Rovelli really says that unification and quantum gravity are definately unrelated. What he says is that one should try to quantize gravity w/o unification b/c it may be possible that they are unrelated and b/c he prefers a step-wise approach (look at the decades it took to develop QED and to formulate GSW model eventually). He does not say how nature really IS but only how he thinks that one could make progress. But LQG is not relevant in the context "PI vs. canonical approach"
 
  • #41
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
 
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  • #42
kith said:
You seem to think that 'Hilbert space + X' is a better set of axioms than 'Hilbert space + Y', where X is 'Schrödinger equation + canonical commutational relations' and Y is 'PI'.
Yes, you can put it this way.

kith said:
The only reasons for this I can think of, are that a) one of them is contained in the other or b) one doesn't contain as much QM as the other. Is one of these options the case?
No. It is
c) X naturally lives together with a Hilbert space, because if you have canonical commutation operators, then it is natural (if not necessary) to represent them on a Hilbert space. On the other hand, PI and Hilbert space together look like a very artificial couple.
 
  • #43
Demystifier said:
PI and Hilbert space together look like a very artificial couple.

This is the part I was too stupid to understnad. I guess I did't see why it's artificial.

/Fredrik
 
  • #44
Fra said:
This is the part I was too stupid to understnad. I guess I did't see why it's artificial.
So you think it isn't artificial? Fine, then try to explain why path integrals and Hilbert space naturally live together as parts of a single theory.
 
  • #45
Fra said:
This is the part I was too stupid to understnad. I guess I did't see why it's artificial.
/Fredrik
Look at my earlier posts. In the PI formalism one studies the object Z which is basically a transition amplitude K. For some reason it turns out that this object contains much more information, e.g. information regarding the energy spectrum. But the idea is that once you have defined (and calculated) Z, you never care about Hilbert spaces. So IFF the PI approach (as a tool!) works nice, Hilbert spaces are history; in that sense the Hilbert space is artificial on the PI context.

But as soon as you want to ask questions which are not encoded in Z, i.e. which cannot be "asked", you have to go back to the Hilbert space context and reformulate the question such that it can be extracted from Z.

Let's look at the energy spectrum. Suppose if I give you some explizit expression for Z and ask you to extract the energy spectrum, then you don't know how to do that w/o referring to the canonical approach. And even if you are able to do that in practice you will not understand WHY it works!

Let's look at some other example: Suppose you have no idea regarding cooking. Suppose I give you recipe named 'pasta', which contains a hidden, encrypted information regarding pizza; suppose you make pasta - and by chance you DO make pizza. You will not understand why the hell you were able to make pizza! You see it, you can eat it, fine. But you cannot explain based on the pasta recipe how you succeeded in making pizza. In order to understand that I have to explain how I managed to hide the encrypted information regarding pizza.

That's what happens with Z. You can extract the energy spectrum, but w/o referring to the encrypted, hidden information, which is the construction of the PI based on the canonical formalism, you do not see why it contains the energy spectrum.

In that sense the canonical approach is more fundamental.
 
  • #46
I didn't comment more on this, but to get back to this just for a second.

I think we misunderstood each other. My picture involves a reconstruction of QM and attacking open problems, meaning that the explicit connection is not in place. What I meant is that I find the PI better.

Ok I figure you mean something like this

Your point got to be that a specific question (=specific hilbert operators) implies a specific partitioning of hilbert space according to the operator spectrum, and and leads to a specific Z that encodes only answers to those questions. Z' different partitioning, can't be logical deduced from Z

We agree on that.

But different Z is attached to different observers, so direct comparasion makes no sense in the first until you actually try to understand observer-observer interactions (which of course I'm trying to do, but canonical approach does not).

Also what I mean by the most general PI, is that it encodes the transition probability from one observer-microstate to another. Ie. it is the probability of the future observer-state conditional on the past observer-state.

So indeed, everything that this observer CAN ask, must be encoded in this. Surely one can sort of imagine that some questions can't be asked but then my conjecture is that these questions AREN't asked byt this observer - and since asking questions is the same as choosin actions, you can tell from how a system behaves that it "can not phrase" certain questions. For example, an electrically neutral system, not coupling with charge, simply can't encode the question "what is the charge of my environment).

This is the idea.

So the usual Z, that exists in current formulation are actually not the complete correspondence to the microcanonial partition function of the observer, a lot of informatin is indeed gone, I agree. But this I see as curable. I can see if you see Z as a tool, as of today that my points does not come across.

I object to the "notion" that an observer CHOOSES what to ask. There is something wrong about this. This refers to a "human experimenter" or a theorist. But I'm talking about an obsever = a system, and here the st ate of the observer in my view encodes that questions that ARE asked. To EXPLAIN WHY different questions are asked, one has to explain why the observer state is changed.

Things that are encoded in "experimental setups" and preparations by HUMANs, does not have a correspondence in this intrinsic picture. Because here only "spontaneous" questions are real.

So, I suppose I see your point, but I'm thinking of the essence of PI as something more general. And this does not yet exist! But in the canonical formalism these issues are hardly phrasable.

So the generalization of the "Z" IMO encodes ALL possible transitions from one observer st ate to another. What can't be encoded in there, are questions this observer WILL not ask.

/Fredrik
 
  • #47
You still miss my point.

Fra said:
Your point got to be that a specific question (=specific hilbert operators) implies a specific partitioning of hilbert space according to the operator spectrum, and and leads to a specific Z that encodes only answers to those questions. Z' different partitioning, can't be logical deduced from Z
No.

I do not introduce a partinioning of the Hilbert space, nor do I ask specific questions regarding specific hilbert space operators. I ask questions regarding some physical entity called 'energy sepctrum'. And I do not encode this is a specific Z. There is just one Z and one 'energy spectrum'. It's the most general approach, nothing beyond it. Nevertheless given this most general Z you CAN derive the energy spectrum but you CAN'T explain why it is contained w/o referring to the CONSTRUCTION of Z.

Fra said:
We agree on that.
No, unfortunately not.

Fra said:
But different Z is attached to different observers, so direct comparasion makes no sense in the first until you actually try to understand observer-observer interactions
No.

There is one Z which encodes everything and which can be used by different observers.

Fra said:
Surely one can sort of imagine that some questions can't be asked but then my conjecture is that these questions AREN't asked byt this observer
The observer (system, apparatus, ...) is free to ask all questions he/she/it likes, and he/she/it will find all answers encoded in Z. Z is something like a transition probability, nevertheless you can you the same Z and extract the energy spectrum. So this is certainly a question you CAN ask, you will get the correct answer, but don't understand WHY.

Fra said:
... and since asking questions is the same as choosin actions, you can tell from how a system behaves that it "can not phrase" certain questions.
No.

Asking questions is not chosing actions, but doing something special (question-specific) with a very general entity Z (or H). One never needs to modify S, Z or H just to ask different questions.

Fra said:
So the usual Z, that exists in current formulation are actually not the complete correspondence to the microcanonial partition function of the observer, a lot of informatin is indeed gone, I agree.
But I don't agree.

No information is gone, as I tried to explain several times. Everything is there, everything can be extracted, nothing is lost. You can extract the information, but you don't understand why you can do that. What's gone in the PI formalism is the EXPLANATION WHY IT WORKS.

Fra said:
But this I see as curable. I can see if you see Z as a tool, as of today that my points does not come across.
Yes.

Sorry to say that but w/o writing down a better Z you will not be able to convince anybody here that what you have in mind IS better. You do not get my point, you see some shortcomings in Z, you propose a new Z which you cannot write down, but you insist on this new Z to be superior to the old one.

Let's look at my idea: no new Z, no new H, nothing but the well-known formalism. The only thing which you have to take seriously is that it is not only Z what matters but 'Z PLUS ITS CONSTRUCTION FROM H'.

Fra said:
... but I'm thinking of the essence of PI as something more general. And this does not yet exist! But in the canonical formalism these issues are hardly phrasable.
As I contunously try to explain is that in the canonical formalism these issues can be addressed more directly whereas in the PI they are hidden. A solution based on a new PI which you cannot present here is not a solution at all.

See what happens: We have two detailed, concrete ideas, let's call them A and B. I (we) present a variety of arguments why A is better than B. Your response is always that A is worse than B, but C will be better than B; but you are not able to tell us what C is. Not a very satisfactory argument.
 
  • #48
tom.stoer said:
A solution based on a new PI which you cannot present here is not a solution at all.

Very fair indeed, I agree! this is why I completely understand that I appear incomprehensible. However, what I propose can't be described in standard formalism.

I suppose I did think that my conceptual point would come across but I'm obviously wrong. Perhaps another time, I could try to elaborate this.

Edit: Just a note:
" There is one Z which encodes everything and which can be used by different observers."
This is exactly what does not make any sense IMHO (ie in the context of open issues) unless you refer to the "external observer" that is all implicit in the classical baggage, but I do understand that I am unable to convey why, and why the assumption of an observer independent theory is the root cause of some open problem OR why a generalized PI picture is a great way to understand this, without working out the full ideas. I'm sorry for the confusion.

/Fredrik
 
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  • #49
We can discuss your conceptual point, but I don't think that it makes sense to mix it with a discussion like PI vs. canonical approach. The obvious problem is that your idea is to go beyond the standard approach whereas our discussion was based on the existing (and complete) PI and canonical formulation.
 
  • #50
tom.stoer said:
See what happens: We have two detailed, concrete ideas, let's call them A and B. I (we) present a variety of arguments why A is better than B. Your response is always that A is worse than B, but C will be better than B; but you are not able to tell us what C is. Not a very satisfactory argument.

Yes you are definitely right.

I overstimated my ability to convey the reasoning. Sorry about that. I realize that we don't get much fruther here until I can be more explicit.

/Fredrik
 
  • #51
I don't know if was already pointed out in thread but PI is very natural to study non purturbative dynamics like instantons, solitons and phenomenon like vacuum tunneling . I am not aware of how to study these using canonical methods.

Fra said:
Yes you are definitely right.

I overstimated my ability to convey the reasoning. Sorry about that. I realize that we don't get much fruther here until I can be more explicit.

/Fredrik
 
  • #52
Prathyush said:
I don't know if was already pointed out in thread but PI is very natural to study non purturbative dynamics like instantons, solitons and phenomenon like vacuum tunneling . I am not aware of how to study these using canonical methods.
In principle, canonical methods are certainly not restricted to perturbative methods, even if in practice it is not easy to include nonperturbative effects in a canonical framework.

But here we do not discuss which method is more practical. We discuss whether the two methods are equivalent IN PRINCIPLE.
 
  • #53
Instantons, theta-angle etc. can be studied in the canonical approach as well; the canonical approach is definately NOT restricted to perturbative treatment!
 
  • #54
I've often wondered about this. A partial answer seems to be that we can regard the path integral as giving the transition between elements of the classical configuration space. The propagator for example is just the transition amplitude
[tex] \langle t_2,x_2| t_1,x_1 \rangle [/tex]
To the extent that this is true, it's just a statement that QM borrows its conceptual framework from classical physics. What's not clear to me is
  • how to incorporate the idea of a superposition of position states into our conceptual framework.
  • How to formalise the idea of observables
One could try arguing as follows, although I'm throwing this out principally for discussion rather than as a claim that our theory is complete:
  • In the absence of a position measurement, the indeterminacy of the time evolution of position implies that we should always begin with some "initial time smearing" of the original position, consistent with the complex exponential form of the contributions from each path. This should of course yield the wavefunction, which satisfies the Schroedinger equation and for which the propagator is a Green function.
  • The question then would be how to justify the implementation of what we would usually think of as 'operators in the position basis' as tools by which we can extract information about observables. You could argue that energy and momentum generate translations in time and space, which would almost immediately identify the corresponding differential operators up to constant factors.
The hardest aspect of the state-vector formalism to incorporate in this framework seems to me to be spin. Classically, one can have "spin vectors" that identify some angular momentum intrinsic to a body as the translation-invariant part of of the total angular momentum; but how you could justify the introduction of spin-1/2 representations (without identifying the space of wavefunctions as a Hilbert space and rederiving the canonical framework) isn't clear to me at all.
 
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  • #55
muppet, I think we agree that the hardest issue for the PI as a conceptual basis is to explain if and why something beyond the specific representation (usually position rep.) can work w/o referring to Hilbert states
 
  • #58

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