Is probability a fundamental aspect of quantum mechanics?

[Ken G]

This is the part where I sharply disagree with Ken G's depiction of the use of reason and science: he seems to equate the use of the strategy "play quantum mechanics" with ignorance that the meta-game exists, and use of the strategy "play science" with ignorance of the meta-meta-game exists.

I followed everything you said, in complete agreement, until we got to this step. I do not recognize anything in these notions attributed to me that resembles my actual thoughts, can you clarify what distinctions you are making here? I do not think people who "play science" are necessarily ignorant that they are playing a game, in fact I think that some are more aware of it than others-- an opinion you appear to share, or else there would not be a need for you to point out that scientists are playing a game of representation and idealization. If you read my last several posts more carefully, perhaps you will see better what my actual thinking is-- my objection is to the idea that physics benefits from being framed as a search for the laws that nature "actually follows," or is a study of the "way nature thinks", when what is demonstrably true is that physics is an attempt for us to think about nature using a set of proven tools (or games) for doing that. In fact, I have found I can usually get many mathematicians quite incensed by suggesting that what they are doing is "playing games like chess", rather than probing the truths of the universe. What's more, I have often espoused that there is a fundamental tension between the certainty of these games, and what constitutes truth in reality, which does not allow any such certainty. Finally, I would point out that to me, this fundamental divide between what is true and what is mathematical is the undercurrent of Godel's theorems.

 

[bhobba]

When you think this way, you find yourself becoming quite skeptical that the universe itself "follows laws" at all, and you think of mathematical physics as a kind of advanced sociology. That doesn't lessen it however-- in some ways, it makes it more exquisite to see that we are really looking very deeply into ourselves when we do physics.

Definitely an element of Kuhn here and even of Wittgenstein.

I recall reading the famous Turing Wittgenstein debates on the foundations of mathematics. At first sight you tend to side with Turing and take the position math reveals objective truth - if not bridges could fall down, all sorts of problems would arise - it has to be more than social convention.

However Wittgenstein's reasoning is quite subtle and can not be dismissed that easily. When you think about it you realize he has a point (just like you do) - it could all be social convention.

The issue here is, I think, the type of people drawn to math, and hence like me are of the mathematical physics ilk. If math is your bag you feel it in your bones this is truth - not social convention - exactly like Turing did. Not that Wittgenstein was simply an ivory tower philosopher - before being drawn to philosophy he was an aeronautical engineer and knew what applied math was about - but he did not seem to have this inbuilt feeling in your gut that this is it - the math is the reality and certainly not taking the extreme view I (and Penrose) do that it actually resides in a Platonic realm and that realm is what is really determining the physical and mental realm. Nothing can prove me, or people like me, correct - its a conviction you have - like Einstein's conviction QM was not complete.

I think you hit on it before - most physicists would not agree with me but many more of those who think of themselves as mathematical physicist would - but I do not think most - I think even amongst those my views are extreme.

Thanks
Bill

 

[bhobba]

In fact, I have found I can usually get many mathematicians quite incensed by suggesting that what they are doing is "playing games like chess", rather than probing the truths of the universe. What's more, I have often espoused that there is a fundamental tension between the certainty of these games, and what constitutes truth in reality, which does not allow any such certainty. Finally, I would point out that to me, this fundamental divide between what is true and what is mathematical is the undercurrent of Godel's theorems.

Very true. I do not agree its a game - but that is a gut feeling I have - not something I can prove. Being incensed about it won't help - thinking a bit deeper about and realising the other side has a point may.

Thanks
Bill

 

[jfy4]

I hope no one minds if I get in on this topic.

My question is also about the use of probability in QM. It pertains to the question of what an observable is, and probability.

My observation is this: Since in QM, and any situation where quantum effects cannot be neglected, individual measurements cannot be predicted, then is it appropriate to define what an observable is based on what can be predicted?

For instance, in a two state system experiment, no individual measurement can be predicted, however, the ensemble averages can be predicted, then, are relative probabilities the only true observable?

To be sure, I mean true in this sense: While we can acquire numbers for individual measurements, those individual numbers where never being tested in the first place, but rather the expectation values were under scrutiny. Then the individual numbers are not true observables, but expectation values are.

This is all in preparation for the question: Does nature care about individual numbers either? Or does nature only worry about large scale relative probabilistic structure also? I mean (I'm going to sound a little philosophical here), we are nature, probing ourselves through experiment, and we can only make conclusions about nature through large scale probabilistic structure.

 

[Jano L.]

Bhobba,

thank you for the links.

I did not want to make an impression that I doubt the possibility to define some path integral.
The existence and meaningfulness of the path integral depends on the situation and the definition one chooses.

My point was that the Feynman field integrals occurring in QFT are heuristic pictures which do not have solid mathematical meaning on a level of classical theory of, say, Riemann's integral. I base this on some limited thinking on the integrals one encounters in statistical physics. One can calculate directly Gaussian field integrals, and with some tinkering with the measure one can even invent what more complicated integrals are supposed to mean, but as far as I understand it, there is no unambiguous general procedure to do it. Different exponent will require different definitions. Please correct me if I am off in this - perhaps the Hida approach solved this?


Can you derive Hamilton's principle from Feynman's sums? Do you have some paper on this?


The book by Omnes is a terrible reading. I have read superficially the 10th and 11th chapter and I can say I have not seen a bit of derivation of classical mechanics. He says that one can introduce operator on Hilbert space which can be used to define classical variable describing ensemble of classical systems.

But in this he stays within a statistical description. Calculating averages in QM is no revolution, it was done already in 1926. This is not a derivation of classical mechanics. I did not find in those chapters any discussion on the wave-packet spreading. How is one supposed to get the particles with determined trajectories from the evolution of some operator? This seems far from complete derivation of classical mechanics to me.

 

[Ken G]

I recall reading the famous Turing Wittgenstein debates on the foundations of mathematics. At first sight you tend to side with Turing and take the position math reveals objective truth - if not bridges could fall down, all sorts of problems would arise - it has to be more than social convention.

However Wittgenstein's reasoning is quite subtle and can not be dismissed that easily. When you think about it you realize he has a point (just like you do) - it could all be social convention.

You are right that I do align closely with Wittgenstein's views, many of which seem quite insightful to me. He said that the point of philosophy was not to discover truth, but rather to make problems "go away", and that if a lion could talk, we wouldn't understand him. But it was in the Turing/Wittgenstein debates that I think his insights really flourished-- he said that if it was discovered that modern arithmetic was not incomplete as hoped, but rather (horrors!) inconsistent, then no one would need to lose faith in bridges, and no one would need to teach mathematics differently to children. Indeed, I hold that essentially nothing would happen anywhere, except in the ivy-covered halls of inquiry at the frontiers of mathematics and philosophy.

But I don't want to get too off track-- we are talking about QM and how it uses probability, I'm just saying that we should frame that as a discussion about this theory we have created and how it works, rather than as a discourse about how reality itself works (like, does God roll dice or not!). I think that's the error, in thinking that this is the kind of question physics is intended to be about. That doesn't mean I completely reject the idea that mathematical physics is a study of laws of the universe, it means I think that is a kind of helpful fantasy that we enter into. If you feel it is true "in your bones", there is certainly no harm in that-- that's one kind of truth, certainly. It's just not the kind of truth that either mathematics or physics deals in, it's the kind of truth that people deal in-- so it is sociological! That is somewhat ironic.

 

[bhobba]

My point was that the Feynman field integrals occurring in QFT are heuristic pictures which do not have solid mathematical meaning on a level of classical theory of, say, Riemann's integral. I base this on some limited thinking on the integrals one encounters in statistical physics. One can calculate directly Gaussian field integrals, and with some tinkering with the measure one can even invent what more complicated integrals are supposed to mean, but as far as I understand it, there is no unambiguous general procedure to do it. Different exponent will require different definitions. Please correct me if I am off in this - perhaps the Hida approach solved this?

It is more sophisticated than the Riemann Integral - sure - its a functional integral which includes other stuff such as stochastic integrals:
http://en.wikipedia.org/wiki/Functional_integration

In fact by using what is called a Wick Rotation you can transform Path Integrals into Stochastic Integrals. The issue here is defining such things in a rigorous manner - that's where Hida Distributions come in - it allows it to be done rigorously. Its a highly technical area that most don't really worry about - you simply accept the formal limits are OK - that's the mathematical issue - defining such limits rigorously. This sort of thing is done in applied math all the time eg the Dirac Delta function. That too has problems at the usual level presented in books - tomes like Gelfland's three volume text on Generalised Functions fix it - but to put it mildly are highly challenging - even for math freaks like me. I have done it (not from Gelfland's books - but others) - it's one reason it took me 10 years part time to get my math up to the level where I was comfortable with this stuff - and I already had a math degree that included two courses on functional analyses. If you want to go down that path expect a long hard slog as well - I can't write a few words to explain it.

Can you derive Hamilton's principle from Feynman's sums? Do you have some paper on this?

Most of the more advanced QM texts do it - eg my reference Ballentine - QM A Modern Development does it on page 116-123.

In fact if you really really want to understand QM that is the book to get. It will take you a while to go through it but when finished you will be amazed what you understand - it really is that good.

However it not hard to see. If you take any path you can always find another path very close to it so that it is 180% out of phase with it and cancels it - except for one exception - where the action is stationary - which means close paths are in phase and reinforce and not cancel. That's the intuitive way of looking at it - if you want greater rigour you would use the method of steepest decent:
http://www.phys.vt.edu/~ersharpe/6455/ch1.pdf

The book by Omnes is a terrible reading. I have read superficially the 10th and 11th chapter and I can say I have not seen a bit of derivation of classical mechanics. He says that one can introduce operator on Hilbert space which can be used to define classical variable describing ensemble of classical systems.

As I said he does not do it - he merely states it can be done. Maybe not in Chapter 11 but elsewhere in the book he explains it requires some very deep math to do it. Again if that's what you want be prepared to some prolonged and deep study.

One thing I want to add - I sit in awe of you mate (that's my Aussi coming out). Its obvious you want to understand this stuff and are willing to do the hard yards - most would simply recoil and say its too hard.

Thanks
Bill

 

[bhobba]

If you feel it is true "in your bones", there is certainly no harm in that-- that's one kind of truth, certainly. It's just not the kind of truth that either mathematics or physics deals in, it's the kind of truth that people deal in-- so it is sociological! That is somewhat ironic.

I just want to be clear what I feel in my bones so to speak - its that the math reveals the hidden truth. Truth in science however is an experimental matter - but many times faith in the underlying mathematical simplicity and beauty of nature triumphed over what at the time seemed contrary experimental data eg the final triumph of gauge theories. It's what motivates the army of string theorists as well. I suspect quite a few of those guys may agree with me - and it has its critics as well precisely because of that.

Thanks
Bill

 

[Ken G]

For instance, in a two state system experiment, no individual measurement can be predicted, however, the ensemble averages can be predicted, then, are relative probabilities the only true observable?

I would say that we don't actually observe probabilities, we use them as an analysis tool to make sense of what we do observe (which is the outcomes of many trials). To have outcomes of many trials, we must first have outcomes of individual trials, so that has to be considered an observable: an outcome of an individual trial. That is how observables are characterized in quantum mechanics-- we take these single-trial observations and call each possible outcome an "eigenvalue" of the observable. Then its off to characterizing states and operators on those states, to characterize the measurement process.

To be sure, I mean true in this sense: While we can acquire numbers for individual measurements, those individual numbers where never being tested in the first place, but rather the expectation values were under scrutiny. Then the individual numbers are not true observables, but expectation values are.

We need more than expectation values, because expectation values are somewhat ancillary to the given preparation of a system that is characterized by some definite superposition of individual measurements giving a definite outcome (often a single such measurement and a single such outcome!). For example, to be able to talk about the expectation value of the energy of a system at some time t, we may need to know the exact energy at t=0, or at least the exact superposition of states of exact energy at t=0. We cannot map an expected energy at t=0 into an expected energy at some later t-- the expectation value for the energy is simply not enough information to determine that. So we need the concept of superposition, not just statistical distribution, and a superposition is more than a statistical distribution, it is an exact state (even though it does not have an exact value of the observable).

This is all in preparation for the question: Does nature care about individual numbers either? Or does nature only worry about large scale relative probabilistic structure also? I mean (I'm going to sound a little philosophical here), we are nature, probing ourselves through experiment, and we can only make conclusions about nature through large scale probabilistic structure.

You are touching on some of the deepest questions of interpreting quantum mechanics. The most orthodox interpretation is due to Bohr, and he held a view a lot like what you are saying-- he held that quantum mechanics was all about what macroscopic systems (like us) can say about microscopic systems. He said "there is no quantum world", meaning that all our language and theories about what we are calling the microscopic world could not exist independently of the macroscopic world of the scientists who are describing that microscopic world. But I don't think we should say that this is what "nature cares about", we need to talk about what we care about. It is normally harmless to anthropomorphize nature, but when you are at the frontiers of understanding what QM actually is or is not, I think it is important to be as precise as possible in the language you choose. You can say that we are nature, but more likely, we are a rather tiny subset of nature, and our cares need not be nature's cares in some larger sense that goes beyond us-- that is again reminiscent of Bohr's basic point.

 

[Ken G]

I just want to be clear what I feel in my bones so to speak - its that the math reveals the hidden truth. Truth in science however is an experimental matter - but many times faith in the underlying mathematical simplicity and beauty of nature triumphed over what at the time seemed contrary experimental data eg the final triumph of gauge theories. It's what motivates the army of string theorists as well. I suspect quite a few of those guys may agree with me - and it has its critics as well precisely because of that.

Yes, string theory, and cosmology of the very early universe, are two places where this issue becomes quite poignant, because these are the places where we are not restricting ourselves to doing proven physics, we are encountering the question of what physics is, or should be. I agree with you that ultimately the successes or failures of physics must be held to an experimental standard, so as long as string theory and the very early universe cannot be held to such a standard, it's kind of hard to know if they are really physics or not!

In that light, I have been seeing string theory seminars for about 20 years now, and I've noticed a very definite change in their tenor-- they started off all sounding like "this is the path to the theory of everything, we just know it in our bones", but now they sound much more like "here is a toy theory we are playing with and we are not really sure what it will lead to but we are hoping we will get some important insights out of it eventually." Indeed, the last one I attended was about using the AdS/CDF correspondence to allow gravitational theories (with weak gravity, i.e., a perturbative approach) to inform the strong force (with strong coupling, i.e., non-perturbative). It certainly is a cute idea to allow a duality to make a theory that does not seem to be conducive to perturbative analysis accessible via a theory that is, but it seemed quite ironic to me that one of the primary accomplishments of string theory to date is as a way to use gravity to understand quantum chromodynamics-- when string theory was initially billed as the means to understand gravity! Winners write the history.

So what does this mean about whether or not math is an ultimate truth? Max Tegmark speculates that every consistent mathematical structure is its own Platonic world that spawns its own universe, taking your views (and Penrose's) to their logical conclusion. My criticism of that view, in addition to its apparent ignoring of Godel's theorems, is that many of those mathematical structures cannot be associated with a universe that can produce intelligence. If the universe cannot produce intelligence, then it cannot calculate, so it cannot know or understand its own mathematical structure. Is a mathematical structure meaningful if it cannot support the concept of a mathematician? I don't think so-- I think that is an internally inconsistent conjecture. Does a chess game mean something without players? Do games exist that no one has ever played or even invented?

I feel we should not separate the mathematics from the mathematician, because there is no way to talk about one without the other, indeed we need the other to carry out any such discussion. Shall we let people who cannot even do algebra debate the truth of mathematics? No, the best we can do is involve the mathematicians-- demonstrating the fundamental fallacy of imagining that the mathematics has a truth that is independent of those who can understand what math is. Just as the players need the rules to play the game, the rules need the players to be a game.

 

[Hurkyl]

I do not recognize anything in these notions attributed to me that resembles my actual thoughts, can you clarify what distinctions you are making here?

As my premise is that the act of reasoning about 'reality' involves writing down a game, following the rules of the game, and interpreting the result in 'reality' -- especially with the notion that doing a good job of it involves coming up with a sufficiently detailed and accurate rule-set so that we don't have to make up additional rules as we go along -- at least superficially fits into your description of "finding the laws that govern nature".

I think willing to chalk things up to natural language simply being rather poor at conveying nuances of topics like this, and I'm reading different emphasis from the words than you're writing than the emphasis you intended to put into them.

 

[Ken G]

As my premise is that the act of reasoning about 'reality' involves writing down a game, following the rules of the game, and interpreting the result in 'reality' -- especially with the notion that doing a good job of it involves coming up with a sufficiently detailed and accurate rule-set so that we don't have to make up additional rules as we go along -- at least superficially fits into your description of "finding the laws that govern nature".

Sure, your way of interpreting the phrase "finding the laws that govern nature" can be a valid interpretation of that phrase, but it certainly isn't the standard one, nor is it the one I aimed my critique at. I have no objection to how you are interpreting it, that is quite demonstrably what we do. But people who hold that nature herself does actually follow laws, and that's the reason it works for us to look for laws, are using the standard meaning of that phrase-- which is that laws are actually part of nature, not something held up to nature as a kind of template or game. They hold that when nature decides what to do, she first says to herself (in effect), "now what do the laws say I must do here." That is what I am talking about, and pointing out pitfalls in. I'm not saying nature does or does not do that, for I have no idea what nature does, I'm saying that it is not in the best interests of physics to frame it in those terms-- at least not when we are probing it as deeply as we can (it's fine when we are speaking colloquially). Your terms, on the other hand, are much more careful, and I would agree are just exactly what we are doing-- and demonstrably so.

I think willing to chalk things up to natural language simply being rather poor at conveying nuances of topics like this, and I'm reading different emphasis from the words than you're writing than the emphasis you intended to put into them.

Yes, communication is the hardest thing, but if we iterate the process I'm sure we will succeed!

 

[Jano L.]

Bhobba,

I am afraid Ballentine does not derive Hamilton's principle at pages you gave. It would be great to discuss this further, but perhaps it is good to move to another thread. I made one here:

https://www.physicsforums.com/showthread.php?p=3937361#post3937361

Jano

 

[yoda jedi]

I am eager to see the derivation of classical mechanics.

and the inverse, quantum mechanics from classical physics.

Schrodinger equation may be derived
from Hamilton-Jacobi equation.
.

 

[Jano L.]

It may be guessed but hardly derived. The Schroedinger equation was constructed with classical mechanics and wave theory in mind, but is new and does not follow from classical mechanics.

 

[yoda jedi]

"The Schrödinger equation is shown to be equivalent to the classical Hamilton-Jacobi equation of motion plus the equation of continuity familiar in classical fluid dynamics and classical electrodynamics, with an additional term added to the potential energy called the quantum potential that is responsible for quantum effects"

"Any time-dependent solution of Schrodinger equation may be always correlated to
a solution of Hamilton equations or to a statistical combination of their solutions"

 

[juzzy]

Well, I have come back to this thread after a few days away and am amazed by the number of posts. From this very intelligent debate (which may be slightly over my head at times), I shall conclude that the statement 'we don't know' was a legitimate one to make. I came seeking an answer and have come away more confused than when I started, which is true of every new thing I learn in quantum mechanics. However, my gut feeling is that the universe is fundamentally probabilistic. By fundamental I mean the point at which our ability to compute breaks down. What is beyond there? Who knows?