The use of probability in QM

bhobba

Well in my case what suckered me into physics from a math/computer science background was Noethers Theorem. Once you understand what it says you realize all this stuff you learnt at school about conservation of energy etc etc that is taken as handed down from God is really saying nothing - its simply a tautological statement about symmetry - energy is the conserved current associated with time symmetry - like Ohms Law it really says nothing - yet has these profound consequences. At first sight the theorems of mathematics seem to say nothing about things out there, but just add a little smidgen of interpretation and profound results quickly emerge eg applying invariance to systems states a vola - QM emerges.

I do feel lucky that my particular math interest at uni was functional analysis which is really handy for QM - but it still took me 10 years of part time study until I was comfortable with all the math involved such as Rigged Hilbert Spaces, The Generalised Spectral Theorem etc - and this is just bog standard QM - QFT is a whole new ball game.

Thanks
Bill

Hurkyl

I had been considering talking about formalism, and this is about as good of a set-up line as I can get, so....


Logic and reasoning is a game. If I have "A" and "A implies B" in some region of play, then I can play the "modus ponens" card to allow me to place "B" in that region. There is no deep reason why "B" should be placed there; it's just the rules of the game.

The basic idea is that you construct the game so that we can interpret the game as something else. For example, I could play a game where I have two dots and a bunch of lines on a sheet of paper, and I look for a short path between the dots. If I've chosen the game board well, I can go out, get in my car, and interpret the path I just drew as a route I should follow in order to get someplace I want!


This method of transforming a problem I care about into another one I can work with is one of the most basic notions of reasoning. In fact, unless we adopt an extreme form of solipsism, it's more or less forced upon us; e.g. I don't actually get to reason about the apple sitting on the table: I am only capable of reasoning about the abstract notion my brain has synthesized from my sense of sight and past experiences with things my brain has called apples and tables. In fact, "the apple sitting on the table" is already part of that abstraction!


But, delving into that topic is somewhat of a tangent. The point is, when we want to reason about something, we create a game, along with an interpretation of that game into something else. Mathematicians often create a game where that "something else" is another sort of mathematical object. Physical scientists play games where the interpretation is into 'reality'. And so forth.

When we do a good job with the level of detail and the rules of the game, we are able to play the game to completion, and our interpretation of the results of the game accurately reflects the thing we were trying to reason about.


For the purposes of reasoning about certain aspects of 'reality', quantum mechanics is a rather good game to play. There is a meta-game that involves deciding which game to play in order to reason about said aspects of 'reality'. Currently, the best known strategy for the meta-game is "play quantum mechanics". There is even a meta-meta-game about how to go about finding strategies for the meta-game. The best known one for that is "play science".

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.

Jano L.

Law of conservation of energy and Ohm's law are not saying nothing! They are not tautologies either.

They are physical laws derived from experience, so-far verified in many circumstances, the first having no known deficiency, the latter being a good but limited description.
The law of conservation of energy is the first law of thermodynamics. Its scope is far broader than that of some theorem of Lagrangian mechanics.

Of course, Noether's theorem is a great theorem :

if the Lagrangian does not depend on time, we can find from it certain mathematical expression E that is conserved in the course of time.

However, the words "if" and "Lagrangian" are necessary parts of that sentence.

Nature can be described differently - in thermodynamics, there is no Lagrangian. But there is energy. Furthermore, tommorow Mr. X may discover that the energy is not conserved in certain special chemical/nuclear reaction. Noether's theorem would not be harmed at all. It is just a mathematical theorem, not a physical law.

bhobba

I beg to differ. Devices like diodes exist that do not obey ohms law. Basically ohms law applies to devices that - well - obey ohms law. All it is saying is devices exist that to a good approximation obey it - hardly a law of nature.

Energy conservation is the same - it does not necessarily apply in non inertial frames - Noethers Theorem basically says its logically equivalent to time symmetry of the Lagrangian - its content is the same as Ohms Law - systems exist which have that symmetry - specifically inertial frames do. Again I don't think the existence of an inertial frame is what people would count as a law of nature - although its something pretty fundamental - but not the type of thing you usually say is a law of nature - its like the existence of atoms is rather fundamental but I don't think it is a law of nature.

The real physical content of the conservation laws implied by Noethers Theorem is what goes into it - namely the principle of least action - that's is the law - physical systems are expressible that way. The reason for that is QM - it follows from Feynmans sum over history approach.

If energy was discovered not to be conserved, and the system had time translation symmetry, it would be profound discovery - basically casting doubt on QM.

Thermodynamics has no Lagrangian - that's news to me - it deals with systems of particles so large you need statistical methods - but in principle the system has a Lagrangian - as you would expect since it uses the concept of phase space.

Noethers Theorem is not a physical law - but it illuminates what actually is a physical law.

Thanks
Bill

Jano L.

Bhobba,

your views are very mathematical and formal. In physics, these are very important, but there are different aspects too. I will try to comment:

Why hardly? Most of laws were formulated for some special situations. We can never be sure that there is a new set-up which will make the law inapplicable.

Ohm's law applies to special situation, current in a metal conductor. Semiconductors behave differently, so we have to formulate a different law for them.

Why not? The fact that it is possible to use position and velocity for mathematical description of body is a general nature of the world that does not presently follow from anything simpler. Sometimes it is called a law - the First Law of mechanics.

Why not? If it were not law of nature, what would it be? Mathematical axiom? theorem? The existence of atoms was supported by experiments, and it required quite an effort. The result is some general knowledge about nature. Atoms are not as clear as rigid body in mechanics, but they explain a lot. They express a way of Nature - why not call it a law?

If you do not take Ohm's law as a physical law, why take this principle? It says that there are physical systems that are expressible that way. It is all the same.

The principle of stationary action has the same content as differential equations of motion. It is just interesting mathematically, but it is no more fundamental.

It is the same as in optics. Fermat's principle is no more fundamental than the laws of reflection and refraction.


Hardly. Feynmans sums usually have no sound mathematical definition. Every once in a while people cheat by subtracting infinities. At present we cannot derive classical physics from it.

I meant classical thermodynamics. There are no particles, statistical methods or Lagrangian. But there is work, heat and energy. We can hope that mechanical explanation of these concepts is possible, but this remains to be achieved or disproved. At present, thermodynamic energy and mechanical energy cannot be said to be the same thing.

bhobba

Come again. I suspect you are thinking of renormalisation which is something different. Yea for quite a while a rigorous definition of the path integral was lacking - but Hida Distributions have now solved that (as it would happen they are also an interest of mine although its been a while since I delved into it).

Also, although I have not seen the details I have read where classical mechanics has now been completely derived from QM - although only recently.

As to the other stuff - I think its an issue of what you count as a law of nature - I simply do not agree the stuff you cite is.

The Principle Of Least Action is not a law of Nature in my way of thinking - it is derivable from more fundamental laws - the real law is the laws of QM.

Yea - my approach is mathematical and formal - its obvious such an approach is not to your liking - which is OK.

Thanks
Bill

Jano L.

Here is what experts say:

http://www.scholarpedia.org/article/...tical_problems

I have never heard of Hida distributions. Do you think these solve the difficulties with divergences?

I am eager to see the derivation of classical mechanics. That would be something. Can you post a link to an article where you have read about it?

It is not only about what we like - I hope the discussion serves more than just an interchange of opinions - the issue can be argued about.

I propose we cannot use mathematics to circumvent physics and explain the world. There are examples of useless mathematics and formalisms gone astray, and there are beautiful explanations in physics that require almost no mathematics.

bhobba

The infinities is the re-normalisation issue - not the existence of the path integral - which as the article explains is another issue.

Its nothing to do with that - its to do with a rigorous definition of the path integral in a mathematically proper way - its difficult and technical - I was simply fortunate I had an interest in it prior to concentrating more on physics - here is a link - but don't be too worried if its obscure:
http://arxiv.org/pdf/0805.3253v1.pdf

The article you linked to also explains about Hida Distributions and how they solve the existence issue so I am scratching my head why you have not heard of them before.

The issue with infinities has to do with QFT. Basically what was obscure when guys like Feynman developed re-normalisation to cope with it has now been clarified - its got to do with a really bad choice of parameter to perturb about - a really lousy choice because it turns out to be infinity - when you replace it with a parameter that is small everything is fine - re-normalisation is basically a trick that allows you to do that. Check out:
http://arxiv.org/pdf/hep-th/0212049.pdf

Understanding Quantum Mechanics - Roland Omnes - Chapter 11.

Mathematics is the language physics and in and of itself explains nothing. However by viewing things in the simplest most transparent mathematical way much greater elegance and a deeper understanding results - to the point those exposed to it think it is this way of looking at it is what's really going on. For example Noethers Theorem, and other stuff, has shown that symmetry is the real key:
http://www.colorado.edu/philosophy/v...thing/Laws.pdf

Thanks
Bill

Ken G

Yes, it's quite an amazing thing. Yet this is also exactly the reason that I reject the idea that "God is a mathematician" explains why mathematics works so well in physics. I take the opposite lesson-- if symmetries in temporal translation allow us to use Newton's laws to identify the proper way to define energy so that it will be conserved, and if symmetries in spatial translation allow us to use Newton's laws to identify the proper way to define momentum so that it will be conserved, then we see the simplicity behind the conservation laws is the simplicity of the idealizations we put into the physics. It's all coming from us, we choose to imagine that we have these symmetries, even though we know we really don't (symmetries were made to be broken). There is no place in the universe where we can really do these translations without any consequence, it's all an idealization that we have built into our "laws." "God" doesn't get to use those idealizations, he/she/it must deal with the actual reality! So we are the mathematicians, not "God"-- we put something simple in, and we get something simple out.

Framed this way, the question is not just why does math work, that's easy (it works because we started with mathematical assumptions, so we can finish with mathematical conclusions), but it is more general: it is why is the universe conducive to idealization? And I think the answer to that might just be that the potential number of situations that are conducive to idealization vastly outnumber the potential number that require a detailed analysis. In other words, perhaps it is easier to come up with universes that separate the scales of the various phenomena, making them conducive to idealization, then it is to come up with universes in which all the phenomena compete on similar scales. Or if that is not generally true, then it might just be that we have learned by experience to automatically ask the kinds of questions that are suitable to idealization-- those that are not are simply not questions that we try to use physics to understand (like human behavior).

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.

Ken G

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

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 realise 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

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

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

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.

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

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

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