Do physics books butcher the math?

[Arsenic&Lace]

Thread was split of from this thread: https://www.physicsforums.com/showthread.php?t=761954

Only read them if you get annoyed at how the physics books butcher the math and if you want to see how the math should actually be done.

Oops, I think you meant to say "Only read them if you are disatisfied with how physicists elegantly reduce the mathematics to be as simple as it needs to be rather than the grotesque overcomplications invented by mathematicians on account of the fact that their subject epistemically misunderstands what mathematics actually is."

 

[micromass]

Oops, I think you meant to say "Only read them if you are disatisfied with how physicists elegantly reduce the mathematics to be as simple as it needs to be rather than the grotesque overcomplications invented by mathematicians on account of the fact that their subject epistemically misunderstands what mathematics actually is."

No, I think butchering is the right word But hey, mathematicians and physicists are different. I totally accept that physicists don't use math very rigorously and that many steps are shady and sometimes criminal. What matters is that they get accurate predictions for experiments. Mathematicians however must rely on logic and they have to make sure that every step is justified and well defined. So both mathematicians and physicists use math, but have different goals. I just prefer the mathematicians perspective on things.

 

[atyy]

But what about Euler summing all the positive integers to -1/12 ?

 

[micromass]

But what about Euler summing all the positive integers to -1/12 ?

Yes, amazing result. But totally wrong and nonrigorous from modern standards. It can be made rigorous however.

This shows the genius of Euler actually. He did a lot of shady things with series which are always a bit wrong from modern standards. But he had an amazing intuition and always did get the right result. Any less good mathematician who would attempt the same thing would get total nonsense.

 

[dextercioby]

I also call it butchering. The most successful theory of physics (quantum electrodynamics) is sadly not mathematically sound.

 

[Arsenic&Lace]

I also call it butchering. The most successful theory of physics (quantum electrodynamics) is sadly not mathematically sound.

The fact that it is so extraordinarily successful speaks volumes about how meaningless mathematical soundness actually is.

 

[micromass]

The fact that it is so extraordinarily successful speaks volumes about how meaningless mathematical soundness actually is.

Mathematical soundness is indeed not necessary for physical theories to work and to be accurate. Physicists can do their job perfectly without being rigorous (and they might even do it better).

However, I think mathematical soundness is a philosophical property of a theory that is desirable. If a theory is mathematically rigorous, then it means we understand it completely and we have reduced it to pure logic. On the other hand, QED is not mathematically rigorous, which means to me that we understand how to do the calculations, but not really why the calculations work

A stupid example, but it is of course well known that we can calculate areas by inversing differentiation. It works fine and gives us all the results. But a theory that stated this principle without any justification, would be incomplete for me. The fundamental theorem of calculus shows us exactly why differentiation and areas are linked. So while the theory would work perfectly well without somebody ever proving the theorem, it would not be philosophically satisfactory.

I know you don't think much of mathematicians and mathematical theory. You are satisfied with knowing you can predict everything. However, you cannot deny that making a theory mathematically rigorous is something that humans should attempt to do. It is in our nature to understand the theory as well as we can, and a nonrigorous theory would not be as well understood as a rigorous one. The rigorization of a theory might not yield any applications, but I think it is wrong to do science only with the applications in mind. One should do it to try and understand nature better.

 

[atyy]

I also call it butchering. The most successful theory of physics (quantum electrodynamics) is sadly not mathematically sound.

Why not just put a high energy cut off, and put the theory in finite volume?

 

[atyy]

Yes, amazing result. But totally wrong and nonrigorous from modern standards. It can be made rigorous however.

This shows the genius of Euler actually. He did a lot of shady things with series which are always a bit wrong from modern standards. But he had an amazing intuition and always did get the right result. Any less good mathematician who would attempt the same thing would get total nonsense.

So physicists are like Euler, except that where he had exceptional intuition to guide him, physicists have experiments - ie. if it predicts experimental results, then the theory is a candidate for being made rigourous. I think doing it the other way round is much harder - if we did not know that QCD was experiemtally successful, it wouldn't make any sense to set rigourous Yang Mills as a Clay problem. So from this point of view, couldn't it be argued that physics is not so different from mathematics, but part of it?

 

[Fredrik]

Oops, I think you meant to say "Only read them if you are disatisfied with how physicists elegantly reduce the mathematics to be as simple as it needs to be rather than the grotesque overcomplications invented by mathematicians on account of the fact that their subject epistemically misunderstands what mathematics actually is."

I couldn't disagree more with this view. Physics books are filled with math that's been "simplified" to the point where it's impossible to understand because the definitions are painfully inadequate. It's been almost 20 years now, but the way physicists explained tensors to me still makes me angry every time I think about it.

 

[micromass]

So physicists are like Euler, except that where he had exceptional intuition to guide him, physicists have experiments - ie. if it predicts experimental results, then the theory is a candidate for being made rigourous. I think doing it the other way round is much harder - if we did not know that QCD was experiemtally successful, it wouldn't make any sense to set rigourous Yang Mills as a Clay problem. So from this point of view, couldn't it be argued that physics is not so different from mathematics, but part of it?

Yes, I agree. There are very intimate links between physics and mathematics. It can even be claimed that one is part of the other. If you haven't read this, you should: http://pauli.uni-muenster.de/~munsteg/arnold.html It's definitely controversial, but he has a point.

But yes. Any discovery in mathematics is usually done in a very informal manner first. It is only later that it is rigorized and that the relevant axioms are invented. So in that sense, mathematics is an experiment science, since we always look at concrete examples first and see what happens there. The way they teach mathematics in high school or university (give a sequence of axioms and definitions and then derive lemmas and useful theorems) is totally inverse to how it was discovered. The theorems are usually discovered first, then the lemmas, then the definitions and axioms are made.

Also, the following quote is nice:

Jean Bourgain, in response to the question, "Have you ever proved a theorem that you did not know was true until you made a computation?" Answer: "No, but nevertheless it is important to do the computation because sometimes you find out that more is there than you realized."

 

[Arsenic&Lace]

Mathematical soundness is indeed not necessary for physical theories to work and to be accurate. Physicists can do their job perfectly without being rigorous (and they might even do it better).

However, I think mathematical soundness is a philosophical property of a theory that is desirable. If a theory is mathematically rigorous, then it means we understand it completely and we have reduced it to pure logic. On the other hand, QED is not mathematically rigorous, which means to me that we understand how to do the calculations, but not really why the calculations work

A stupid example, but it is of course well known that we can calculate areas by inversing differentiation. It works fine and gives us all the results. But a theory that stated this principle without any justification, would be incomplete for me. The fundamental theorem of calculus shows us exactly why differentiation and areas are linked. So while the theory would work perfectly well without somebody ever proving the theorem, it would not be philosophically satisfactory.

I know you don't think much of mathematicians and mathematical theory. You are satisfied with knowing you can predict everything. However, you cannot deny that making a theory mathematically rigorous is something that humans should attempt to do. It is in our nature to understand the theory as well as we can, and a nonrigorous theory would not be as well understood as a rigorous one. The rigorization of a theory might not yield any applications, but I think it is wrong to do science only with the applications in mind. One should do it to try and understand nature better.

This implies that it might be possible to make QFT mathematically rigorous.

I am suspicious that this is possible. To you (I think? I do not want to caricature you mistakenly) y' = a*y is an object which has an existence independent of its applications. Because it is an independent object, it makes sense to climb the ladder of abstraction and think about such objects generally.

To me however it is meaningless until one begins to talk about an application. The equation is in the image of something found in reality. Fascinatingly this particular equation applies to backterial growth rates as well as it applies to nuclear decay, which are very different things, but this does not imply to me that this symbolic representation of real phenomena should suddenly spring to life and have a meaningful independent existence.

This explains why it is extremely challenging, as an example, to derive meaningful abstract relations between non-linear ODE's. They are symbolic representations of real world phenomena, and once the restriction of linearity is removed they have significantly greater freedom, as a symbolic language, to describe real phenomena, and as real phenomena from astrophysics to economics are varied and lack sweeping general principles so too do nonlinear ODE's. It also explains why it is easy to pen down an equation that describes nothing at all, and by describing nothing at all, is meaningless.

That is why I think it is plausible that no rigorous formulation may ever be composed for QFT, and also why rigorous formulations of physical theories typically add little to the progress of physics. Since mathematics is made in the image of real world phenomena and extremely high energy scales are so absurd to our intuition, that mathematics, which is the product of an object (the human brain) which evolved for and exists in a low energy environment, could be invented which is sensible to us seems unlikely. If not impossible, it is at least unnecessary. Again, in my
universe mathematics is only ever a reflection of something in reality; if that reality is fundamentally incomprehensible to us, the mathematics probably will be as well.

You may point out that non-relativistic quantum mechanics is completely rigorous or can be formulated in such a manner (to my knowledge anyway, I've never investigated it). But much of it is done in non-rigorous ways; think of the horrible delta function in everything from projection operators to the physicist's functional derivative. Even more troubling, the mathematics makes no intuitive sense, even in cases as basic as how physicists combine probabilites with the squared norm.

 

[micromass]

This implies that it might be possible to make QFT mathematically rigorous.

I am suspicious that this is possible. To you (I think? I do not want to caricature you mistakenly) y' = a*y is an object which has an existence independent of its applications. Because it is an independent object, it makes sense to climb the ladder of abstraction and think about such objects generally.

No, I don't think that at all. A mathematical object should not be divorced from its applications. The applications add a lot to the mathematics and give us ways to deal with it.

But every mathematics there is comes from some kind of application (even if the link is not very clear anymore). If there is no link with known applications or other known mathematics, then only a very few people will bother to study it. I personally am not interested in studying something that has no interesting applications.

You may point out that non-relativistic quantum mechanics is completely rigorous or can be formulated in such a manner (to my knowledge anyway, I've never investigated it). But much of it is done in non-rigorous ways; think of the horrible delta function in everything from projection operators to the physicist's functional derivative. Even more troubling, the mathematics makes no intuitive sense, even in cases as basic as how physicists combine probabilites with the squared norm.

Sure, much of quantum mechanics is done in very nonrigorous ways. From a mathematical point of view, most QM books are horrible. Quantum Mechanics has even been discovered in a very nonrigorous way. Still, we were able to completely rigorize it. And I think that rigorization is very interesting and reveals quite a lot. Indeed, if one does not know the rigorous version of quantum mechanics, then one will easily fall into various traps: http://arxiv.org/abs/quant-ph/9907069

So I am almost certain that the same thing will happen to QED.

Also, the mathematics makes perfect intuitive sense to me. I don't virtually nothing of physics (and certainly not of quantum mechanics). But it has happened often that my knowledge of math alone was sufficient to talk about quantum mechanics to physicists and even help them with issues they had. So the mathematical intuition I created while studying the math is not so far off from the physical intuition of the physicists. Not that I say that every physicist should start learning pure math, but I thought that was rather remarkable.

 

[Arsenic&Lace]

"Also, the mathematics makes perfect intuitive sense to me."

Fascinating. I'd love an intuitive explanation of quantum mechanics. Can we start with the Schroedinger equation? Why is that the equation describing the time evolution of a wave function?

EDIT: You've said the math makes sense. Well, sure, once you start with the fact that it is a PDE and you know what a PDE is everybody who understands PDE's "understands" the Schroedinger equation. But I would venture to guess that a true understanding of the mathematics would coincide with an understanding of why the mathematical superstructure was built the way it is.

 

[micromass]

"Also, the mathematics makes perfect intuitive sense to me."

Fascinating. I'd love an intuitive explanation of quantum mechanics. Can we start with the Schroedinger equation? Why is that the equation describing the time evolution of a wave function?

EDIT: You've said the math makes sense. Well, sure, once you start with the fact that it is a PDE and you know what a PDE is everybody who understands PDE's "understands" the Schroedinger equation. But I would venture to guess that a true understanding of the mathematics would coincide with an understanding of why the mathematical superstructure was built the way it is.

Sure. And the Schrodinger equation can be very neatly understood by using mathematics. For example, C*-algebras and operator theory gives me a very intuitive explanation of quantum mechanics and the Schrodinger equation. It shows that it's just classical mechanics which is made noncommutative.

 

[Arsenic&Lace]

Why would you want to make classical mechanics commutative (I'm also baffled that it would be commutative; momentum and position operators commute in classical mechanics but not in quantum mechanics, but I know nothing of C* algebras)?

 

[micromass]

Why would you want to make classical mechanics commutative (I'm also baffled that it would be commutative; momentum and position operators commute in classical mechanics but not in quantum mechanics, but I know nothing of C* algebras)?

Sorry, it should have been noncommutative. Classical mechanics is just the commutative version of QM in the C*-algebra formalism.

 

[Arsenic&Lace]

Sorry, it should have been noncommutative. Classical mechanics is just the commutative version of QM in the C*-algebra formalism.

Sure, but what about nature at the quantum scale necessitates non-communativity of the position/momentum operators?

One possible axiom is the uncertainty principle, but this just raises a deeper question: why is there some fundamental limit on the uncertainty of these operators? Why propose it in the first place?

 

[Fredrik]

Fascinating. I'd love an intuitive explanation of quantum mechanics. Can we start with the Schroedinger equation? Why is that the equation describing the time evolution of a wave function?

The problem is that these things are only intuitive to people who understand the mathematics really well. I'm only half-way there myself, but I understand enough to say that a person who understands topology, measure theory, integration theory, Hilbert spaces and operator algebras well enough to understand exactly in what sense QM is a generalization of probability theory, has a far better understanding of QM than a typical physicist.

 

[Fredrik]

Sure, but what about nature at the quantum scale necessitates non-communativity of the position/momentum operators?

To ask this is to ask why the universe is such that quantum theories makes more accurate predictions than classical theories. The only thing that can answer that is a better theory. (Better than all quantum theories). If we had such a theory, you'd probably be asking the same question about that theory instead.

 

[atyy]

Regarding QFT: why not make it rigourous just by putting it on a lattice in finite volume?

According to the Copenhagen interpretation, in QM we always divide the world into classical and quantum parts. Since the quantum part is only a subset of the universe, finite volume should be ok. Then by putting it on a lattice, we ensure that there are no ultraviolet divergences, since everything is now QM. So in this way, couldn't one make QFT rigourous?

I do understand that in 2+1 D one can make QFT rigourous without putting it on a lattice, and it is intellectually interesting to ask whether one can do the same for any 3+1 D QFTs, such as Yang Mills, which non-rigourous asymptotic freedom calculations suggest should be possible. But given that our experiments can't tell whether there is a fine lattice or not, why not just use the lattice and stop saying that QED is not rigourous?

 

[micromass]

Sure, but what about nature at the quantum scale necessitates non-communativity of the position/momentum operators?

One possible axiom is the uncertainty principle, but this just raises a deeper question: why is there some fundamental limit on the uncertainty of these operators? Why propose it in the first place?

It can be seen as an experimental fact, can't it?
I'm not claiming you can use math to invent new kinds of physics without ever needing experiments. I'm just saying that the mathematics of QM is quite intuitive to me and makes many parts of the physics of QM also intuitive to me. I do not claim to be an expert in QM or that I could invent it.

 

[Fredrik]

Regarding QFT: why not make it rigourous just by putting it on a lattice in finite volume?
...
But given that our experiments can't tell whether there is a fine lattice or not, why not just use the lattice and stop saying that QED is not rigourous?

This type of approach seems reasonable to me too. I don't know why exactly it's so important to find a rigorous version of QED on $\mathbb R^4$. I guess one reason is that if we e.g. use a box of finite volume with periodic boundary conditions to remove the infinities, we'd have one version of QED for each box size. This would be pretty ugly, but I think each of those theories would be fully rigorous, and experimentally indistinguishable from each other, if we only consider large enough box sizes.

On the other hand, I think a rigorous QED on $\mathbb R^4$ might turn out to be pretty awesome. It could e.g. improve our understanding of QFTs in a way that's similar to how non-commutative probability theory has improved our understanding of QM.

 

[WannabeNewton]

This thread is getting extremely off topic and I think it is very unfair to the OP.

Not everyone does physics for reproducible experimental results. A good number of people do it to work with, discover, and appreciate mathematical beauty and elegance in nature. As much as it might sting to admit it, pure math in the hands of mathematicians is absolutely indispensable for this if only to put a physical theory in a more coherent, compactified, and elegant theoretical framework. This is probably why GR is orders of magnitude more elegant than QFT even though QED is infinitely superior in experimental accuracy and reproducible/testable results. This does matter to some people and you can even find articles on it. As far as learning physics goes it is probably comletely irrelevant. Micromass' statements in post #2 are sound in that regard and there is quite literally nothing more to say than what he already said.

 

[micromass]

This thread is getting extremely off topic and I think it is very unfair to the OP.

Yes, I'll split the thread in two parts.

 

[MathematicalPhysicist]

Why not just put a high energy cut off, and put the theory in finite volume?

I second dexter on that, although I am still a novice, I feel there's too much handwaving there; and I am only at chapter 14 of Srednicki.

 

[MathematicalPhysicist]

I'll just add my 2 cents.

A physical theory should use the math rigourously, the axioms and hypotheses need not be logical or make sense; but the inference should be sound and logical, otherwise we can deduce anything we wish and our predictions for the experiments might as well be a gamble.

 

[atyy]

This type of approach seems reasonable to me too. I don't know why exactly it's so important to find a rigorous version of QED on $\mathbb R^4$. I guess one reason is that if we e.g. use a box of finite volume with periodic boundary conditions to remove the infinities, we'd have one version of QED for each box size. This would be pretty ugly, but I think each of those theories would be fully rigorous, and experimentally indistinguishable from each other, if we only consider large enough box sizes.

On the other hand, I think a rigorous QED on $\mathbb R^4$ might turn out to be pretty awesome. It could e.g. improve our understanding of QFTs in a way that's similar to how non-commutative probability theory has improved our understanding of QM.

Yes, it'd be ugly, but hopefully it would make physical and mathematical sense. I don't actually know how to get the QED perturbation series rigourously given UV and finite volume cut-offs, but DarMM's comments (linked below in my reply to MathematicalPhysicist) suggest it is possible. I agree it would be interesting to have rigourous QFT on $\mathbb R^4$. QED is suspected not to exist on $\mathbb R^4$ because of the Landau pole, but this belief could be wrong if QED is asymptotically safe. Most people guess that Yang-Mills is the best candidate for such a construction because it is asymptotically free.

I second dexter on that, although I am still a novice, I feel there's too much handwaving there; and I am only at chapter 14 of Srednicki.

I don't know whether the theory really exists, but from the comments of DarMM, a good lead to understanding may be:

https://www.physicsforums.com/showpost.php?p=4611431&postcount=45
https://www.physicsforums.com/showpost.php?p=4613526&postcount=47

1) There are rigrourous ways to produce formal perturbation series for QED. However, these formal perturbation series do not have non-formal, physical meaning unless the theory exists.

2) If a cutoff is put in place (maybe finite volume is also needed), the theory exists for low energies, and assiging the perturbation series a physical meaning makes rigourous mathematical sense.

3) This leaves the problem of the UV completion of QED open. In order to be practical, such a theory should probably also be a UV completion of gravity, since QED only fails above the Planck scale. Other UV completions such as asymptotic safety would also be conceptually interesting.

This seems very in line with the Wilsonian "effective field theory" point of view. I think it would be interesting to discuss this more properly. If anyone would like to, open a thread in QM, and hopefully dextercioby and DarMM can help out.

 

[WannabeNewton]

A physical theory should use the math rigourously, the axioms and hypotheses need not be logical or make sense; but the inference should be sound and logical, otherwise we can deduce anything we wish and our predictions for the experiments might as well be a gamble.

I see no reason as to why a physical theory should use math rigorously. It can that's no doubt but it's certainly not a requirement by any stretch. Rigor should be left to the mathematicians. I think physics would be extremely boring if it used math rigorously. I mean this is just my personal opinion but there is nothing more boring to me in physics than formal QM for exactly that reason. All the incessant talk of math obscures the physics and in the end it's the physics that is interesting, not the math. Math is just a tool. Learning QM would be so much more fun if books focused more on applications to Fermi statistics of metals, Bose statistics of radiation, the theory of classical lattice vibrations, magnetism at low temperatures etc. and not on the absolutely mind-numbingly boring mathematics of linear algebra, Hilbert spaces and such.

Physics books don't butcher math. They abuse it. There's a stark difference. It's needlessly elitist to assume that any math that isn't done at the same level of rigor as a pure math book is butchered math. Math done at that level is not necessarily useful it's just precise. More often than not it isn't useful.

 

[Arsenic&Lace]

The problem is that these things are only intuitive to people who understand the mathematics really well. I'm only half-way there myself, but I understand enough to say that a person who understands topology, measure theory, integration theory, Hilbert spaces and operator algebras well enough to understand exactly in what sense QM is a generalization of probability theory, has a far better understanding of QM than a typical physicist.

I submit that my mind will change very rapidly if you can make a short list of achievements by these mathematically enlightened individuals. I am but a humble undergraduate student; thinking through all of the major advances in quantum mechanics of which I am aware, I could think of none which relied on any of the disciplines you mentioned*.

*You might argue that the formalism in terms of Hilbert spaces was an achievement in and of itself. I concede that this increased the organization of quantum mechanics into a neater package; however, I do not consider this to be a great achievement or to be something which advanced our knowledge of the physical world.

 

[MathematicalPhysicist]

I see no reason as to why a physical theory should use math rigorously. It can that's no doubt but it's certainly not a requirement by any stretch. Rigor should be left to the mathematicians. I think physics would be extremely boring if it used math rigorously. I mean this is just my personal opinion but there is nothing more boring to me in physics than formal QM for exactly that reason. All the incessant talk of math obscures the physics and in the end it's the physics that is interesting, not the math. Math is just a tool. Learning QM would be so much more fun if books focused more on applications to Fermi statistics of metals, Bose statistics of radiation, the theory of classical lattice vibrations, magnetism at low temperatures etc. and not on the absolutely mind-numbingly boring mathematics of linear algebra, Hilbert spaces and such.

Physics books don't butcher math. They abuse it. There's a stark difference. It's needlessly elitist to assume that any math that isn't done at the same level of rigor as a pure math book is butchered math. Math done at that level is not necessarily useful it's just precise. More often than not it isn't useful.

Well I must say the clarity that I would like in physics textbooks, that it will be clear what follows from what and why, it's not always clear why for example eq 14.40 pops in when the equation before that had a term which doesn't appear in 14.40, whilst there is no explanation as to why is that. Sometimes it feels like equations are popping out of thin air without any explanation how did they come here, or are they arbitrary.

I myself don't like Bourbaki kind of books, but it's important to understand the derivation of stuff, cause otherwise what's the difference between believing the bible or physics book?

Mind you, also there are math book which are written badly or there's the option that the field in question is plagued with wrong proofs, this also can happen (quite often as well), quite worrying times we live...

 

[WannabeNewton]

Well I must say the clarity that I would like in physics textbooks, that it will be clear what follows from what and why, it's not always clear why for example eq 14.40 pops in when the equation before that had a term which doesn't appear in 14.40, whilst there is no explanation as to why is that. Sometimes it feels like equations are popping out of thin air without any explanation how did they come here, or are they arbitrary.

Well I don't disagree with you there at all, if that's what you meant by "rigorously". I was using the term to mean rigorous mathematics as one would find in a typical pure math book.

 

[WannabeNewton]

The problem is that these things are only intuitive to people who understand the mathematics really well. I'm only half-way there myself, but I understand enough to say that a person who understands topology, measure theory, integration theory, Hilbert spaces and operator algebras well enough to understand exactly in what sense QM is a generalization of probability theory, has a far better understanding of QM than a typical physicist.

This is an insane claim I must say. I don't personally know how much of the formal mathematics of QM Lev Landau knew but he uses no formal math in his QM book and that book clearly shows a brilliant mastery of QM at an incredibly intuitive level, more so probably than any other pedagogical QM book out there. I highly, highly doubt people who "understand exactly in what sense QM is a generalization of probability theory" have a better understanding of QM than Lev Landau. They simply know the precise mathematical structure of QM and all of the rigorous details behinds its constructions and mathematical subtleties. That's far from understanding the physics of QM and understanding it better than a proper physicist. That's like saying someone who understands exactly in what sense classical dynamics is a theory of a certain symplectic form on a configuration space manifold has a far better understanding of mechanics than a mechanical engineer. It's obviously not true in any stretch of the imagination.

 

[micromass]

This is an insane claim I must say. I don't personally know how much of the formal mathematics of QM Lev Landau knew but he uses no formal math in his QM book and that book clearly shows a brilliant mastery of QM at an incredibly intuitive level, more so probably than any other pedagogical QM book out there. I highly, highly doubt people who "understand exactly in what sense QM is a generalization of probability theory" have a better understanding of QM than Lev Landau. They simply know the precise mathematical structure of QM and all of the rigorous details behinds its constructions and mathematical subtleties. That's far from understanding the physics of QM and understanding it better than a proper physicist. That's like saying someone who understands exactly in what sense classical dynamics is a theory of a certain symplectic form on a configuration space manifold has a far better understanding of mechanics than a mechanical engineer. It's obviously not true in any stretch of the imagination.

Lev Landau was hardly your typical physicist.

Also, there are different levels of understanding. The understanding of mechanics by a mechanical engineer is different from understanding mechanics in the differential geometry sense.

 

[AlephZero]

Also, there are different levels of understanding. The understanding of mechanics by a mechanical engineer is different from understanding mechanics in the differential geometry sense.

Or in some cases, there is the same level of understanding, but different terminology.

It's entertaining to trace some of the things understood by engineers back in the 1950s that were "discovered" much later in papers on functional analysis, written in notation the engineers couldn't understand.

 

[atyy]

The problem is that these things are only intuitive to people who understand the mathematics really well. I'm only half-way there myself, but I understand enough to say that a person who understands topology, measure theory, integration theory, Hilbert spaces and operator algebras well enough to understand exactly in what sense QM is a generalization of probability theory, has a far better understanding of QM than a typical physicist.

I can rigourously prove that this is untrue (I think). If this is based on bhobba's claim that he frequently posts in the QM forum, then although he often links to Scott Aaronson's blog post, I believe he is thinking of http://arxiv.org/abs/quantph/0101012. But everything there is just finite dimensional Hilbert spaces, so no mathematical sophistication is needed. If one can understand Newtonian mechanics, one can understand bhobba's claim.

 

[Fredrik]

This is an insane claim I must say. I don't personally know how much of the formal mathematics of QM Lev Landau knew but he uses no formal math in his QM book and that book clearly shows a brilliant mastery of QM at an incredibly intuitive level, more so probably than any other pedagogical QM book out there. I highly, highly doubt people who "understand exactly in what sense QM is a generalization of probability theory" have a better understanding of QM than Lev Landau. They simply know the precise mathematical structure of QM and all of the rigorous details behinds its constructions and mathematical subtleties. That's far from understanding the physics of QM and understanding it better than a proper physicist. That's like saying someone who understands exactly in what sense classical dynamics is a theory of a certain symplectic form on a configuration space manifold has a far better understanding of mechanics than a mechanical engineer. It's obviously not true in any stretch of the imagination.

I said "a typical physicist", and you try to use Lev Landau as a counterexample? I was thinking of the people who taught QM and QFT at my university (and others like them). What I said is very different from suggesting that someone who understands symplectic geometry understands classical mechanics better than a mechanical engineer. If you're going to be rude, you should at least try to understand the comment you're responding to.

 

[atyy]

I submit that my mind will change very rapidly if you can make a short list of achievements by these mathematically enlightened individuals. I am but a humble undergraduate student; thinking through all of the major advances in quantum mechanics of which I am aware, I could think of none which relied on any of the disciplines you mentioned*.

*You might argue that the formalism in terms of Hilbert spaces was an achievement in and of itself. I concede that this increased the organization of quantum mechanics into a neater package; however, I do not consider this to be a great achievement or to be something which advanced our knowledge of the physical world.

For a start, if you don't know the Hilbert space formulation, you could never get to QFT. You'd be lost in Dirac's sea of antiparticles which is brilliant, confusing and wrong.

You would also never understand the Feynman path integral, whose derivation was again brilliant, confusing and wrong.

I think this "either-or" thinking is harmful. Mathematics itself often has non-rigourous beginnings. Newton's calculus and Fourier's decomposition are celebrated examples. But if science is to understand our world, and part of our world is our understanding, then understanding our understanding is part of science. You can see this interplay between rigour and natural language in Goedel's theorem, which is certainly rigourous, yet requires the intuitive natural numbers (or if one uses the natural numbers from ZFC, ZFC itself needs natural language to be defined).

 

[atyy]

While we are talking rubbish here , let me rigourously prove that the real fight is not between rigour and non-rigour, but between algebra and geometry.

Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine. —Sir Michael Atiyah, 2002 http://divisbyzero.com/2010/07/26/algebra-the-faustian-bargain/

 

[WannabeNewton]

I said "a typical physicist", and you try to use Lev Landau as a counterexample? I was thinking of the people who taught QM and QFT at my university (and others like them). What I said is very different from suggesting that someone who understands symplectic geometry understands classical mechanics better than a mechanical engineer. If you're going to be rude, you should at least try to understand the comment you're responding to.

Your claim was that people who knew rather esoteric aspects of QM formalism understood it better than working physicist. That is arguably a blanket assertion with only personal anecdotes serving as the nexus. If you want more mundane examples then I can confidently say that many of the HEPT and condensed matter theorists at my university understand QM and QFT much better than the people who pride themselves solely in delving into needlessly abstract formulations of said physical theories. There is nothing wrong with this of course as learning is learning and these people aren't necessarily claiming to know the subjects better than people who actually publish papers solving actual physical problems in their respective fields. A person can learn all they want about the background mathematical abstractions of a physical theory but that doesn't mean they can even remotely solve relevant physics problems in the theory and when I say problems I mean publishable ones. Anyways I didn't mean to come off as rude and apologize if I did.

Actually a very relevant example is my analysis 2 professor. He loves physics and knows quite an extensive amount of the formal mathematics behind both GR and QFT. But he never once claimed to understand QFT better than a physicist. In fact during our conversations he would always mention one of the HEP theorists at my university, Csaba, as the go to man of supreme QFT knowledge and intuitive understanding. That being said, "understanding" itself is an ambiguous term in this context as micromass rightly pointed out.

 

[WWGD]

The fact that it is so extraordinarily successful speaks volumes about how meaningless mathematical soundness actually is.

Why do you assume mathematics is/should be bound by the needs of physics (or biology, etc)? That may be the case for mathematical physics, but not for math as a whole, which has much broader scope?

 

[Arsenic&Lace]

For a start, if you don't know the Hilbert space formulation, you could never get to QFT. You'd be lost in Dirac's sea of antiparticles which is brilliant, confusing and wrong.

You would also never understand the Feynman path integral, whose derivation was again brilliant, confusing and wrong.

Well I don't want to get hoist by my own petard (if that's the right expression), but unless I'm mistaken, the preferred modern formulation of QFT is not the canonical formulation with Hilbert spaces and what have you, but in terms of path integrals.

The path integral formalism is still "wrong"; only for the real time case has it been put on a rigorous footing, a relatively long time after the "confusing and wrong" intuitive argument (which is hardly confusing) was used to generate it.

Why do you assume mathematics is/should be bound by the needs of physics (or biology, etc)? That may be the case for mathematical physics, but not for math as a whole, which has much broader scope?

Earlier in the thread I argued that mathematics is meaningless outside the context of applications. Mathematics is a tool humans invent to solve problems. If a group of people call themselves experts in this subject are ignored without consequence by everyone else, they can hardly be called experts can they? Indeed, this empirical fact calls into question their entire academic enterprise. Granted one can find various fruits produced by individuals who just want to think about differential equations and not their applications, but these are generally ancient (1-2 hundred years ago!) and often in the spirit of applied, not pure mathematics.

I don't know, I'm not one to say that we should stop funding all math departments, collect all of the the wrinkly math professors and throw them unceremoniously from the top of their ivory towers to a mob of torches and pitch forks below. However I think any academic discipline should be subjected to criticism about its relevance.

 

[atyy]

Well I don't want to get hoist by my own petard (if that's the right expression), but unless I'm mistaken, the preferred modern formulation of QFT is not the canonical formulation with Hilbert spaces and what have you, but in terms of path integrals.

The path integral formalism is still "wrong"; only for the real time case has it been put on a rigorous footing, a relatively long time after the "confusing and wrong" intuitive argument (which is hardly confusing) was used to generate it.

Yes, you are mistaken. The modern formulation of QFT is in terms of Hilbert spaces etc. The path integral is good for calculation, but it is because it can be related to the Hilbert space formulation (eg. via the Osterwalder-Schrader conditions) that the path integral is quantum mechanics. Take a look at http://www.rivasseau.com/resources/book.pdf (p17).

 

[Arsenic&Lace]

Yes, you are mistaken. The modern formulation of QFT is in terms of Hilbert spaces etc. The path integral is good for calculation, but it is because it can be related to the Hilbert space formulation (eg. via the Osterwalder-Schrader conditions) that the path integral is quantum mechanics. Take a look at http://www.rivasseau.com/resources/book.pdf (p17).

Hang on, so the farthest I've gotten was a graduate course in non-relativistic quantum mechanics, and the professor (who is both a mathematician and physicist) stated that the entirety of non-relativisitic quantum mechanics can be formulated in terms of path integrals with no reference whatsoever to esoteric Hilbert spaces.

It seems to me that you have confused the equivalence of two formulations of the same thing with the idea that they are the same. Lagrangian and Newtonian mechanics are not the same, much as path integral quantum mechanics is not the same as canonical quantum mechanics, although they are equivalent.

Honestly I'm too stupid to understand that eloquent math jargon in the rivasseau page you linked to, so somebody will have to explain whether or not it is merely showing that they are equivalent or identical.

 

[atyy]

Hang on, so the farthest I've gotten was a graduate course in non-relativistic quantum mechanics, and the professor (who is both a mathematician and physicist) stated that the entirety of non-relativisitic quantum mechanics can be formulated in terms of path integrals with no reference whatsoever to esoteric Hilbert spaces.

It seems to me that you have confused the equivalence of two formulations of the same thing with the idea that they are the same. Lagrangian and Newtonian mechanics are not the same, much as path integral quantum mechanics is not the same as canonical quantum mechanics, although they are equivalent.

Honestly I'm too stupid to understand that eloquent math jargon in the rivasseau page you linked to, so somebody will have to explain whether or not it is merely showing that they are equivalent or identical.

The professor was wrong. People used to say things like the path integral is a new formulation of QM, analogous to Lagrangian and Newtonian mechanics. But that is untrue. The Hilbert space formulation is the primary formulation of QFT. See eg. Weinberg's QFT text.

If you are not learning the Hilbert space formulation of QM, you are not learning QM.

 

[Arsenic&Lace]

Well all of my QFT knowledge is self-taught and therefore dubious, but my understanding is as follows:

Step 1: Pen down the Lagrangian of your theory what has the right units and is Lorentz invariant and gets you the correct equations of motion.
Step 2: The path integral with this action as argument gives you the propagator for whatever reaction your interested in; add Dirac terms for the particles which are interacting and then prepare for the tedious process of actually computing it.
Step 3: Carry out the perturbative expansion as a weighted sum over histories of fields.

This formalism, which according to Wikipedia at least is distinct from the canonical formalism, does not depend upon knowledge of what a Hilbert space is. Of course you can object and say that I'm just computing a single matrix element of the S-Matrix, but this picture of what the result actually refers to does not displace the alternate (and frankly much prettier) picture of the sum over histories.

Coincedentally, I have my copy of Feynman's book on QM and path integrals, and the index does not have an entry for Hilbert spaces.

EDIT: That said Weinberg is a bright fellow, and I don't have his text. Can you paraphrase or quote him about why he thinks the Hilbert space formalism is the "true" formalism?

 

[atyy]

Well all of my QFT knowledge is self-taught and therefore dubious, but my understanding is as follows:

Step 1: Pen down the Lagrangian of your theory what has the right units and is Lorentz invariant and gets you the correct equations of motion.
Step 2: The path integral with this action as argument gives you the propagator for whatever reaction your interested in; add Dirac terms for the particles which are interacting and then prepare for the tedious process of actually computing it.
Step 3: Carry out the perturbative expansion as a weighted sum over histories of fields.

This formalism, which according to Wikipedia at least is distinct from the canonical formalism, does not depend upon knowledge of what a Hilbert space is. Of course you can object and say that I'm just computing a single matrix element of the S-Matrix, but this picture of what the result actually refers to does not displace the alternate (and frankly much prettier) picture of the sum over histories.

Coincedentally, I have my copy of Feynman's book on QM and path integrals, and the index does not have an entry for Hilbert spaces.

Yes, Feynman didn't understand path integrals and QM as well as we do now. The path integral is a very powerful formalism, and one can use its power without understanding its Hilbert space underpinnings, just like one can drive a car without knowing how the engine works.

EDIT: That said Weinberg is a bright fellow, and I don't have his text. Can you paraphrase or quote him about why he thinks the Hilbert space formalism is the "true" formalism?

He basically says everything from QM carries over to QFT, then proceeds to lay down the standard axioms including states in Hilbert space, observables as operators, wave function collapse etc.

 

[George Jones]

Coincedentally, I have my copy of Feynman's book on QM and path integrals, and the index does not have an entry for Hilbert spaces.

Feynman took great pleasure in doing things in non-standard ways. This doesn't mean that one should ignore the standard techniques, nor does it mean that one should ignore Feynman's techniques. Picking one extreme or the other is just being too simplistic and naive.

EDIT: That said Weinberg is a bright fellow, and I don't have his text. Can you paraphrase or quote him about why he thinks the Hilbert space formalism is the "true" formalism?

For Weinberg's somewhat nuanced views, read carefully all of the attached two pages,

 

[Arsenic&Lace]

Feynman took great pleasure in doing things in non-standard ways. This doesn't mean that one should ignore the standard techniques, nor does it mean that one should ignore Feynman's techniques. Picking one extreme or the other is just being too simplistic and naive.



For Weinberg's somewhat nuanced views, read carefully all of the attached two pages,

Hm, that is very interesting indeed, thank you for posting. The final few paragraphs are most important since they stress the complementary nature of the canonical and path integral formalisms.

I will stress then that I do not deem Feynman's non-standard approach to be superior or even to entirely supplant the canonical approach, but rather that the argument that the canonical formalism is somehow the deeper of the two formalisms seems too simplistic and naive a view.

 

[micromass]

If a group of people call themselves experts in this subject are ignored without consequence by everyone else, they can hardly be called experts can they? Indeed, this empirical fact calls into question their entire academic enterprise. Granted one can find various fruits produced by individuals who just want to think about differential equations and not their applications, but these are generally ancient (1-2 hundred years ago!) and often in the spirit of applied, not pure mathematics.

I hope you realize that by far most theoretical physics research done today will end up being ignored without consequence by engineers. So your remark doesn't only apply to mathematics but physics also. Please don't think that just because physicists study nature, that their work is actually useful.

rather that the argument that the canonical formalism is somehow the deeper of the two formalisms seems too simplistic and naive a view.

Why? Because it doesn't agree with your hate of pure mathematics and everything to do with it?

Also, Hilbert spaces are not esoteric. They're a very standard object.