Do physics books butcher the math?

[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 :tongue: 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.

 

[atyy]

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.

That is unfortunately wrong. I am saying this as clearly as I can, because I too thought the same way as you did due to the misinformation that is out there. Quantum mechanics is about operators, Hilbert spaces, wave function collapse. One should know the path integral as a powerful and less fundamental tool.

There are some research approaches in which the path integral is considered primary, but these are research approaches, and not yet textbook. What I am saying is the textbook understanding.

 

[atyy]

If you wish to have an argument about needless mathematical sophistication in the foundations of QM, I suggest you pit C* algebras versus Hilbert spaces

 

[micromass]

If you wish to have an argument about needless mathematical sophistication in the foundations of QM, I suggest you pit C* algebras versus Hilbert spaces

Ah come on The C*-algebra approach is extremely elegant and beautiful. It clarifies a lot of why certain things are done the way they're done in QM. I'm not saying we should teach into physicists, but for theoretical purposes, the C*-algebra approach is the most fundamental.

 

[WannabeNewton]

If you wish to have an argument about needless mathematical sophistication in the foundations of QM, I suggest you pit C* algebras versus Hilbert spaces

The C* algebra approach isn't any more sophisticated than the Hilbert space approach. It's just non-standard. But as micromass says it is much more elegant and certainly infinitely better in relating the state space structure of classical mechanics to that of QM.

The amount of effort one would have to put into a functional analysis class to understand QM at a deep mathematical level serves the double purpose of preparing one to jump right into the C* algebra formulation.

 

[atyy]

Well, I didn't say which was more sophisticated or pointless :tongue:

 

[Arsenic&Lace]

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.



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.

Theoretical physics is more than particle theory. For instance, I am doing theoretical/computational physics; the problem of interest is Brownian motion on a network. It turns out that the "statistical mechanics" of complex networks has far reaching applications on everything from systems biology to social networks.

To get more esoteric, look at the history of condensed matter theory. There are numerous cases where very esoteric physics and somewhat less esoteric physics is directly plugged into major applications, from transistors to quantum computers. The culture of materials engineering, for instance, looks a bit like the culture of physics relative to mathematics; when they are working on a project, with market pressures etc they are perfectly content to do linear regressions on massive piles of data, not understand what is fundamentally going on, and push out a (perfectly good) product. But periods of incremental growth are punctuated by critical advancements which require basic research in materials science and yes, materials physics.

On the extremely esoteric side, one can conceive of applications all the same. The barrier to using knowledge of particle physics (which already has industrial applications!) is that it is not easy to build a particle accelerator that can reach high energies; yet recent and continuing advancements in things such as plasma wakefield generators and competitors could drastically decrease the size of these devices.

And yeah, General Relativity, that most esoteric of creatures, has a very important application; faster than light travel. Stop laughing! If you want to determine if faster than light travel is possible, and if it is, implement it, you need general relativity to do so. I think it even has lower level applications such as to sattelites, although engineers often ignore the fancy math and just use some kind of Newtonian hybrid.

I don't actually hate pure math, I've taken many such courses, some of which I hated, some of which I enjoyed. It's like a series of interesting puzzles. I am signed up for a course in algebraic topology next semester which I fully expect to be completely useless but which I hope I will enjoy; I've already worked some problems from the book and they were fun!

That is unfortunately wrong. I am saying this as clearly as I can, because I too thought the same way as you did due to the misinformation that is out there. Quantum mechanics is about operators, Hilbert spaces, wave function collapse. One should know the path integral as a powerful and less fundamental tool.

Did you read the Weinberg paper posted by George Jones? His first concern was that the unitarity of the S matrix is not apparent from the path integral formalism. That no one has derived this fact does not mean that it cannot be done. Even still, while it is mathematically important, if the path integral formalism agrees with experiment, I'm not sure how much I care whether or not you prove that the S-matrix is unitary until you run out of experiments and begin to speculate, say in quantum gravity.

The second point he makes is that a naive, simplistic application of the Feynman rules can produce wrong results for one model, the non-linear sigma model and presumably others. However he never claims that the Feynman rules cannot produce correct results.

To me this looks like two powerful, complementary views, one which is more rigorous (but still nowhere near what a mathematician would find satisfactory I'd wager) and one which is less, with neither subsuming the other.

 

[atyy]

Did you read the Weinberg paper posted by George Jones? His first concern was that the unitarity of the S matrix is not apparent from the path integral formalism. That no one has derived this fact does not mean that it cannot be done. Even still, while it is mathematically important, if the path integral formalism agrees with experiment, I'm not sure how much I care whether or not you prove that the S-matrix is unitary until you run out of experiments and begin to speculate, say in quantum gravity.

The second point he makes is that a naive, simplistic application of the Feynman rules can produce wrong results for one model, the non-linear sigma model and presumably others. However he never claims that the Feynman rules cannot produce correct results.

To me this looks like two powerful, complementary views, one which is more rigorous (but still nowhere near what a mathematician would find satisfactory I'd wager) and one which is less, with neither subsuming the other.

What do you even mean by a unitary S matrix if there is no Hilbert space formulation? The Hilbert space formulation is fundamental, because every path integral formulation that is "equivalent to QM" is equivalent because it can be shown to have a Hilbert space formulation.

 

[Arsenic&Lace]

What do you even mean by a unitary S matrix if there is no Hilbert space formulation? The Hilbert space formulation is fundamental, because every path integral formulation that is "equivalent to QM" is equivalent because it can be shown to have a Hilbert space formulation.

The S-Matrix is an object which belongs to the canonical formulation; the observable of interest is usually the differential crossection. You can obtain this via the path integral formalism without ever reference the concept of the S-Matrix. It would seem that Weinberg is judging the path integral formalism through the lens of canonical quantum field theory, which seems to be a mistake.

The equivalence of the path integral formalism to QM is more accurately demonstrated by its agreement with experiments in QM.

 

[atyy]

The S-Matrix is an object which belongs to the canonical formulation; the observable of interest is usually the differential crossection. You can obtain this via the path integral formalism without ever reference the concept of the S-Matrix. It would seem that Weinberg is judging the path integral formalism through the lens of canonical quantum field theory, which seems to be a mistake.

The equivalence of the path integral formalism to QM is more accurately demonstrated by its agreement with experiments in QM.

I think we shall have to disagree at least temporarily.

But since the path integral has no abstract maths, could you please explain to me in an intuitive way how the path integral deals with fermions?

 

[Arsenic&Lace]

I think we shall have to disagree at least temporarily.

But since the path integral has no abstract maths, could you please explain to me in an intuitive way how the path integral deals with fermions?

What do you mean? Are you saying "How can one derive spin-statistics from the path integral formalism?" or are you asking "How do you calculate a propagator for spin 1/2 particles?"

 

[WannabeNewton]

What do you mean? Are you saying "How can one derive spin-statistics from the path integral formalism?" or are you asking "How do you calculate a propagator for spin 1/2 particles?"

Well no I think atyy is simply asking how you would even define fermions without the notion of a Hilbert space. The elementary ones are mode excitations of spinor fields that can only be defined abstractly by multi-particle states in a Fock space.

Also I don't kind quite understand, atyy, in what sense the canonical formulation of qft is more fundamental than the path integral formulation. Weinberg does mention the manifest unitarity of the canonical formalism vs. its non-triviality in the path integral formulation but not impossibility.

 

[Arsenic&Lace]

Well no I think atyy is simply asking how you would even define fermions without the notion of a Hilbert space. The elementary ones are mode excitations of spinor fields that can only be defined abstractly by multi-particle states in a Fock space.

I had to ponder this one for a while. The weakest reply I can make is that the Wikipedia articles on http://en.wikipedia.org/wiki/Fermionic_field[/URL] and [URL="the path integral formalism"]http://en.wikipedia.org/wiki/Path_integral_formulation[/URL] either make no reference to Fock spaces/Hilbert spaces or only reference them to point out that there is an alternative formulation.

The stronger reply is that it is easier to see in the canonical formalism that the Dirac equation has solutions which correspond to spin 1/2 particles. Once you are aware of this, you know that the field solution to this equation is what you need to quantize. Perhaps if you were really bored on a rainy day you could try to see if the fermionic nature of its solutions could be extracted without ever thinking about the canonical formalism (if this is obvious, feel free to point it out, I gave up after lazily thinking about it for 5 minutes). The mathematical technology of Hilbert spaces/Fock spaces does not ever need to be mentioned when solving the Dirac equation. Ah but gamma matrices obey a Clifford algebra, and they've got a basis which consists of Pauli matri--hold on a minute! It may be true that there is a rich underlying algebraic structure to these objects, but that is a feature of the symmetries of the theory, which applies equally to the canonical formalism as it does to the path integral formulation.

Peskin and Schroeder has a somewhat compact section on the functional quantization of the Dirac field which makes no reference to Hilbert Space/Fock space technology. For this reason, I would venture to say that one can define a Dirac fermion as a solution to the Dirac equation. I can then proceed without ever thinking about these more esoteric mathematical objects (and yes Micromass, I agree that Hilbert spaces are really not that exotic, but you must understand that to a physics major like me they were once pretty bizarre)

There seems to be some confusion as to whether or not the notion of a Hilbert space is equivalent to the canonical formalism. To me the canonical formalism is an algebraic approach to QFT; it is an algebraic perspective, where as the path integral formalism is a more analytic perspective. The propagator between two states is often sandwiched between two kets, which are the vanguards of a Hilbert space if anything. Functions are often expanded in terms of orthonormal eigenfunctions; another concept of linear algebra. To me it is not whether or not these concepts are used which makes the Hilbert space more fundamental; it is whether or not the algebraic perspective is adopted wholesale. The overwhelming majority of the work done in the path integral formalism makes little to no reference to this algebraic alternative, and does not leverage the advantages of this point of view.

 

[atyy]

Well no I think atyy is simply asking how you would even define fermions without the notion of a Hilbert space. The elementary ones are mode excitations of spinor fields that can only be defined abstractly by multi-particle states in a Fock space.

Also I don't kind quite understand, atyy, in what sense the canonical formulation of qft is more fundamental than the path integral formulation. Weinberg does mention the manifest unitarity of the canonical formalism vs. its non-triviality in the path integral formulation but not impossibility.

What I mean is that not every path integral defines a quantum theory. We only accept a path integral theory as a quantum theory if it has an equivalent Hilbert space (or C* algebra) formulation. If one could write a path integral formulation of a quantum theory which has no Hilbert space formulation, then one can say that the path integral formulation is more fundamental. At this stage, since the Hilbert space formulation is needed to define an acceptable path integral quantum theory, the Hilbert space formulation is more fundamental.

 

[atyy]

What do you mean? Are you saying "How can one derive spin-statistics from the path integral formalism?" or are you asking "How do you calculate a propagator for spin 1/2 particles?"

What I like about the path integral formulation is everything is quite classical. For QM it's classical particle trajectories, for bosonic QFT it's classical field configurations. Then it's just classical statistical mechanics. So it is very intuitive, maybe just a bit weird that you go to D+1 dimensions.

But for fermions, the variables are not classical variables, they are Grassmann numbers, and the integral is a brand new object called the Berezin integral. So I would like to know why you think this is more intuitive than the "abstract" Hilbert space formalism. It's abstract enough that Feynman was not able to derive the path integral for fermions.

 

[atyy]

Incidentally, it has been conjectured that some quantum theories have no Lagrangian formulation - I don't understand this work at all - just thought I'd bring it up in case someone can explain it.

https://www.ictp.it/media/101047/schwarzictp.pdf
http://db.ipmu.jp/seminar/sysimg/seminar/842.pdf
http://arxiv.org/pdf/hep-th/9609161v1.pdf
http://arxiv.org/pdf/1004.4735.pdf

 

[martinbn]

http://www.ams.org/notices/201009/rtx100901121p.pdf

 

[ZombieFeynman]

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.

Meh, I highly doubt they would have a far better understanding than a typical physicist. I think the typical physicist (read: postdoc or professor) understands physics just as much or more than someone who has diddled with some math.

If I'm wrong, show me the results! There are many high impact papers in physics coming out from labs full of people with little knowledge of (and often great disdain for) the advanced mathematics you speak of. Where are all of the breakthroughs from the folks who have only the mathematical tools?

 

[disregardthat]

Meh, I highly doubt they would have a far better understanding than a typical physicist. I think the typical physicist (read: postdoc or professor) understands physics just as much or more than someone who has diddled with some math.

If I'm wrong, show me the results! There are many high impact papers in physics coming out from labs full of people with little knowledge of (and often great disdain for) the advanced mathematics you speak of. Where are all of the breakthroughs from the folks who have only the mathematical tools?

I don't think there's many physicists with disdain for advanced mathematics that come up with breaktroughs in theoretical physics.

 

[Arsenic&Lace]

I don't think there's many physicists with disdain for advanced mathematics that come up with breaktroughs in theoretical physics.

"If all mathematics disappeared, it would set physics back precisely one week."-Feynman
To paraphrase Einstein (i can't find the original quote), "General relativity became unrecognizable after the mathematicians got their hands on it.", implying that he was not particularly fond of (or perhaps just not clever enough to understand) pure mathematics.

A cursory glance at the modern theoretical physics literature would probably further help you to disabuse yourself of this absurd notion. Beware of citing articles in string theory or topological matter, for instance, since neither of these fields constitute breakthroughs yet.

http://www.ams.org/notices/201009/rtx100901121p.pdf

This article does indeed sum up a few of my complaints with mathematics.

What I mean is that not every path integral defines a quantum theory. We only accept a path integral theory as a quantum theory if it has an equivalent Hilbert space (or C* algebra) formulation. If one could write a path integral formulation of a quantum theory which has no Hilbert space formulation, then one can say that the path integral formulation is more fundamental. At this stage, since the Hilbert space formulation is needed to define an acceptable path integral quantum theory, the Hilbert space formulation is more fundamenta

This conception of a quantum theory is so ludicrous that I can only conclude that a mathematician came up with it and not a physicist. It is a definition munged from the Platonist's world view and not from those who just want to figure out how the world works.

But for fermions, the variables are not classical variables, they are Grassmann numbers, and the integral is a brand new object called the Berezin integral. So I would like to know why you think this is more intuitive than the "abstract" Hilbert space formalism. It's abstract enough that Feynman was not able to derive the path integral for fermions.

For field theory the path integral formalism becomes murkier and more difficult to grasp. But it retains its key advantages, such as the fact that it handles Lorentz invariance with much greater ease than the canonical formalism. The fundamental intuitive picture, best communicated in terms of particles, remains elegant, as does the least action principle.

As for the complexity of the mathematical objects involved, my own experience of the Berezin integral and Grassman variables is that the former is barely remarked upon in field theory texts (it is not usually given that name) and important integrals are computed very intuitively; in the latter case, the relevance of Grassman variables is challenging to communicate but the rules which govern them are hardly sophisticated. I would even argue that the canonical formalism is really not that sophisticated as far as pure mathematics is concerned, it's just very slightly more abstract.

Incidentally, it has been conjectured that some quantum theories have no Lagrangian formulation - I don't understand this work at all - just thought I'd bring it up in case someone can explain it.

A reasonable way to define a quantum field theory would be one which actually describes nature. None of the theories you described actually describe nature, so far as we know. Therefore, I am unsure why anybody would be impressed by the fact that they might not admit themselves to a Lagrangian formulation; this may merely be an artifact of faulty assumptions about nature.

Of course if they could compute interesting experimental results, that would make it very interesting indeed!

 

[micromass]

"If all mathematics disappeared, it would set physics back precisely one week."-Feynman

If we are going to cherry pick quotes:

"To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in." - Feynman

"Our experience hitherto justifies us in trusting that nature is the realization of the simplest that is mathematically conceivable. I am convinced that purely mathematical construction enables us to find those concepts and those lawlike connections between them that provide the key to the understanding of natural phenomena. Useful mathematical concepts may well be suggested by experience, but in no way can they be derived from it. Experience naturally remains the sole criterion of the usefulness of a mathematical construction for physics. But the actual creative principle lies in mathematics. Thus, in a certain sense, I take it to be true that pure thought can grasp the real, as the ancients had dreamed." - Einstein

"One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts." - Einstein

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature." - Einstein

And I think the Feynman quote is apocryphical, I cannot find any source for it.