Do physics books butcher the math?

[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

 

[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.

 

[Arsenic&Lace]

The Einstein quote:
Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
Quoted in P A Schilpp, Albert Einstein, Philosopher-Scientist (Evanston 1949).

Source for Feynman quote:
http://www.mathteacherctk.com/blog/2011/09/physics-minus-mathematics-the-week-of-creation/

I would hate for this thread to devolve into a debate over what dead physicists thought of mathematics, but I must point out to you micromass that mathematics is not wholly trademarked by the math department. The Feynman path integral, for instance, even though it was invented in the physics department and is non-rigorous, is still mathematics. Therefore, it is wholly possible for a physicist to praise mathematics and not be referring to the Platonic culture.

However I'm not trying to argue with these quotes that Feynman and Einstein were not Platonists, but rather that they made major contributions to physics without knowing much pure mathematics. The first quote implies disdain for pure mathematics, which, coupled with Feynman's very intuitive physical arguments and lack of concern for rigor, imply either a lack of knowledge of pure mathematics or at least a very low opinion of it; it certainly wasn't necessary for his major discoveries. In Einstein's case this is just one piece of evidence that he knew little of pure mathematics even if he was a Platonist of sorts with his views on grand unification etc.

 

[jbunniii]

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

http://www.mathteacherctk.com/blog/2011/09/physics-minus-mathematics-the-week-of-creation/

To that outrageous comment, Kac shot back with that yes, he knew of that week; it was "Precisely the week in which God created the world."

 

[disregardthat]

Not all mathematicians are platonists, and platonism has little to do with mathematics, really. I don't know why you lump in this false dichotomy between physicists and mathematicians in your arguments.

 

[disregardthat]

I would also point out that much of modern mathematics is, in a sense, non-rigorous, meaning that it has no well-established and agreed upon foundation. As far as I understand, in categorical homotopy theory, (and historically in algebraic geometry), much of the work lies in establishing foundations as well as actually doing mathematics (but I'm no topologist). Questions which often arises, in mathematics as well as in physics, is what foundations does these concepts require?. There are many suggestions as to what formalism one would need, and would want, which varies depending on the situation you are in. In this sense, mathematics is independent of foundations and a set formal system. So as much as physicists push the boundaries of formality, mathematicians does as well.

 

[George Jones]

To paraphrase Einstein (i can't find the original quote), "General relativity became unrecognizable after the mathematicians got their hands on it."

The Einstein quote:
Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
Quoted in P A Schilpp, Albert Einstein, Philosopher-Scientist (Evanston 1949).

I don't know about math, but please do not butcher the context and accuracy of quotes. I was fairly certain that I knew the Einstein quote more accurately, and that I knew the context better, but, to make sure, I went to the source. Obviously, you didn't. Here is more extended excerpt from Schilpp:

When, later on, Minkowski built up the special theory of relativity into his "world-geometry," Einstein said on one occasion: "Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore." But soon thereafter, at the time of the conception of the general theory of relativity, he readily acknowledged the indespensability of the four-dimensional scheme of Minkowski.

It is rather ironic that you chose this quote in an effort to help your position. Einstein's quote refers not to general relativity, but to Minkowski's 1908 paper on special relativity. Einstein then used this, along with his outstanding physical intuition, to help formulate general relativity.

I can't say that I am a fan of your passive-aggressive style of posting.

 

[micromass]

The Einstein quote:
Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
Quoted in P A Schilpp, Albert Einstein, Philosopher-Scientist (Evanston 1949).

Source for Feynman quote:
http://www.mathteacherctk.com/blog/2011/09/physics-minus-mathematics-the-week-of-creation/

I would hate for this thread to devolve into a debate over what dead physicists thought of mathematics, but I must point out to you micromass that mathematics is not wholly trademarked by the math department. The Feynman path integral, for instance, even though it was invented in the physics department and is non-rigorous, is still mathematics. Therefore, it is wholly possible for a physicist to praise mathematics and not be referring to the Platonic culture.

However I'm not trying to argue with these quotes that Feynman and Einstein were not Platonists, but rather that they made major contributions to physics without knowing much pure mathematics. The first quote implies disdain for pure mathematics, which, coupled with Feynman's very intuitive physical arguments and lack of concern for rigor, imply either a lack of knowledge of pure mathematics or at least a very low opinion of it; it certainly wasn't necessary for his major discoveries. In Einstein's case this is just one piece of evidence that he knew little of pure mathematics even if he was a Platonist of sorts with his views on grand unification etc.

Did you even read Einstein's quote on pure mathematics that I posted.

And yes, I fail to see what Platonism has to do with any of this.

 

[WannabeNewton]

What is this thread even about anymore?

This whole thing is reminiscent of a PS3 vs. Xbox 360 fanboy argument.

 

[Student100]

What is this thread even about anymore?

This whole thing is reminiscent of a PS3 vs. Xbox 360 fanboy argument.

Sony > Microsoft

 

[Arsenic&Lace]

What is this thread even about anymore?

This whole thing is reminiscent of a PS3 vs. Xbox 360 fanboy argument.

I would tend to agree were it not for the fact that there is a disdainful and pernicious attitude presented by mathematicians towards physicists/physics students which ought to be quashed.

When someone states that "physicists butcher the math" as if mathematicians actually know something about math (they know as much about it as psychologists know about human behavior, I'd wager) I feel it necessary to point out just how absurdly wrong this point of view is. It's extremely misleading; one might think that opening up a book on advanced calculus would teach you something useful about calculus, but it's debatable whether such a book even constitutes additional knowledge about mathematics at all.

But the thread has spun a bit out of control I will admit.

 

[disregardthat]

How is it debatable whether a book on advanced calculus constitutes additional knowledge about mathematics?

 

[micromass]

I would tend to agree were it not for the fact that there is a disdainful and pernicious attitude presented by mathematicians towards physicists/physics students which ought to be quashed.

So you claim that mathematicians have a bad attitude towards physicists, and you follow that up with:

When someone states that "physicists butcher the math" as if mathematicians actually know something about math

Really?

I'm sorry, but I really don't see a "disdainful and pernicious attitude towards physicists" anywhere (well obviously it exists, but a disdainful attitude from physicists or engineers towards math also exists, and I think you're the most extreme example of that that I've ever met). I don't know where you got that from. Maybe you should just try to be a bit less sensitive.

And I'm sure mathematicians know as much about math as physicists know something about physics.

 

[Student100]

I would tend to agree were it not for the fact that there is a disdainful and pernicious attitude presented by mathematicians towards physicists/physics students which ought to be quashed.

When someone states that "physicists butcher the math" as if mathematicians actually know something about math (they know as much about it as psychologists know about human behavior, I'd wager) I feel it necessary to point out just how absurdly wrong this point of view is. It's extremely misleading; one might think that opening up a book on advanced calculus would teach you something useful about calculus, but it's debatable whether such a book even constitutes additional knowledge about mathematics at all.

But the thread has spun a bit out of control I will admit.

What attitude? I've never encountered such a thing. That's an awful large brush you're painting with there.

 

[rubi]

I don't see how this is even worth being discussed. Mathematics is the science of exact reasoning. It should be obvious that exact reasoning is desirable in physics. At what point are we allowed to be sloppy? Is $\sqrt{x^2+y^2} = x + y$ okay already? I remember the time back in high school when some kids were always asking: "What are integrals good for?" This discussion is the same, just on a slightly higher level. Now people ask: "What are Hilbert spaces good for?"

 

[dextercioby]

Fortunately, there are physics books which don't butcher the math: the Quantum Mechanics 2-volume set of the Spanish fellows Gallindo and Pascual. A must have for anyone claiming to know QM.

 

[micromass]

Fortunately, there are physics books which don't butcher the math: the Quantum Mechanics 2-volume set of the Spanish fellows Gallindo and Pascual. A must have for anyone claiming to know QM.

It sadly has no problems, but I've heard that they're working on that.

 

[atyy]

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.

So Martin Lüscher is a mathematician and not a physicist?
http://en.wikipedia.org/wiki/Martin_Lüscher

How about Rajan Gupta, another abstract mathematician, not interested in getting numbers for comparison to experiment?
http://cnls.lanl.gov/~rajan/

Are Stefano Capitani or Karl Jansen head-in-the-clouds abstractionists?
http://inspirehep.net/search?ln=en&p=stefano+capitani&of=hb&action_search=Search
http://www-zeuthen.desy.de/~kjansen/

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.

Great now you are almost retracting your position that the Hilbert space formulation is more abstract than the path integral. No one here was arguing that the Hilbert space formalism was sophisticated or abstract - that was you.

Now, can you explain to me why you have to rotate to imaginary time to compute anything? Why is imaginary time intuitive and not abstract?

 

[Arsenic&Lace]

How is it debatable whether a book on advanced calculus constitutes additional knowledge about mathematics?

Mathematics is a tool. If some "mathematics" never permeates into other disciplines, especially given that mathematics is a tool, it is questionable whether or not it is mathematics. One could argue that it is a tool in the same way a square wheel is a tool, but then you're in the unflattering position of having invented a useless tool. As an example, the epsilon delta formalism for proving that a limit exists is not something people actually use in other disciplines**Papers in which they are "used" are often written by mathematicians disguised as engineers or physicists. I'd be delighted if someone could demonstrate, as an example, a case in engineering or physics where the epsilon delta formalism is actually necessary to learn something about nature or build a device.

Really?

I'm sorry, but I really don't see a "disdainful and pernicious attitude towards physicists" anywhere (well obviously it exists, but a disdainful attitude from physicists or engineers towards math also exists, and I think you're the most extreme example of that that I've ever met). I don't know where you got that from. Maybe you should just try to be a bit less sensitive.

And I'm sure mathematicians know as much about math as physicists know something about physics.

I don't think my choice of words was appropriate. It's not really an attitude of condescension or something, although that exists it's not the real problem and it is present in all disciplines. Rather, it's the pretensions mathematicians have about being experts in mathematics when there is no evidence that they are in fact experts in mathematics. The additional "math" that they know that say, an engineer doesn't, is not necessary for anything in applied disciplines, to the point where they are never called upon for their expertise. Nor is it the same as a scientific discovery, such as, say, learning an interesting fact about duck penises: humans did not invent ducks for a purpose. When it comes to math that is shared between disciplines, that expertise is still never needed; collaborations between theoretical physicists and mathematicians are unheard of outside of anything other than purely speculative physics (e.g. string theory). The example I gave about advanced calculus is a case in point; the supposedly advanced, deeper knowledge of calculus is never consulted by anybody other than mathematicians, and provides no actual insight into using calculus. People calling themselves experts for whom there is no empirical evidence that this so called expertise exists get my goat, I'm afraid.

That said I believe that the enterprise of studying the tool can be fruitful, but it would need to look much more like 18th/19th century mathematical culture than 20th/21st centure culture.

In this regard mathematicians remind me very much of grammarticians; they cringe and say that "My bad" is terrible grammar since bad is an adjective, but the humans who actually use the language get along just fine without there being any evidence that, as far as communication is concerned, this phrase is anything but understandable and useful.

I don't see how this is even worth being discussed. [\QUOTE]

Why join the discussion then? Your commentary seems incredible ignorant of the actual discussion at hand. The concept of abstraction using a very formal language is obviously useful and even necessary to do physics. The culture which has emerged in mathematics departments is not, and had you bothered to read anything I've written, you'd have figured out that this culture is what I'm addressing.

Great now you are almost retracting your position that the Hilbert space formulation is more abstract than the path integral. No one here was arguing that the Hilbert space formalism was sophisticated or abstract - that was you.[\QUOTE]
Firstly, many theoretical physicists such as Martin Luscher are increasingly adopting this mathematical culture (which I think can be described as Platonist, as it is in the article posted by martinbn). I suspect this will retard the development of theoretical physics, although many problems at its precipices seem fundamentally intractable from an observational/experimental standpoint anyway. Of course if this attitude bears fruit (e.g if the string theory gubbins ever get around to actually making a testable prediction which turns out to be true) I will change my mind.

You seem to have become interested in a relatively small point that I made previously in the thread, which is that one can formulate quantum mechanics in a "more" intuitive manner without reference to objects such as Hilbert spaces. We should probably just agree that intuitive is a subjective point of view. I can write an essay on why I think the path integral formalism is more intuitive and it would probably still not convince you. Since nobody understands quantum mechanics, of course you will find bizarre qualities such as complex time; I am making a relative, not absolute argument.

 

[micromass]

Mathematics is a tool. If some "mathematics" never permeates into other disciplines, especially given that mathematics is a tool, it is questionable whether or not it is mathematics.

Then what is it?
And what do you call physics that never permeates into other disciplines?

Rather, it's the pretensions mathematicians have about being experts in mathematics when there is no evidence that they are in fact experts in mathematics.

Then who is the expert in mathematics? You? The physicists? The engineers? Everybody but the mathematicians?

The additional "math" that they know that say, an engineer doesn't, is not necessary for anything in applied disciplines, to the point where they are never called upon for their expertise.

Quote some assumptions here. Why do you think that mathematicians are never called upon for their expertise? I've seen it happen many times.

When it comes to math that is shared between disciplines, that expertise is still never needed;

Evidence? Or did you just make it up to prove your point?

collaborations between theoretical physicists and mathematicians are unheard of outside of anything other than purely speculative physics (e.g. string theory). The example I gave about advanced calculus is a case in point; the supposedly advanced, deeper knowledge of calculus is never consulted by anybody other than mathematicians, and provides no actual insight into using calculus. People calling themselves experts for whom there is no empirical evidence that this so called expertise exists get my goat, I'm afraid.

And what is the empirical evidence that physicists have expertise on physics? Many physicists are never called upon their expertise either (hey if you can make stuff up, so can I), so they're not experts on physics? Somebody like Witten doesn't know physics and math according to you? Somebody like Wald doesn't know physics? I guess the experts on both physics and math are the engineers then, because they actually apply it to things which are useful.

Why join the discussion then? Your commentary seems incredible ignorant of the actual discussion at hand. The concept of abstraction using a very formal language is obviously useful and even necessary to do physics. The culture which has emerged in mathematics departments is not, and had you bothered to read anything I've written, you'd have figured out that this culture is what I'm addressing.

Mod note: please refrain from calling other people ignorant.

 

[Fredrik]

Let me guess:

A n-tuple of numbers that behave according to a "transformation law," and no mention of linear algebra concepts?

Right. Some of my teachers didn't even use terms like n-tuple. Instead they said e.g. that a tensor is something that transforms according to the tensor transformation law, without explaining what sorts of "something" the definition can be applied to, or what they mean by "transforms". Some of them said that a vector is 3 numbers that transform according to the tensor transformation law. I could have forgiven them if they had at least defined the vector as the function that associates triples with coordinate systems, but no, they made it sound like it's a specific triple...if it "transforms" correctly.

The thread title talks about "butchering" mathematics. This sort of thing is a great example of what that means to me. I have no problem with non-rigorous arguments. But stupid definitions like this really irritate me.

Is (1,2,3) a tensor according to this "definition"? How about {1,2,3}? How about a cow? (Yes, the farm animal). Does the definition tell us that a cow isn't a tensor?

I eventually found the multilinear algebra definition of "tensor" in Schutz, and the definition of tensor field in Spivak. When I read the extremely clear presentation of tensors in Schutz, I couldn't believe that all my teachers had been peddling that painfully inadequate definition in class.

 

[1MileCrash]

So basically, if some topic of math cannot be applied to other disciplines, then it is not truly math, because mathematics is a tool, because it's math.

But if some topic of physics cannot be applied to other disciplines, then it's still physics, because physics is not a tool, because it's physics.

 

[atyy]

Firstly, many theoretical physicists such as Martin Luscher are increasingly adopting this mathematical culture (which I think can be described as Platonist, as it is in the article posted by martinbn). I suspect this will retard the development of theoretical physics, although many problems at its precipices seem fundamentally intractable from an observational/experimental standpoint anyway. Of course if this attitude bears fruit (e.g if the string theory gubbins ever get around to actually making a testable prediction which turns out to be true) I will change my mind.

Well, then your point is lost. Luscher and the others I mentioned mostly study the path integral, discretizing it and numerically studying to see if it gives results that match experiment. I thought that's what you said physicists do. Now physicists are mathematicians?

Here's Jansen's lectures:
http://www-zeuthen.desy.de/~kjansen/lattice/qcd/talks/dubna1.pdf
http://www-zeuthen.desy.de/~kjansen/lattice/qcd/talks/dubna2.pdf

Hopefully it's obvious these are physics questions by the standard you've been proposing, approached by path integral you favour.

You seem to have become interested in a relatively small point that I made previously in the thread, which is that one can formulate quantum mechanics in a "more" intuitive manner without reference to objects such as Hilbert spaces. We should probably just agree that intuitive is a subjective point of view. I can write an essay on why I think the path integral formalism is more intuitive and it would probably still not convince you. Since nobody understands quantum mechanics, of course you will find bizarre qualities such as complex time; I am making a relative, not absolute argument.

I'll take this as a concession from you.

Incidentally, I do understand quantum mechanics - an increasing number have since 1952

 

[1MileCrash]

Right. Some of my teachers didn't even use terms like n-tuple. Instead they said e.g. that a tensor is something that transforms according to the tensor transformation law, without explaining what sorts of "something" the definition can be applied to, or what they mean by "transforms". Some of them said that a vector is 3 numbers that transform according to the tensor transformation law. I could have forgiven them if they had at least defined the vector as the function that associates triples with coordinate systems, but no, they made it sound like it's a specific triple...if it "transforms" correctly.

The thread title talks about "butchering" mathematics. This sort of thing is a great example of what that means to me. I have no problem with non-rigorous arguments. But stupid definitions like this really irritate me.

Is (1,2,3) a tensor according to this "definition"? How about {1,2,3}? How about a cow? (Yes, the farm animal). Does the definition tell us that a cow isn't a tensor?

I eventually found the multilinear algebra definition of "tensor" in Schutz, and the definition of tensor field in Spivak. When I read the extremely clear presentation of tensors in Schutz, I couldn't believe that all my teachers had been peddling that painfully inadequate definition in class.

I learned the same definition initially, and had the same experience as you (though you are vastly more well-read).

I'm certainly extremely interested in whether or not the two definitions are actually even equivalent. I've heard that they are, but when I consider that the "transformation" definition leads to pseudotensors (which do not meet the transformation definition), I start to think the definition is just flat out wrong. But it's just a passing curiosity for another thread.

 

[jergens]

Mathematics is a tool. If some "mathematics" never permeates into other disciplines, especially given that mathematics is a tool, it is questionable whether or not it is mathematics. One could argue that it is a tool in the same way a square wheel is a tool, but then you're in the unflattering position of having invented a useless tool. As an example, the epsilon delta formalism for proving that a limit exists is not something people actually use in other disciplines*

A much greater portion of modern mathematics (including the stuff developed by and for mathematicians) has applications in other disciplines than you give it credit for. Just to give a few examples:

1. Donaldson and Freedman made some revolutionary advances in 4-manifold theory during the 1980s and these have found a number of applications in understanding gauge theories. More recent advances like those in Seiberg-Witten theory fall into the same category.
2. Homotopy groups of spheres and of the classical groups seem to come up occasionally in condensed matter theory publications, which means these researchers are relying on computations from algebraic topology.
3. Stochastic calculus is another relatively recent subfield of mathematics with some pretty serious applications in statistical physics. Lots of early results in this direction came from physicists, but increasingly important things like the Ito rule and the connection with random walks have come from mathematicians.
4. Algebraic topology and functional analysis (in the form of fixed-point theorems) are important in economics and finance. Stochastic calculus is also super prominent in this field.
5. Stochastic calculus underlies much of machine learning algorithms and data analysis. Algebraic topology is finding applications in this area too with things like persistent homology.

Now while the epsilon-delta definition of limits will likely appear nowhere in the textbooks on 4-manifold theory and algebraic topology and stochastic calculus, this definition underlies all of those fields. A rigorous footing for analysis was absolutely crucial for developing these results.

It is also worth mentioning how bizarre your conception of mathematics as purely a tool really is---honestly it is much like the people who slag modern math with the tagline that it has gotten away from its roots in computation (the irony there being that huge swaths of it are actually devoted to computation in some form or another). But getting back to the point this notion of yours has no real basis in the historical record since even the Greeks considered mathematical problems of intrinsic interest. This has remained unchanged through the rise of physics in the 17th century onwards as evidenced by the continued interest in solutions to polynomial equations and Fermat's Last Theorem (among other things) during that time. Now you are certainly entitled to conceive of mathematics however you wish, but you have to recognize your ideas on this issue are not widely held by anyone.

 

[WannabeNewton]

Obviously a lot of pure math is useless but so is quite a bit of physics. Right now I am doing a project involving the pole-dipole approximation of spin evolution in GR with external fields. Suffice to say it will probably not be of any use in applications that don't care about extremely precise measurements. More illustrative is the case of Marek Abramowicz and the series of papers he coauthored on a covariant formulation of inertial forces in curved spacetimes. As fun as it is to read the papers, it is safe to say that they will have no applications whatsoever. With that said, math and physics are certainly different in that most of physics can have applications in principle, regardless of whether or not it is realizable or of any interest, which is certainly not true of math by any stretch of the imagination. But that's ok. It's clear from the ridiculously abstract problems tackled by pure mathematicians of various modern fields that even the slightest of application is the farthest thing on the agenda.

In the end, I find physics infinitely more interesting than pure math, except for differential topology of course, while others are of the opposite persuasion. I wish we could just leave it at that.

 

[AnTiFreeze3]

I would tend to agree were it not for the fact that there is a disdainful and pernicious attitude presented by mathematicians towards physicists/physics students which ought to be quashed ...

And the following denotes your... friendly disposition towards mathematicians?

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.

Beneath your pseudo-philosophy of mathematics done by mathematicians not being mathematics unless the mathematics is used by physicists as a tool (i.e. masonry done by masons forging bricks isn't masonry unless that brick is picked up by a construction worker who uses it for a building), I sense just a little bit of overwhelmingly distasteful solipsism and arrogance.

No one cares if you don't like mathematicians. But when you're making claims that mathematicians don't know math, or that math isn't math unless a physicist blesses it with his reverent hands and uses it for a failed theory, it becomes hard to take you seriously.

And I'm sure you're probably smarter than I am, and as of now the rigorous discussions of mathematics and physics and their entwined existence has flown beyond my comprehension. Regardless, it doesn't take a genius to see your brooding chauvinism and your distaste for anything or anyone outside your own domain. The topic was, and has been, whether physicists, in their textbooks, "butcher" and overly simplify mathematics. They do. No one ever claimed this was an inherently bad thing. As WBN points out, this allows physics texts to hone in more so on the important physical theory and intuition than on usually unnecessary logical formulations. But this fact does not nullify the importance of the existence of rigorous mathematics, nor the usefulness of understanding rigorous mathematics when approaching your oft-repeated "esoteric" physical theories.

As Razumikhin says in C&P:

Talk nonsense, but talk your own nonsense, and I'll kiss you for it. To go wrong in one's own way is better than to go right in someone else's. In the first case you are a man, in the second you're no better than a bird.

 

[ZombieFeynman]

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

you ought to do some more glancing. Bernevig Hughes and Zhang decidedly predicted the quantum spin hall effect in HgTe quantum wells in their landmark 2006 science paper. The molenkamp groups subsequent observations convincingly made HgTe the first 2D Topological insulator.

Topological quantum matter definitively is a breakthrough, one guided by rather esoteric field theory along with some astoundingly creative physical insights.

 

[Char. Limit]

You know, now that I think about it... where are the applications for things like high-energy particle physics? I mean, you can't make power plants out of it, you can't design new materials from it. It's almost like people are doing it for the sake of... knowledge... you know, just like mathematics...

 

[1MileCrash]

You know, now that I think about it... where are the applications for things like high-energy particle physics? I mean, you can't make power plants out of it, you can't design new materials from it. It's almost like people are doing it for the sake of... knowledge... you know, just like mathematics...

Stop being so reasonable.

 

[AnTiFreeze3]

You know, now that I think about it... where are the applications for things like high-energy particle physics? I mean, you can't make power plants out of it, you can't design new materials from it. It's almost like people are doing it for the sake of... knowledge... you know, just like mathematics...

To be fair, though, most new fields of research are deemed too abstract to be applied to anything. It's not until a bit later that some intelligent engineers and entrepreneurs figure out how to give the research a meaningful role in society.

Take, for instance, our journeys to space. Sure, we do some relatively important experiments up at the ISS, but landing on the moon was largely done just to do it (not that it's a bad thing at all). We didn't learn anything really important except that the moon's rocks were just like the Earth's rocks.

But the technologies developed to reach the moon have had a huge impact on modern civilization. As with HEP, when we're forced to create novel machinery and techniques that greatly strain our abilities, there are often more implementations for these technologies than what their original design might indicate. With Fermi and CERN, lots of medical applications have arisen due their research, i.e. neutron therapy at FermiLab.

When, for a current events class sophomore year in HS, you choose NASA as your subject of study, you're forced to find reasons why it deserves funding, and when you have an internship with a minor role in HEP at Fermi/CERN, you tend to find ways to combat the "Well, it's cool, but why is it useful?" question

 

[Char. Limit]

To be fair, though, most new fields of research are deemed too abstract to be applied to anything. It's not until a bit later that some intelligent engineers and entrepreneurs figure out how to give the research a meaningful role in society.

Take, for instance, our journeys to space. Sure, we do some relatively important experiments up at the ISS, but landing on the moon was largely done just to do it (not that it's a bad thing at all). We didn't learn anything really important except that the moon's rocks were just like the Earth's rocks.

But the technologies developed to reach the moon have had a huge impact on modern civilization. As with HEP, when we're forced to create novel machinery and techniques that greatly strain our abilities, there are often more implementations for these technologies than what their original design might indicate. With Fermi and CERN, lots of medical applications have arisen due their research, i.e. neutron therapy at FermiLab.

When, for a current events class sophomore year in HS, you choose NASA as your subject of study, you're forced to find reasons why it deserves funding, and when you have an internship with a minor role in HEP at Fermi/CERN, you tend to find ways to combat the "Well, it's cool, but why is it useful?" question

That's a very good point, Anti, but it's actually similar to the point I was making. Just as you said that most new fields of research are deemed to be too abstract to really be applied to anything, the same could be said about new fields of mathematics when they are first introduced. While subjects like calculus and differential equations saw use in physics and other disciplines as soon as they were created, I'd be willing to bet large sums of money that many of the more modern mathematics were of the style to see no real practical use at first, but then physics "caught up", if you will. And the same could be said about many of the highest features of theoretical mathematics today.

The point I was trying to make was that it's folly to discard a discipline because "there aren't any applications for it right now", especially if you're going to embrace things like high-energy particle physics.