There was a postdoc in my research group in grad school who had a sister in college. She called him once to ask for help with a math assignment dealing with series expansions. He checked a book to refresh his memory, and then told her how to generate the various series needed for her homework assignment.
A week or so later, he asked how she'd done. "Terrible," she said. It seems that he had just plunged ahead with generating series terms without doing the convergence tests and other proofs that a mathematician would do for the same problems. She told him, "My professor said I answered all the questions like a physicist!"
I was reminded of this in the comments to a recent post, where "alkali" suggested getting around the problem of algebra-based physics by "teaching a really degenerate form of single-variable calculus (no derivations, basically "here it is") and then go[ing] on to teach an actually useful kind of physics." David responds, saying:
If you do this, you will get students whose knowledge of calculus has analagous flaws to the physical knowledge Chad describes above. And this will hurt them in the end. When I took introduction to electricity and magnetism, I watched my fellow students destroyed because they didn't know what div, grad and curl actually meant. In mechanics, there were similar problems with respect to single variable calculus--why do you integrate to get an average pressure, why is acceleration a double derivative--although I don't remember it being so bad.
There's always a certain tension regarding the degree to which we need to emphasize mathematical correctness in teaching physics. In an ideal world, of course, we would do everything precisely correctly. But then, if you approach physics derivations like a mathematician, you end up spending so much time proving theorems that you never get anywhere...
Physicists, particularly experimental ones, tend to be a little cavalier about mathematics. We have a tendency to neglect strict formalism, and fall back on physical intuition to justify steps in our derivations. In lecture last week, I used a couple of tricks that tend to give math majors hives-- things like taking a "dx" dividing it by a "dt," and calling it a velocity. The steps are all correct, but they're justified not by first-principles mathematical reasoning and proof, but by a sort of hand-waving reference to basic physics concepts. When you've got an infinitesimal distance divided by an infinitesimal time, of course that's a velocity. What else would it be?
We get away with this sort of thing because we've always got comparisons to physical reality to fall back on (at least in the low-energy physics world). As my colleague pointed out regarding his sister's homework, we don't usually bother checking the validity of series expansions because they work very well for the vast majority of physically interesting cases. And when a series expansion fails, it usually shows up as a physically impossible prediction, in which case you know to go back and check your assumptions. You don't have that luxury in pure math, so you need to be a lot more careful about following formal procedure. (More after the cut.)
Personally, I'm perfectly happy with this state of affairs. I'm not all that mathematically inclined, and I have a much easier time understanding things when I can construct a concrete physical picture of what's going on. This is why my understanding of solid state physics is absolutely abysmal-- two chapters in, they started doing everything in terms of "reciprocal lattice vectors," and lost me completely. I couldn't see where the atoms were, and what the electrons were doing, and I never really recovered.
As a result, I tend to emphasize intuition over formalism when I'm teaching. I spend a good deal of time working on physically appealing justifications for how things work, and try to gloss over the mathematical details as much as I can. That's how I learn best, and that's what I'm comfortable with teaching.
I was amused to learn, when this subject came up at a New Year's party thrown by a math professor (we're wild and crazy in academia-- earlier in the party, I had spent about half an hour talking about the BEC-BCS crossover...), that there are people who have the opposite problem. One of the math faculty said that she had bombed introductory physics back in college, because of exactly the dilemma suggested by my commenters. Her professor was trying to teach a bastard version of calculus on the fly, and she struggled with it because she wanted to see proofs and theorems, and was very disturbed by the frequent use of "proof by physical plausibility" and "proof by dimensional analysis."
I've got a couple of students this term who might be in that boat, and I'll do my best to accomodate them (even if I have to fire up the Mathematica Integrator to fill in some of the details). Still, part of my job is to teach students what it means to "think like a physicist," so you give me a length and a time, and I'll give you a velocity, and theorems be damned...
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Well, this reminds me of the time when one of my juniors cornered me with the question - "what are these reciprocal vectors anyway ?". I struggled for a while before I came up with this answer- "see - if you take a crystal and shine a light of one particular colour(i.e, a monochromatic light), you will see that the light scattered off forms a beautiful pattern with sharp spots. Now that means, the momentum that the crystal gives to/takes away from light is quantised. This set of momenta that a crytal can lose/gain (apart from a factor of hbar) form what are called the reciprocal vectors of a crystal."
I like this "explanation" because electrons in the crystal are also bouncing in the crystal lattice and momentum transfer is quantised in the same way. This explains why electron's momentum is always conserved modulo a reciprocal lattice vector (the law of conservation of quasimomentum). Further, "defining" reciprocal lattice this way, we can explain why the set of reciprocal vectors should be closed under addition etc.
The downside is of course that to get to the usual definition you have to appeal to Born approximation or some such thing....
This reminds me of one my favorite nerdy science/math jokes. It's a test to see whether you think more like a physicist or a mathematician:
Imagine you're in a cabin in the woods. You have a gas stove, a box of matches, an empty pan, and a cold water tap. How do you get hot water?
Don't tax your brain -- a physicist and mathematician would both say "fill the pan with cold water, light the stove, put the pan on the stove, wait a while."
Let's try again:
Imagine you're in a cabin in the woods. You have a gas stove, a box of matches, a pan full of cold water, and a cold water tap. How do you get hot water?
Well?
If I remember the joke correctly, the physicist says:
"Well, I'll light the stove with the matches, and put the pan on the stove."
and the mathematician says:
"I'll empty out the pan. Now I've reduced it to a previously solved problem."
Exactly so, Dennis. Good on you!
My favorite version of that joke involves a firehose:
A physicist and a mathematician are walking down the street, and they come across a burning house. The physicist quickly grabs a nearby firehose, attaches it to a hydrant, and put out the fire.
The next day, the mathematician is walking down the same street, and sees the firehose, still attached to the hydrant. He disconnects it, and sets a nearby house on fire, thereby reducing it to a previously solved problem.
There are some mathematicians who like infinitesimals too - I think this is what they pointed me to for an intro.
Oh, and my favorite physicist/mathematician joke? (Actually, it brings in an engineer also):
A mathematician and an engineer are sitting in a talk on string theory. The engineer turns to the mathematician and says "I just don't get this stuff about 11 dimensional space. How do you even begin to understand something like that?" The mathematician replies "11 dimensional space? Oh, it's easy! You just think about an n-dimensional space, and let n go to 11."
I take David's comments, with two reservations:
1) A lot of high school physics students will never take college physics.
2) I'm not suggesting that hardcore calculus not be taught to anyone, ever -- presumably students will proceed in the math curriculum regardless of what the physics teachers are up to -- but in the meantime, you can teach high school students how to solve a lot of single variable calculus problems, enough to do a fair bit of simple mechanics and E&M.
An engineer, a physicist, and a mathematician are staying in a hotel. A fire breaks out, spreading to each of their rooms.
The engineer grabs the ice bucket, goes to the bathtub, fills the bucket with water, and puts out the fire.
The physicist calculates the amount of water necessary to put the fire out, fills the bucket with exactly that much water, and puts out the fire.
The mathematician sees the ice bucket, sees the bathtub, and quickly does some calculations. "A solution exists!" he says, and goes back to sleep.
An engineer, a physicist and a mathematician find themselves in an anecdote, indeed an anecdote quite similar to many that you have no doubt already heard. After some observations and rough calculations the engineer realizes the situation and starts laughing. A few minutes later the physicist understands too and chuckles to himself happily as he now has enough experimental evidence to publish a paper.
This leaves the mathematician somewhat perplexed, as he had observed right away that he was the subject of an anecdote, and deduced quite rapidly the presence of humor from similar anecdotes, but considers this anecdote to be too trivial a corollary to be significant, let alone funny
Oh dear, Aaron's joke nearly made me fall off my chair.
Now that was funny, Aaron. I almost embarrasses me into not telling this one (which again is at least somewhat related to the original blog post).
A physicist, engineer, and mathematician are riding in a train through the Scottish countryside. They see a lone black sheep on a hill. Each writes home:
Physicist - "Today we saw a black sheep in Scotland."
Engineer - "Today I learned that Scottish sheep are black."
Mathematician - "Today in Scotland we saw a sheep, at least one side of which is black."
I wish I could take credit, but I found it here.
Physics only made sense to me when all the "physical plausability" and Heuristic nonsense was REMOVED. For example, General Relativity made sense when I realized that all the phenomena were just simply derived out of the Einstein Field equations. No falling elevators, no Mach principle etc.
Special rel made sense when I realized that it was just the fact that the wave equation has the Lorentz group as part of its invariance group, and when I plotted the causal diagrams ( ala Penrose).
Quantum mechanics made sense when instead of all the argument over paradox, and all the idiotic philosophy, I just learned the axioms-- and thought of it in terms of operators in Hilbert Spaces.
I wish physics teachers would just drop all that heuristic baloney ( a confusing turnoff) and just assume that SOME students are math talented.
After all, Newton, Maxwell, Euler, and even; Dirac,Bardeen and Born ,Heisenberg, Pauli, Penrose, and Hawking were mathematicians: Dirac and Hawking ,Lucsian Profs of math. Born and Heisenberg studied under Sommerfield, who wrote a great series on mathematical physics. Born was David Hilbert's Postdoc, and received a PHD in MATH ( elastic stablility) under Schwartzchild ( yes, also a mathematician). Pauli received his PHD under Sommerfield as well.
By the way, it irks me that these people are almost never represented as mathematicians--especially to physics students in their formative years, but are essentially always presented as PHD physicists. THAT'S A LIE.
The mathematician Bardeen ( Phd Math: Princeton) is the only person to win TWO Nobel prizes in PHYSICS.
Want to be a great theoretical physicist--then get a phd in MATH, and learn to think in THEOREMS. Even Einstein ( phd in PHYSICS) was a protege and class student of Minkowski ( mathematician).
Math rules.
Penny
p.s " It's not a little black pith ball, It's a section of a Spinor bundle."--Penny, as undergrad.
p.s." And I am not living on a math cloud, as I child, I built a Betratron and ground telescope mirrors and was very interested in experimental physics.
But, it only made real sense, when at 14, I discovered that one could do physics with theorems. That's what Newton called "Natural Philosophy".
penny: Physics only made sense to me when all the "physical plausability" and Heuristic nonsense was REMOVED. For example, General Relativity made sense when I realized that all the phenomena were just simply derived out of the Einstein Field equations. No falling elevators, no Mach principle etc.
{...}
I wish physics teachers would just drop all that heuristic baloney ( a confusing turnoff) and just assume that SOME students are math talented.
There are absolutely some people for whom pure math is the most effective way to think about physics. (Many of them seem to be string theorists, and many of them probably think that I'm not a real physicist because I don't find group theory illuminating.)
I have this argument from time to time with a professor in the Math department, who always argues that we turn off a bunch of students by de-emphasizing or disparaging math in class. He may very well be right, but I'm not sure there's much I can do about it.
It's a matter of what I can and can't do in the classroom, and I'm just not comfortable with presenting material entirely in terms of theorems. I don't think I could teach a class without the heuristic stuff, any more than I could teach a class without cracking jokes or being sarcastic (I've tried that, and it was a disaster).
My personal feeling is also that the best physicists I know, and the best physicists I know of have an excellent grasp of the heuristic stuff, beyond the theorems. Einstein's Special Relativity was a success not because of the brilliant mathematics, but because he provided ingenious physical arguments to get people to believe the math. Schwinger's formulation of QED is arguably more mathematically elegant than Feynman's, but Feynman's version is pre-eminent these days because he provided an intuitive way to think about what the math means.
You can't be a great physicist without understanding theorems. But I also don't think you can be a truly great physicist by only understanding theorems.
But these are all jokes about mathematicians. What do jokes by mathematicians look like?
Example 1: What do you get if you cross a mountaineer with a mosquito?
Answer: Nothing, you can't cross a scaler with a vector
Example 2: Why did the chicken cross the Mobius strip?
Answer: To get to the same side
Example 3: Let epsilon be less than zero...
Example 4: A mathmo song
Corollary: Fear the mathmo jokes. FEAR THEM!
A student asked his physics professor, "Dr., who came first, the chicken or the egg?"
The professor replied, "Well, first we must assume a perfectly spherical chicken."
There's always the classic "What's purple and commutes?"
Check out Jokes With Einstein for a Flash movie of one of the "cross product" jokes. My personal favorite of the set is #2, but YMMV.
Thanks Chad for posting! Some of the math talented students in physics become general relativists.
Personally, as a mathematician, I share the opinion of most mathematicians about String theory--STRING THEORY IS TRASH.
There are no really rigorous theorems in String Theory and no experimental results either. String theorists claim lots of mathematical progress due to strings--such as the Seiberg-Witten equations in topology--but these equations really don't come from string theory--physically they come from supergravity--which is quite another thing.
In the same way, mirror manifolds are a natural mathematical outgrowth of Hodge theory, and never really needed physical motivation. However, when strings were a big newspaper type thingy, math people were happy to use the term. Personally, I find that trendiness disgusting.
Feynman's approach ( Dyson-Feynman Diagrams)for the Kac-Volterra-Feynman expansion is easier for the not very mathematical experimentalists to sort of use. But, the best approach was really by Tomanaga--who worried about the causal structure.
Feynman's later approach: Feyman-Kac's version of the Wiener integral ( cylinder measures)with complex time, is much more mathematically elegant than Schwinger's. Though, of course, it is only well defined for quadratic potentials and slight generalizations ( Work of De-witt ( promeasures) and Daubechies ( wavelet path integrals).
I still think theorems are key. Especially since, it was proved in the 1980's that the only Q.F.T. satisfying the Whiteman axioms in four dimensions is the free scalar field.
Thus Feynman's very appealing Q.E.D. doesn't even exist!
Penny
p.s. As Eliot Lieb has said, most of the so-called great accuracy in Q.E.D. calculations of spectra are actually due to a careful analysis of the standard Dirac Equation, and QED gives about three decimal places more. Since a condtionally convergent series can be rearranged to converge to anything--it's clearly a fudge.
Which again points to theorems, as without the theorems the question of the consistancy of QED can't even be raised in a clear way.
p.s. The way I found special rel easiest to understand--as a teenager was: Take the fourier transform of the wave equation. One gets the Lorentz form. Thus this form's invariance group ( aka: Lorentz transformations+ conformal group) is key. Now, all of special rel is just the study of this quadratic form and its group.
Previously, I had read Einstein's popular books and found them dreadful. All that clock synchronization.
You know that it is impossible to set up those clocks? You would have to instantly set an infinite system of such clocks and rulers into space. Only God could do that.
It was a relic of his interest in the defunct philosophy of Logical Positivism, by the way.
Don't you find students get really confused with all the clock sych stuff?
It is so much simple to ignore all that, and just introduce space time diagrams from the quadratic form.
Dear corkscrew,
How do you cage a tiger in the desert?
You open the door and enter the cage and close the door. Now the tiger has to go through the cage to get to you.
--topologist
Open the door and wait for the random tiger movement to put the tiger into the cage--probabilist
Invert the universe with center in the cage.--analyst
Enclose the desert in a wall, and build a bisecting wall.
Iterate on the side where the tiger is. Eventually, tiger is pierced by wall.--topologist
penny
p.s. Are all odd integers prime?
Engineer: 1,3,5,7. 9 --Sure.
Physicist: 1,3,5,7, 9, 11, 13... Yes, up to experimental error.
Mathematician: Prime or Irreducible?
(Why have I appointed myself blogdom defender of string theory? Ah well.)
These are a bit out of order.
There are no really rigorous theorems in String Theory and no experimental results either. String theorists claim lots of mathematical progress due to strings--such as the Seiberg-Witten equations in topology--but these equations really don't come from string theory--physically they come from supergravity--which is quite another thing.
Actually, the Seiberg-Witten equations come from supersymmetric gauge theory.
In the same way, mirror manifolds are a natural mathematical outgrowth of Hodge theory, and never really needed physical motivation.
Too bad no mathematician had the idea, then, huh?
Personally, as a mathematician, I share the opinion of most mathematicians about String theory--STRING THEORY IS TRASH.
You know, I was going to post a long list of mathematicians who work on string related stuff, but that seemed rather silly. It's just factually incorrect to claim that there has not been significant, interesting mathematics that have come out of string theory. If you like, I'd be happy to provide a list. Strings might not have succeeded as physics yet, but it's done quite well lending inspiration to mathematicians.
Especially since, it was proved in the 1980's that the only Q.F.T. satisfying the Whiteman axioms in four dimensions is the free scalar field.
I assume you're referring to Haag's theorem here? As I understand it, this theorem uses an inappropriate definition of a QFT and that, I think, this was pointed out by Wightman soon after it came out. Regardless,
Thus Feynman's very appealing Q.E.D. doesn't even exist!
it is true that QED does not exist because of the Landau pole. It is generally expected, however, that asymptotically free theories (like QCD) do exist although a rigorous formulation is still lacking.
As Eliot Lieb has said, most of the so-called great accuracy in Q.E.D. calculations of spectra are actually due to a careful analysis of the standard Dirac Equation, and QED gives about three decimal places more. Since a condtionally convergent series can be rearranged to converge to anything--it's clearly a fudge.
The failure to converge of the perturbation expansion in QED (and in other, simpler, examples) is an example of aysmptotic convergence. In fact, the well-known accuracy of the g-2 calculation for the electron includes more than just QED contributions.
On the other hand, I do agree that too much emphasis is given to the axiomatic formulation of SR. The essence of SR is the Lorentz group. The rest is just metaphysical baggage.
And, ObScienceJoke: A physicist, an engineer and a computer scientists are in a car driving throught the desert when the car breaks down. Immediately they begin discussing how to deal with the problem. The physicist says, "It's clear that the car needs more energy in order to travel; let's check the gas tank." The engineer says, "Look, there's all sorts of mechanical things that could go wrong. Pop open the hood, and I'll start tinkering around." The computer scientists looks at them both and says, "Let's not be so hasty guys. Why don't we all just get out of the car, get back in again, turn the key and see what happens?".
Did you hear the one about the constipated mathematician?
He worked it out with a pencil!
My husband is an engineer and I'm a mathematician. We had this discussion about 11-dimensional space a while ago, with the exact wording as the so-called joke.
My, this is embarrassing :-)
But here's another true-life story, which some would find funny, especially considering how engineers deal with broken down cars.
I bought a new cheese cutter which has a thin wire and a roller. I tried to get through a piece of Gouda, but somehow the cheese refused to get between the wire and the roller, as it should. I was mightly annoyed, tried it upside down, swore at the cutter, even brought out the directions. It still wouldn't work. When I was close to throwing the thing through the window, my husband came in, took the cutter out of my hands and examined it. He discovered two screws, loosened them and moved the roller down. "How thick would you like your slices, dear?" he said. Needless to say that I was close to murdering him :-)
Ooops, forgot to post two jokes.
A physicist, an engineer and a mathematician are caught by aliens. Each gets his own cell, a pencil and a closed can with soup. The next day, the aliens check in on the three.
In the cell of the physicist, the walls are full with equations and there's a single dent in the wall. The physicist is happily gulping his (hopefully self-warming) soup.
In the engineer's cell, there's not a single equation to be seen and the walls are full with dents. But still, the engineer is eating his soup.
In the mathematician's cell, the man is sitting on the floor and looks at the can with a happy smile. On the wall there's written: Let: can=open.
Here's another:
A man is driving a balloon and loses his way. He manages to get low above a garden, where someone is sitting and admiring the scenery.
"Good sir, can you tell me where I am?", the balloonist asks.
The man thinks a bit, then nods. "In a balloon."
"Good gracious, you must be a mathematician!"
"How did you know?"
"That answer was absolutely correct and not in the least helpful."
Thanks for a great post Aaron!!
Supergravity is a supersymmetric gauge theory ( with a different gauge group). Anyway, the point is that it is NOT string theory.
As to mirror manifolds, there is a history of Hodge theory and Calabi-Yau theory. Calabi-Yau manifolds were fairly new ( since Yau's theorem) and it was too early to see what would have developed. But,the Hodge diamond was well known and the Hodge conjectures were much studied.
As to Strings motivating math, well, one must be careful to make sure that the claimed thing was really string theory and not some other physics--as in the case of the Seiberg-Witten equations or Topological QFT.
Sure, bad physical theories can create math motivation--knot theory was created by mathematicians interested in Lord Kelvin's theory of the atom as a vortex. The original derivation of the heat equation came from the flow of Caloric, and classical elastodynamics igores the existence of atoms.
But, just as we still believe in atoms, most mathematicians do not yet believe that string theory is not garbage physics--even if some of us find T-duality motivating.
Yes, there are many mathematicians who work on string theory.
In a certain sense, you might even include me.
My co-author David Johnson and I have spent more than a decade developing a theory of area minimizing sections of high co-dimension in fiber bundles and giving rigorous existence and regularity theorems. This is a natural bundle generalization of the string energy functional at the classical level. ( Do a math arxiv search on Penny ( or Penelope Smith) and David Johnson.) You might find it useful in string theory, and I would be happy.
However, although you give a long list of mathematicians who work on string theory related questions, it is STILL TRUE that the majority of mathematicians ( say differential geometers) think that String theory IS NONSENSE. And the majority of differential geometers and geometric analysts do NOT work in string theory.
As to Whiteman axioms and the nonexistence of QED, I asked Whiteman in the early nineties and he still thought that the theorem of nonexistence was true. He was willing to say that perhaps QCD was ok ( NOTICE, NO THEOREM YET). He gave me some Italian notes that attempted to give a rigorous axiomatic derivation of gauge theory but they somehow had to assume both positive and negative probability distributions.
Anyway, I am grateful for your excellent post.
Penny
p.s. Of course, if suddenly an experimental verification of string theory ( not some related thing claimed as string theory) actually appeared--I would be thrilled.
p.s. No need to be the defender of string theory to me, as a mathematician I am happy to let anyone work on anything--that is our culture. But, it would be nice if the claims of physicists about string theory were less full of hype.
There are other approaches to unified field theory ( ask the loop gravity people) and phrases like "Theory of Everything" don't help.
I would like to see more theorems in physics, especially in string theory, because lacking experiment to keep people honest, rigorous proof is a good substitute.
Supergravity is a supersymmetric gauge theory ( with a different gauge group).
Gravity is more than just a gauge theory with the diff group.
Anyway, the point is that it is NOT string theory.
It's the low energy limit of string theory. Regardless, the SW equations come from gauge theory, not from gravity.
As to Strings motivating math, well, one must be careful to make sure that the claimed thing was really string theory and not some other physics--as in the case of the Seiberg-Witten equations or Topological QFT.
I was thinking of things like Gromov-Witten invariants, Donaldson-Thomas theory, Gopakumar-Vafa invariants, homological mirror symmetry, etc.
But, just as we still believe in atoms, most mathematicians do not yet believe that string theory is not garbage physics--even if some of us find T-duality motivating.
My impression is that most mathematicans don't really care one way or another although I remember a quote by someone reasonably famous that string theory had to be true because there was no other way to explain how it led to so much surprising mathematics. I don't happen to agree with that sentiment, but I'm happy to pass it along.
As to Whiteman axioms and the nonexistence of QED, I asked Whiteman in the early nineties and he still thought that the theorem of nonexistence was true.
It's Wightman. As I said, the it's widely believed that QED doesn't exist because of a phenomenon called the Landau pole. This is a different issue than Haag's theorem.
He was willing to say that perhaps QCD was ok ( NOTICE, NO THEOREM YET).
QCD lacks a Landau pole (and is, in fact, asymptotically free), so it is much more likely that it exists. As you say, there is no rigorous construction of it -- you can get a million bucks from the Clay institute if you manage it, though. Unfortunately, the rigorous pursuit of quantum field theory has been a spectacularly unfruitful endeavor. In the meantime, physicists use it to great effect in predicting the results of experiment. In that sense, proving theorems seems to be overrated for the purposes of doing physics.
I would like to see more theorems in physics, especially in string theory, because lacking experiment to keep people honest, rigorous proof is a good substitute.
Well, you can always check out my last two papers for rigorous theorems. I do owe much of the rigor to my collaborator, though.
Thanks again for a great post Aaron,
The Seiberg-Witten
invariants were again motivated by Supergravity ( or even, if you like, by supersymmetric gauge theory)--not by a To me, a gauge theory is a system of field equations on a bundle with an automorphism group. That includes gravity. It is true that gravity is not of the form of say the Yang-Mills equation ( but check out the work of Ashtekar!).
The Gromov-Witten invariants are NOT from string theory. Gromov was certainly not motivated by string theory. His work was done using Pseudoholomorphic curves. Witten's string theory.
Donaldson's original work was motivated by Gauge theory. I have a small paper on this sort of thing: " Removable Singularities for the Yang-Mills-Higgs equations in two dimensions" ( actually the hardest case). Donaldson-Thomas is motivated by Sieberg-Witten which AGAIN doesn't come from string theory.
We have already discussed Mirror symmetry.
Yes. I misspelled Whitman. There are other spelling error in my posts here. Oops.
You are correct on Vafa etc. That IS motivated by string theory. Of course, there were people in Von-Neumann algebra theory who were grateful for the sudden interest in their field. Similarly, for the Russians working in what we now call Super Lie Algebras, who had discovered it from deformation theory--pure math.
I gave lots of examples on how theorems ARE the most important thing in top level physics--starting with Newton, Maxwell, Born, Einstein etc.
I am glad that YOU prove theorems. Don't mimimize your contribution because of some baloney brainwashing you got from physics classes.
Penny
p.s. I have enjoyed your posts st Sci.Physics.Research. You, Squark and Baez were alway the most interesting posters.
Aaron,
I wrote:
//
The Seiberg-Witten
invariants were again motivated by Supergravity ( or even, if you like, by supersymmetric gauge theory)--not by a
//
And it got scrambled. I meant to say:
//not by String Theory
Penny
Aaron:
More scrambling in my post.
I wrote:
//
The Gromov-Witten invariants are NOT from string theory. Gromov was certainly not motivated by string theory. His work was done using Pseudoholomorphic curves. Witten's string theory.
//
I meant: "Witten's invariants were motivated by supergravity and ( if you like) supersymmetric gauge theory--but NOT by string theory.
Penny
Aaron,
You papers look interesting. I will read them. Thanks for the reference. I may have to navigate a steep learning curve.
Again, if you can find a String Theory application ( which must exist) for my work with Johnson, I would be quite happy.
Here is a conjecture: Just as gauge theory ultimately needed to replace Space-time with a vector bundle, so too will string theory eventually need to study the area functional of sections of fiber bundles. This would be the Hamiltonian in the Path Integral.
We have generalized Geometric Measure Theory to this context, and we have proved existence, and sharp regularity results in this setting.
best
Penny
The Seiberg-Witten invariants were again motivated by Supergravity
No. They weren't. The Seiberg and Witten showed how there was a certain duality in an N=2 supersymmetric Yang-Mills (SYM) theory. Witten applied these results to his construction of Donaldson invariants via a topologicall twisted N=2 SYM theory and got Seiberg-Witten invariants out of it. The original work certainly had little to do with string theory although you can see the relevant dualities arise in a string theory context.
The Gromov-Witten invariants are NOT from string theory. Gromov was certainly not motivated by string theory. His work was done using Pseudoholomorphic curves. Witten's string theory.
I don't know the Gromov history, but on the string theory side, Gromov-Witten invariants arise in certain topological sigma models which are the worldsheet theory for the topological string.
Donaldson's original work was motivated by Gauge theory.
Donaldson invariants are. Donaldson-Thomas theory is something else. As far as I can tell, it has absolutely nothing to do with Seiberg-Witten theory. You can read the introduction to the paper for the string theory motivation (in particular, the citation of references 18 and 34).
Dear Aaron,
Ok. I will accept that the Seiberg-Witten theory came from Supersymmetric Gauge theory and not from Supergravity. Thanks. The main point is that it didn't come from String theory.
A good reference for the Gromov Theory ( to start out) is the book by Dusa Macduff on Pseudoholomorphic curves. Gromov was counting intersection points on such curves. Pure math. She also has a great book on Sympletic topology.
I will check out your reference on Donaldson-Thomas theory.
However, I am sure that Donaldson ( a rigorous topologist and student of Atiyah) didn't use "String theory" as a motivation. But thanks for the reference. I have learned a lot from you discussion here.
Thanks
Penny
I think of Sigma models as motivated by harmonic maps--aka an approximation to the area functional ( so ok, I can see how a physicist might think that is string theory). And, I can see that Witten would think of this as a string theory.
In any event his math was not correct( not rigorous), and the Gromov theory was correct math--with real proofs.
Aaron,
Aha, Donaldson-Thomas is related to Chern-Simon Gauge theory. Now, THAT, I know something about. It's a natural Pure Mathematical gauge theory used in topology.
OF course, a physicist might think of it in a physicial manner--at one point it was related by physicists to the topological Field theory of anyons.
Thanks for the reference. Another great paper to read. This has been VERY stimulating for me.
Your fan
Penny
Penny
Aaron,
Chern Simons, of course. Spelling at typing speed---Yeech.
The point is that Chern-Simons are a secondary charactistic class that is quite important in topology, and it was very natural to look at a Gauge theory based on them.
Penny
Off to get chinese food and write up a paper. Thanks for a fun morning.
Gromov? Rigorour? My (second-hand) impression has always been that Gromov was rather famous for being sketchy. Anyways, IIRC, Witten's conjectures included more than just Gromov's original work. I think it was Kontsevich who came up with the way to make things rigorous, the moduli space of stable maps, which was finally constructed, again IIRC, by Fulton and Pandharipande. But the history of these things was never my strong point.
Anyways, the point with Chern-Simons theory is that Witten showed that it is dual to a certain open topological string theory. If you look at the closed strings in the same theory, you get something that computes GW invariants. The idea, at least as I understand it is that
Chern-Simons Theory : open toplogical strings :: Donaldson-Thomas Theory : closed topological strings
and, thus the DT theory should compute Gromov-Witten invariants.
Us physicists quite like Chern-Simons theory. It shows up all over the place. It also can compute HOMFLY knot invariants. Tres cool.
(Sorry about hijacking the thread, Chad....)
Aaron,
Gromov may be sketchy, but not like Witten. Gromov doesn't always give best constants and he sometimes leaves out elementary but tedious things--but his proofs are correct-and brilliant!
Turning Witten's conjectures and "path integral handwaving" into real math has built several people's careers.
I am glad that you physicists like Chern-Simons because it is tres cool. I agree. However, that it can compute knot invariants is physics?
I have always suspected that you are a closet mathematician
( take it as an earned compliment).
Anyway, sorry Chad that Aaron and I hijacked the thread. But, it was delightful to me, and I learned a lot from Aaron.
Off the thread to give others a chance.
Penny
p.s. Another math joke: It was not a large Transgression to spell Chern Simons badly.
There is, of course, the "You Might Be a Phyics Major..." stuff.
(Sorry about hijacking the thread, Chad....)
Hijack away. I like having an active comment section.
Just don't expect me to contribute much, because I have no idea what you're talking about...
The experimentalist comes running excitedly into the theorist's office, waving a graph taken off his latest experiment. "Hmmm," says the theorist, "That's exactly where you'd expect to see that peak. Here's the reason."
A long logical explanation follows. In the middle of it, the experimentalist says "Wait a minute", studies the chart for a second, and says, "Oops, this is upside down." He fixes it. "Hmmm," says the theorist, "you'd expect to see a dip in exactly that position. Here's the reason...".
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While it was interesting to follow, and amusing to read the sequence of time stamps of the preceeding interaction, I must take issue. Penny (Dr. Smith?), physics is not math. The difference lies not in the hueristics or the approach, for you are right that physicists have been great mathematicians as often as we've played fast and loose with math, and that mathematicians have made for top notch physicists as well. The difference lies in the fact that, protestations of theorists aside, if we do something in an analysis and it matches what we measure, then until such time as some better explanation comes along, that is valid. For physics, the test is not analytical rigour but as much consistency as possible with what we can observe of reality.
I think the misidentification comes out of assuming that because a person is wearing or has worn one hat, the same person cannot wear the other. One can be both a mathematician and a physicist, just as one can be both a chef and a diner, or a musician and a dancer. These are not mutually exclusive identities.
amusing to read the sequence of time stamps of the preceeding interaction
Watching football and occasionally clicking reload on the laptop. Not an awful way to spend an afternoon, really, although things got a little nuts over at Woit's place. I wish the games were better, though.
Anyways, Dr. Penny's not the only person to think me a closet mathematician, and I certainly don't take it poorly. Now if I can only figure out more about those A-infinity algebras....
(re: #18)
After all these years, the Abelian grape joke still makes me laugh so hard my sides hurt.
Then there's always what's yellow and equivalent to the axiom of choice ...