In the Fall I’ll be starting teaching again, after a semester away on sabbatical and then enjoying teaching relief during my first semester at Penn. I’ll be teaching a course that I truly love, and that I’ve taught a number of times before – Mathematical Methods of Physics I, to a class of beginning graduate students, and some interested seniors.
The backbone of this course, as I teach it, is rather traditional, since the topics involved are things that form the basis of the toolbox that professional physicists need. From year to year I have added various extra topics (some differential geometry, some topology, some group theory, …), but I always cover
- Analysis of Complex Functions
- Exact and Approximate Evaluation of Sums and Integrals
- Exact and Approximate Solution of Ordinary Differential Equations
- Transform Calculus
- Sturm-Liouville Theory
- The Calculus of Variations
One challenge in a course like this is to maintain the connection with actual applications of the techniques one is covering. Since I was originally taught this material in a set of courses as a mathematics undergraduate, my own take on the material can be rather formal, and I have worked over the years to balance this out. However, as you might guess, my own examples are predominantly drawn from those areas of physics with which I am most familiar – for example, supersymmetry, and the restrictions that holomorphy places on superpotentials, is a nice illustration of the power of complex analysis.
But this course is supposed to provide a basis for all graduate students, including those with interests in other branches of theoretical physics or, indeed, experimental physics or observational astrophysics and cosmology. There are, of course, rather general things that one can do that should be of use to everyone, such as the use of Fourier and Laplace transforms in solving heat, diffusion, and other equations. And the calculus of variations appears everywhere already. There are also, incidentally, lots of cute things one can do in the opposite direction, like cooking up examples of oscillating systems in which the sum over all modes gives the total energy, which is easy to calculate another way, and using this to provide a way to compute infinite sums. Nevertheless, what I really yearn for are even more examples illustrating the use of some of the above topics from other branches of physics.
I could, of course, annoy my colleagues with this question, but I thought that opening it up to Cosmic Variance readers might provide some novel suggestions. So, if you have some unusual example, brief enough to be useful in a class, of the use of any of the above in any branch of physics (even particle physics and cosmology – there’s plenty I don’t know there also), I’d appreciate you filling me in in the comments.
And if any of my students-to-be are reading this – beware; it’s possible that good suggestions you see here, that don’t make it into class, may turn up on exams – who knows?
June 17th, 2009 at 7:49 pm
You could discuss how approximations can lead to divergent series. This is very typical in physics, so it shouldn’t be difficult to find a suitable example. Then you can discuss how you can resum such a series using e.g., Borel resummation. This then illustrates how the non-perturbative part that was lost when you did the perturbative expansion was regained: It was hidden in the divergent tail of the summation.
You can then refer interested students who want to learn more about this to good books like this one:
http://www-personal.umich.edu/~jpboyd/boydactaapplicreview.pdf
June 17th, 2009 at 8:28 pm
Since you’ll be touching on group theory, and all of your students will have had at least brief exposure to special relativity, you could demonstrate the isomorphism between the (restricted) Lorentz group and the Möbius group, and Möbius transformations of the night sky.
June 17th, 2009 at 9:22 pm
My class on mathematical physics when I was an undergraduate was deeply dissappointing to me, and it covered the same material as your syllabus (I gather its pretty standard). Maybe it was simply poorly taught so that im biased, but then I think many of the books titled ‘mathematical physics’ suffer from the same sets of problem.
So much so that I ended up simply taking the mathematics classes that correspond (so grad complex analysis, real analysis, functional analysis, group theory, algebraic and differential geometry etc).
The problem typically is that you don’t learn enough about the nitty gritty mathematics to really understand the material deeply. Which would be fine (we are mostly physicists not mathematicians) but then in practise you more or less have to relearn the entire equivalent but idiosyncratic notation and concepts over again when it comes to special subfields where the material is applicable. Read a lot of redundant information. Also, the material gets caught in a sort of never never land between the exact mathematics and what physicists really need in practise, and for time reasons also have to cut corners (which can be confusing)
I went in there believing that I was going to learn a lot of shortcuts and tricks that theorists use to save time relative to pedantic mathematicians, but in practise it ended up saving very little (and in some cases simply adding or wasting time if you already have the relevant mathematics course).
June 17th, 2009 at 9:28 pm
I loved my first mathematical methods course. Only book I still own from undergraduate (Boas: Mathematical methods in theoretical Physics). Hardest course I have ever had the pleasure of taking.
For practical purposes, a good refresher on statistics is among the most useful pieces. It is so often shunted aside, but getting a fundamental feeling for statistics is incredibly important. Usually dull, and hard to visualize for just about everyone.
I also find control (PID and equivalent) a really nice way to visualize the use of complex function analysis. And you get to tell the poles on the right hand side of the plane joke.
If you are doing group theory, basic crystallography and diffraction is useful. Raman spetroscopy is also a nice trick. There are lots of nice bits of math that you can go through, and more important, you cna realyl improve visualization skills.
Comparisons of finite element and spectral methods of solving various types of mechanical equations is good. Details of finite elements actually touch on a whole range of interesting and often underappreciated equations. It also touches on just about every major field.
Greens finctions as applied to crystalline dislocations (which you can actually see and test) is an excellent approach
And finally, heat and mass transport (say, simple fluid dynamics) are realyl sueful topics.
I have a strong bias towards learning pieces that give physical intuition. Thi sdevelops that all-important sense of physical intuition that most undergraduat courses steal away (you can much more easily plug and chug your way to success).
June 17th, 2009 at 9:36 pm
Well, two topics I can think of:
1. Evaluating partition functions in statistical mechanics. There are a number of little mathematical tricks that can be used to make these integrals manageable, and I remember having a rather difficult time evaluating them in my statistical mechanics courses (normally I have no problem with this sort of thing…). It might be nice for at least some students to see these sorts of integrals more than once.
2. Calculating geodesics in General Relativity. I’ve found it to be quite useful to know how to compute geodesics, particularly for light, in my work in cosmology. Some examples that are used are computing the horizon scale in a de Sitter universe, or computing the deflection of light rays by a lensing potential.
Just two ideas.
June 17th, 2009 at 9:42 pm
I definitely agree about adding a day or two of statistics. Its one of those subjects that is often put aside in physics curriculums (until you take either stat mech or a laboratory class where you have to do error analysis).
Another big one, add a day of numerical methods (so Runge-Kutta, etc). Most of us have to learn that in the first week of research (heres a book, learn it) and even a day of class would have saved a lot of time if for no other reason that it acclimates the student to the language.
The heat equation would have been nice (if a bit advanced) and you can always throw in a non run of the mill subject (20 minutes of wavelet analysis after the fourier analysis is done for instance to spice things up)
June 17th, 2009 at 9:55 pm
For the love of God, tell your students about what a gauge theory is. I’m constantly astonished by the number of people, many of whom are even very senior graduate students, who aren’t aware of the link between gauge theories and singular Lagrangian systems. This is really little more than elementary linear algebra, but for some reason it’s never taught. It’s all the more appalling when you realise that it should be covered in any introductory course on analytical mechanics and variational methods.
By the way, unless you’re going to treat those subjects you mentioned in the post at an enormously advanced and rigorous level, I can’t imagine why graduate students would gain from them. Surely transforms, variational calculus, Sturm-Liouville et al are covered in excruciating detail during the first year or two of their undergraduate degrees?
June 17th, 2009 at 10:37 pm
If you want to go a little outside of the box, there are a lot of neat applications that come up in accelerator physics. For example: you can talk about hamiltonian systems and the consequences of symplecticity, which is in a lot of accelerator physics textbooks, and also Liouville’s theorem and how it results in the Vlasov equation, which being quasi-first order is subject to techniques like this one:
http://people.math.gatech.edu/~loss/07falltea/6341/notes/quasilinear.pdf
There’s also a wealth of trickery with studying analytic properties, Laplace transforms and the like that arises out of Landau’s treatment of the Vlasov equation. I don’t have the citation, and my copy of the paper is sadly about a thousand miles away right now, but he gets into a lot of the tricks that also arise in evaluating Feynman diagrams, but in a different context. Laplace transforms seem to be popular amongst Russians for solving these types of things. Another example can be grabbed by the solution of the 1D FEL equations in Saldin Schneidmiller and Yurkov’s textbook on the subject.
Finally, there are transfer matrices and the like, but the course is “math methods” not “classical mechanics” and that is actually a rather trivial result.
June 17th, 2009 at 11:19 pm
There’s always the classic use of complex analysis and conformal maps to deal with two-dimensional fluid flows.
Digital filters provide a nice way to link Fourier transforms, z-transforms, and complex rational functions. Convolution and deconvolution, with their natural Fourier interpretation, also turn up all the time in imaging; if you want a real monster headache you can talk about the black art of radio telescope interferometry.
Quantum mechanics could be a really nice way to get at Sturm theory: it lets you get the qualitative behaviour of eigenfunctions that you can’t actually solve for.
June 18th, 2009 at 1:06 am
The physics of music might be a good way of presenting a number of topics.
June 18th, 2009 at 2:17 am
I definitely think you should include a bit of probability and statistics. They are neglected in most physics undergraduate courses, but need to be understood by every working physicist…there are some neat connections to other things on your list but it’s good to come at things from another direction to reinforce understanding. Plus, you get to say funny things like “Karhunen–Loève theorem”…
June 18th, 2009 at 2:50 am
Our department offers a graduate-level probability and statistics course.
I took traditional math methods courses when I was in school, but I have to admit as an experimentalist I’ve never used any of this stuff since. Numerical methods and statistics however are used ALL THE FREAKING TIME.
June 18th, 2009 at 5:33 am
I agree with 12. It is really unfortunate that numerical methods are never really covered very well in these sorts of courses, when in fact they are some of the most important tools in the modern scientist’s toolbox. While one can debate endlessly the merits of being able to solve equations analytically, numerical methods deserve at least some coverage–while there is a small set of “soluble” differential equations, a knowledge of numerical methods allows students to gain an understanding of almost any differential equation. Moreover, the graphing and plotting of those solutions that often accompanies their numerical computation is very good for building intuition.
Statistics probably belongs in another course, though…
June 18th, 2009 at 6:34 am
Thanks to everyone for the comments so far. I appreciate them all, and will think about some of them. Just to focus things a little more though, I’m not really looking for curiculum suggestions – I’ve thought already about what I want to go into a course of this length, and while I will include different bits of other things from year to year, when time allows, I don’t expect to include large new topics in my particular version of the course (and yes Martin, the level will be significantly more rigorous than one sees in typical undergraduate treatments, and there will actually be some nitty-gritty mathematics Haelfix – sorry to hear that your course was so disappointing). Not to say that the perspectives provided here aren’t useful – they are.
What I would really like though is more stuff like Anne, Steve, Jason, Nick and Brennan have provided – examples of the topics being applied in a variety of fields.
June 18th, 2009 at 6:40 am
I like conformal mapping both in 2d but also in 3d where spheres go to planes. That let’s you solve problems on, say, when a charged oil drop breaks up into two charged oil drops. As for complex analysis, how about infinite products that evaluate to trig functions (or elliptic functions)? That’s a nice example of the uniqueness of meromorphic functions. I know an experimentalist that could have used that knowledge a few weeks ago. Wavelets as an example of orthogonal functions?
June 18th, 2009 at 7:21 am
Hi Mark,
When I taught this class, one of the best pieces of advice I was given for a course philosophy was to emphasize that it was all about techniques for evaluating Green’s functions. That is something that comes up in all aspects of physics, and something that all students can relate to, whether you’re talking about QM, E&M or propagators in QFT.
Best,
Steve
June 18th, 2009 at 9:43 am
One thing I really liked learning about in my graduate E&M class was solving E&M problems using conformal mappings. It not only helps justify a lot of the complex analysis techniques, but really highlights a lot of the structure of E&M that most people don’t normally think about when solving electrodynamics problems (or at least, highlights it in a different way).
I don’t know how far you’re going to go with geometry, but one thing I always found very beautiful is the connections between classical E&M and differential forms. Of course, it’s even nicer to look at gauge theories in general like this, but learning about this was one of the first things that made me realize how much amazing structure went behind this seemingly simple theory.
Of course, if you’re going that far, it’s nice to see Newtonian mechanics done in that language as well
.
If you’re doing the calculus of variations, once nice discussion is on when the action is a min/max/saddle point. This isn’t something I’ve seen discussed in too many mechanics books, but there is some interesting behavior here to look at.
It’s also nice, and an easy exercise, to derive the action for SR and think about its nice geometric interpretation. It’s not too bad to go from this to some discussion of the GR action and geodesics.
I also really like the idea of discussing asymptotic series, and some specific instances where these arise, and how to deal with them and get sensible answers.
June 18th, 2009 at 10:04 am
First of all, with respect to the first comment, the following quote belongs in every set of notes that mentions divergent series
“Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.”
–Niels Hendrik Abel
So, this term I’m teaching a similar course for undergrads, and a few of them have been chomping at the bit for something more involved than the standard fare. Since we’re spending a lot of time with special functions, I have one of these students working on an inflationary power spectrum calculation. One you introduce a few basic concepts, it becomes a straightforward application of the material on Bessel functions. Best of all, it makes immediate contact with the sorts of things physicists are interested in right now, which is a different sort of reward than what a student normally gets in this sort of class.
June 18th, 2009 at 11:12 am
Sounds good; please post course notes on the blog if you prepare them.
June 18th, 2009 at 11:30 am
Since most of your students will end up doing experimental work, you should ask some of your experimental colleagues if there are examples they can think of.
For transforms, you can discuss the relation of laplace transforms to filters. This is a nice and concrete application, and very useful!
June 18th, 2009 at 1:02 pm
If you read the pdf I linked to in my first posting, you find many wonderful applications of asymptotic methods in physics. It may be the case that Abel’s quote mentioned by Robert above led to mathematicians not developing the hyperasymptotic methods until the 20th century. It is often the case that physicists develop a dodgy mathematical techique which mathematicians do not like and then later it turns out to lead to a new branch of mathematics. E.g., think about Dirac delta function and the subsequent development of the theory of distributions.
Anyway, as far as physics is concerned this quote may be more relevant:
George F. Carrier
(it is also mentioned in the book I linked to in my first posting above)
June 18th, 2009 at 1:21 pm
Lie groups, Group Representations, Hibert & Banach spaces are v. useful topics as well. I don’t know about the academic background of your students, but I’m guessing that some of them would have a stronger training in Physics, while others would have come to your class via a more Math intensive track. The primary difficulty I envision in a class like yours is that (in general) Math people tend to like the theorems presented in a more generalized, abstract sense while the Physics folks tend to like the applications based approach. Given the time constraints, I presume it would be hard to appeal to both crowds, though.
June 18th, 2009 at 1:52 pm
I’ve had oodles of maths that I never really learnt to connect to the real world (my fault – it didn’t interest me at the time – real problem was that I never really understood it).
So I’d love to see this course on Youtube!
Or at least lecture notes of some sort if at all possible.
June 18th, 2009 at 2:32 pm
Hi Mark,
Like you I learned mathematical physics in a math dept. Since you are from the UK, you might already be aware of Walter Appel’s mathematical physics book.
http://press.princeton.edu/titles/8452.html
The exercises are lacking, but I really like the content of this book. It presents everything as it would appear in a math setting without being rigorous with proofs. This helps me communicate with mathematicians while staying focused on the physics. For instance, my physics professors always taught me to remember that the Dirac Delta function doesn’t exist outside an integral and that it is properly defined in terms of a distribution without telling me anything further. Appel breaks it down, explaining the relevant measure theory, to give a satisfying definition as a linear form on a space of test functions.
June 18th, 2009 at 6:01 pm
If you’re looking for ODE examples, calculating deflections of beams came up over and over again during my PhD, which included part designing and building an STM.
http://en.wikipedia.org/wiki/Euler-Bernoulli_beam_equation
Using this stuff to crudely estimate resonance frequencies of parts of my instrument was crucial both in design and in diagnostics. The math here might not be advanced enough for your class though.
June 18th, 2009 at 8:18 pm
Wow, where do I begin…?
Seems many responses were not unlike someone ordering a dinner guest their recommendations for the tastiest dishes off a french menu…”Escargot?” Non, merci !!!
LETS GET PHYSICAL & PRACTICAL…Mobius groups ? Borel summation ? SUSY ? This is supposed to be a nuts & bolts `Arfken-like’ course in MM, which covers bread & butter topics like standard model groups, tensor and vector calculus, complex analysis, Fourier/Laplace transforms, adv.diff-eq., distribution theory,etc., so that beginning grad students are equipped early-on with the basic mathematical techniques to understand the content of their core-courses & go on to comprehend the literature. If time permits, maybe a smattering of differential geometry, lie group reps, and other more advanced topix at the discretion of the instructor.
The bread & butter topix are those which a physics grad student MUST get from physicists, as the mathematics depts. neither care nor want to muddy their mittens,and as a result cater mainly to mathematicians, leaving most experimental and theoretical physicists out in the cold.
Hopefully, Mark will pick a good established text,and augment it a bit here & there, but for the most part adhere to the author’s plan so the class knows this is a standardized course, and not a pedestrian tour thru exotica which will only confuse, and not empower them.
Sadly, many physics depts. are no longer requiring a full year of math methods, and Arfken-like courses may already be an endangered species. Part of the failure is Arfken itself, the `Goldstein-Jackson-Sakurai’ of MM texts, which as Mark laments, short-schrifts the student on the depth & background of this & that topic. Alternatives tb considered range from the practical to the esoteric, such as: Hobson/Riley to Hassani to Lam to Szekeres.
If I were an experimentalist, I would demand Hobson/Riley.
But I’m a theorist, and for them, its hard to beat Szekeres.
June 18th, 2009 at 9:38 pm
Oops, I forgot, thank-U Twaters !
Walter Appel’s book, `Mathematics for Physics and Physicists’ is nothing short of perfection.
Our library at Oregon has a copy of it, that’s how I came to be familiar with it, for neither Amazon nor Google have examinable pages, but Amazon shows good reviews.
I emailed the author for a list of errata, which he rtn’d & it’s very short.
June 19th, 2009 at 7:02 am
Steve, in 8, mentioned accelerator physics. An important part of accelerator physics are storage rings- think either LHC or a synchrotron light source. There are lots of interesting physics associated with storage rings besides the primary goals of colliding particles or creating light.
Basically you want to know about the long term dynamics- what happens over many turns, to these ~speed of light particles. Near the main orbit, one can use standard linear optics methods to analyze the dynamics. Away from the orbit, one needs non-linear dynamics, and it gets into chaos and such.
A big step forward in understanding how to analyze rings was the Lie Algebra methods of Alex Dragt. He has a book online at
http://www.physics.umd.edu/dsat/dsatliemethods.html
From the mathematical physics angle, one could also look at a much neglected book by L. Michelotti called “Intermediate Classical Dynamics with Applications to Beam Physics”.
One learns here about manifolds, entropy, Hamiltonians etc.
This example is an awesome one for getting pretty deeply into a variety of classical mechanics topics- KAM theorem, Nekhoroshev theorem, and other topics in chaotic dynamics all apply.
And, like any good mathematical physics topic, it has fed into entirely different systems: the topic of quantum accelerator modes in cold kicked atoms:
http://arxiv.org/abs/0805.2040
or
http://massey.dur.ac.uk/research/qchaos/qam.html
Anyway, I think this is a fascinating, rich topic that deserves a more central place in mathematical physics courses.
I’d be glad to think of more references if interested.
Cheers,
Boaz
June 19th, 2009 at 7:40 am
Are you familiar with the problem of finding the resistance between two nodes in an infinite grid of resistors? See here for instance: http://www.geocities.com/frooha/grid/node2.html
The problem has the virtue of being really easily stated and readily understood by anyone with even a semester of undergraduate E&M . . . but solving it is quite tricky, until it occurs to you to use a Fourier transform.
Plus, it gives you an excuse to show them this comic: http://www.xkcd.com/356/
June 19th, 2009 at 9:00 am
Looking back, I would have preferred more on statistics and on Lie groups.
June 19th, 2009 at 2:20 pm
How about variational methods for waveguide discontinuities? Show your students that Schwinger was a rockstar even before QED!
Waveguides as an example of modal expansions?
Far-field antenna patterns using fourier transforms?
June 19th, 2009 at 2:40 pm
The perspective of a non-theorist (observational cosmologist):
I had one quarter of mathematical methods first quarter of my first year of grad school. I had one quarter of statistics and numerical methods at the end of my second year. After my qualifying exam, I have never, ever used anything I learned in MM. Statistics and programming is my bread and butter (although the class came too late and was taught too poorly to be of much use).
So I’m actually all for some more esoteric, fun examples. I love math (got an undergrad degree in it), and would have enjoyed such examples. No offense, but the experimentalists/observationalists will probably not use anything you teach them. They don’t actually care much about your class. So throw in some entertaining examples at least.
June 19th, 2009 at 3:04 pm
What’s Randy doing now? His class last fall was a total blast! You’re going to have to beat having a whole roasted pig for the end-of-class lunch, Mark.
That being said, ask Ron Donagi for good Sturm-Liouville examples. When I took his Math 240 class he did a neat example of vibrations in a metal beam.
June 19th, 2009 at 3:27 pm
Charon’s experience matches this observer’s quite well for my undergrad MM class, though I see this as a systemic problem. My undergrad class used the “traditional” curriculum and text (Arfken) and was mostly useless and incredibly badly taught be a well-respected prof. My two semester grad class however was great: didn’t use a text, was very practical rather than being designed for math majors (if I liked math that much, I would have majored in it instead of physics), and it was taught by a great lecturer. So, most good suggestions here have already been shot down with a few notable exceptions like deconvolution which could lead (but probably won’t) into other non-unique problems.
June 20th, 2009 at 4:53 am
Speaking of the ODEs: as long as you throw some group theory in, it would be great to discuss the symmetry-based methods for solving ODEs. There is an excellent book on the subject: P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, N.Y. 2000. You also can use this book for covering Hamiltonian systems and the Noether theorem in the calculus of variations.
June 20th, 2009 at 5:18 am
Keep getting the “Load comment failed” error, so will continue here. This is pretty standard but just in case: for the complex analysis, one could mention the applications in QFT and scattering theory (analytic S-matrix and all that). As for the approximate solution of the ODEs, how about including some small bits of KAM theory and applications to celestial mechanics? Also, you might consider discussing some basic aspects of soliton equations (like Korteweg–de Vries, nonlinear Schrödinger and sine-Gordon) and the inverse scattering transform. In particular, considering the KdV equation could be a nice extension of the Sturm-Liouville theory.
June 22nd, 2009 at 5:34 am
Mark, I hope you won’t be offended if I throw in a general suggestion on teaching, from the perspective of a student who just recently finished his degree. You may already be a great teacher, but there are definitely some professors I had who would benefit from this advice.
In my experience, the more mathematical the subject matter, the greater the temptation for the professor to basically spend the class just kind of working through the algebra and writing one equation after the next. This adds very little to what can be gained from just reading the textbook. What I wish professors would do is plan each lecture around a set of concepts they want the students to learn, and structure the lectures with the goal that, if nothing else, the students will understand X at the end of the hour. And, moreover, that it will be clear to the students themselves that X is the point of the lecture. I can’t tell you how many lectures I attended that consisted of just furiously scribbling down equations, at the end of which I understood how one equation followed from the next but had no idea which equations were important or why.
E.g., instead of diving into “here is how you evaluate a contour integral”, one can start with some bullet points like “contour integrals are integrals along paths in the complex plane”, “there are a number of useful theorems for evaluating them, such as the residue theorem or the Cauchy integral formula”, and “we can translate some real integrals into contour integrals, which allows us to evaluate them by means of these tricks”. It might be good to actually start the lecture by writing these bullet points on the board, because (as some professors seem not to have realized) students who are busy taking notes won’t necessarily hear every word that comes out of your mouth. Alternatively, one could hand out a paper (or post one online) with “Here are the key points you should learn from this lecture.”
Maybe that’s obvious advice, but believe me when I say that not everyone follows it.
June 24th, 2009 at 8:01 am
This is a minor point, but if you’re covering Sturm-Liouville theory etc, a subtlety that doesn’t seem to be well-known is the use of non-orthogonal functions in solutions of the Schroedinger equation. Merzbacher (Quantum Mechanics) is the only book I know that covers it. Arfken just straight up dismisses the technique. I used it this year for a Master’s project in quantum mechanics, and the convergence I found was much better than using orthogonal functions for the same function.
I’m not saying it merits major discussion, but I think it’s something to be pointed out, and the tensor formulation of the theory is really nice – you end up with an object a bit like a metric representing the overlap integral between the non-orthogonal functions.
June 26th, 2009 at 4:57 am
Maybe only in the ‘neat’ category:
There’s a neat way to introduce 4D twistors through standard 2D complex analysis & group theory.
- on CP^1 there are homogeneous and inhomogeneous coordinates. If you study the transformations the conformal transformations are only linear on the homogeneous coordinates; same is not true for the inhomogeneous ones. You can derive the transformations.
key equation: z_1 = w z_2
So far so good.
- The key observation is that ‘twistors’ are to the usual x^\mu as what homogeneous coordinates are compared to the inhomogeneous ones: twistors transform linearly under the conformal group.
Now you will need 2 vectors on C^2 (2 spinors) and the w is just x^{\mu} \sigma_\mu. The above key equation turns in to the coincidence relation, you can discuss SO(4) \sim SU(2) x SU(2), spinors, etc, etc