From: toby@ugcs.caltech.edu (Toby Bartels) Subject: Re: Geometric quantization Date: Wed, 12 Jul 2000 03:20:39 GMT Newsgroups: sci.physics.research Summary: [missing] John Baez wrote: >Toby Bartels wrote: >>John Baez wrote: >>>I want to finish up what we've been talking about, and say a bit >>>about how it relates to Cech cohomology and other stuff, and then >>>get back to talking about geometric quantization. >>Yes, do that please. >Okay, I'll do this at the end of this post... IF you do well enough >on these questions. [rest cut] Agh! So many "Eh?"s! I'm so ashamed! I'm going to fail topology if I don't get my head on straight. Let me do it all over again from the beginning. >>>1. Classify complex line bundles over the n-sphere for all different n. [S^(n-1), U(1)] = pi_{n-1} (S^1) = delta_{2,n} Z. >>>2. Classify real line bundles over the n-sphere for all different n. [S^(n-1), O(1)] = pi_{n-1} (S^0) = delta_{1,n} Z_2. >>>3. Classify principal bundles with structure group SU(2) over the n-sphere >>> for n = 0,1,2,3,4. (Hint: SU(2) is a sphere in its own right.) {0 if n = 0 {0 if n = 1 [S^(n-1), SU(2)] = pi_{n-1} (S^3) = {0 if n = 2 {0 if n = 3 {Z if n = 4 This is because, as you said, >All maps from a sphere to a higher-dimensional sphere are homotopic. But I'm afraid I don't know enough homotopy theory to say offhand what pi_4 (S^3) is. Do I have to calculate it? Do you think I can do that? -- Toby toby@ugcs.caltech.edu ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Geometric quantization Date: 12 Jul 2000 20:40:15 GMT Newsgroups: sci.physics.research In article <8k3ptk$i19@gap.cco.caltech.edu>, Toby Bartels wrote: >Agh! So many "Eh?"s! I'm so ashamed! Hey, don't feel bad - we all have our off days. C'est la vie. >>>>1. Classify complex line bundles over the n-sphere for all different n. >[S^(n-1), U(1)] = pi_{n-1} (S^1) = delta_{2,n} Z. Right. In other words, there are nontrivial complex line bundles over the 2-sphere - but not over spheres of any *other* dimension! What does this mean for physics? Well, as I explained in http://math.ucr.edu/home/baez/braids.html , nontrivial complex line bundles over the n-sphere correspond to magnetic monopoles of the Dirac sort in (n+1)-dimensional space. So we see that this sort of magnetic monopole can only exist in 3d space! Moreover, since complex line bundles over the 2-sphere are classified by *integers*, the magnetic charge of such a monopole is *quantized*. And if two monopoles could collide and form a single, their charges would *add* - that's where the group structure on the integers comes into the picture. >>>>2. Classify real line bundles over the n-sphere for all different n. >[S^(n-1), O(1)] = pi_{n-1} (S^0) = delta_{1,n} Z/2. Right. In other words, there are nontrivial real line bundles over the 1-sphere - but not over spheres of any *other* dimension! What does this mean for physics? Well, if we were studying a gauge theory where the gauge group was O(1) instead of U(1), it would mean *this* sort of theory would have "magnetic monopoles" only when space was *2-dimensional*. And the "magnetic charge" of such a monopole could only take on two different values: present or absent! In other words, the charge is Z/2-valued. In fact, I bet that such a theory is of interest in some corner of condensed matter physics. The great thing about condensed matter physics is that materials can be modelled by all sorts of funky models. Somewhere there are thin films of some stuff with degrees of freedom corresponding to an O(1) gauge theory, and this stuff will have spots called "defects": when you march around a defect, you get a nontrivial holonomy in the O(1) gauge field. Since O(1) = Z/2, the holonomy around a loop is either 0 or 1. If it's 0 there's not a defect inside that loop; if it's 1 there is. Defects can cancel when they meet - that's because 1 + 1 = 0 in Z/2. The calculation you did also shows that there are just 2 kinds of [0,1] bundle over the circle: the Moebius strip (where a twist is present) and the ordinary untwisted strip (where it's absent). If we think of these as strips made of paper, we can imagine a "strip-splicing operation" where we cut open two such strips and splice them together. With this operation, isomorphism classes of [0,1] bundles over the circle form the group Z/2. E.g., "Moebius strip spliced with Moebius strip gives untwisted strip", and so on. >>>>3. Classify principal bundles with structure group SU(2) over the n-sphere >>>> for n = 0,1,2,3,4. (Hint: SU(2) is a sphere in its own right.) > > {0 if n = 0 > {0 if n = 1 >[S^(n-1), SU(2)] = pi_{n-1} (S^3) = {0 if n = 2 > {0 if n = 3 > {Z if n = 4 > >This is because, as you said, >>All maps from a sphere to a higher-dimensional sphere are homotopic. Right. In other words, there are nontrivial quaternionic line bundles over the 4-sphere - but not over spheres of any *lower* dimension! What does this mean for physics? Well, if we were studying SU(2) gauge theory in 4d *space*, we would get monopoles with integral charge. But since we usually study SU(2) gauge theory in 4d *spacetime*, we instead get "instantons" with integral "instanton number". I should talk more about instantons sometime, but not today. Suffice it to say that they're a big deal in particle physics - so all the calculations I forced you to do are positively *dripping* with applications. >But I'm afraid I don't know enough homotopy theory >to say offhand what pi_4 (S^3) is. >Do I have to calculate it? Do you think I can do that? Probably not - that's why I didn't ask you to classify SU(2) bundles over the 4-sphere! Since I am a wizard, I know that [S^4, SU(2)] = pi_4 (S^3) = Z/2 so that there are exactly two SU(2) bundles over the 5-sphere. And I know that [S^5, SU(2)] = pi_5 (S^3) = Z/2 so there are also exactly two SU(2) bundles over the 6-sphere. And on a good day I even know that [S^6, SU(2)] = pi_6 (S^3) = Z/12 so there are exactly twelve SU(2) bundles over the 7-sphere. In fact, this is related to the mysterious appearance of the number 24 throughout string theory! But at this point I'm just showing off. It's more important to remember that nobody knows all the homotopy groups of any spheres except S^0 and S^1. So classifying quaternionic line bundles over spheres is infinitely more difficult than it is for real or complex line bundles! - except in the low-dimensional cases I asked you about. Okay, enough of this.... let's get back to geometric quantization per se.