From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Nilpotent fuzz Date: 11 Oct 2000 14:23:00 GMT Newsgroups: sci.physics.research Summary: [missing] In article <8rssa4$qr$1@news.acs.ttu.edu>, Jeff Lee wrote: >Dan uses the term nilpotent fuzz to describe what "it looks like" at a point >on a supermanifold. I have to admit that I am both annoyed and baffled. > >1. Can anyone talk me through some ideas that will help clarify what this >"nilpotent fuzz" is exactly (or even inexactly)? >1.a Physical intuitions behind this? >1.b Is the math rigorous? Yes, it's rigorous. Consult your neighborhood algebraic geometer! Unless the work of Grothendieck & Co. has completely passed them by, they should be able to give you some intuition for "nilpotent fuzz". This is one big reason why algebraic geometers switched from working with varieties to working with schemes. The basic idea is simple. Consider a ring like C[x]/ When c is nonzero, this is the ring of algebraic functions on the 2-point space {c, -c} - an affine variety in the complex plane. As a ring, this ring is isomorphic to C + C. No nilpotent elements! But as the number a approaches zero, the two points c and -c get closer together, and when c *equals* zero, we get C[x]/ This is *not* the ring of algebraic functions on the one-point variety {0} - that ring would be C[x]/. This ring is different! It's bigger! It remembers that we started with two points c and -c and let them approach each other. Heuristically, we should think of it as the ring of algebraic functions on "two points separated by an infinitesimal distance" - a slightly "fuzzed-out" point, as it were. Mathematically, this is manifested by the fact that C[x]/ has a nilpotent element, namely x. A typical element of C[x]/ looks like a + bx i.e., a Taylor series truncated after the first two terms, which tells us the value and first derivative of a function at x = 0. People have been thinking about this for ages, at least ever since they tried to understand the distinction between the usual sort of root of a polynomial equation and a "repeated root". Broadening the notions of algebraic geometry to include rings with nilpotents allow us to think of a "repeated root" as a different space - or technically, a different "scheme" - than an ordinary root. Not a mere point! A point with nilpotent fuzz! If you read good expository books on algebraic geometry you'll sometimes even see desperate attempts to *draw* this nilpotent fuzz by taking a point and drawing little rings around it. >2. Why can't we just use exterior forms to handle fermions? We can - but we want to think about them geometrically, too. Yes, one can say we've just got a bundle of exterior algebras over an ordinary manifold, but calling it a "supermanifold" and learning to treat it like a manifold with fermionic and bosonic coordinates helps us see what we're supposed to do with it. >Why try to make this supermanifold thing "modeled on a superalgebra" >with all the weird sign conventions etc. etc? The sign conventions are not weird, they are god-given and natural! Odd things anticommute with each other; even things commute with everything; the world was always thus. >Doesn't ordinary differential geometry, bundle theory >etc. have the recources to handle the physics of supersymmetry? Not if one wants to really *understand* supersymmetry. It's just like learning to work with vectors instead of lists of numbers. The pain you are experiencing is the pain of coming to understand a new way of looking at things. As usual, it's actually fun once you get over the hump. ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Nilpotent fuzz Date: 12 Oct 2000 08:08:22 GMT Newsgroups: sci.physics.research In article <8s09cr$ebp$1@mortar.ucr.edu>, John Baez wrote: >The basic idea is simple. Consider a ring like > >C[x]/ > >When c is nonzero, this is the ring of algebraic functions on the >2-point space {c, -c} - an affine variety in the complex plane. As >a ring, this ring is isomorphic to C + C. No nilpotent elements! > >But as the number a approaches zero, the two points c and -c get >closer together, and when c *equals* zero, we get > >C[x]/ Sorry: "the number a" should read "the number c". I should also point out a couple of other things. This algebra C[x]/ is just the exterior algebra generated by x, which is a nice simple example of a supermanifold. Every supermanifold has an underlying manifold called its "body", and here the "body" is a single point. However, not all examples of "nilpotent fuzz" correspond to supermanifolds. For example, if we start with the 3-point algebraic variety whose ring of algebraic functions is C[x]/<(x-a)(x-b)(x-c)> and we let all of a,b,c become zero, we get C[x]/ In this ring x is still nilpotent, since x^3 = 0. But this ring is not the ring of functions on a supermanifold - basically since the presence of nilpotent elements has nothing to do with exterior algebras here. Conversely, most supermanifolds don't have the same relationship to algebraic geometry that my original example did - since the ring of functions on a supermanifold is rarely commutative. Still, one can get a little intuition about supergeometry from algebraic geometry, since they share this concept of "nilpotent fuzz". ============================================================================== From: james dolan Subject: Re: Nilpotent fuzz Date: 11 Oct 2000 17:37:23 GMT Newsgroups: sci.physics.research jeff lee wrote: -I am a mathematician who wants to get more understanding of -supergeometry. I have looked at the liturature written for -mathematicians such as Dan Freeds book "Lectures on supersymetry" as -well as the famous two volumes on strings and quantum field theory for -mathematicians (can't remember the publisher....AMS??...). - -Dan uses the term nilpotent fuzz to describe what "it looks like" at a -point on a supermanifold. I have to admit that I am both annoyed and -baffled. - -1. Can anyone talk me through some ideas that will help clarify what -this "nilpotent fuzz" is exactly (or even inexactly)? -1.a Physical intuitions behind this? -1.b Is the math rigorous? - -2. Why can't we just use exterior forms to handle fermions? Why try to -make this supermanifold thing "modeled on a superalgebra" with all the -weird sign conventions etc. etc? Doesn't ordinary differential -geometry, bundle theory etc. have the recources to handle the physics -of supersymmetry? "nilpotent fuzz" is originally an idea from "ordinary" geometry (as opposed to "super" geometry). as such it is an extremely fruitful and intuitive idea with both rigorous and non-rigorous aspects. i'll try to explain the original idea here a bit, while remaining somewhat noncommital about whether some "super" version of it is a good idea. around the late 1800's, mathematicians proved some theorems (most notably "hilbert's nullstellensatz") of algebraic geometry giving nice correspondences between certain geometric objects (called "equational varieties in complex affine n-space") and certain algebraic objects (called "ideals in complex polynomial rings"). in these theorems, "nilpotent fuzz" made its first appearance as an annoying bug in the otherwise tasty soup. in order to get the nice correspondence between the geometric "varieties" and the algebraic "ideals", you first had to throw away those ideals which possessed "nilpotent fuzz" (meaning that the quotient ring of the ideal contained nilpotent elements, which are nonzero quantities some finite power of which is zero). in the conceptual picture of late 1800's algebraic geometry, nilpotence was an algebraic phenomenon which lacked a good geometric interpretation. over the following century or so, mathematicians gradually realized how foolish it was to think of nilpotent fuzz as a "bug" rather than as a "feature". it turned out that nilpotence _did_ have a geometric interpretation after all, and that the geometric interpretation of nilpotence had secretly been a central theme (perhaps even _the_ central theme) of mathematics for hundreds of years. another name for "the geometric interpretation of nilpotence" is "differential calculus". another name for "nilpotent quantity" is "leibnizian infinitesimal". find a real cheap 1950's style calculator. load a really small number (say .00000001 or so) into it. square it. the answer is zero (or your calculator isn't cheap enough). a really small number on a cheap 1950's style calculator is nilpotent. cheap 1950's style calculators were invented by pascal and leibniz, the same people who invented differential calculus; that's not just a coincidence. real small nonzero numbers get smaller and smaller if you keep squaring them, but as long as you can continue to afford more and more expensive calculators they never quite become zero. nilpotence is a comic exaggeration of the real properties of real small nonzero numbers. ("my number is _so_ small, it's nonzero but when you square it you get zero!") how do you calculate the differential of a function f at an input x? the dainty way is to consider the difference quotient formula (f(x+d)-f(x))/d for tinier and tinier real values of d, taking the limit as d goes to zero. the smart way is, to hell with limits, you just plug a nilpotent quantity d (satisfying d^2=0) into the difference quotient formula and evaluate it. where do you get a nonzero element d satisfying d^2=0? you just invent it. or buy a 1950's style calculator. more comic exaggerations: "my hometown was so small, the other side of the sign that said 'welcome to smallville' said 'welcome to smallville'". "the geometric variety corresponding to the ideal of one-variable functions whose taylor series vanish up to first order at a point p on the line contains the point p, and it also contains a non-trivial tangent vector- but it doesn't contain any macroscopic points other than p." a geometric variety, a "space", that's so small that it contains a tangent vector that "doesn't go anywhere". the whole space is just this disembodied "tangent vector to nowhere". a sort of "walking tangent vector". "nilpotent fuzz". the disembodied tangent vector itself embodies the very concept of tangent vector. other, "macroscopic" spaces embody macroscopic concepts; the walking tangent vector embodies a "microscopic", "infinitesimal" concept. other microscopic spaces embody other important microscopic concepts. quotient rings with nilpotent elements were thought not to correspond to geometric concepts, but they do; they correspond to geometric concepts with important infinitesimal aspects. how can infinitesimal things be important? who knows; we only know that they _are_ important. nature likes comic exaggerations. "nilpotent fuzz" makes sublime sense in "ordinary" geometry. what about in "super" geometry? i have no idea. i've seen attempts to explain supergeometry that try to claim that the "nilpotent fuzz" idea from ordinary geometry can provide intuitive motivation for supergeometry, but that just end up sounding stupid and ugly and unconvincing. maybe i just haven't run across a good explanation of supergeometry yet. maybe i will eventually, or maybe no one ever will. Sent via Deja.com http://www.deja.com/ Before you buy.