[continued from previous message]   
      
   This algebra is actually a strong r-commutative algebra in a unique   
   manner such that   
      
   R(p x h) = h x p   
      
   R(q x h) = h x q   
      
   and   
      
   R(p x q) = q x p - i h x h.   
      
   Note what these say: the first two say that you switch h with p and q in   
   the usual way, and the third one says that when you switch p and q, you   
   get the expected term AND THEN a piece, -i h x h, which comes from the   
   fact that p and q don't commute.  The point is that while the Heisenberg   
   algebra is not commutative, the canonical commutations relations do tell   
   you exactly what to do when you switch p and q, which is just as good!   
      
   If I may digress... I happen to have the paper by Heisenberg, Born, and   
   Jordan with me, "Zur Quantenmechanik. II.", published in 1926, in which   
   the canonical commutation relations are introduced.  I quote:   
      
   Das Rechnen mit den quantentheoretischen Groessen wuerde wegen der   
   Nichtgueltigkeit des kommutativen Gesetzes der Multiplikation in   
   gewissem Sinne unbestimmt bleiben, wenn nicht der Wert von pq - qp   
   vorgeschrieben wuerde.  Wir fuehren daher als fundamentale   
   quantenmechanische Relation ein:   
      
                          pq - qp = (h/2 pi i)1.   
      
   ("Due to the failure of the commutative law for   
   multiplication, computations with quantum-theoretical quantities are   
   ambiguous in a certain sense, unless one prescribes a value for $pq-qp$.   
   We introduce accordingly the following as the fundamental   
   quantum-mechanical relation...  pq - qp = (h/2 pi i)1.")   
      
   If we take the Heisenberg algebra as a strong r-commutative algebra and   
   define the differential forms on this algebra by the rules 1), 2) and   
   3'), we get the following standard-looking relations:   
      
   h dp = dp h,   h dq = dq h,   p dh = dh p,   q dh = dh q,   
   h dh = dh h,   p dp = dp p,   q dq = dq q,   
      
   and a bunch of similar ones involving two differentials:   
      
   dh dp = -dp dh,   dh dq = -dq dh,   dp dh = -dh dp,   dq dh = -dh dq,   
   dh dh = dp dp = dq dq = 0,   
      
   but then some more interesting ones:   
      
   p dq - dq p = -i h dh,   
   q dp - dp q = i h dh,   
   dp dq = -dq dp.   
      
   (It's a nice little exercise to see that these really *do* follow from 1)-4')   
   One can have fun doing various things with these "quantized differential   
   forms" (and their generalizations), basically by taking your favorite   
   facts from the differential geometry of phase space and trying to   
   "quantize" them, but let me just briefly run through the basics.   (Now I   
   will let myself be more mathematical.)   One may form a quotient of the   
   Heisenberg algebra by specializing the variable h to some value; this is   
   called the Weyl algebra.  The corresponding differential forms on the   
   Weyl algebra were discovered by I. Segal some time ago and called   
   "quantized differential forms;" their cohomology gives a nice way of   
   understanding the Wick product in quantum field theory.  (He also dealt   
   with a fermionic version using the Clifford algebra.)  In my paper on   
   r-commutative geometry I recommend an approach in which one views the   
   Heisenberg algebra as a "bundle" over the "line" whose coordinate is   
   given by h.  A given "fiber" (at which h has some numerical value) is then a   
   Weyl algebra (except for the "classical fiber" at h = 0, which is just   
   the algebra of polynomials on phase space).  I put quotes around the   
   words "fiber bundle" because only the classical fiber is really a   
   manifold; the rest are "quantum manifolds," i.e. noncommutative   
   algebras.  But the fiber bundle viewpoint gives good insight into the   
   relation between the differential forms on the base (the line), the   
   fibers (Weyl algebras) and the total space (the Heisenberg algebra).   
   This viewpoint also works for noncommutative tori (and other cases).   
   Neither of these examples are exciting from the point of view of braid   
   invariants, since they are "strong".  The differential forms as defined   
   above do not work very well for non-strong r-algebras.  I have no idea   
   whether there is a good general definition of differential forms for   
   r-commutative algebras; there is a definition that works for the quantum   
   plane and other quantum vector spaces with Hecke-type relations   
   (i.e., not R^2 = 1, but  R^2 = (1-q)R + q.)  I just finished a paper,   
   "Hochschild Homology in a Braided Tensor Category," in which I define a   
   generalization of Hochschild homology for r-algebras and relate it to   
   the r-commutative differential forms defined above.  (I have not yet   
   computed this homology theory in non-strong cases.)  This paper should   
   appear in Transactions of the American Mathematical Society shortly   
   after hell freezes over.  In the meantime, you may content yourself with   
   ``R-commutative geometry and quantization of Poisson algebras,''   
   in Adv. Math. 95 (1992), 61 - 91.   
      
      
   The Noncommutative Torus.   
      
   I'm beginning to tire out but there is one loose end (out of many)   
   that I'd like to nail down.  I've mentioned noncommutative tori a couple   
   of times but haven't defined them or said what they have to do with   
   physics.   
      
   Okay: recall that $L^2({\bf R})$ means the Hilbert   
   space of square-integrable complex function on the real line.   If we   
   define the unitary operators u and v on L_2(R) given by   
      
   u = translation by the amount s   
   v = multiplication by  exp(ix)   
      
   we can see that they don't commute, but instead satisfy   
      
   uv = qvu   
      
   where q = exp(is).  (Note: I didn't say WHICH WAY to translate by the   
   amount s -- if you pick this correctly things will work out.)   
      
   The algebra of operators on L^2(R) generated by u and v and their   
   inverses is called the noncommutative torus T_q.  (If you know how, it's   
   better to take the C*-algebra generated by two unitaries u and v   
   satisfying   
      
   uv =  qvu.   
      
   This is actually quite a bit bigger for q = 1.)   This is clearly   
   a natural sort of thing because it's built up out of simple translation   
   and multiplication operators, and all of Fourier theory is based on the   
   interplay between translation and multiplication operators.   
      
   Why is called a "torus"?  Note that it depends on the parameter q.  If   
   we take q = 1, T_q is the C*-algebra generated by two unitaries u and v   
   that *commute*.   This may identified the algebra of functions on a   
   torus if we think of u as multiplication by exp(i theta) and v as   
   multiplication by exp(i phi), where theta and phi are the two angles on   
   the torus.  So we've got a one-parameter family of algebras T_q   
   such that when q = 1, it's just the algebra of (continuous) functions   
   on a torus, but for q not equal to one we have some sort of   
   noncommutative analog thereof.  The parameter q measures   
   noncommutativity or "quantum-ness", and one can relate it to Planck's   
   constant (which also measures "quantum-ness") by   
      
   q = exp(i hbar).   
      
   This example is actually the tip of an iceberg called called   
   "deformation theory".   One can read more about it Marc Rieffels' paper   
   "Deformation quantization and operator algbras," Proc. Symp. Pure Math.   
   51, or (from a different viewpoint) "Deformation theory and   
   quantization' by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer,   
   in Ann. Physics 111, p. 61-151.   As you can see from these titles, it   
   has a lot to do with quantization, since in quantization one is trying   
   to start with a commutative algebra of functions (observables) on phase   
   space and "quantize" it, or make it noncommutative, somehow.   
      
   Now it shouldn't be too surprising that the noncommutative torus is an   
   r-commutative algebra, since the commutation relations uv = qvu tell you   
   exactly how to "switch" u and v.  I've shown that there is a unique way   
      
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