So...let's say we have an r-algebra. That's an algebra with an invertible linear map , which we draw as a right-crossing:
which satisfies 1) the Yang-Baxter equations:
2) and ,
and 3) the quasitriangularity conditions; writing multiplication as the joining of strands these are
We say an r-algebra is ``r-commutative'' if
and ``strong'' if is the identity, or equals its inverse, i.e.:
(Note that a left-crossing
is the inverse of a right-crossing.)
All sorts of noncommutative analogs of manifolds are r-commutative algebras: quantum groups, noncommutative tori, quantum vector spaces, the Weyl and Clifford algebras, certain universal enveloping algebras, supermanifolds, etc. It seems that the ones with direct relevance to quantum theory in 4 dimensions are ``strong,'' while the non-strong ones, like quantum groups, are primarily relevant to 2- and 3-dimensional physics. I would now like to describe an analog of differential forms for strong r-commutative algebras, and illustrate it for the case of the Heisenberg algebra -- i.e., the algebra defined by the canonical commutation relations: .
What are differential forms? Of course, they're the basis of a lot of differential geometry, and there are lots of equivalent ways of defining them, but let me take an algebraic viewpoint. Let be the algebra of smooth functions on a manifold . We define differential forms as follows. Each function in has a ``differential'' , and the functions and their differentials generate an algebra in which we impose the following relations:
This algebra is called the algebra of differential forms on .
That's all, folks! If you've taken a course in differential geometry you were probably exposed to tangent planes and all that stuff, but if you want to get calculating with differential forms as soon as possible this is all you need to start with.
Note that rule 4) is the only one in which we switch elements of -- moving to the right of . This is the rule we'll need to modify for an r-commutative algebra. You could, of course, just leave out rule 4): given any algebra , the algebra whose relations are just given by 1)-3) is called the ``universal differential calculus'' for . It's a reasonable substitute for differential forms when is any old noncommutative algebra, and (from one viewpoint) it's the basis of Alain Connes' approach to noncommutative differential geometry. But when you have a strong r-commutative algebra one can replace rule 3) with
4) Commutation relations: and .
Here I should explain that I'm writing as the sum over of tensor products of the form (one can always do this), and I'm using the Einstein summation convention (sum over repeated indices) to avoid writing the summation sign. What we're doing in rule 4) is just what we should do: use to ``switch'' and instead of ``naively'' switching them as in 4).
Now let me show what this buys you in the case of the Heisenberg algebra.
This is the algebra over generated by 3 formal variables, , , and
, subject to the relations that
This algebra is actually a strong r-commutative algebra in a unique
manner such that
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 Grössen würde wegen der Nichtgültigkeit des kommutativen Gesetzes der Multiplikation in gewissem Sinne unbestimmt bleiben, wenn nicht der Wert von vorgeschrieben würde. Wir führen daher als fundamentale quantenmechanische Relation ein:or if your German is rusty:
`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 . We introduce accordingly the following as the fundamental quantum-mechanical relation:
Ain't that cool?
Anyway, 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:
But if actually you want to read that, you probably should read this first:
© 1992 John Baez