Both direct sum and tensor product are standard ways of putting together little Hilbert spaces to form big ones. They are used for different purposes. Suppose we have two physical systems A and A', with Hilbert spaces H and H'. Roughly speaking, if we have a physical system whose states are either states of A OR states of A', its Hilbert space will be the direct sum H+H'. If we have a physical whose states are states of A AND states of A', its Hilbert space will be the tensor product HxH'.
MEASURE THEORY: disjoint union Cartesian product HILBERT SPACE THEORY: direct sum tensor productNathan Urban:
I'm kind of confused here. Isn't the direct sum of two vector spaces V and V' just the space of all pairs (v,v') where v is in V and v' is in V'? If so, isn't that just the Cartesian product of V and V'?
Yes, the direct sum of two Hilbert spaces is just their Cartesian product, made into a Hilbert space in a fairly obvious way.
However, this doesn't conflict with anything I wrote above. The point of that post was to explain the correspondence between operations on classical configuration spaces and operations on Hilbert spaces of quantum states. One way to "quantize" is to take a classical configuration space and form the Hilbert space of L2 functions on it. For this, the classical configuration space needs to be a measure space. So we have some process taking any measure space X to the Hilbert space L2(X). This process takes disjoint unions of measure spaces to direct sums of Hilbert spaces:
L2(X U Y) = L2 (X) + L2 (Y),
since an L2 function on X U Y is just a pair consisting of an L2 function on X together with an L2 function on Y. Similarly, it takes Cartesian products of measure spaces to tensor products of Hilbert spaces:
L2 (X x Y) = L2 (X) x L2 (Y)
since every L2 function on X x Y is a linear combination of those of the form f(x)g(y), which corresponds to the tensor product f x g over in L2 (X) x L2 (Y).
So you see the analogy works as I said: Cartesian product of measure spaces is not analogous to direct sum of Hilbert spaces, despite the fact that the direct sum of two Hilbert spaces is, as a set, just the Cartesian product.
Part of your puzzlement might be explained by noting that the process of going from a set to the space of complex-valued functions on that set is a contravariant functor. In other words, given sets X and Y and a function F: X -> Y, and letting Fun(X) be the vector space of functions on X, we get a linear map F*: Fun(Y) -> Fun(X) by
F*(f)(x) = f(F(x))
This is just the "pullback" operation you know and love.
Now, since we've got a contravariant functor
Fun: Set -> Vect
we might hope for it to take coproducts in Set to products in Vect. After all, in any category, both products and coproducts are defined using commutative diagrams, the only difference between the two being that the arrows are turned around. And a contravariant functor turns around arrows.
What does this mean? Well, the coproducts in Set are just disjoint unions. The products in Vect are just direct sums. So we might hope for our functor to take disjoint unions to direct sums --- and indeed it does:
Fun(X U Y) = Fun(X) + Fun(Y) !
So you see, category theory tells us that the analogy here is between disjoint unions in Set and direct sums in Vect. And as we've already seen, a similar thing works when we consider the category of measure spaces and the category of Hilbert spaces.
WARNING: THE FOLLOWING PASSAGE IS VERY CONFUSING AND IS FOR EXPERTS ONLY!
DO NOT READ THE REST OF THIS WEBPAGE!
I should add that there are all sorts of extra twists and subtleties in the picture which I deliberately avoided mentioning. For example, Hilb is a *-category: we can turn around any morphism F: H -> H' in Hilb by taking its adjoint F*: H' -> H. And this lets us turn any contravariant functor from or to Hilb into a covariant functor. This is very handy, but it makes things a bit more confusing. Also, when discussing the category of measure spaces, we should really be more precise about what sort of morphisms we're going to use in this category. Furthermore, you might naively hope that, just as
Fun: Set -> Vect
took coproducts in Set to products in Vect, it would take products in Set to coproducts in Vect. But it doesn't: the product in Set is just the Cartesian product, and Fun carries this to the tensor product in Vect, which is not the coproduct in Vect. The coproduct of two vector spaces is just their direct sum! It may seem weird that the direct sum of vector spaces is both the product and coproduct, but it's true: this is part of Vect being an "abelian category". The same is true of Hilb.
Next: About this document... Up: Schmotons Previous: Appendix: Notation