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'?

John Baez:

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 L^{2}
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 L^{2}(X). This process takes disjoint unions of
measure spaces to direct sums of Hilbert spaces:

L^{2}(X U Y) = L^{2} (X) + L^{2} (Y),

since an L^{2} function on X U Y is just a pair consisting of an
L^{2}
function on X together with an L^{2} function on Y. Similarly, it
takes Cartesian products of measure spaces to tensor products of
Hilbert spaces:

L^{2} (X x Y) = L^{2} (X) x L^{2} (Y)

since every L^{2} 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 L^{2} (X) x L^{2} (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