A common theme in quantum mechanics and multi-particle systems is manipulation of vector spaces - usually spaces of physical states of a system.

I've come across a little confusion between outer products or tensor products (specifically tensor direct products) and direct sums in this area in particular, so just a quick comparison to make the difference clear:

• Notation: U, V two (for simplicity finite-dimensional) vector spaces with bases u(1), ..., u(l) and v(1), ..., v(m).
Tensor productDirect sum
What it looks like:
Basis elements:
Typical element:
Dimensions: basis elements basis elements
Physics:

You need to know information (of different types) from both U and V to describe the system

U and V represent two alternative (groups of) possibilities for the system, so that you can have a system definitely in a U-type state, for example

QM example:

U specifies spin of a particle, V specifies spin of a second (or the position of the first)

• U contains all states which are symmetric under interchange of two particles, V contains all states which are antisymmetric
• U contains states in which the particle is in {x ≤ 0}, V those in which it is in {x > 0}

### More Confusion: Direct Products

Something I fell for when I was writing this down the first time is that there is both a direct product and a tensor (direct) product/outer product . A direct product is - incredibly confusingly - basically the same as a direct sum in this case.

A direct product is - so each element involves specifying one element of U and one element of V. In the specific case described above - where u and v can be added together - this looks exactly the same as (is isomorphic to) the direct sum. We just write (u,v) ≡ u + v. This is quite general, as you can see in the Wolfram article on direct products.

If you're wondering what the relationship between a tensor product and a direct product is, it's slightly subtle. The tensor product arises from the free vector space on the direct product . The idea is that we define one basis vector ('direction') per element of , so there is a basis . This is a huge space. (If non-trivial, it is of uncountably infinite dimension here, since each real multiple of one vector corresponds to a new dimension.)

Then we identify basis vectors in a particular way (discussed properly in the Wikipedia article on tensor products) so that addition and scalar multiplication of the elements of corresponds to addition and multiplication in - that is

This gets rid of a huge load of basis elements, and leaves us (it turns out!) with the tensor product.

See also: On Wikipedia, the tensor product of Hilbert spaces article provides some more formalism.