Tensor (Direct) Products vs. Direct Sums
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 product||Direct sum|
|What it looks like:|
|Dimensions:||basis elements||basis elements|
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
U specifies spin of a particle, V specifies spin of a second (or the position of the first)
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.