I have some struggles with the concepts mentioned in the header of this subject.
I thought I'd know what direct sums and direct products are, but there are always incidents where I am confused as to how these
concepts are used. Let me start with direct sums.
If one has two subspaces, say and , of a vector space V, then one can form the direct sum vector space
, which is a larger vector space, containing all linear combinations of those two vector spaces. This is completely clear to me.
However, there are many cases where one doesn't have a priori a "big" vector space V where the subspaces are embedded. So one only has vector spaces W and U, say, of different dimension, and then forms the space . Now here would be my question: What does that mean, then? Do I have to think of a "big" vector space V where these two are embedded beforehand? But then there'd be more than one possibility of combining these two vector spaces W and U by direct sum (with more or less "zeros" in the vectors, if one thinks about it in such a way).
In an infinite-dimensional Hilbert space , which is actually more abstract, the concept is - curiously - perfectly clear again.
Now direct products. I always thought of direct products as some sort of tensor products, that is, if , then
. One can represent this as tuple (v,u) (in which the order is important). One can also define some sort of operation. Let's consider two groups and . Then one might form the direct product with
some direct product group operation defined by with .
Applying this to vector spaces, for example, one might set the operation between g and h as g+h.
However there is once again some point which confuses me: In quantum mechanics, one considers states of a system, namely |n> and |m>, being in the Hilbert space . Now one can form the product , or shortly written, (yes, one does such things in quantum mechanics).
But then, sometimes one just considers as being a "normal" product between functions (states) in the Hilbert space, an AT THE SAME TIME as direct products, being in the product Hilbert space!
One possible explanation for me could be that one can once again consider the states as tuple , equipped with the multiplication of functions. But firstly this would be arbitrary, and secondly, the question would arise as what the difference between the cartesian product and the direct product would be.
I'd be thankful if someone clarifies the above concepts for me and maybe give some reference where this is explained in an adequate way.