I wasn't sure where to put this since this is under group theory. I am having a little bit trouble understand when to use direct product vs direct sum. One question I have about this is is that if you have two vector spaces that are orthogonal to each other (and example of this might be two different Hilbert spaces in quantum mechanics, where one space is all the spacial wavefunctions of bosons and the other is all the spatial wave functions of fermions due to the fact that one is symmetric and the other is antisymmetric) will the direct product of these two spaces be 0 and what will the direct sum yield (continuing on with the quantum mechanics example, will is yield all possible linear combinations of states since any state can be broken up into a symmetric and antisymmetric part). Also if anyone could state their answer to my dilema in terms of quantum mechanics states that would be most appreciated. I thank all the people who reply to this post in advance. (Also let me know if I have any terminology in the wrong context).