# Direct products and direct sums in QM

Dear forumers,

I have a question about taking direct sums and products of state spaces in QM. Picture I have a state space that describes two (indistinguishable) particles which is a direct sum of two one-particles spaces:

$$\epsilon_t = \epsilon_1 \oplus \epsilon_2$$

Furthermore, picture that both one-particle state spaces can be represented as products of two space-states for different observables.

$$\epsilon_1 = \epsilon_{o1} \otimes \epsilon_{o2}$$

and

$$\epsilon_2 = \epsilon_{o3} \otimes \epsilon_{o3}$$

that is to say

$$\epsilon_t = (\epsilon_{o1} \otimes \epsilon_{o2}) \oplus (\epsilon_{o3} \otimes \epsilon_{o4})$$

Now picture that the spaces for o1 (which refers to particles 1) and o3 (which refers to particle 2) are actually the same (they contain all the same states). Is it a stretch to say that since these particles are indistinguishable:

$$(\epsilon_{o1} \otimes \epsilon_{o2}) \oplus (\epsilon_{o1} \otimes \epsilon_{o4}) = \epsilon_{o1} \otimes (\epsilon_{o2} \oplus \epsilon_{o4})$$

?????

I'm new to this fock-space, many particle thing, so pardon me if what I'm doing is totally wrong.