Why the tensor product (historical question)?

[greypilgrim]

Hi.

Why did the founding fathers of QM know that the Hilbert space of a composite system is the tensor product of the component Hilbert spaces and not a direct product, where no entanglement would emerge? I mean today we can verify entanglement experimentally, but this became technologically possible far later than for example the EPR paper.

My knowledge about functional analysis is pretty limited, is there a mathematical reason to exclude the direct product?

 

[Truecrimson]

I'm not sure of the historical reason but it is at least sufficient for the Born rule to work. If ##P_1## and ##P_2## are transition probabilities for independent systems, the joint transition probability ##P## is the product $$ P = P_1 P_2 = | \langle \varphi_1 |\psi_1 \rangle |^2 |\langle \varphi_2 |\psi_2 \rangle |^2 = | ( \langle \varphi_1 | \otimes \langle \varphi_2 |)(|\psi_1 \rangle \otimes |\psi_2 \rangle |^2$$ If we use the direct product instead (which I think is the same as direct sum for finite-dimensional vector spaces), states of independent systems will be orthogonal and the probabilistic interpretation won't work.

See also this post https://www.physicsforums.com/threads/tensor-products-and-subsystems.360264/#post-2473399.