I Why the tensor product (historical question)?

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1. Jun 22, 2016

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?

2. Jun 22, 2016

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.