So it's not quite that you need two projectors, but rather that you need one projector which is taken from a complete basis for the space - which takes the form of a tensor product of two single-particle states. In bra-ket notation, is an inner product, x>x> is a tensor product. $$|\psi\rangle = \mathbb I |\psi\rangle = \int dx |x\rangle\underbrace = \int dx dy \Psi(x,y) |x,y\rangle$$ Therefore, the identity operator takes the form $\mathbb I = \int dx |x\rangle\langle x|$, and any state $|\psi\rangle\in\mathcal H$ can be expanded as $L^2(\mathbb R)$), the generalized position eigenvectors $|x\rangle$ form a continuous basis of the space. Given some single-particle Hilbert space $\mathcal H$ (e.g.
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