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Let's start with a single particle Hilbert space $H$ with basis $\{|\alpha\rangle\}$, where $\alpha$ represents a complete set of quantum numbers. If we now take two particles of the same type, we must consider the space $$H_2=H_1\otimes H_1,$$ and more generally $n$ particles live in $$H^{\otimes n}=H_1\otimes...\otimes H_1.$$ The (free) Fock space is now defined by taking the direct sum $$\mathcal{F(H)}:=\bigoplus_{n\ge 0} H^{\otimes n}.$$ I understand why we take the tensor products: we wish to describe the whole system of $n$ particles at the same time. However, it is not at all clear to me why we also need to take the direct sum. What is its physical interpretation?

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Consider identical particles. Consider also first just a single particle in your system. This "1-particle sector" is described by Hilbert space $H_1$. This is the first term, $n=1$, in the direct sum you wrote. ($n=0$ can be probably ignored.) Consider now two particles. This system's Hilbert space is $H_1 \otimes H_1$. This is the second term in the direct sum. And so on. The sectors with different numbers of particles are independent and thus you take the direct sum over them.

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  • $\begingroup$ This is all nice and well, but what is the need to actually consider such "$m$-particles sectors" in the first place? $\endgroup$ – Mr. Mister Sep 29 at 10:10

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