# Tensor product of Hilbert space and Fock space

If we have system consist of two or more than two subsystems, we can write hilbert space of a system in terms of tensor product of Hilbert spaces of subsystems and we can write state of a system. But also in Fock space for same system, when we write state of a system, Fock space is direct sum of hilbert spaces of subsystems. So can we use both formalisms for a system at the same time?

## 1 Answer

There are no different formalisms here. When you have two subsystems $$S_1$$ and $$S_2$$, the Hilbert space is the tensor product $$\mathcal H = \mathcal H_1 \otimes \mathcal H_2$$.

Let $$\mathcal H_1$$ be the Hilbert space of a single particle. The Hilbert space describing $$N$$ (distinguishable) particles is then $$\mathcal H_N = \mathcal H_1 \otimes \cdots \otimes \mathcal H_1 = \mathcal H_1^{\otimes N} .$$ If you can have any number of particles, then the total state lies in the direct sum $$\mathcal H = \mathcal H_1 \oplus \cdots \oplus \mathcal H_N \oplus \cdots .$$ (To get the Fock space, you now have to take the symmetrization / antisymmetrization, because the particles are indistinguishable.)

• So they are different, the Hilbert space has a definite number of particles, and the tensor product goes up to N. Focker space adds all hilbert spaces with different number of partiocles in it. They are two different structures, and as such they should be considered different formalisms.
– user65081
Jan 14 '19 at 13:20
• @Wolphramjonny My reading of the question is: "how is it possible that there are two different mathematical descriptions of one physical situation, isn't that inconsistent?" I think the most helpful thing to say is: no, these are two different physical situations, each of which have only one model. These two models are derived from the same formalism and therefore closely related. (I might be misunderstanding the question though?) Jan 14 '19 at 19:55
• @glS Direct sum and direct product are the same thing, except for subtleties in the case of an infinite number of summands / factors. The Fock space is written as a direct sum to exclude superpositions of an infinite number of states with different particle numbers. Jan 14 '19 at 19:59
• @glS Maybe you should ask this as a separate question. In short: the tensor product describes a system made up of different subsystems, the direct sum describes different states the same system can be in. Note that e.g. $\lambda |+\rangle + |-\rangle$ is different from $|+\rangle + \lambda |-\rangle$ which does actually make sense :) Jan 14 '19 at 22:16
• @Noiralef ah, right, I see what you meant now. I still think it's worded a bit weirdly though. In your second equation you are essentially decomposing the Hilbert spaces of the first equation then? Wouldn't it be clearer to write it as something like $\mathcal H_k=\bigoplus_{j=1}^\infty \mathcal H^{(1)}$ then? Or even better just $\mathcal H_k=\bigoplus_{j=1}^\infty \mathbb C$, as you are simply saying that the space is infinite dimensional and nothing more here
– glS
Jan 15 '19 at 9:13