# What exactly is a Fock state?

I am a bit confused by the way a Fock state is defined and hope to find some clarification.

The Fock space is defined as the direct sum of all $$n$$-particle Hilbertspaces $$H_i$$

$$F = H_0 \oplus H_1 \oplus H_2 \oplus ...$$

Let $$|\Phi \rangle$$ be a $$m$$-particle state. Obviously $$|\Phi \rangle \in H_m$$ is a true statement. However I am wondering wether $$|\Phi \rangle$$ itself is a Fock state

$$|\Phi \rangle \in F$$

or a state that looks like

$$|\Phi \rangle_F = 0 \oplus 0 \oplus ... \oplus |\Phi \rangle \oplus ...$$

• I changed $\bigoplus$ to $\oplus$ because it improved formatting imo. If there is any significance to \bigoplus that I'm not aware of, feel free to change it back. Nov 7, 2021 at 13:40
• @Jonas \bigoplus is often used as the direct sum variant of the standard summation symbol (i.e. $\displaystyle \bigoplus_{n=0}^\infty H_n=H_0 \oplus H_1 \oplus \ldots$) so your edit is good. Nov 7, 2021 at 13:49
• the previous to last belong to hilbert space, the last one to focker space
– user65081
Nov 7, 2021 at 13:53
• The last equation doesn't make much sense to me. What do you mean with the direct sum of zeroes and a state $\vert \Phi\rangle$? Nov 7, 2021 at 14:02
• The most general form of a Fock state is some vector like this: $| \Psi \rangle = |0\rangle \oplus \sum_i a_i | \phi_i \rangle \oplus \sum_{i,j} | a_{i,j} \phi_i, \phi_j \rangle \oplus ...$. The "direct sum of zeros" should emphazise, that all coefficients $a_i, a_{i,j} ,...$ are equal to zero except for the ones the state $| \Phi \rangle$ is made up. Nov 7, 2021 at 14:11

The direct sum $$\oplus$$ maps a pair of Hilbert spaces to a larger Hilbert space. It doesn't directly act on individual states in the Hilbert space.
A generic Fock space state would look like $$$$|\Psi \rangle = a_0|0 \rangle + \sum_i a_i |\phi_i\rangle + \sum_{ij} a_{ij} |\phi_i, \phi_j \rangle + \cdots$$$$ where $$|\phi_i \rangle$$ are a basis for the one particle states.
You could write a generic $$n$$ particle state as $$$$|\Psi \rangle = 0 |0\rangle + \sum_i 0 |\phi_i\rangle + \cdots + \sum_{i_1, \cdots, i_n} a_{i_1 \cdots i_n} |\phi_{i_1} \cdots \phi_{i_n}\rangle + \cdots$$$$ But most people don't write terms that are identically zero unless they are making some kind of point. Therefore a more common way to write an $$n$$ particle state is simply $$$$|\Psi \rangle = \sum_{i_1, \cdots, i_n} a_{i_1 \cdots i_n} |\phi_{i_1} \cdots \phi_{i_n}\rangle$$$$