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

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

  • 1
    $\begingroup$ 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. $\endgroup$
    – jng224
    Nov 7, 2021 at 13:40
  • 3
    $\begingroup$ @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. $\endgroup$
    – J. Murray
    Nov 7, 2021 at 13:49
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    $\begingroup$ the previous to last belong to hilbert space, the last one to focker space $\endgroup$
    – user65081
    Nov 7, 2021 at 13:53
  • $\begingroup$ The last equation doesn't make much sense to me. What do you mean with the direct sum of zeroes and a state $\vert \Phi\rangle$? $\endgroup$ Nov 7, 2021 at 14:02
  • $\begingroup$ 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. $\endgroup$
    – maxxam
    Nov 7, 2021 at 14:11

1 Answer 1


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 \begin{equation} |\Psi \rangle = a_0|0 \rangle + \sum_i a_i |\phi_i\rangle + \sum_{ij} a_{ij} |\phi_i, \phi_j \rangle + \cdots \end{equation} where $|\phi_i \rangle$ are a basis for the one particle states.

You could write a generic $n$ particle state as \begin{equation} |\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 \end{equation} 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 \begin{equation} |\Psi \rangle = \sum_{i_1, \cdots, i_n} a_{i_1 \cdots i_n} |\phi_{i_1} \cdots \phi_{i_n}\rangle \end{equation}


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