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  "body": "\n## Description\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]].\nFor $n \\ge 1$, let $H^{\\otimes n}$ be the [[tensor product|EY0X2NKA3S]] of $n$ copies of $H$, and let $H^{\\otimes 0} = \\C$.\nThe *full Fock space* over $H$ is the [[Hilbert sum|WJ7PDMGQQW]]\n$$\\mathcal{F}(H) = \\widehat{\\bigoplus}_{n \\ge 0} H^{\\otimes n}.$$\nLet $S_+$ and $S_-$ be the orthogonal projections of $H^{\\otimes n}$ onto its symmetric and antisymmetric tensors.\nThe *bosonic Fock space* is $\\mathcal{F}_+(H) = \\widehat{\\bigoplus}_{n \\ge 0} S_+ H^{\\otimes n}$, and the *fermionic Fock space* is $\\mathcal{F}_-(H) = \\widehat{\\bigoplus}_{n \\ge 0} S_- H^{\\otimes n}$.\nEach of them is a Hilbert space.\nIndeed, each summand is a closed subspace of $H^{\\otimes n}$, and a Hilbert sum of Hilbert spaces is a Hilbert space.\n\n## Remarks\n\nThe summand of degree $n$ is the space of states with exactly $n$ particles; the vector $1 \\in H^{\\otimes 0}$ is the vacuum.\nFor $H = \\C$, every symmetric power $S_+ \\C^{\\otimes n}$ has dimension $1$, so $\\mathcal{F}_+(\\C)$ is isomorphic to $\\ell^2(\\N)$.\nFor the same $H$, the antisymmetric part vanishes for $n \\ge 2$, so $\\mathcal{F}_-(\\C) = \\C \\oplus \\C$; this reflects the Pauli exclusion principle.\nCreation and annihilation operators act on Fock space by raising and lowering the degree $n$.\n\n## Explanations\n\n### L1 human\n\nFock space is the state space for a quantum system in which the number of particles can change.\nIt stacks the spaces for $0, 1, 2, \\ldots$ particles side by side.\nFor identical bosons one keeps only symmetric states, and for fermions only antisymmetric ones.\n",
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