Editing Second quantization (section)

=== First-quantized many-body wave function ===
Consider a complete set of single-particle wave functions <math>\psi_{\alpha}(\mathbf{r})</math> labeled by <math>\alpha</math> (which may be a combined index of a number of quantum numbers). The following wave function
:<math>\Psi[\mathbf{r}_i]=\prod_{i=1}^{N}\psi_{\alpha_i}(\mathbf{r}_i)\equiv \psi_{\alpha_1}\otimes\psi_{\alpha_2}\otimes\cdots\otimes\psi_{\alpha_N}</math>
represents an ''N''-particle state with the ''i''th particle occupying the single-particle state <math>|{\alpha_i}\rangle</math>. In the shorthanded notation, the position argument of the wave function may be omitted, and it is assumed that the ''i''th single-particle wave function describes the state of the ''i''th particle. The wave function <math>\Psi</math> has not been symmetrized or anti-symmetrized, thus in general not qualified as a many-body wave function for identical particles. However, it can be brought to the symmetrized (anti-symmetrized) form by operators <math>\mathcal{S}</math> for symmetrizer, and <math>\mathcal{A}</math> for [[antisymmetrizer]].

For bosons, the many-body wave function must be symmetrized,
:<math>\Psi_{\rm B}[\mathbf{r}_i]=\mathcal{N}\mathcal{S}\Psi[\mathbf{r}_i]=\mathcal{N}\sum_{\pi\in S_N}\prod_{i=1}^{N}\psi_{\alpha_{\pi(i)}}(\mathbf{r}_i)=\mathcal{N}\sum_{\pi\in S_N}\psi_{\alpha_{\pi(1)}}\otimes\psi_{\alpha_{\pi(2)}}\otimes\cdots\otimes\psi_{\alpha_{\pi(N)}};</math>
while for fermions, the many-body wave function must be anti-symmetrized,
:<math>\Psi_{\rm F}[\mathbf{r}_i]=\mathcal{N}\mathcal{A}\Psi[\mathbf{r}_i]=\mathcal{N}\sum_{\pi\in S_N}(-1)^\pi\prod_{i=1}^{N}\psi_{\alpha_{\pi(i)}}(\mathbf{r}_i)=\mathcal{N}\sum_{\pi\in S_N}(-1)^\pi\psi_{\alpha_{\pi(1)}}\otimes\psi_{\alpha_{\pi(2)}}\otimes\cdots\otimes\psi_{\alpha_{\pi(N)}}.</math>
Here <math>\pi</math> is an element in the ''N''-body permutation group (or [[symmetric group]]) <math>S_{N}</math>, which performs a [[permutation]] among the state labels <math>\alpha_i</math>, and <math>(-1)^\pi</math> denotes the corresponding [[parity of a permutation|permutation sign]]. <math>\mathcal{N}</math> is the normalization operator that normalizes the wave function. (It is the operator that applies a suitable numerical normalization factor to the symmetrized tensors of degree ''n''; see the next section for its value.)

If one arranges the single-particle wave functions in a matrix <math>U</math>, such that the row-''i'' column-''j'' matrix element is <math>U_{ij}=\psi_{\alpha_{j}}(\mathbf{r}_i)\equiv \langle\mathbf{r}_i|\alpha_j\rangle</math>, then the boson many-body wave function can be simply written as a [[Permanent (mathematics)|permanent]] <math>\Psi_{\rm B}=\mathcal{N}\operatorname{perm} U</math>, and the fermion many-body wave function as a [[determinant]] <math>\Psi_{\rm F}=\mathcal{N}\det U</math> (also known as the [[Slater determinant]]).<ref name="Koch2013">{{cite book
 | author = Koch, Erik
 | chapter = Many-electron states
 | title = Emergent Phenomena in Correlated Matter
 |editor=Pavarini, Eva |editor2=Koch, Erik |editor3=Schollwöck, Ulrich
 | series = Modeling and Simulation
 | volume = 3
 | publisher = Verlag des Forschungszentrum Jülich
 | location = Jülich
 | pages = 2.1–2.26
 | url = http://hdl.handle.net/2128/5389
 | year = 2013
 | hdl = 2128/5389
 | isbn = 978-3-89336-884-6
}}</ref>