Editing Second quantization (section)

==Quantum field operators==

Defining <math>a^{\dagger}_{\nu}</math> as a general annihilation (creation) operator for a single-particle state <math>\nu</math> that could be either fermionic <math>(c^{\dagger}_{\nu})</math> or bosonic <math>(b^{\dagger}_{\nu})</math>, the [[Position and momentum space|real space representation]] of the operators defines the [[quantum]] field [[operator (physics)|operator]]s <math> \Psi(\mathbf{r})</math> and <math>\Psi^{\dagger}(\mathbf{r})</math> by
:<math> \Psi(\mathbf{r})=\sum_{\nu} \psi_{\nu} \left( \mathbf{r} \right) a_{\nu}</math>
:<math> \Psi^{\dagger}(\mathbf{r})=\sum_{\nu} \psi^*_{\nu} \left( \mathbf{r} \right) a^{\dagger}_{\nu}</math>

These are second quantization operators, with coefficients <math>\psi_{\nu} \left( \mathbf{r} \right)</math> and <math> \psi^*_{\nu} \left( \mathbf{r} \right)</math> that are ordinary [[first quantization|first-quantization]] [[wavefunctions]]. Thus, for example, any expectation values will be ordinary first-quantization wavefunctions. Loosely speaking, <math>\Psi^{\dagger}(\mathbf{r})</math> is the sum of all possible ways to add a particle to the system at position '''r''' through any of the basis states <math>\psi_{\nu}\left(\mathbf{r}\right)</math>, not necessarily plane waves, as below.

Since <math> \Psi(\mathbf{r})</math> and <math>\Psi^{\dagger}(\mathbf{r})</math> are second quantization operators defined in every point in space they are called [[quantum field]] operators. They obey the following fundamental commutator and anti-commutator relations,

:<math> \left[\Psi(\mathbf{r}_1),\Psi^\dagger(\mathbf{r}_2)\right]=\delta (\mathbf{r}_1-\mathbf{r}_2) </math> boson fields,
:<math> \{\Psi(\mathbf{r}_1),\Psi^\dagger(\mathbf{r}_2)\}=\delta (\mathbf{r}_1-\mathbf{r}_2) </math> fermion fields.

For homogeneous systems it is often desirable to transform between real space and the momentum representations, hence, the quantum fields operators in [[Fourier transform|Fourier basis]] yields:
:<math> \Psi(\mathbf{r})={1\over \sqrt {V}} \sum_{\mathbf{k}} e^{i\mathbf{k\cdot r}}a_{\mathbf{k}}</math>
:<math> \Psi^{\dagger}(\mathbf{r})={ 1\over \sqrt{V}} \sum_{\mathbf{k}} e^{-i\mathbf{k\cdot r}}a^{\dagger}_{\mathbf{k}}</math>