Editing Fock state (section)

===Boson creation and annihilation operators===
We should be able to express the same symmetric property in this new Fock space representation. For this we introduce non-Hermitian bosonic [[creation and annihilation operators]],<ref name="TIFR"/> denoted by <math>b^{\dagger}</math> and <math>b</math> respectively. The action of these operators on a Fock state are given by the following two equations:

* Creation operator <math display="inline">b^{\dagger}_{{\mathbf{k}}_l} </math>:
*: <math>b^{\dagger}_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}} +1 } |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}+1 ,...\rangle </math><ref name="TIFR" />
* Annihilation operator <math display="inline">b_{{\mathbf{k}}_l} </math>:
*: <math>b_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}}} |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}}, n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}-1 ,...\rangle </math><ref name="TIFR" />

[[File:Action of operator on bosonic fock state.jpg|center|''The Operation of creation and annihilation operators on Bosonic Fock states.'']]

====Non-Hermiticity of creation and annihilation operators====
The bosonic Fock state  creation and annihilation operators are not [[Self-adjoint operator|Hermitian operators]].<ref name="TIFR"/>
{{math proof|title= Proof that creation and annihilation operators are not Hermitian.
|proof=
For a Fock state, <math>|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l}, \dots \rangle</math>, 
<math display="block">\begin{align}
    \left\langle n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1, \dots
       \left| b_{\mathbf{k}_l} \right|
       n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l}, \dots \right\rangle
  &= \sqrt{n_{\mathbf{k}_l}}\left\langle n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1, \dots
       | n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1, \dots \right\rangle \\[6pt]
     \left(\left\langle n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l}, \dots
       \left| b_{\mathbf{k}_l} \right|
       n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1, \dots \right\rangle\right)^*
  &= \left\langle n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1 \dots
     \left| b_{\mathbf{k}_l}^\dagger \right|
       n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l}, \dots \right\rangle \\
  &= \sqrt{n_{\mathbf{k}_l} + 1}\left\langle n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} - 1 \dots
      | n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} \dots n_{\mathbf{k}_l} + 1 \dots \right\rangle
\end{align}</math>

Therefore, it is clear that adjoint of creation (annihilation) operator doesn't go into itself. Hence, they are not Hermitian operators.

But adjoint of creation (annihilation) operator is annihilation (creation) operator.<ref name="Altland">{{Cite book | last1 = Altland | first1 = Alexander | last2 = Simons | first2 = Ben | title = Condensed Matter Field Theory | publisher = Cambridge University Press | date = 2006 | url = https://books.google.com/books?id=0KMkfAMe3JkC&pg=PA39 | isbn = 0521769752 }}</ref>{{rp|45}}
}}

====Operator identities====
The commutation relations of creation and annihilation operators in a [[Boson|bosonic system]] are
: <math>\left[b^{\,}_i, b^\dagger_j\right] \equiv b^{\,}_i b^\dagger_j - b^\dagger_jb^{\,}_i = \delta_{i j},</math><ref name="TIFR"/>
: <math>\left[b^\dagger_i, b^\dagger_j\right] = \left[b^{\,}_i, b^{\,}_j\right] = 0,</math><ref name="TIFR"/>
where <math>[\ \ , \ \ ]</math> is the [[commutator]] and <math>\delta_{i j}</math> is the [[Kronecker delta]].