A.2 Creation and Annihilation Operators

For each single-particle state $\vert\lambda\rangle$ of the single-particle Hilbert space $\mathcal{H}$ a Boson or Fermion creation operator $a^{+}_{\lambda}$ is defined by its action on any symmetrized or antisymmetrized state $\vert\lambda_{1}...\lambda_{N}\}$ of the Hilbert space of $N$ Bosons, $\mathcal{B}_{N}$, or $N$ Fermions, $\mathcal{F}_{N}$, as follows:

$$a_{\lambda}^{+}\vert\lambda_{1}...\lambda_{N}\}=\vert\lambda\lambda_{1}...\lambda_{N}\}$$ (A.4)

The creation operators $a^{+}_{\lambda}$ do not operate within one space $\mathcal{B}_{n}$ or $\mathcal{F}_{n}$, but rather operate from any space $\mathcal{B}_{n}$ or $\mathcal{F}_{n}$ to $\mathcal{B}_{n+1}$ or $\mathcal{F}_{n+1}$. It is useful to define the Fock space as the direct sum of the Boson or Fermion spaces

$$\mathcal{B}=\oplus_{n=0}^{\infty}\mathcal{B}_{n},$$ (A.5)

$$\mathcal{F}=\oplus_{n=0}^{\infty}\mathcal{F}_{n},$$

where by definition:

$$\mathcal{B}_{0}=\mathcal{F}_{0}=\vert\rangle,$$ (A.6)

$$\mathcal{B}_{1}=\mathcal{F}_{1}=\mathcal{H}.$$

It can be easily shown that for Bosons the creation operators commute:

$$a_{\lambda}^{+}a_{\mu}^{+}-a_{\mu}^{+}a_{\lambda}^{+}=0,$$ (A.7)

whereas they anticommute for Fermions:

$$a_{\lambda}^{+}a_{\mu}^{+}+a_{\mu}^{+}a_{\lambda}^{+}=0.$$ (A.8)

The annihilation operators $a_{\lambda}$ are defined as the adjoints of the creation operators $a_{\lambda}^{+}$. The commutation and anticommutation relations of annihilation operators follow from (A.7) and (A.8), respectively. They commute for Bosons:

$$a_{\lambda}a_{\mu}-a_{\mu}a_{\lambda}=0,$$ (A.9)

whereas they anticommute for Fermions:

$$a_{\lambda}a_{\mu}+a_{\mu}a_{\lambda}=0.$$ (A.10)

The action of the annihilation operator on a many particle state is given for Bosons as

$$a_{\lambda}\vert\alpha_{1}...\alpha_{n}\}=\sum_{i=1}^{n}\delta_{\lambda\alpha_{i}}\vert\alpha_{1}...\hat{\alpha}_{i}...\alpha_{n}\},$$ (A.11)

while for Fermions it reads:

$$a_{\lambda}\vert\alpha_{1}...\alpha_{n}\}=\sum_{i=1}^{n}(-1)^{i-1}\delta_{\lambda\alpha_{i}}\vert\alpha_{1}...\hat{\alpha}_{i}...\alpha_{n}\}.$$ (A.12)

Here $\hat{\alpha}_{i}$ shows that the state $\alpha_{i}$ has been removed from the mani-particle state $\vert\alpha_{1}...\hat{\alpha}_{i}...\alpha_{n}\}$.

The commutation rules for the creation and annihilation operators are:

$$a_{\lambda}a_{\mu}^{+}-a_{\mu}^{+}a_{\lambda}=\delta_{\lambda\mu}\thickspace ( )$$ (A.13)

$$a_{\lambda}a_{\mu}^{+}+a_{\mu}^{+}a_{\lambda}=\delta_{\lambda\mu}\thickspace ( ).$$

If the orthonormal basis $\{\alpha\}$ transforms into another basis $\{\widetilde{\alpha}\}$, the creation and annihilation operators transform as follows:

$$a_{\widetilde{\alpha}}^{+}=\sum_{\alpha}\langle\alpha\vert\widetilde{\alpha}\rangle a_{\alpha}^{+},$$ (A.14)

$$a_{\widetilde{\alpha}}=\sum_{\alpha}\langle\widetilde{\alpha}\vert\alpha\rangle a_{\alpha}.$$

Of particular importance is the coordinate basis $\{\vert\vec{x}\rangle\}$. In this case the creation and annihilation operators are traditionally denoted by $\hat{\psi}^{+}(\vec{x})$ and $\hat{\psi}(\vec{x})$ and are called field operators. From (A.15) it follows:

$$\hat{\psi}^{+}(\vec{x})=\sum_{\alpha}\phi_{\alpha}^{*}(\vec{x})a_{\alpha}^{+},$$ (A.15)

$$\hat{\psi}(\vec{x})=\sum_{\alpha}\phi_{\alpha}(\vec{x})a_{\alpha},$$

where $\phi_{\alpha}(\vec{x})$ is the coordinate representation wave function of the state $\vert\alpha\rangle$.

It can be shown that $n$-body operators (A.3) can be expressed through the creation and annihilation operators in a simple form:

$$\hat{U}=\frac{1}{n!}\sum_{\lambda_{1}...\lambda_{n}}\sum_{\mu_{1}... ......\mu_{n}) a_{\lambda_{1}}^{+}...a_{\lambda_{n}}^{+}a_{\mu_{n}}...a_{\mu_{1}}.$$ (A.16)

For example using the coordinate representation, the kinetic energy operator $\hat{T}$

$$\hat{T}=\sum_{i}\frac{\hat{\vec{p}}^{2}_{i}}{2m}$$ (A.17)

may be rewritten in second quantized form as:

$$\hat{T}=-\frac{\hbar^{2}}{2m}\int\,d^{3}x\hat{\psi}^{+}(\vec{x})\nabla^{2}\hat{\psi}(\vec{x}).$$ (A.18)

S. Smirnov: