String Theory

Sunday, 9 June 2013

Quantum field operators

Defining ${a^{(\dagger)}}_{\nu}$ as a general annihilation(creation) operator that could be either fermionic $({c^{(\dagger)}}_{\nu})$ or bosonic $({b^{(\dagger)}}_{\nu})$ , the real space representation of the operators defines the quantum field operators $\Psi(\bold{r})$ and $\Psi^{\dagger}(\bold{r})$ by

$\Psi(\bold{r})=\sum_{\nu} \psi_{\nu} \left( \bold{r} \right) a_{\nu}$

$\Psi^{\dagger}(\bold{r})=\sum_{\nu} {\psi^*}_{\nu} \left( \bold{r} \right) {a^{\dagger}}_{\nu}$

Second quantization operators, while the coefficients $\psi_{\nu} \left( \bold{r} \right)$ and ${\psi^*}_{\nu} \left( \bold{r} \right)$ are the ordinary first quantization wavefunctions. Loosely speaking, $\Psi^{\dagger}(\bold{r})$ is the sum of all possible ways to add a particle to the system at position r through any of the basis states $\psi_{\nu}\left(\bold{r}\right)$ . Since $\Psi(\bold{r})$ and $\Psi^{\dagger}(\bold{r})$ 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,

$\begin{align} \left[\Psi(\bold{r}_1),\Psi^\dagger(\bold{r}_2)\right]=\delta (\bold{r}_1-\bold{r}_2) &\text{ boson fields,}\\ \{\Psi(\bold{r}_1),\Psi^\dagger(\bold{r}_2)\}=\delta (\bold{r}_1-\bold{r}_2)&\text{ fermion fields.} \end{align}$

In homogeneous systems it is often desirable to transform between real space and the momentum representations, hence, the quantum fields operators in Fourier basis yields:

$\Psi(\bold{r})={1\over \sqrt {V}} \sum_{\bold{k}} e^{i\bold{k\cdot r}}a_{\bold{k}}$

$\Psi^{\dagger}(\bold{r})={ 1\over \sqrt{V}} \sum_{\bold{k}} e^{-i\bold{k\cdot r}}{a^{\dagger}}_{\bold{k}}$

String Theory

Sunday, 9 June 2013

Quantum field operators

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