# Utilizator:Victor Blacus/Texte

## Introductory review on quantum field theory with space–time resolution

State in quantum field theory: ${\displaystyle ||\Phi \left(t\right)\rangle \rangle }$.

Electron-positron field operator: ${\displaystyle {\hat {\Psi }}\left(t\right)}$. It is a hybrid between a quantum field theoretical operator and a quantum mechanical state. As a hybrid, it fulfills simultaneously the quantum field theoretical Heisenberg equation of motion with the field Hamiltonian ${\displaystyle {\hat {H}}}$

${\displaystyle i{\frac {\partial {\hat {\Psi }}}{\partial t}}=\left[{\hat {H}}\left(t\right),{\hat {\Psi }}\right]}$

and the ordinary Dirac equation

${\displaystyle i{\frac {\partial {\hat {\Psi }}}{\partial t}}=h\left(t\right){\hat {\Psi }}}$

with the usual Dirac time-evolution generator

${\displaystyle h\equiv c\alpha \left(p-A/c\right)+c^{2}\beta +V}$.

From now on we use atomic units, for which ${\displaystyle m=e=\hbar =1\,a.u.}$ and ${\displaystyle c=137.036\,a.u.}$

If we choose an arbitrary basis set of quantum mechanical single-particle states ${\displaystyle |\alpha \rangle }$, the field operator can be expanded as

${\displaystyle {\hat {\Psi }}=\sum _{a}{\hat {b}}_{\alpha }|\alpha \rangle =\sum _{p}{\hat {b}}_{p}|p\rangle +\sum _{n}{\hat {d}}_{n}^{\dagger }|n\rangle }$.

The operator ${\displaystyle b_{p}^{\dagger }}$ would create an excitation of the mode ${\displaystyle |b\rangle }$, while ${\displaystyle d_{n}^{\dagger }}$ would excite the mode labelled ${\displaystyle |n\rangle }$. One often uses the energy eigenstates of the force-free Hamiltonian ${\displaystyle h_{0}\equiv c\alpha p+c^{2}\beta }$. Here the states ${\displaystyle \alpha }$ with positive energy (${\displaystyle >c^{2}}$) are denoted by ${\displaystyle p}$, whereas those with negative eigenvalue (${\displaystyle ) are denoted by ${\displaystyle |n\rangle }$. Note that it is nontrivial to associate the negative part of the spectrum with energies. In order to associate the charge-conjugated states of ${\displaystyle |n\rangle }$ with positrons, the annihilation operators ${\displaystyle {\hat {b}}_{n}}$ have been renamed into ${\displaystyle {\hat {d}}_{n}^{\dagger }}$. This crucial step is an important additional postulate. It also re-arranges the energy levels in the corresponding field theoretical Hamiltonian ${\displaystyle {\hat {H}}\equiv {\hat {\Psi }}^{\dagger }h{\hat {\Psi }}}$. This equation also relates the quantum field theoretical Hamiltonian ${\displaystyle {\hat {H}}}$ (acting on states denoted ${\displaystyle ||\Phi \rangle \rangle }$) to its limiting case, the quantum mechanical Hamiltonian ${\displaystyle h\,}$ (acting on states denoted by ${\displaystyle \alpha }$).

### Referințe

• T. Cheng, Q. Su and R. Grobe: Introductory review on quantum field theory with space–time resolution, Contemporary Physics, 51 (4), 315–330 (2010)