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

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

Electron-positron field operator: ${\hat {\Psi }}\left(t\right)$ . 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 ${\hat {H}}$ $i{\frac {\partial {\hat {\Psi }}}{\partial t}}=\left[{\hat {H}}\left(t\right),{\hat {\Psi }}\right]$ and the ordinary Dirac equation

$i{\frac {\partial {\hat {\Psi }}}{\partial t}}=h\left(t\right){\hat {\Psi }}$ with the usual Dirac time-evolution generator

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

From now on we use atomic units, for which $m=e=\hbar =1\,a.u.$ and $c=137.036\,a.u.$ If we choose an arbitrary basis set of quantum mechanical single-particle states $|\alpha \rangle$ , the field operator can be expanded as

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

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

### 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)