Introductory review on quantum field theory with space–time resolution

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State in quantum field theory:  .

Electron-positron field operator:  . 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  

 

and the ordinary Dirac equation

 

with the usual Dirac time-evolution generator

 .

From now on we use atomic units, for which   and  

If we choose an arbitrary basis set of quantum mechanical single-particle states  , the field operator can be expanded as

 .

The operator   would create an excitation of the mode  , while   would excite the mode labelled  . One often uses the energy eigenstates of the force-free Hamiltonian  . Here the states   with positive energy ( ) are denoted by  , whereas those with negative eigenvalue ( ) are denoted by  . Note that it is nontrivial to associate the negative part of the spectrum with energies. In order to associate the charge-conjugated states of   with positrons, the annihilation operators   have been renamed into  . This crucial step is an important additional postulate. It also re-arranges the energy levels in the corresponding field theoretical Hamiltonian  . This equation also relates the quantum field theoretical Hamiltonian   (acting on states denoted  ) to its limiting case, the quantum mechanical Hamiltonian   (acting on states denoted by  ).

Referințe

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

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