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