The Dirac Equation

Schrödinger and Klein-Gordon equation

  1. Non-relativistic quantum mechanics: The Schrödinger equation
  2. Relativistic formulation: The Klein-Gordon equation
  3. Equations of motion, once more
  4. Equations for a complex scalar field

    THE FIELDS AND E.o.M:

    For the case of a complex scalar field
    the same reasoning applies and similar formulae emerge. In this case the Lagrangian reads
    quite similar to the previous one. Differentiating now w.r.t. one of the two fields (the field or its complex conjugate) results in equations of motion for the other one, which look identical to the Klein-Gordon equation.

    QUANTISATION

    However, there is one difference: Since the field is complex now, the corresponding field operator is not self-adjoint (Hermitean) any longer. Therefore there are two kinds of creation and annihilation operators, here denoted with indices + and -. The field operator for φ can be expressed in terms of them as
    where the operator for the complex conjugate field is nothing but the Hermitean conjugate of the field operator. The commutator relations for the two sectors decompose and read
    and there are two number operators, one for each charge α=±,
    The propagator is given by the vacuum expectation value of the time-ordered product of a field and its complex conjugate,
    Other propagators, e.g. between two (identical) fields, both with or without complex conjugation, vanish.

    CHARGES

    Obviously, there are two types of particles, labelled with their charge and three-momentum. Accordingly, the particle states can be constructed by applying the corresponding creation operators,
    The four-momentum of such a state can be determined by appling the momentum-operator
    The charge is given by the charge operator. It is given as the integral over the probabilty density, i.e.
    From these two operators, it is clear that the particles have the same mass, but an opposite charge, Q = ±1. Specific examples for such pairs of oppositely charged particles are the charged pions or the electrically neutral kaons, in the latter case, the charge clearly is not the electric one.