Adjoint Dirac Equation

In standard quantum field theory, particles are identified as (positive frequency) solutions ijj of the Dirac equation (p — m) fj = 0, with p = y p, m is the rest mass and p the four-momentum operator, and antiparticles (the CP conjugates, where P is parity or spatial inversion) as positive energy (and frequency) solutions of the adjoint equation (p + m) fi = 0. This requires Cq to be linear e u must be transformed into itself. Indeed, the Dirac equation and its adjoint are unitarily equivalent, being linked by a unitary transformation (a sign reversal) of the y matrices. Hence Cq is unitary. 

We will start by setting up a simple 2x2 matrix that (without interaction) displays perfect symmetry between the particle and its antiparticle image. Note that it is well known that the Klein-Gordon and the Dirac equation can be written formally as a standard self-adjoint secular problem (see e.g. [11,12]), based on the simple Hamiltonian matrix (in mass units)... 

The Dirac equation (42) and its adjoint can also be derived by requiring that... 

The equation above is written using the units h c 1. The quantity y is a vector of Dirac matrices, m is the electron mass multiplied by a Dirac matrix. Fermi level. With this definition the energy functional is... 

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