Klein-Gordon formalism

The formalism can be extended for a quantum Jield with the TFD Lagrangian density given by t = — , where is a replica of for the tilde fields so leading to similar equations of motion. For the purpose of our applications, we shall restrict our analysis to free massless fields. Thus, considering the free-massless boson (Klein-Gordon) field, the two-point Green function in the doubled space is given by... 

As already discussed at the end of Section 2.2.3, we derived a universal superposition principle from a complex symmetric ansatz arriving at a Klein-Gordon-like equation relevant for the theory of special relativity. This approach, which posits a secular-like operator equation in terms of energy and momenta, was adjoined with a conjugate formal operator representation in terms of time and position. As it will be seen, this provides a viable extension to the general theory [7, 82]. We will hence recover Einstein s laws of relativity as construed from the overall global superposition, demonstrating in addition the independent choice of a classical and/or a quantum representation. In this way, decoherence to classical reality seems always possible provided that appropriate operator realizations are made. 

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 Schrodinger equation with this Hamiltonian is called the square-root Klein-Gordon equation because of its formal similarity to a square-root of the Klein-Gordon equation (Oscar Klein and Walter Gordon in fact had little to do with the square-root equation). Unfortunately, the meaning of the square-root Klein-Gordon equation is obscured by the following points. 

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