Klein-Gordon-like equation

The present account has been published before, see Refs. [7, 82, 83], and references therein. Using our preference for complex symmetric forms we will proceed directly to derive a Klein-Gordon-like equation as follows. Consider the nonpositive square root, cf. the Dirac equation, from the simple ansatz of the Hamiltonian H... 

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. 

The extension of a Klein-Gordon-like equation, see Eqs. (65)-(73), to include gravitational interactions is quite straightforward in our present theory. Since general relativity associates gravity with tensor fields, we need to incorporate the operator and their conjugate counterparts simultaneously. To accomplish the first part of the conjugate pair formulation, we will attach to our previous model in the basis m,m), the interaction... 

As promised we will present a simple model that displays both the fundamental nature of the concept of resonances as well as the quality of relativistic principles. A Klein-Gordon-like equation will be derived with specific restrictions and interpretations. Although the representation turns up somewhat naive and ad hoc, we will appreciate some general and surprising features that reveal the underlying fundamental principles. 

To summarize the discussion in previous reports [6-9], we will employ a simple complex symmetric ansatz obtaining a Klein-Gordon-like equation with precise restrictions and constraints. Although not explicitly written out here it is easy to identify the present construction with a standard arrangement in terms of a nonpositive definite metric. Specifically, we write... 

From the secular equation, based on the complex symmetric matrix 3 [, we obtain the eigenvalues X = mo via the Klein-Gordon-like equation. 

P. Gueret and J. P. Vigier, Non-linear Klein-Gordon equation carrying a non-dispersive soliton-like singularity, Lett. Nuovo Cimento 35(8) (Ser. 2), 256—259 (1982). 

Because of the unphysical feature of the Klein-Gordon density and the fact that spin does not emerge naturally (but would have to be included a posteriori as in the nonrelativistic framework) we are not able to deduce a fundamental relativistic quantum mechanical equation of motion for a freely moving electron. However, we may wonder which results of this section may be of importance for the derivation of such an equation of motion for the electron. Certainly, we would like to recover the plane wave solutions of Eq. (5.8) for the freely moving particle, but in order to introduce only a single integration constant (or the choice of a single initial value) for a positive definite density distribution we need to focus on first-order differential equations in time. These must also he first-order differential equations in space for the sake of Lorentz co-variance. 

Now, in a wave equation analogous to the Schrodinger equation, the time-like variable E/c and the space-like momenta p should appear raised to the same power, since space and time have to be treated on the same footing. Obviously, the expression (3) as it stands leads to an equation which is of second order in both E/c and p, a relativistic wave equation known as the Klein-Gordon equation, valid for particles with spin 0. A linearization of the equation, i.e., a factorization according to... 

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