Fock-Klein-Gordon equation

This equation has at least one advantage over the Schrodinger equation ct and x, y, z enter the equation on equal footing, whieh is required by special relativity. Moreover, the Fock-Klein-Gordon equation is invariant with respect to the Lorentz transformation, whereas the Schrodinger equation is not. This is a prerequisite of any relativity-consistent theory, and it is remarkable that such a simple derivation made the theory invariant. The invariance, however, does not mean that the equation is accurate. The Fock-Klein-Gordon equation describes a boson particle because vk is a usual scalar-type function, in contrast to what we will see shortly in the Dirac equation. 

Paul Dirac used the Fock-Klein-Gordon equation to derive a Lorentz transformation invariant equation for a single fermion particle. The Dirac equation is solvable only for several very simple cases. One of them is the free particle (Dirac), and the other is an electron in the electrostatic field of a nucleus (Charles Darwin-but not the one you are thinking of). 

None of the many fathers of the Fock-Klein-Gordon equation dared to take into account another possibility, the one with the negative square root in Eq. (3.40), a step made by Paul Dirac. In this case the Dirac s argument about the electron sea and the Pauli exclusion principle would not work, since we have to do with the bosons We would have an abyss of negative energies, a disaster for the theory. 

An analysis of the transformation properties of the Fock-Klein-Gordon equation and of the Dirac equation leads to the conclusion that satisfaction of the first of these equations requires the usual (i.e., scalar) wave function, whereas the second equation requires a bispinor character of the wave function. Scalar functions describe spinless particles (because they cannot be associated with the Pauli matrices), while bispinors in the Dirac equation are associated with the Pauli matrices, and describe a particle of 1/2 spin. 

Fermi contact term (p. 148) Fock-Klein-Gordon equation (p. 122) inertial sjfstem (p. 108)... 

Indeed, using the anticommutation relations, one recovers the Fock-Klein-Gordon equation ... 

In a moment, we wiU see the determination with which Dirac attacked the Fock-Klein-Gordon operator equation, which we will write a little differently here ... 

Schrodinger amplitude relation to Klein-Gordon amplitude, 500 Schrodinger equation, 439 adiabatic solutions, 414 as a unitary transformation, 481 for relativistic spin % particle, 538 for the component a, 410 in Fock representation, 459 in the q representation, 492 Schrodinger form of one-photon equation, 548... 

In this section we describe the general approach to constructing conformally invariant ansatzes applicable to any (linear or nonlinear) system of partial differential equations, on whose solution set a linear covariant representation of the conformal group 0(1,3) is realized. Since the majority of the equations of the relativistic physics, including the Klein-Gordon-Fock, Maxwell, massless Dirac, and Yang-Mills equations, respect this requirement, they can be handled within the framework of this approach. 

Note that n — N/2 corresponds to the independent particle model analogous to the celebrated Hartree-Fock equations in atomic and molecular physics. We also observe that the fundamental interaction mentioned above is unitarily connected with the electromagnetic interactions between the particle m0 and the antiparticle —m0. Since we do not make any distinctions between the Klein-Gordon and the Dirac equation, we are not able here to integrate the electro-weak theory although in principle this should be possible. 

What, then, should be inserted as the operator H of the energy E This was done by Schrddinger (before by Fock, Klein, and Gordon, which is why it is also known as the "equation of many fathers ). Schrddinger inserted what he had on the right side of his time-dependent equation i.e., H = ih. ... 

---