Square-Root Klein-Gordon Equation

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. 

For the same reason it is not clear, how to modify the equation for the inclusion of external fields. The principle of minimal coupling p —> p — A, E E + V for the (scalar) square-root Klein-Gordon equation was critizised by J. Sucher [4], who states that there are solutions ip x) and electromagnetic potentials, such that the Lorentz transformed solution is not a solution of the equation with the Lorentz-transformed potentials. Moreover, the nonlocal nature of the equation means that the value of the potential at some point influences the wave function at other points and it is not clear at all how one can interpret this. 

The spin of elementary particles is not described by the square-root Klein-Gordon equation. The solutions of the square-root Klein-Gordon equation are scalar wave functions. Real electrons have spin and should be described by a matrix-wave equation. 

A significant point here is that it is not the squared invariant ds2 that is to underlie the covariance of the laws of nature. It is rather the linear invariant ds that plays this role. How, then, do we proceed from the squared metric to the linear metric That is to say, how does one take the square root of ds2l The answer can be seen in Dirac s procedure, when he factorized the Klein-Gordon equation to yield the spinor form of the electron equation in wave mechanics -the Dirac equation. Indeed, Dirac s result indicated that by properly taking the square root of ds2 in relativity theory, extra spin degrees of freedom are revealed that were previously masked. 

Our objective is to And a complex symmetric formulation that contains the seed of the relativistic frame invariants. The trick is to entrench an apposite matrix of operators whose characteristic equation mimics the Klein-Gordon equation (or in general the Dirac equation). Intuitively, one might infer that we have realised the feat of obtaining the negative square root of the aforementioned operator matrix. Thus, the entities of the formulation are operators and furthermore since they permit... 

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. 

The square-root operator is difficult to evaluate in position space because of the square root to be taken of a differential operator that would represent p. We have already discussed this issue in the context of the Klein-Gordon equation in section 5.1.1. Hence, the action of the X-operator is most conveniently studied in momentum space, where the inverse operator may be applied in closed form without expanding the square root. 

It should be recalled that, because of the presence of the external potential and the nonlocal form of Ep given by Eq. (11.11), all operators resulting from these unitary transformations are well defined only in momentum space (compare the discussion of the square-root operator in the context of the Klein-Gordon equation in chapter 5 and the momentum-space formulation of the Dirac equation in section 6.10). Whereas So acts as a simple multiplicative operator, all higher-order terms containing the potential V are integral operators and completely described by specifying their kernel. For example, the... 

Considering the derivation of DKH Hamiltonians so far, we are facing the problem to express all operators in momemtum space, which is somewhat unpleasant for most molecular quantum chemical calculations which employ atom-centered position-space basis functions of the Gaussian type as explained in section 10.3. The origin of the momentum-space presentation of the DKH method is traced back to the square-root operator in Sq of Eq. (12.54). This square root requires the evaluation of the square root of the momentum operator as already discussed in the context of the Klein-Gordon equation in chapter 5. Such a square-root expression can hardly be evaluated in a position-space formulation with linear momentum operators as differential operators. In a momentum-space formulation, however, the momentum operator takes a... 

This equation is usually referred to as the Klein-Gordon equation < By using the square of an energy relation, we have introduced a negative-energy root... 

We will not re-derive this formulation, see e.g. [12] and references therein, except make a brief outline of the main results. Rather than going through the construction through apposite complex symmetric forms, we will here proceed directly via the observation that the classical-quantum equations of relativity cf. the Klein-Gordon equation, is a quadratic form in the actual observables. Considering the non-positive square root from the simple ansatz of the Hamiltonian below... 

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... 

The quantities defined in Eq. (3) must in general be identified as operators for example p, which is conventionally a self-adjoint operator, looses this property in its extended form. Nevertheless, it will be consistent with the relationship p = mu (v is the velocity relative a system in rest, wherever the rest mass of the particle is mo) with appropriate modifications made for a particle in an electromagnetic or other field. We also note that p should be a vector operator although we treat it here temporarily as scalar operator, the square root of p. Our operator-secular equation generates as expected a Klein-Gordon-type equation, with the resulting eigensolutions. 

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