**Authors:** John R. Smith

The square-root Klein-Gordon operator, √m^2 − ∇^2 , is a non-local operator with a natural scale inversely proportional to the mass (the Compton wavelength). The fact that there is a natural scale in the operator as well as the fact that the single particle theory for the Coulomb potential, V (r) = −Ze2/r, yields a different eigenvalue spectrum from either the Dirac Hamiltonian or the Klein-Gordon Hamiltonian indicates that this operator is truly distinct from either of the other two Hamiltonians (all three single-particle Hamiltonians have eigenspectra for the 1s states that converge at small atomic numbers, Z → 0, but diverge from each other at large Z). We see no fundamental reason to exclude negative energy states from a “square-root” propagation law and we find several possible Hamiltonians associated with √m2 − ∇2 which include both positive and negative energy plane wave states. Depending on the specific Hamiltonian, it is possible to satisfy the equations of motion with commutators or anticommutators. However, for the scalar case considered, only the Hamiltonian that requires commutation rules has a stable vacuum. We investigate microscopic causality for the commutator of the Hamiltonian density. Also we find that despite the non-local dependence of the energy density on the field operators, the commutators of the physical observables vanish for space-like separations. This result extends the application of Pauli’s1 result to the non-local case. Pauli explicitly excluded √m2 − ∇2 because this op- erator acts non-locally in the coordinate space. We investigate the problems with applying minimal coupling to the square-root equation and why this method of interactions is inconsistent with the exponential shift property of the square-root operator and the demand for gauge-invariance. The Mandelstam representation offers the possibility of avoiding the difficulties inherent in minimal coupling (Lorentz invariance and gauge-invariance). We also compute the propagators for the scat- tering problem and investigate the solutions of the square-root equation in the Aharonov-Bohm problem.

**Comments:** 60 Pages.

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[v1] 2017-07-10 13:29:52

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