Quantum Physics


Second Quantization of the Square-Root Klein-Gordon Operator

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). There is no fundamental reason to exclude negative energy states from a “square-root” propagation law. We find several possible Hamiltonians associated with √(m^2− ∇^2) which include both positive and negative energy plane wave states. It is possible to satisfy the equations of motion with commutators or anticommutators. For the scalar case, only the canonical commutation rules yield a stable vacuum. We investigate microscopic causality for the commutator of the Hamiltonian density. 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. Hence, Pauli’s result can be extended to the non-local case. Pauli explicitly excluded √(m^2− ∇^2) because this operator acts non-locally in the coordinate space. 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 scattering problem and investigate thesolutions of the square-root equation in the Aharonov-Bohm problem.

Comments: 58 Pages.

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Submission history

[v1] 2017-07-06 20:04:58

Unique-IP document downloads: 32 times

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