Stationary bound states of Dirac particles in collapsar fields
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tationary Bound States of Dirac Particles in Collapsar Fields1 M. V. Gorbatenko and V. P. Neznamov* Russian Federal Nuclear Center—AllRussian Research Institute of Experimental Physics, pr. Mira 37, Sarov, Nizhegorodskaya oblast, 607188 Russia *email: [email protected] Abstract—It is the first time stationary bound states of elementary spin 1/2 particles that do not decay with time are obtained for a Schwarzschild gravitational field using a selfconjugate Hamiltonian with a flat scalar product in a wide range of gravitational coupling constant. In order to obtain a discrete energy spectrum, we introduce a boundary condition such that the current density of Dirac particles near the “event horizon” is zero. DOI: 10.1134/S1063779614010377 1
The classical Schwarzschild solution is character ized by a point spherically symmetrical source of grav itational field of mass M and “event horizon” (gravita tional radius). 2GM r 0 = . 2 c
(1)
In (1), G is the gravitational constant, and c is the speed of light. For a particle of mass m, the dimension less gravitational coupling constant equals r0 Mm = α = GMm = . 2 បc 2l c mp
(2)
បc = 2.2 × G ប is the 10 ⎯5 gram is the Planck mass, and lc = mc Compton wavelength. For an electron, α ⯝ 1 corresponds to M = 0.5 × 1015 kg. The value of the coupling constant corre sponding to a source having the mass of the Sun (M = M䉺 ≈ 2 × 1030 kg) for an electronmass particle is α ≅ 4 × 1015. Despite the evident electromagnetic analogy in atomic physics, bound states of Dirac particles in the Schwarzschild field have been investigated compara tively scantily. For the gravitational case, there is a belief that bound states have complex energies. In this case, these states decay exponentially with time. The existence of resonant Schwarzschild states for massive scalar particles using the Klein–Gordon equation was discussed in [1–4]. The same problem for massive Dirac particles was discussed in [5–9]. In these papers, a hydrogenlike spectrum with relativistic cor rections was obtained for α Ⰶ 1 by direct solution of In (2), ប is the Planck constant, mp =
the Dirac equation in a weak Schwarzschild field for the real part of energy. In papers [10–12], we develop a method for deriv ing selfconjugate Dirac Hamiltonians with a flat sca lar product within the framework of pseudoHermi tian quantum mechanics for arbitrary, including time dependent, gravitational fields. Apparently, such sta tionary Hamiltonians, if there are square integrable wave functions and if corresponding boundary condi tions are specified, will provide existence of stationary bound states of Dirac particles with a real energy spec trum. We suppose that for these cases Hamiltonians are Hermitian ((Ψ, Hϕ) = (HΨ, ϕ)). This report is devoted to defining stationary bound states of a Dirac particle in a Schwarzschild field. Below we will use the system of units G = ប = c = 1, signature η α β = diag [1, − 1, − 1, − 1]
(3)
and notations, γα, γα are Dirac matrices wi
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