Semiclassical Quantization Condition for a Relativistic Bound System of Two Equal-Mass Fermions
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EMENTARY PARTICLES AND FIELDS Theory
Semiclassical Quantization Condition for a Relativistic Bound System of Two Equal-Mass Fermions Yu. D. Chernichenko* International Center for Advanced Studies, Pavel Sukhoi Gomel State Technical University, Gomel, Republic of Belarus Received November 27, 2019; revised December 12, 2019; accepted December 12, 2019
Abstract—New relativistic semiclassical quantization conditions are obtained for a system of two equalmass fermions interacting via nonsingular confining quasipotentials and quasipotentials of the funnel type. The quantization conditions are specified in the pseudoscalar, pseudovector, and vector cases. The respective analysis is performed within the Hamiltonian formulation of quantum field theory via a transition to the relativistic configuration representation for the case of a bound system formed by two relativistic spin particles of equal mass. DOI: 10.1134/S1063778820030047
1. INTRODUCTION The method that was used most frequently—and quite successfully—to describe the mass spectrum of ¨ mesons is based on the nonrelativistic Schrodinger equation with a linear potential in the form Vlin (r) = σr,
σ > 0.
However, the nonrelativistic model turned out to be inappropriate in describing the mass spectrum of substantially relativistic systems, since the contribution of relativistic corrections for higher radial excitations becomes large (v 2 /c2 ≈ 0.4); for light vector rho and omega mesons, it is even commensurate with the contribution of the nonrelativistic Hamiltonian, which one chooses to be the starting point [1–3]. An alternative approach to determining the mass spectrum of mesons is based on the application of the fully covariant two-particle three-dimensional relativistic quasipotential (RQP) approach developed by Logunov and Tavkhelidze within quantum field theory [4]. In the present study, we use that version of the RQP approach [5] to the problem of composite systems of two relativistic spin particles which relies on the Hamiltonian formulation of quantum field theory [6]. It is of importance that the approach in question takes into account a three-dimensional character of the problem from the outset; moreover, all particles, including the intermediate ones, turn out here to be physical—that is, they are on-shell particles. Thereby, the two-particle problem being considered is reduced to a single-particle one described in terms *
E-mail: [email protected];[email protected]
of the RQP wave function characterizing one relativistic particle and satisfying a fully covariant threedimensional RQP equation in momentum space (see, for example, [7–10]). Further, it is noteworthy that, in the case of interaction between two relativistic spin particles of equal mass (m1 = m2 = m), the RQP approach developed in [5, 6] makes it possible to go over from the momentum formulation in Lobachevsky space to the three-dimensional relativistic configuration representation introduced in [11]. For spherically symmetric potentials, the finite-difference form of the RQP equation for the wave funct
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