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Semiclassical Quantization Condition for a Relativistic Bound System of Two Equal-Mass Fermions

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Abstract

New relativistic semiclassical quantization conditions are obtained for a system of two equal-mass 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.

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Notes

  1. We recall that, here, \(\lambda_{\mathcal{Q}}=(\lambda_{\mathcal{Q}}^{0};{\boldsymbol{\lambda}_{\mathcal{Q}}})=\mathcal{Q}/\sqrt{\mathcal{Q}^{2}}\) is the 4-velocity of a composite particle that has the 4-momentum \(\mathcal{Q}=q_{1}+q_{2}\), all of the 4-momenta involved belonging to the upper sheets of the mass hyperboloids \(\Delta_{q,m\lambda_{\mathcal{Q}}}^{2}=\Delta_{q,m\lambda_{\mathcal{Q}}}^{02}-c^{2}{\boldsymbol{\Delta}}_{q,m\lambda_{\mathcal{Q}}}^{2}=m^{2}c^{4}\), where \(\Delta_{q,m\lambda_{\mathcal{Q}}}^{0}\) and \({\boldsymbol{\Delta}}_{q,m\lambda_{\mathcal{Q}}}\) are, respectively, the time and spatial components of the 4-vector \(\Lambda^{-1}_{\lambda_{\mathcal{Q}}}q=\Delta_{q,m\lambda_{\mathcal{Q}}}\) from Lobachevsky’s space (for more details, see [12]).

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ACKNOWLEDGMENTS

I am grateful to O.P. Solovtsova for discussions on the results obtained in the present study, valuable comments, and technical support and to A.E. Dorokhov, Yu.A. Kurochkin, I.S. Satsunkevich, V.V. Andreev, and A.V. Kiselev for discussions on the results quoted above, their comments, and stimulating discussions in course of this investigation.

Funding

This work was supported by the program for international cooperation of Republic of Belarus with Joint Institute for Nuclear Research (JINR, Dubna) and by the state research program for the period spanning 2016 and 2020 Convergence-2020, Microscopic World, Plasma, and Universe Subprogram.

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  1. International Center for Advanced Studies, Pavel Sukhoi Gomel State Technical University, Gomel, Republic of Belarus

    Yu. D. Chernichenko

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Correspondence to Yu. D. Chernichenko.

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Chernichenko, Y.D. Semiclassical Quantization Condition for a Relativistic Bound System of Two Equal-Mass Fermions. Phys. Atom. Nuclei 83, 488–494 (2020). https://doi.org/10.1134/S1063778820030047

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