Upper limit on the critical strength of central potentials in relativistic quantum mechanics

Abstract

In the context of relativistic quantum mechanics, where the Schrödinger equation is replaced by the spinless Salpeter equation, we show how to construct a large class of upper limits on the critical value, g c ͑ᐉ͒ , of the coupling constant, g, of the central potential, V͑r͒ =−gv͑r͒. This critical value is the value of g for which a first ᐉ-wave bound state appears.

On the Relativistic Canonical Quantum Mechanics and Field Theory of Arbitrary Spin

“…Other concrete physical effects and the problem of description of (inter)-particle interactions in frames of the RCQM are described by other authors, see the references [24][25][26][27][28][29][30][31][32][33][34][35][36].…”

“…Note that relations (29) are valid only for the dimensions, where the Γ 2N are defined (where the matrix representation of the Clifford algebra exists). Therefore, the relations (29) can be useful not for all multiplets of RCQM.…”

“…As well as in (29) among the generators of (42) only the 4+2=6 matrices (41) are independent and form the basis of the algebra. Therefore, the found above algebra over the field of real numbers is defined as Cl R (4,2) and the dimension of this algebra is 2 6 = 64.…”

“…Therefore, the relations (29) can be useful not for all multiplets of RCQM. Nevertheless, existing relations (29) are very useful in order to operate with spins, with standard FW transformation and in order to formulate the FW transformation generalizations for the dimensions 2(2s+1)>4.…”

“…The anti-commutation relations of the Clifford-Dirac algebra generators (29) and corresponding Γ 2N matrices must be introduced in the FW representation. Therefore, in order to apply (29) in RCQM one must transform the Γ 2N matrices from the FW representation into the RCQM representation. Corresponding operator transformation is given by…”