Facets of the Spinless Salpeter Equation

The spinless Salpeter equation represents the simplest, and most straightforward, generalization of the Schrödinger equation of standard nonrelativistic quantum theory towards the inclusion of relativistic kinematics. Moreover, it can be also regarded as a well-defined approximation to the Bethe–Salpeter formalism for descriptions of bound states in relativistic quantum field theories. The corresponding Hamiltonian is, in contrast to all Schrödinger operators, a nonlocal operator. Because of the nonlocality, constructing analytical solutions for such kind of equation of motion proves difficult. In view of this, different sophisticated techniques have been developed in order to extract rigorous analytical information about these solutions. This review introduces some of these methods and compares their significance by application to interactions relevant in physics. PACS numbers: 03.65.Ge, 03.65.Pm, 11.10.St ∗ E-mail address: [email protected] † E-mail address: [email protected] 1 Bethe–Salpeter Formalism in the “Instantaneous Approximation” Within quantum field theory, the appropriate framework for the description of bound states is the Bethe–Salpeter formalism [1]. Therein, all bound states of two particles (in fact, of any two fermionic constituents) are governed by the homogeneous Bethe–Salpeter equation. Here we are interested in a particular well-defined approximation to this formalism, obtained by several simplifying steps: 1. The instantaneous approximation, neglecting any retardation effect, considers all interactions of the (two) bound-state constituents in their static limit. 2. The additional assumption that all the bound-state constituents propagate as free particles with some effective mass m yields the Salpeter equation [2]. 3. A disregard of all of their spin degrees of freedom focuses on the treatment of scalar bound particles. 4. In technical respect, the canonical transformation x → λx , p → p λ (1) of position (x) and momentum (p) variables casts in the case of particles of equal mass m for a scale factor λ = 2 this approach into one-particle form. (For more details of the derivation, consult, for instance, Refs. [3–5] and references therein.) Refraining from the nonrelativistic limit, we get the (nonlocal!) Hamiltonian H = T + V . (2) This operator is composed of the “square-root operator” T of the relativistically correct expression for the kinetic or free energy of a particle of mass m and momentum p, T = T (p) ≡ √ p2 +m2 , (3) and a (coordinate-dependent) static interaction potential