Bethe–Salpeter Formalism Overview

• Bethe–Salpeter formalism is a relativistic framework that models two-particle bound states by rigorously incorporating both spatial and temporal dynamics.
• The approach uses a hierarchy of semirelativistic approximations, such as the Salpeter and spinless Salpeter equations, to simplify complex four-dimensional integrals.
• Variational techniques and strict spectral bounds ensure reliable predictions for bound-state spectra and stability under physically meaningful potentials.

The Bethe–Salpeter formalism provides a fully relativistic, Poincaré-covariant framework for describing two-particle bound states (such as conventional mesons) in quantum field theory. While the original four-dimensional, homogeneous Bethe–Salpeter equation rigorously encodes the dynamics of these systems, including their relative timelike degrees of freedom, the full formalism is analytically and numerically complex. To make exploration of bound-state spectra more practical, a series of semirelativistic approximations have been developed, each reducing the complexity of the original equations while aiming to retain essential relativistic features relevant for hadronic and quantum systems.

1. The Bethe–Salpeter Equation: Covariant Bound-State Framework

The Bethe–Salpeter equation (BSE) in its e-covariant, homogeneous four-dimensional form is the foundational equation for relativistic two-particle bound states. It encompasses both spatial and time-like (relative energy) coordinates of the two-body problem and includes all relevant interactions via the two-particle irreducible kernel and full propagators. In explicit terms, the bound-state mass and amplitude are determined nonperturbatively by locating the poles of the two-particle Green’s function in the complex energy plane.

Handling this four-dimensional integral equation directly is formidable due to the coupling of spatial and temporal degrees of freedom and the nontrivial analytic structure of the propagators and the kernel. The equation’s full solution sets the reference for assessing the fidelity of reduced or approximate forms.

2. Hierarchy of Approximations: From Bethe–Salpeter to Semirelativistic Reductions

To render analysis tractable, a systematic hierarchy of approximations is invoked:

1. Instantaneous Bethe–Salpeter Reduction: The kernel is assumed to depend only on spatial coordinates—effectively discarding the timelike relative coordinate ("instantaneous approximation").
2. Salpeter Equation: Bound-state constituents are assumed to propagate freely, leading to a three-dimensional reduction; the equation still contains positive and negative energy components.
3. Reduced Salpeter Equation: Negative-energy components are omitted, further simplifying the structure.
4. Spinless Salpeter Equation: Spin degrees of freedom are ignored. The resulting eigenvalue equation involves a Hamiltonian with exact relativistic kinetic energy operators,

$$H = \sqrt{p^2 + m_1^2} + \sqrt{p^2 + m_2^2} + V(x),$$

and, for equal-mass constituents ($m_1 = m_2 = m$),

$\widehat{H} = 2 \sqrt{p^2 + m^2} + V(x).$

Here, $V(x)$ is the effective static potential, typically motivated by phenomenological considerations or by matching with nonrelativistic and field-theoretic limits.

This layered reduction approach preserves critical relativistic features (notably the exact kinetic energy) while eliminating the most analytically intractable elements of the original BSE.

3. Analysis and Estimation of the Bound-State Spectrum

Within the semirelativistic framework, the quantum bound-state problem reduces to an eigenvalue problem for the nonlocal Hamiltonian. Rigorous mathematical methods, primarily drawn from functional analysis and variational calculus, are used to analyze the spectrum:

• Variational Principle & Minimum–Maximum Theorem: The eigenvalues of the spinless Salpeter Hamiltonian can be estimated by restricting $H$ to a finite-dimensional trial space. For central potentials, a basis of trial functions is constructed as

$$\psi_{k, \ell}(x) \propto r^{\ell+\beta-1} e^{-\mu r} L_k^{2\ell+2\beta}(2\mu r) Y_\ell(\Omega_x),$$

where $L_k$ are generalized Laguerre polynomials and $Y_\ell$ are spherical harmonics. The variational parameters $(\mu, \beta)$ are optimized to obtain upper bounds $\bar{E}_k$ on the physical eigenvalues $E_k$ of $H$:

$E_k \leq \bar{E}_k, \quad k=0,1,2,\ldots,d-1.$

• Nonrelativistic Limit: For weak potentials, the spectrum can be approximated using nonrelativistic quantum mechanics, yielding a trivial upper bound on the binding energy for the ground state.

These techniques provide concrete upper and lower bounds on binding energies and allow estimation of the number of bound states for specific potentials.

4. Rigorous Constraints: Stability and Spectral Bounds

The formalism imposes strict constraints on the potential parameters to guarantee physical spectra:

• Boundedness from Below: The Hamiltonian must be bounded from below for stability. For instance, for the Coulomb potential,

$V_C(r) = -\frac{\alpha}{|r|}, \qquad \alpha > 0,$

the Hamiltonian is bounded from below only if $\alpha < 4/\pi \approx 1.273239\ldots$. A rigorous lower bound on the spectrum is then given by

$$\sigma(\widehat{H}) \geq 2m \sqrt{1 - \left(\frac{\pi\alpha}{4}\right)^2}.$$

• Upper Bounds on the Number of Bound States: For potentials $V \in L^{3/2}$ and $L^3$, Lebesgue-space analysis provides rigorous upper limits on the number of discrete eigenvalues.
• Classification of Potentials by Singularity and Decay: The spectrum’s behavior depends crucially on whether the potential is singular (as with Coulomb) or regular (short-range).

These constraints ensure that the semirelativistic models yield physically meaningful spectra and delineate the permissible parameter ranges for phenomenological applications.

5. Applications to Benchmark Potentials

The practical methodology is illustrated by treating several canonical potentials:

Potential Type	Form	Boundedness Criterion
Coulomb	$V_C(r) = -\alpha/|r|$	$\alpha < 4/\pi$
Hulthén	$V_S(r) = -\eta/(e^{br} - 1)$	$\eta/b < 4/\pi$
Generalized HeLLMann	$V_H(r) = -\kappa/r - \upsilon\,e^{-br}/r$	$(\kappa+\upsilon)/b < 4/\pi$

• Coulomb: Exhibits a critical coupling threshold; for $\alpha > 4/\pi$, the spectrum is unbounded from below.
• Hulthén: Approaches Coulomb-like behavior near the origin but decays exponentially for large $r$; the nonrelativistic eigenvalue for $\ell = 0$ states is (with reduced mass $\widehat{m}$)

$$E_n^{\text{(NR)}} = m_1 + m_2 - \frac{(2\widehat{m}\eta - n^2 b^2)^2}{8\widehat{m} n^2 b^2}, \quad n \leq \frac{\sqrt{2\widehat{m}\eta}}{b}.$$

• Generalized HeLLMann: Combines Coulomb and Yukawa tails; spectral type (bounded or singular) is set by the coefficient $\kappa+\upsilon$.

These analyses demonstrate how semirelativistic reductions support qualitative and even quantitative spectral predictions for physically relevant systems while maintaining mathematical rigor.

6. Interpretation and Significance of the Semirelativistic Approach

The examined approach provides a physically transparent, mathematically controlled scheme for extracting spectral information from complex relativistic bound-state equations. By systematically reducing the BSE to a one-body semirelativistic eigenproblem, one obtains nonlocal Hamiltonians that retain the exact relativistic kinetic energy operator while implementing binding through phenomenologically or microscopically motivated static potentials.

The hierarchy of mathematical constraints, particularly the variational and boundedness results, ensures that any predictions for spectra or number of bound states remain consistent with fundamental requirements of relativistic quantum mechanics. This discipline serves to test the physical relevance and practical reliability of solutions generated through further approximations to the original four-dimensional BSE.

7. Outlook and Extensions

While the semirelativistic reduction substantially streamlines the analysis of two-body bound states, care must be taken in interpreting its predictions for strongly (or ultra-relativistically) bound systems, especially for singular potentials at critical coupling. The framework admits further extensions, such as systematic incorporation of spin, the paper of more general potential forms, or the development of improved variational bases. Additionally, the formalism provides a reference point for validating numerical or analytic treatments that seek to approximate full four-dimensional BSE solutions for systems of phenomenological or theoretical interest.