Bethe-Salpeter Equation Overview

Updated 6 December 2025

The Bethe-Salpeter Equation is a covariant integral equation that defines the quantum dynamics of two-body systems, capturing bound state formation and scattering phenomena.
It systematically resums two-particle irreducible interactions to predict excitonic effects, meson spectra, and other collective excitations in QED, QCD, and condensed matter.
Advanced numerical techniques like the Nakanishi integral representation and matrix eigenvalue reductions enable efficient solutions of the equation for complex physical systems.

The Bethe-Salpeter equation (BSE) is a canonical, covariant integral equation describing the quantum dynamics of two-body (and, by generalization, few-body) systems in relativistic quantum field theory and many-body perturbation theory. It encompasses bound-state formation, scattering, and collective excitations in diverse areas such as nuclear and particle physics, quantum electrodynamics (QED), quantum chromodynamics (QCD), and ab initio electronic-structure theory in condensed matter and quantum chemistry. The BSE formally resums the infinite series of two-particle irreducible kernel insertions in the relevant channel, yielding dynamical correlation functions and excitation spectra inaccessible by single-particle approaches.

1. Fundamental Structure of the Bethe-Salpeter Equation

The homogeneous BSE for a two-body (e.g., scalar or Dirac) system is typically expressed as

$\Phi(k;P) = G_0(k,P)\int\frac{d^4k'}{(2\pi)^4}i\mathcal{K}(k,k';P)\Phi(k';P)$

where $\Phi(k;P)$ is the Bethe-Salpeter amplitude as a function of the total four-momentum $P$ and relative four-momentum $k$ , $G_0$ is the disconnected product of dressed one-particle propagators, and $\mathcal{K}$ is the two-particle irreducible interaction kernel. The condition for the existence of a nontrivial solution gives the bound-state mass $M$ via $P^2 = M^2$ (Frederico et al., 2013).

In the context of many-body electronic systems, projection onto an electron–hole basis (valence and conduction orbitals) leads to a matrix eigenproblem for neutral (exciton) energies $\Omega$ . Neglecting coupling between resonant and antiresonant terms (Tamm-Dancoff approximation, TDA) and adopting the static screening limit, the excitonic problem reduces to

$(\varepsilon_c - \varepsilon_v)A_{vc} + \sum_{v'c'}K_{vc,v'c'}A_{v'c'} = \Omega A_{vc}$

where $A_{vc}$ are excitonic amplitudes, $\varepsilon_p$ are single-particle quasiparticle energies, and $K_{vc,v'c'}$ is the electron–hole interaction kernel (Loos et al., 2020).

2. Kernel Construction and Approximations

The BSE kernel $K$ encodes all relevant two-particle-irreducible interactions. Its exact form depends on the underlying field theory:

QED bound states: $K$ comprises the sum of photon-exchange diagrams (ladder and crossed-ladder skeletons) plus self-energy and vertex corrections (Mátyus et al., 2022, Owen et al., 2015).
QCD (mesons, glueballs, tetraquarks): $K$ includes gluon exchange, quark–gluon and three-gluon vertices, and, in practical truncations, only subsets such as the rainbow-ladder (one-gluon exchange) or including corrections required by Ward–Takahashi identities (Qin, 2016, Eichmann et al., 2015, Sanchis-Alepuz et al., 2015).
Many-body electronic structure: $K$ incorporates the repulsive exchange (bare Coulomb) and attractive screened direct terms, with the screened interaction $W$ often approximated via the static limit, plasmon-pole model, or computed exactly via the random-phase approximation (RPA) (Loos et al., 2020, Nguyen et al., 2019).

The frequency dependence of $W$ leads to a dynamical BSE kernel. Commonly, the static approximation is enforced ( $W(\omega)\to W(0)$ ), but systematic improvements are possible via plasmon-pole models or by evaluating $W$ exactly in RPA and applying perturbative dynamical corrections, as in Loos and Blase's approach for molecular excitations (Loos et al., 2020).

3. Methodologies for Solving the BSE

Two principal strategies dominate the solution of the BSE:

Direct Integral Equation Methods: In Minkowski or Euclidean space, the amplitude is expanded using the Nakanishi integral representation (NIR), leading to uniquely-defined weight functions $g(\gamma, z)$ , which can be extracted via light-front projections or by solving eigenequations obtained by LF projection (Frederico et al., 2013, Gigante et al., 2017). The Nakanishi uniqueness theorem has been verified nonperturbatively.
Reduction to Matrix Eigenvalue Problems: In many-body applications, discretization on a finite set of transitions allows recasting of the BSE as a large, dense, structured eigenvalue problem, often of the $2\times2$ block form. Specialized algorithms exploit the product structure and scalar-product symmetries, enabling efficient and stable computation of the spectrum (Cholesky-based, SVD-based methods) (Benner et al., 2020).

Other techniques include: perturbative correction schemes for dynamical kernels (Loos et al., 2020); finite-field approaches eliminating explicit dielectric matrices (Nguyen et al., 2019); advanced numerical linear algebra for high-dimensionality crystalline problems (Benner et al., 2020); and diagrammatic Monte Carlo with resummation to solve BSE directly on the real-frequency axis at finite $T$ (Tupitsyn et al., 2023). In nuclear physics, asymptotic approximations allow reduction to one-dimensional kernels with explicit analytic control for long-range interactions (Kinpara, 2013).

4. Physical Contexts and Applications

The BSE is central to a wide array of theoretical and computational frameworks:

Excitons and Optical Spectra in Condensed Matter/Chemistry: The BSE provides quantitatively accurate excitation energies and response spectra for molecules, solids, and nanostructures, superseding TDDFT in systems where electron–hole correlation is critical. Dynamical corrections to the kernel improve agreement with high-level coupled-cluster benchmarks, especially for valence transitions (Loos et al., 2020). BSE predictions for extended systems such as disordered water and ice, or large fullerenes, are feasible via finite-field and localized-orbital methodologies (Nguyen et al., 2019).
Bound States in QED and QCD: Relativistic two- and four-body bound states such as positronium, kaonic atoms, mesons, tetraquarks, and glueballs are described. The BS framework includes recoil and relativistic corrections nonperturbatively. For instance, in Landau-gauge QCD, coupled BSEs for gluon and ghost-antighost amplitudes predict glueball masses consistent with lattice gauge theory and reveal sensitivities to symmetry content and vertex structure (Sanchis-Alepuz et al., 2015, Eichmann et al., 2015, Watson et al., 2012, Qin, 2016).
Scattering and Response: The inhomogeneous BSE and the related K-matrix facilitate analysis of two-nucleon elastic scattering and inclusion of inverse-square corrections (e.g., via pion-exchange) in phase shifts and polarization observables (Kinpara, 2015).
Generalized Classical Limits: The classical BSE derived for gravitationally-bound two-body systems shows connections to Hamilton–Jacobi theory, resummation of eikonal diagrams, and the relativistic generalization of Sommerfeld enhancement, clarifying the analytic structure of classical bound and scattering observables (Adamo et al., 2022).

5. Structural Features and Theoretical Properties

The BSE possesses formal properties crucial for its consistent physical interpretation:

Uniqueness and Covariance: The solution in Nakanishi integral form is unique (Nakanishi theorem) (Frederico et al., 2013), and the equation is manifestly Lorentz covariant.
Symmetry Preservation: The kernel must be constructed to preserve key symmetries, such as gauge invariance, chiral symmetry (ensuring the validity of Goldstone's theorem), and crossing symmetry. These requirements are manifest in systematic, Ward–Takahashi identity-respecting truncation schemes (e.g., rainbow-ladder, DCSB-improved kernels) (Qin, 2016).
Spectrum and Stability: The analytic structure of the Salpeter equation (instantaneous-approximation BSE) reveals that true confining kernels must couple via the time component of a Lorentz vector for the spectrum to be real and discrete; scalar or pseudoscalar kernels induce unphysical instabilities (Lucha, 2010).

The role of the Dirac–Lorentz structure in the kernel is thus as fundamental as the functional form of the interaction.

6. Computational Techniques and Algorithmic Developments

The practical solution of the BSE for realistic systems has advanced through several key innovations:

Efficient Eigenproblem Solvers: Block structures, product-eigenvalue reductions, and use of scalar-product symmetries yield algorithms that reduce the cost of standard approaches by up to a factor of two while restoring full backward stability (e.g., Cholesky and SVD-based algorithms) (Benner et al., 2020).
Finite-Field Approaches: Direct evaluation of the screened Coulomb interaction via finite perturbations and orbital localization circumvents explicit inversion of dielectric matrices and avoids virtual-state summations, achieving competitive scaling and precision for large molecules/condensed-phase systems (Nguyen et al., 2019).
Matrix Compression and Exponential Convergence: Intermediate representation (IR) basis techniques and sparse sampling of Matsubara frequencies enable $O(L^8)$ computational time scaling (with $L$ logarithmic in system scale and accuracy), making BSE solutions tractable at low temperature and for impurity models (Wallerberger et al., 2020).
Direct Real-Frequency Solutions: Ladder-type finite-temperature BSEs can be solved directly on the real axis via diagrammatic Monte Carlo with series resummation, eliminating artifacts from analytic continuation and accurately capturing dynamical effects such as Landau damping (Tupitsyn et al., 2023).

7. Extensions, Generalizations, and Hierarchies

The BSE admits natural generalizations:

Few-Body and Many-Body Generalization: Four-body BSEs for tetraquarks or glueballs, or higher $n$ -PI effective actions, generate coupled hierarchies of integral equations for multi-point correlations (Carrington et al., 2013, Eichmann et al., 2015).
Hierarchies from nPI Effective Actions: Systematic derivation from the 4PI or higher effective action leads to a closed set of coupled BS-type equations for all multi-point functions, ensuring correct diagrammatic resummation and symmetry properties (Carrington et al., 2013).
Dimensional Reduction and Condensed-Matter Applications: In 2+1D, the Minkowski-space BSE with Nakanishi representation is validated as numerically equivalent to Euclidean approaches, underpinning applications to 2D materials (graphene, transition-metal dichalcogenides) (Gigante et al., 2017).
Nonperturbative Phenomena: The BSE enables exploration of dynamical chiral symmetry breaking, heavy-quark symmetry (scaling $f_{PS}\sqrt{M_{PS}}=$ const), and the emergence of collective modes or new phases in strongly-correlated systems (Watson et al., 2012, Qin, 2016, Wallerberger et al., 2020).

In summary, the Bethe-Salpeter equation is the fundamental tool for the covariant, nonperturbative description of two-body (and few-body) phenomena in quantum field theory and many-body physics. Its effectiveness depends critically on the structure of the interaction kernel, the preservation of symmetries, and on innovations in analytical and numerical solution techniques, which enable quantitatively reliable description of bound states, resonances, and excitation spectra across disciplines (Loos et al., 2020, Frederico et al., 2013, Mátyus et al., 2022, Nguyen et al., 2019, Benner et al., 2020, Qin, 2016, Eichmann et al., 2015, Lucha, 2010, Carrington et al., 2013).