Källén-Lehmann Superposition

• Källén-Lehmann superposition is a spectral decomposition in quantum field theory that expresses the exact two-point function as an integral over free-particle propagators weighted by a positive spectral density.
• It employs principles of spectral positivity and Hilbert space structure to distinguish discrete particle poles from multiparticle branch cuts, thereby ensuring unitarity and causality.
• The framework extends to curved spacetimes, lattice models, and nonlocal theories, providing a versatile tool for spectral analysis and practical computations in high-energy physics.

The Källén-Lehmann superposition, also known as the Källén–Lehmann representation or spectral decomposition, is a fundamental result in quantum field theory and statistical field theory. It exhibits the exact two-point function (propagator) of a quantum field as a continuous superposition (integral) of free-particle propagators over a spectral density, thereby encoding both particle poles and multiparticle branch cuts in a model-independent framework. The construction generalizes naturally to curved spacetimes, lattice theories, nonlocal models, and various operator representations.

1. Mathematical Structure of the Superposition Representation

Given a quantum field $\phi(x)$, the Källén-Lehmann superposition for its time-ordered two-point function is

$$D_F(x-y)=\int_0^\infty d\mu^2\,\rho(\mu^2)\int\frac{d^4p}{(2\pi)^4}\frac{i\,e^{-ip\cdot(x-y)}}{p^2-\mu^2+i\epsilon}$$

with the spectral density $\rho(\mu^2)\geq 0$ a non-negative function, determined non-perturbatively by the full quantum dynamics via insertion of a complete set of intermediate eigenstates. In momentum space this reads

$$D_F(p^2) = \int_0^\infty d\mu^2\,\frac{\rho(\mu^2)}{p^2-\mu^2+i\epsilon}$$

which holds for scalar field theory and generalizes to fermions and higher-spin fields by appropriate operator-valued numerator structures and matrix-valued densities (Potting, 2011).

The spectral density separates into discrete and continuous contributions: $$\rho(\mu^2) = Z\,\delta(\mu^2 - m^2) + \theta(\mu^2 - M^2)\,\sigma(\mu^2)$$ where $Z$ is the pole residue and $M^2$ marks the onset of multiparticle continuum contributions.

2. Spectral Positivity, Hilbert Space, and Reflection Positivity

The non-negativity of $\rho(\mu^2)$ is a reflection of Hilbert space positivity (unitarity) and is guaranteed by both canonical quantization in Minkowski signature and by reflection positivity (Osterwalder-Schrader) in Euclidean or lattice formulations (Usui, 2012). For lattice models, the key axioms ensuring reconstruction of the Källén-Lehmann representation are:

• (A1): Hermiticity under reflection.
• (A2): Translational invariance.
• (A3): Reflection positivity.
• (A4): Polynomial boundedness of correlations.

Violation of reflection positivity (A3), as in the overlap scalar boson model, leads to negative regions in $\rho(\lambda,\mathbf{p})$ and signals lack of a physical Hilbert space interpretation (Usui, 2012). Conversely, the existence of a spectral representation with positive $\rho$ is both necessary and sufficient for these axioms to hold.

3. Generalizations: Curved Spacetimes, Lattices, and Operator Structure

3.1. Anti-de Sitter and de Sitter backgrounds

In maximally symmetric curved backgrounds, the Källén-Lehmann decomposition persists with spectral densities defined with respect to the representation theory of the isometry group. For AdS$_d$,

$$W^{(d)}_\lambda(z_1, z_2) = \int_0^\infty dm^2\,\rho_\nu(m^2)\,\Delta_M^{(d-1)}(x_1, x_2; m^2)$$

where $\Delta_M^{(d-1)}$ is the $(d-1)$-dimensional Minkowski-space Wightman propagator, and $\rho_\nu(m^2)$ is a Bessel-kernel spectral weight determined by radial mode decomposition in the bulk (Moschella, 21 Dec 2025). This diagonalization enables transparent Wick rotation and analytic continuation between Lorentzian and Euclidean AdS, clarifies the spectral structure of boundary correlators, and underlies the AdS/CFT correspondence at the level of two-point functions.

For de Sitter (dS$_{d+1}$), the decomposition involves the principal series of $SO(d+1,1)$, with bulk two-point functions

$$G^{(J)}(Y_1, W_1; Y_2, W_2) = \sum_{\ell=0}^J \int_{-\infty}^{+\infty} d\lambda\,\rho^{\mathcal P, \ell}_{\mathcal O^{(J)}}(\lambda)\,[\ldots]\,G_{\lambda, \ell}(Y_1, Y_2; W_1, W_2)$$

where $G_{\lambda,\ell}$ are free harmonic modes, and $\rho^{\mathcal P, \ell}$ is non-negative by unitarity (Loparco et al., 2023). In the Minkowski limit ($R\to\infty$), these densities reduce to standard spectral densities of flat-space QFT.

3.2. Lattice Gauge Theories and SU(2) Glueball Example

In lattice gauge theory, the two-point function for a gauge-invariant glueball operator $O(x)$ can be written in the Källén-Lehmann form after appropriate UV subtractions: $$G(p^2) = \int_0^\infty \frac{2\omega\,\rho(\omega)\,d\omega}{\omega^2 + p^2}$$ The Laplace inversion of the Euclidean-time Schwinger function, performed under positivity and using regularization such as Tikhonov with nonnegativity constraints, yields a spectral density with sharp ground-state peaks and broader structures for excited states, robustly matching conventional large-time exponential fits (Dudal et al., 2021).

3.3. Nonlocal Quantum Gravity

For nonlocal gravity models with entire form factors in the propagator, the time-ordered two-point function remains expressible in a Källén-Lehmann superposition: $$\tilde G_{\rm to}(-k^2) = \int_0^\infty ds\,\frac{\rho(s)}{s+k^2-i\epsilon}$$ with spectral density

$$\rho(s) = e^{-\ell_*^2(s-m^2)} \sum_\lambda \delta(s - m_\lambda^2) |\langle\Omega|\tilde\phi(0)|\lambda_0\rangle|^2$$

The nonlocality scale $\ell_*$ smoothly deforms the spectral weight but preserves positivity and allows recovery of the standard local theory as $\ell_*\to0$ (Briscese et al., 22 May 2024).

4. Spectral Representation Beyond Lorentz Invariance

In Lorentz-violating models, the spectral density becomes a function not only of $s=p^2$ but also other momentum invariants $u_i$ built from background tensors $c^{\mu\nu}$: $$\rho(s; \{u_i\}) = (2\pi)^{-3} \sum_n \delta^{(4)}(p - p_n) |\langle n | \phi(0) | 0 \rangle|^2$$ The resulting propagator is

$$D_F(x-y) = \int_{-\infty}^{+\infty} ds\int \frac{d^4p}{(2\pi)^4} \frac{i \rho(s; \{u_i\})}{p^2-s+i\epsilon} e^{-ip\cdot(x-y)}$$

with sum rules and dispersion relations modified accordingly, but again reducing to the standard Källén-Lehmann superposition in the $c^{\mu\nu}\to 0$ limit (Potting, 2011).

5. Superposition Principle and Spectral Inversion

The superposition captures the fact that any two-point function in a QFT with an invariant vacuum can be expanded as a continuous sum over free-field correlators weighted by a non-negative measure: $$G(x-y) = \int d\mu^2\,\rho(\mu^2)\,G_{\rm free}(x-y; \mu^2)$$ This principle is ubiquitously exploited in:

• The spectral analysis of composite operators (e.g. anomalous dimensions in conformal field theory via the boundary operator expansion (Loparco et al., 2023)),
• Loop diagram reductions (e.g. spinor-scalar bubbles and one-loop self-energies in curved space via harmonic analysis (Altshuler, 10 Aug 2025)),
• Wick rotations and analytic continuations between Euclidean/Lorentzian signatures,
• Operator product expansions and identification of physical excitations.

Manifest positivity of $\rho$ ensures that such decompositions are physically meaningful, encoding unitarity, causality, and analyticity, and are compatible with perturbative and nonperturbative treatments.

6. Applications and Computational Implementations

• Glueball and hadron spectroscopy: Extraction from lattice QCD two-point functions using inversion of the Laplace transform, regularized and enforced to yield positive densities, providing simultaneous access to ground and excited spectra (Dudal et al., 2021).
• Curved-space quantum field theory: Explicit harmonic decomposition of correlators and loop diagrams, analytic continuation between AdS, dS, and flat backgrounds, revealing thresholds, discrete contributions, and bound states (Loparco et al., 2023, Moschella, 21 Dec 2025, Altshuler, 10 Aug 2025).
• Form-factor models in quantum gravity: Spectral positivity and smooth interpolations between local and nonlocal effective field theories (Briscese et al., 22 May 2024).
• Lorentz-violating theories: Modified but still positive-definite spectral decompositions respecting new observer invariants (Potting, 2011).
• Reflection positivity diagnostics: Lattice reconstructions directly test for pathologies (e.g., negative spectral density) as in overlap or non-nearest-neighbor models (Usui, 2012).

7. Outlook and Theoretical Significance

Källén-Lehmann superpositions remain a cornerstone of modern field theory. They provide a bridge from microscopic Lagrangian or Hamiltonian dynamics to observable spectral features and constrain viable models through positivity, sum rules, and analytic structure. Their generalizations accommodate emergent symmetries, topologies, and novel UV/IR dynamics, and their utility extends to nonperturbative analysis, finite-temperature theory, and the deep structure of quantum fields on arbitrary spacetimes. Fundamental advances in spectral inversion (analytic, numerical, and harmonic-analytic) continue to illuminate nontrivial spectra, anomalous dimensions, and dynamical properties in both high-energy and condensed-matter contexts.