Completing Nambu's general-order spectral representation of Green's functions: existence, canonical uniqueness, and constructive determination of the real spectral functions in interacting quantum field theory

A 1956 paper of Yoichiro Nambu (Phys. Rev. 101, 459) proposed representing the exact Green's functions of relativistic quantum field theory — vacuum expectation values of time-ordered products of arbitrarily many Heisenberg fields — as parametric integrals over invariant energy denominators with real spectral functions, and showed consistency with the renormalized equations of motion for neutral scalar fields. The paper explicitly left open: the non-uniqueness of the spectral-function decomposition; the extension to spinor and tensorial fields; the systematic incorporation of the polynomial numerators required by renormalized self-interactions such as $\lambda\varphi^4$; and a constructive scheme for obtaining the spectral functions of an interacting theory. The subsequent literature reconstructs the fate of the program: Nakanishi's perturbation-theory integral representation settled existence and uniqueness for scalar, non-derivative theories to all orders in perturbation theory, and modern work solves Bethe–Salpeter and gap equations (the equations determining the fully dressed propagators) in Minkowski space via Nakanishi-type representations — but only within ladder/rainbow-type truncations, while the all-order representation for nonzero spin or derivative couplings has, per a 2006 review, never been proved, and no non-perturbative constructive determination or refuting counterexample exists. The surviving open core is the completion of Nambu's program for general interacting renormalizable theories.

Background

In relativistic quantum field theory, a Green's function is the vacuum expectation value of a time-ordered product of Heisenberg-picture field operators, for example $G(x,y)=\langle 0|T\varphi(x)\varphi(y)|0\rangle$; in momentum space these functions are the building blocks from which scattering amplitudes are extracted. For the two-point function of a scalar field, the Källén–Lehmann representation (Lehmann, 1954) writes the exact propagator as an integral over free propagators,

$G(p)=\int ds\,\frac{\rho(s)}{p^2-s+i\epsilon}$

with a real spectral density $\rho(s)$ that is non-negative when the theory has a positive Hilbert-space metric. The representation encodes causality and spectrum: the imaginary (absorptive) part of $G$ is concentrated where $p^2$ equals the squared mass of a state the field can create — a real process.

In two 1955–56 papers, Yoichiro Nambu asked what replaces the Källén–Lehmann representation for Green's functions of arbitrary order. In the second paper, 'Structure of Green's Functions in Quantum Field Theory. II' (Phys. Rev. 101, 459 (1956)), he proposed that a general-order Green's function of a renormalizable theory (one whose ultraviolet divergences can be absorbed into finitely many redefined parameters) can be written as a sum of parametric integrals whose integrand is a product of invariant energy denominators $1/(m^2+\sigma+i\epsilon)$ — where $m$ ranges over partial sums of the external four-momenta and $\sigma$ over auxiliary mass-squared variables — multiplied by real spectral functions $\nu$ of the remaining variables. A maximal set of such partial sums (at most $N-1$ for an $N$-point function) characterizes a set of real processes that can occur simultaneously; the $\nu$'s are defined as the absorptive parts of the Green's function associated with those sets. For neutral scalar fields Nambu showed that the renormalized equations of motion, which relate Green's functions of different order, are converted into finite, 'manifestly renormalized' relations among the $\nu$'s, so the representation contains no inherent infinities.

The paper itself lists what it does not settle: (i) the $\nu$-decomposition is not unique (different $\nu$'s can generate effectively the same product of denominators, and a maximal set of partial sums can sit inside different diagrams), and no canonical resolution is given — Nambu remarks only that non-uniqueness of this nature 'will not be physically significant'; (ii) only a brief sketch (Section 5) treats Dirac spinor fields, with no complete derivation for general tensorial field content; (iii) a Note added in proof records that a renormalized self-interaction of $\lambda\varphi^4$ type forces an energy denominator $(p_1+p_2)^2+\sigma$ to be accompanied by a polynomial numerator $(p_1+p_2)^2+u$, and withdraws a related statement of Part I, without incorporating the modification systematically; (iv) no constructive algorithm determines the $\nu$'s of an interacting theory — the paper's own consistency check of the representation against the equations of motion is, by its own footnote, perturbation-theoretic in character.

The subsequent literature transformed this program in two stages. First, Noboru Nakanishi developed a graph-theoretic Perturbation Theory Integral Representation (PTIR), systematized in his 1971 monograph: for scalar theories with non-derivative couplings, every $n$-point amplitude admits, to all orders of perturbation theory, an integral representation over real weight functions with a fixed denominator structure, and the weight function is unique for unrenormalized amplitudes (the uniqueness theorem); renormalization enters only as scheme-dependent subtraction polynomials. Second, from the mid-1990s the representation became a working computational tool: the Bethe–Salpeter equation (the bound-state analog of the equations of motion for a two-body amplitude) and later the Schwinger–Dyson equations (SDEs — the tower of exact equations of motion for all Green's functions) have been solved directly in Minkowski space — without a Wick rotation, the standard analytic continuation to imaginary energies that removes singular denominators at the price of losing direct access to timelike physics — by expanding amplitudes in Nakanishi-type integral representations and solving for the real weight functions, often combined with light-front projection (projection onto a null plane, which removes singularities from one momentum component of the integration). These constructions are established and solved only within ladder/rainbow-type truncations (retaining only single-boson exchange in the kernel and its counterpart in propagators) of specific models, and the all-order existence and uniqueness for theories with nonzero spin or derivative couplings has, per the 2006 review of Šauli, 'never been proved'.

Problem Statement

Complete, or refute, Nambu's general-order spectral representation program for interacting relativistic quantum field theory beyond the settled scalar sector. Concretely, for a renormalizable theory with general field content — including spinor fields, vector fields, derivative couplings, and renormalized self-interactions of $\lambda\varphi^4$ type — establish the following two linked components.

(1) Existence and canonical form: prove, or disprove by explicit counterexample, that the exact n-point Green's functions of such theories admit a Nambu–Nakanishi-type integral representation — a sum of parametric integrals over real weight functions with invariant energy denominators reflecting the possible simultaneous real processes, including the polynomial numerators the interactions require, generalizing the settled scalar non-derivative PTIR template documented in Nakanishi's 1971 monograph (Reference 4) — establishing at minimum its all-orders perturbative validity together with a uniqueness (canonical-form) theorem that either fixes the weight functions or precisely characterizes the residual non-uniqueness, and determining its non-perturbative status (validity beyond any fixed order, or a precise obstruction).

(2) Constructive determination: provide a scheme that actually determines the real spectral weight functions of an interacting theory beyond the settled perturbative scalar sector directly from the renormalized equations of motion (the Schwinger–Dyson hierarchy), not confined to an uncontrolled ladder/rainbow-type truncation, with the mode and locus of the demonstration left open; or demonstrate a principled obstruction (forced complex weights, unremovable non-uniqueness, or a positivity/complex-singularity obstruction) showing why the program cannot be completed.

The two components are jointly posed: a positive resolution must establish both the existence and canonical-form results of component (1) and the constructive determination or obstruction of component (2), and the target class must cover the general field content named in the scope sentence above — spinor fields, vector fields, derivative couplings, and renormalized self-interactions of $\lambda\varphi^4$ type — as the required minimum generality rather than as optional examples. Refuting the existence statement of component (1) also answers the constructive question of component (2), since a representation that does not exist admits no determination of its weight functions.

References

1. Y. Nambu, 'Structure of Green's Functions in Quantum Field Theory. II', Physical Review 101, 459–467 (1956). DOI: 10.1103/PhysRev.101.459, https://doi.org/10.1103/PhysRev.101.459. Source paper.
2. Y. Nambu, 'Structure of Green's Functions in Quantum Field Theory', Physical Review 100, 394–411 (1955). DOI: 10.1103/PhysRev.100.394, https://doi.org/10.1103/PhysRev.100.394. Part I.
3. Y. Nambu, 'Parametric representations of general Green's functions', Il Nuovo Cimento 6, 1064–1083 (1957). DOI: 10.1007/BF02747390, https://doi.org/10.1007/BF02747390.
4. N. Nakanishi, Graph Theory and Feynman Integrals, Gordon and Breach, New York (1971). ISBN 9780677029504 (ISBN-10 0677029500); Library of Congress control number 71123483; https://lccn.loc.gov/71123483.
5. V. Šauli, 'Implications of Analyticity to the Solution of Schwinger-Dyson Equations in Minkowski Space', Few-Body Systems 39, 45–99 (2006). DOI: 10.1007/s00601-006-0156-0; arXiv:hep-ph/0412188; https://arxiv.org/abs/hep-ph/0412188.
6. K. Kusaka, K. Simpson, A. G. Williams, 'Solving the Bethe-Salpeter equation for bound states of scalar theories in Minkowski space', Physical Review D 56, 5071–5085 (1997). DOI: 10.1103/PhysRevD.56.5071; arXiv:hep-ph/9705298; https://arxiv.org/abs/hep-ph/9705298.
7. J. Carbonell, V. A. Karmanov, 'Solutions of the Bethe-Salpeter Equation in Minkowski Space and Applications to Electromagnetic Form Factors', Few-Body Systems 49, 205–222 (2010). DOI: 10.1007/s00601-010-0133-5; arXiv:1012.0246; https://arxiv.org/abs/1012.0246.
8. T. Frederico, G. Salmè, M. Viviani, 'Nakanishi representation onto the null plane and the solution of the Bethe-Salpeter equation', Acta Physica Polonica B Supplementum 6, 303 (2013). DOI: 10.5506/APhysPolBSupp.6.303, https://doi.org/10.5506/APhysPolBSupp.6.303; article page https://www.actaphys.uj.edu.pl/S/6/1/303.
9. W. de Paula, T. Frederico, G. Salmè, M. Viviani, 'Advances in solving the two-fermion homogeneous Bethe-Salpeter equation in Minkowski space', Physical Review D 94, 071901(R) (2016). DOI: 10.1103/PhysRevD.94.071901; arXiv:1609.00868; https://arxiv.org/abs/1609.00868.
10. G. Salmè, W. de Paula, T. Frederico, M. Viviani, 'Two-Fermion Bethe-Salpeter Equation in Minkowski Space: The Nakanishi Way', Few Body Systems 58, 118 (2017). DOI: 10.1007/s00601-017-1286-2; arXiv:1703.07127; https://arxiv.org/abs/1703.07127.
11. C. Mezrag, G. Salmè, 'Fermion and photon gap-equations in Minkowski space within the Nakanishi integral representation method', The European Physical Journal C 81, 34 (2021). DOI: 10.1140/epjc/s10052-020-08806-x; arXiv:2006.15947; https://arxiv.org/abs/2006.15947.
12. S. Jia, 'Direct solution of Minkowski-space Bethe-Salpeter equation in the massive Wick-Cutkosky model', Physical Review D 109, 036020 (2024). DOI: 10.1103/PhysRevD.109.036020; arXiv:2312.08698; https://arxiv.org/abs/2312.08698.
13. S. Jia, 'Nakanishi integral representation of pseudoscalar fermion-antifermion bound states in the Bethe-Salpeter equation' (2024). arXiv:2402.01125, https://arxiv.org/abs/2402.01125.
14. L. D. Landau, 'On analytic properties of vertex parts in quantum field theory', Nuclear Physics 13, 181–192 (1959). DOI: 10.1016/0029-5582(59)90154-3, https://doi.org/10.1016/0029-5582(59)90154-3.
15. N. N. Bogoljubov, B. V. Medvedev, M. K. Polivanov, 'Probleme der Theorie der Dispersionsbeziehungen', Fortschritte der Physik 6, 169–245 (1958). DOI: 10.1002/PROP.19580060402, https://doi.org/10.1002/PROP.19580060402.
16. C. Brouder, 'The Structure of Green Functions in Quantum Field Theory with a General State', book chapter in Quantum Field Theory (Birkhäuser, Basel), 163–175 (2009). DOI: 10.1007/978-3-7643-8736-5_10, https://doi.org/10.1007/978-3-7643-8736-5_10.
17. H. Lehmann, 'Über Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder', Il Nuovo Cimento 11, 342–357 (1954). DOI: 10.1007/BF02783624, https://doi.org/10.1007/BF02783624.