In 1955 Yoichiro Nambu showed that the Green's functions of an interacting relativistic quantum field theory (treated up to four field operators, with the symmetric pseudoscalar meson-nucleon theory — the charge-symmetric Yukawa model whose parity-odd coupling of the pseudoscalar pion isospin triplet (the charged and neutral pions) to nucleons made it the standard idealization of the pion-nucleon interaction — as prototype) admit parametric spectral representations, and he translated the operator equations of motion into coupled, nonlinear integral equations for the spectral functions in manifestly renormalized form: only observed masses and coupling constants appear, and causality - reconstruction of dispersive parts from absorptive parts, with asymptotic boundary normalization - replaces the usual subtraction procedure. Nambu himself stated that the framework lacked a mathematical foundation: it is not known whether the coupled equations admit physically acceptable solutions at all (existence, uniqueness, analytic properties); whether the causality-based renormalization is completely equivalent to conventional subtractive renormalization beyond lowest order; how to prevent approximate solutions from developing spurious causality-violating singularities; and how the formulation extends to Green's functions of all orders and to bound states, in light of Källén's observation that the renormalization constants cannot all be finite in a nontrivial theory. Nambu's own 1957 sequel supplied the all-orders parametric representations only perturbatively; the equations of motion for all n-point functions form the Dyson-Schwinger hierarchy, whose rigorous solvability remains unresolved; constructive quantum field theory has constructed interacting models only in two and three spacetime dimensions, with the four-dimensional construction problem open (a Clay Millennium Problem in its Yang-Mills form); and four-dimensional triviality results (rigorous for scalar models) show the existence question can also have a negative answer. The surviving open core is the nonperturbative solvability question for the spectral-function system of an interacting 3+1-dimensional theory, together with its satellite questions in nonperturbative form.

Background

In quantum field theory (QFT), the Green's functions are vacuum expectation values of time-ordered products of Heisenberg field operators; the Heisenberg picture lets the interacting dynamics act on the operators themselves, and time-ordering arranges them chronologically. These functions encode the propagation, interaction and scattering of the observed quanta, and a nonperturbative handle on them would bypass the limitations of perturbation expansion in the coupling. For the two-point function, a general representation independent of the details of the interaction was established before 1955 by Källén, by Lehmann, and by Gell-Mann and Low: Lorentz invariance plus completeness of positive-energy states force the propagator to be an integral over a real, nonnegative spectral weight divided by invariant energy denominators - the Källén-Lehmann or spectral representation.

Nambu's 1955 paper extended this structure to three- and four-point functions (Green's functions of up to four field operators). His prototype was the symmetric pseudoscalar meson-nucleon theory - the charge-symmetric Yukawa model in which the pseudoscalar pion isospin triplet (the charged and neutral pions) couples to the nucleon field through a parity-odd ('pseudoscalar') interaction, the standard idealization of the pion-nucleon force of the time. Each Fourier transform is expressed as a parametric integral over real, scalar weight functions - the spectral functions - with products of energy denominators whose thresholds coincide with the energies at which real intermediate processes become possible; stable particles appear as point spectra, i.e., delta-function contributions to the spectral functions at the particle masses. Differentiating the Heisenberg equations of motion yields an infinite chain of relations between Green's functions of different order (known today as the Dyson-Schwinger equations); substituting the spectral representations turns this chain into coupled, nonlinear integral equations for the spectral functions.

Renormalization - the removal of ultraviolet divergences - is conventionally performed by subtraction schemes that introduce unobservable bare masses, bare couplings, and infinite renormalization constants. Nambu instead observed that the absorptive (imaginary) parts of the Green's functions are finite, since they arise only from real energy-conserving processes, and that causality - the field-theoretic analogue of the Kramers-Kronig relation between the real and imaginary parts of a response function, ultimately expressing commutativity of observables at spacelike separation (microscopic causality) - then reconstructs the dispersive (real) parts. Boundary normalization at asymptotic times fixes the observed masses and couplings, and no bare quantity or infinite constant enters the final equations: they are manifestly renormalized. Physically acceptable solutions must then satisfy a set of spectral conditions: reality and positivity of the weights, the correct point spectra and reaction thresholds, analyticity compatible with microscopic causality, and finite observed parameters.

Two clouds hung over the framework, both acknowledged in the paper. First, Källén (1954) had shown, via sum rules for the coupling-constant renormalization, that not all renormalization constants can be finite in a nontrivial theory; with Pauli he demonstrated rigorously in the Lee model - an exactly solvable toy renormalizable field theory introduced by T. D. Lee in 1954, in which some renormalization constants can be computed in closed form - that a finite constant with nontrivial coupling forces negative-norm ghost states, the ancestor of today's Landau-pole problem (the divergence of the perturbatively defined effective coupling at a finite ultraviolet momentum scale) and of the triviality question, whether a nontrivial continuum interacting theory exists at all in four spacetime dimensions. Second, the mathematical analysis of the coupled equations themselves - whether they admit any solution meeting the spectral conditions - was beyond reach: constructing such a solution would amount to constructing the interacting quantum field theory itself, and constructive QFT in the rigorous sense (building fields satisfying the Wightman axioms: relativistic covariance, positive energy, unique vacuum, and locality) has succeeded only in two and three spacetime dimensions. In parallel, the Lehmann-Symanzik-Zimmermann (LSZ) formalism (1955) supplied the reduction of general n-point functions to scattering amplitudes, and the Dyson-Schwinger program treats the hierarchy of equations of motion as the definition of the theory, solving it in practice only with truncations whose artifacts (violations of positivity and of analyticity) are an active methodological concern to this day.

Problem Statement

The problem, as left open by Nambu (1955) and surviving today, is to determine whether the spectral-function formulation of an interacting relativistic quantum field theory is mathematically viable: whether the coupled, nonlinear, manifestly renormalized integral equations for the spectral functions actually admit physically acceptable solutions, and whether the framework's renormalization and approximation theory is sound. The common scientific objective uniting the components is the validation (or definitive refutation) of a nonperturbative solution theory for an interacting 3+1-dimensional theory - the symmetric pseudoscalar meson-nucleon theory (the charge-symmetric Yukawa model of the pseudoscalar pion isospin triplet (the charged and neutral pions) coupled to nucleons through a parity-odd, 'pseudoscalar' interaction) as prototype - in which the spectral functions are subject to the physical conditions of the source: reality; nonnegativity of spectral weights; correct stable-particle point spectra and reaction thresholds; analyticity of the reconstructed Green's functions compatible with microscopic causality; and observed masses and couplings entering only through the boundary conditions. Specifically:

(a) Core: prove or disprove that the full coupled system admits nontrivial physically acceptable solutions - establishing all of existence, uniqueness up to the boundary constants, and the analytic properties of the spectral functions - or prove that no nontrivial physically acceptable solution exists (leaving only the free/trivial theory), and determine under what conditions on the couplings each alternative holds. Here 'the full coupled system' means the finite closed system of the 1955 paper - the coupled integral equations for the two-, three- and four-point spectral functions together with their asymptotic boundary conditions - not a truncation of it; the extension beyond four-point functions is item (d).

(b) Establish whether the causality-based renormalization procedure (dispersive reconstruction of real parts from absorptive parts plus asymptotic boundary normalization) is fully equivalent to conventional subtractive renormalization. The lowest-order agreement is known, and the purely perturbative part of this question is regarded as settled by general renormalization theory; the open content is the nonperturbative statement.

(c) Provide both a principled prescription under which truncations or approximations of the self-energy and vertex spectral kernels cannot generate spurious singularities that violate causality, and a proof that the required analyticity and positivity properties are preserved.

(d) Complete the formulation nonperturbatively to Green's functions of all orders (the perturbative, per-Feynman-diagram parametric representations were supplied by Nambu's 1957 sequel), requiring the treatment of bound states as point spectra below the thresholds, and the rigorous incorporation of possible direct local four-body (contact) interactions, which enter as point-spectrum delta contributions at infinite spectral variable.

(e) Determine, within the spectral framework, the status of the renormalization constants in light of Källén's finiteness constraint; for this fermionic model class the question is open (four-dimensional triviality is rigorous only for scalar models).

The problem comprises the open points (a)-(e), posed jointly; in the paper's own framing, 'to make a complete study of all these points mentioned above, it would be necessary to extend our formulation to the Green's functions of all orders which are connected by the equations of motion'. Items (a), (c) and (d) belong to the problem whichever way its existence question is answered: (c) and (d) are methodological construction demands - respectively a prescription for truncation and approximation schemes, and the all-orders nonperturbative formulation - whose content does not depend on whether a nonperturbative solution exists, so a proof that none exists leaves them standing. Items (b) and (e) belong to the problem only when the existence question is answered in favour of a nontrivial solution: with no nontrivial physically acceptable solution there is no interacting solution for the causality-versus-subtractive renormalization equivalence (b) to relate, and the status of the renormalization constants (e) is then that of the trivial theory. The finite four-point existence question (a) is therefore not the whole of the problem. The framework is thereby either furnished with a genuine nonperturbative solution theory for the theory or proved not to admit one.

References

  1. Yoichiro Nambu (1955). Structure of Green's Functions in Quantum Field Theory. Physical Review 100, 394-411. DOI 10.1103/PhysRev.100.394. https://doi.org/10.1103/PhysRev.100.394
  2. M. Gell-Mann, F. Low (1951). Bound States in Quantum Field Theory. Physical Review 84, 350-354. DOI 10.1103/PhysRev.84.350. https://doi.org/10.1103/PhysRev.84.350
  3. G. Källén (1952). On the definition of the Renormalization Constants in Quantum Electrodynamics. Helvetica Physica Acta 25, 417. https://inspirehep.net/literature/900589
  4. H. Lehmann (1954). Über Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder. Il Nuovo Cimento 11, 342-357. DOI 10.1007/BF02783624. https://doi.org/10.1007/BF02783624
  5. G. Källén (1954). The coupling constant in field theory. Il Nuovo Cimento 12, 217-225. DOI 10.1007/BF02781176. https://doi.org/10.1007/BF02781176
  6. T. D. Lee (1954). Some Special Examples in Renormalizable Field Theory. Physical Review 95, 1329-1334. DOI 10.1103/PhysRev.95.1329. https://doi.org/10.1103/PhysRev.95.1329
  7. H. Lehmann, K. Symanzik, W. Zimmermann (1955). Zur Formulierung quantisierter Feldtheorien. Il Nuovo Cimento 1, 205-225. DOI 10.1007/BF02731765. https://doi.org/10.1007/BF02731765
  8. G. Källén, W. Pauli (1955). On the Mathematical structure of T.D. Lee's model of a renormalizable field theory. Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser 30, No. 7, 1-23. https://inspirehep.net/literature/43934
  9. Y. Nambu (1957). Parametric representations of general Green's functions. Il Nuovo Cimento 6, 1064-1083. DOI 10.1007/BF02747390. https://doi.org/10.1007/BF02747390
  10. K. Johnson, M. Baker, R. Willey (1964). Self-Energy of the Electron. Physical Review 136, B1111-B1119. DOI 10.1103/PhysRev.136.B1111. https://doi.org/10.1103/PhysRev.136.B1111
  11. Y. Nambu (1964). Quasi-elementary massless bosons associated with the quantum electrodynamics of Johnson, Baker and Willey. Physics Letters 9, 214-216. DOI 10.1016/0031-9163(64)90150-7. https://doi.org/10.1016/0031-9163(64)90150-7
  12. Eugene R. Speer (1968). Analytic Renormalization. Journal of Mathematical Physics 9, 1404-1410. DOI 10.1063/1.1664729. https://doi.org/10.1063/1.1664729
  13. M. Baker, K. Johnson (1971). Asymptotic Form of the Electron Propagator and the Self-Mass of the Electron. Physical Review D 3, 2516-2526. DOI 10.1103/PhysRevD.3.2516. https://doi.org/10.1103/PhysRevD.3.2516
  14. M. Aizenman (1982). Geometric analysis of phi^4 fields and Ising models. Parts I and II. Communications in Mathematical Physics 86, 1-48. DOI 10.1007/BF01205659. https://doi.org/10.1007/BF01205659
  15. J. Fröhlich (1982). On the triviality of lambda phi^4 theories in d dimensions and the approach to the critical point in d > 4 dimensions. Nuclear Physics B 200, 281-296. DOI 10.1016/0550-3213(82)90088-8. https://doi.org/10.1016/0550-3213(82)90088-8
  16. David J. E. Callaway (1988). Triviality pursuit: Can elementary scalar particles exist? Physics Reports 167, 241-320. DOI 10.1016/0370-1573(88)90008-7. https://doi.org/10.1016/0370-1573(88)90008-7
  17. C. D. Roberts, A. G. Williams (1994). Dyson-Schwinger equations and their application to hadronic physics. Progress in Particle and Nuclear Physics 33, 477-575. DOI 10.1016/0146-6410(94)90049-3. https://doi.org/10.1016/0146-6410(94)90049-3 (preprint arXiv:hep-ph/9403224, https://arxiv.org/abs/hep-ph/9403224)
  18. R. Alkofer, W. Detmold, C. S. Fischer, P. Maris (2005). Analytic structure of the gluon and quark propagators in Landau gauge QCD. Nuclear Physics B Proceedings Supplement 141, 122-127. DOI 10.1016/j.nuclphysbps.2004.12.019. https://doi.org/10.1016/j.nuclphysbps.2004.12.019 (preprint arXiv:hep-ph/0309078, https://arxiv.org/abs/hep-ph/0309078)
  19. Christian Brouder (2009). The Structure of Green Functions in Quantum Field Theory with a General State. In: B. Fauser, J. Tolksdorf, E. Zeidler (eds.), Quantum Field Theory, pp. 163-175. Birkhäuser Basel. DOI 10.1007/978-3-7643-8736-5_10. https://doi.org/10.1007/978-3-7643-8736-5_10 (preprint arXiv:0710.5652, https://arxiv.org/abs/0710.5652)
  20. Stephen J. Summers (2012). A Perspective on Constructive Quantum Field Theory. arXiv:1203.3991. https://arxiv.org/abs/1203.3991
  21. E. L. Solis, C. S. R. Costa, V. V. Luiz, G. Krein (2019). Quark propagator in Minkowski space. arXiv:1905.08710. https://arxiv.org/abs/1905.08710
  22. Tobias Frederico, Dyana C. Duarte, Wayne de Paula, Emanuel Ydrefors, Shaoyang Jia, Pieter Maris (2019). Towards Minkowski space solutions of Dyson-Schwinger Equations through un-Wick rotation. arXiv:1905.00703. https://arxiv.org/abs/1905.00703
  23. Michael Aizenman, Hugo Duminil-Copin (2021). Marginal triviality of the scaling limits of critical 4D Ising and phi^4 models. Annals of Mathematics 194, 163-231. arXiv:1912.07973. https://arxiv.org/abs/1912.07973