Consistent infinite-component wave equations with hydrogenlike mass spectra: existence of a theory free of spacelike solutions, PCT asymmetry, spin-statistics loosening, and nonlocal currents

Nambu (1967) showed that relativistic wave equations with infinitely many components, built from a unitary infinite-dimensional representation of the dynamical group SO(4,2) combined with Dirac spinors, can reproduce discrete hydrogenlike mass spectra, but every explicit model was afflicted with unphysical features: redundant spacelike (imaginary-mass) solutions, missing negative-frequency solutions (breaking PCT symmetry), a loosened spin-statistics connection, indefinite energy or charge sign, and no established realization as a local quantum field theory rather than a mere S-matrix scaffold. Nambu left open at the end of that paper whether an infinite-component wave-equation theory in this sense can be constructed that simultaneously avoids all of these pathologies, or whether a general impossibility theorem rules it out. Later literature established a no-go theorem (Grodsky-Streater 1968) showing that no acceptable local infinite-component field theory exists under stated axioms, but its known escape routes are themselves unphysical features, and the Barut school contested the axioms; no construction satisfying all desiderata and no loophole-free impossibility theorem is known. The modern review literature treats the question as unresolved and the program as dormant, with string theory's infinite tower of finite-dimensional representations regarded as the only known consistent route to an infinite mass spectrum.

Background

A relativistic wave equation for a free particle is traditionally built on a finite-dimensional representation of the Lorentz group: the Dirac equation uses the four-dimensional spinor representation, the Klein-Gordon equation the trivial one. In 1932 Majorana instead wrote a wave equation based on an infinite-dimensional unitary representation of the Lorentz group. A unitary representation is one in which the Lorentz transformations act through operators that preserve a positive-definite norm; such representations are necessarily infinite-dimensional, and a single wave equation of this type describes an infinite tower of particle spins with different masses, which is why these are called infinite-component wave equations. The program was revived in the 1960s for hadron physics: with the proliferation of resonances (short-lived excited states of strongly interacting particles), an infinite tower of states in one equation was attractive, and it was hoped that the bilinear currents built from such fields could saturate the current-algebra sum rules of the day (consistency conditions derived from the postulated commutation relations among the electromagnetic and weak current densities).

The appropriate algebra turned out to be SO(4,2), the dynamical group of the hydrogen atom (the group generated by the angular-momentum and Runge-Lenz symmetries — the Runge-Lenz vector being the hidden conserved vector specific to the Coulomb potential — enlarged by ladder operators that connect the hydrogen energy levels). Barut and Kleinert and, independently, Fronsdal showed in 1967 that the exact SO(4) degeneracy structure of the bound hydrogen atom is organized by SO(4,2). Nambu's 1967 paper, the source of the present question, then constructed explicit relativistic models by taking the product of an infinite-dimensional unitary SO(4,2) representation (realized on bosonic oscillator ladders) with the finite Dirac spinor representation, and writing first- and second-order wave equations linear in four-vector generators $\Gamma_\mu$ of SO(4,2) and the four-momentum $p^\mu$. Diagonalization in the rest frame gives discrete hydrogenlike spectra of the form $m_N = m_0(1 - \mathrm{const}/N^2)^{1/2}$ with a principal quantum number $N$, and in the nonrelativistic limit the equations become exactly equivalent to the Schrödinger equation of the hydrogen atom. One model (Nambu's model No. 5, a second-order equation) even admits a charge density of definite sign, suggesting Fermi quantization with positive energy.

The catch, in Nambu's own words, is that this class of equations "still suffers from various diseases". First, the equations admit redundant or unphysical solutions, in particular solutions with spacelike four-momenta, i.e. $p_\mu p^\mu < 0$, corresponding to imaginary mass (tachyonic) continua. Second, many models lack negative-frequency (antiparticle) solutions, so the mass spectrum is asymmetric between positive and negative frequencies, spoiling the PCT theorem (the theorem asserting that any local, Lorentz-invariant quantum field theory is invariant under the combined particle-antiparticle conjugation, parity, and time reversal). Third, the sign of the energy or charge density is indefinite in most models. Fourth, the connection between spin and statistics appears loosened: fields of integer spin may have to be quantized with Fermi statistics and vice versa, as Fronsdal and Feldman and Matthews had pointed out. These features threaten local commutativity, the axiom that fields (anti)commute at spacelike separation, which underlies causality in quantum field theory.

The reason the standard theorems offer no protection is representation-theoretic: the PCT and spin-statistics theorems of axiomatic quantum field theory are proved for fields transforming under finite-dimensional representations of the Lorentz group, and their proofs require an analytic continuation of the Lorentz boost through imaginary angles; for unitary infinite-dimensional representations the relevant boost operator has poles at that angle, so both proofs break down. Indeed, Abers, Grodsky, and Norton exhibited explicit local covariant infinite-component field theories that violate PCT and the spin-statistics connection without contradicting any theorem. Grodsky and Streater then proved a no-go theorem: assuming covariance, a mass spectrum of timelike mass shells with finite degeneracy, an infinite-dimensional representation possessing a four-vector operator, and a completeness relation with polynomially bounded positive-energy projector, no such system of single-particle functions exists — every mass shell is forced to be infinitely degenerate, so no acceptable local infinite-component field theory or wave equation exists under those axioms. The theorem can be evaded in more than one way — for instance by allowing spacelike solutions into the spectrum or by accepting infinite mass-shell degeneracy. Nambu himself, at the end of his paper, asked what becomes of the spin-statistics connection, what the positive-negative asymmetry of the mass spectrum implies for P, C, and T invariance, whether the spacelike solutions are really dangerous, and whether these features imply a breakdown of local commutativity and causality between fields, with the possibility that the currents (bilinear forms of the fields) might still be causal; he also noted the lack of a clear-cut principle for choosing an equation, and that it was unknown whether such equations could be realized as a field theory at all or merely as an S-matrix theory (a formalism specifying only scattering amplitudes between asymptotic states, not local fields).

Problem Statement

Determine whether there exists a relativistic wave-equation theory in Nambu's infinite-component sense — that is, built on unitary infinite-dimensional representations of a dynamical group containing Lorentz four-vector generators (such as SO(4,2), possibly tensored with finite Dirac structure), producing a discrete hydrogenlike spectrum of timelike masses — that simultaneously satisfies all of the following, and either construct one or prove that none can exist:

1. Spectrum: the physical solutions form a discrete, correctly ordered, hydrogenlike spectrum of timelike masses (the degeneracy of each mass shell is left unconstrained), with no spacelike (imaginary-mass) solutions, or with a proof that any spacelike sector decouples from all observables.
2. PCT symmetry: the mass spectrum is symmetric between positive and negative frequencies and the interactions admit PCT invariance, so the theory does not violate the PCT theorem.
3. Statistics and positivity: the field admits a quantization with the standard spin-statistics connection and a positive-definite energy (or charge) on the physical subspace.
4. Locality and causality: the quantized fields, or at least the observable currents (bilinear forms of the fields), obey local commutativity, i.e. (anti)commutators vanish at spacelike separation.
5. Realization as a field theory: the construction is a complete Lagrangian field theory, not merely a generator of Born-approximation (first-order perturbative) vertices or an S-matrix, and the wave equation must be derived from a stated symmetry or representation-theoretic principle, not merely chosen to reproduce the Nambu mass formula; extending that principle to systems whose rest-frame symmetry group is not that of hydrogen, such as the harmonic oscillator with its SU(3) degeneracy group, is illustrative and not a required deliverable.

A proof that no such theory exists must rule out the known evasion classes of the Grodsky-Streater no-go theorem — equations with spacelike solution sectors, infinitely degenerate mass shells, nonlocal wave operators, and S-matrix-only or current-only realizations — which are exactly the loopholes through which every existing construction passes. This is the question Nambu left open at the end of his 1967 paper: whether the pathologies are incidental features of particular models or forced properties of the infinite-component approach, i.e. whether 'the possibility of constructing a theory that simultaneously avoids all these pathologies' can be realized.

References

1. Y. Nambu, Infinite-Component Wave Equations with Hydrogenlike Mass Spectra, Physical Review 160, 1171 (1967). DOI: 10.1103/PhysRev.160.1171, https://doi.org/10.1103/PhysRev.160.1171
2. E. Majorana, Teoria relativistica di particelle con momento intrinseco arbitrario, Il Nuovo Cimento 9, 335 (1932). DOI: 10.1007/BF02959557, https://doi.org/10.1007/BF02959557
3. A. O. Barut and H. Kleinert, Transition Probabilities of the Hydrogen Atom from Noncompact Dynamical Groups, Physical Review 156, 1541 (1967). DOI: 10.1103/PhysRev.156.1541, https://doi.org/10.1103/PhysRev.156.1541
4. C. Fronsdal, Infinite Multiplets and the Hydrogen Atom, Physical Review 156, 1665 (1967). DOI: 10.1103/PhysRev.156.1665, https://doi.org/10.1103/PhysRev.156.1665
5. G. Feldman and P. T. Matthews, Multimass Fields, Spin, and Statistics, Physical Review 154, 1241 (1967). DOI: 10.1103/PhysRev.154.1241, https://doi.org/10.1103/PhysRev.154.1241
6. E. Abers, I. T. Grodsky, and R. E. Norton, Diseases of Infinite-Component Field Theories, Physical Review 159, 1222 (1967). DOI: 10.1103/PhysRev.159.1222, https://doi.org/10.1103/PhysRev.159.1222
7. I. T. Grodsky and R. F. Streater, No-Go Theorem, Physical Review Letters 20, 695 (1968). DOI: 10.1103/PhysRevLett.20.695, https://doi.org/10.1103/PhysRevLett.20.695
8. G. Bisiacchi, G. Calucci, and C. Fronsdal, On the infinite component wave equations, Acta Physica Academiae Scientiarum Hungaricae 26, 21 (1969). DOI: 10.1007/BF03156798, https://doi.org/10.1007/BF03156798
9. A. O. Barut, The physical interpretation of spacelike and ghostlike solutions of infinite-multiplet wave equations, Lettere al Nuovo Cimento 1, 601 (1969). DOI: 10.1007/BF02753966, https://doi.org/10.1007/BF02753966
10. H. Bebié, F. Ghielmetti, V. Gorgé, and H. Leutwyler, Current Algebra and Infinite-Component Fields, Physical Review 177, 2133 (1969). DOI: 10.1103/PhysRev.177.2133, https://doi.org/10.1103/PhysRev.177.2133
11. R. Casalbuoni and G. Longhi, A nonlocal infinite-component wave equation with a physical mass spectrum, Il Nuovo Cimento A 70, 329 (1970). DOI: 10.1007/BF02725378, https://doi.org/10.1007/BF02725378
12. A. O. Barut and W. Rasmussen, The hydrogen atom as a relativistic elementary particle. I. The wave equation and mass formulae, Journal of Physics B: Atomic and Molecular Physics 6, 1695 (1973). DOI: 10.1088/0022-3700/6/9/015, https://doi.org/10.1088/0022-3700/6/9/015
13. H. Bacry and N. P. Chang, A causal ghost-free covariant harmonic oscillator model for bosons respecting CPT and all that, Physics Letters B 44, 286 (1973). DOI: 10.1016/0370-2693(73)90228-1, https://doi.org/10.1016/0370-2693(73)90228-1
14. O. Hara and S. Naka, An Example of Infinite-Component Wave Equation without Space-Like Solution, Progress of Theoretical Physics 53, 1194 (1975). DOI: 10.1143/ptp.53.1194, https://doi.org/10.1143/ptp.53.1194
15. A. O. Barut and J. Nagel, Interpretation of Space-Like Solutions of Infinite Component Wave Equations and Grodsky-Streater No-Go Theorem, Journal of Physics A: Mathematical and General 10, 1233 (1977). DOI: 10.1088/0305-4470/10/7/020, https://doi.org/10.1088/0305-4470/10/7/020
16. R. Casalbuoni, Majorana and the infinite component wave equations, PoS(EMC2006)004 (2007). DOI: 10.22323/1.037.0004, https://pos.sissa.it/037/004/. Also available as arXiv:hep-th/0610252, https://arxiv.org/abs/hep-th/0610252
17. P. A. Horvathy, M. S. Plyushchay, and M. Valenzuela, Bosonized supersymmetry from the Majorana-Dirac-Staunton theory and massive higher-spin fields (2007). arXiv:0710.1394, https://arxiv.org/abs/0710.1394
18. L. M. Slad, Some field-theoretical aspects of two types of the Poincare group representations (2014). arXiv:1401.0393, https://arxiv.org/abs/1401.0393
19. P. Schuster and N. Toro, A Gauge Field Theory of Continuous-Spin Particles (2013). arXiv:1302.3225, https://arxiv.org/abs/1302.3225