Hypersurface independence of stationary states and the physical consistency of Heisenberg's 1950 non-Hamiltonian S-matrix formalism

In a 1950 paper, Werner Heisenberg derived stationary states of a relativistic quantum field theory directly from the S-matrix (the scattering matrix, which relates the free asymptotic states before and after a scattering process), using the unitarizing ansatz $S=T(T^*T)^{-1/2}$ and an everywhere-regular free-field commutation function, and left two questions unresolved: (i) whether the restrictions his differential-equation route imposes (a discrete, finite free-particle spectrum and interaction densities local in the fields at a single spacetime point) are physically justified relative to the more general integral-equation S-matrix formulation, and (ii) whether the stationary-state kernel is independent of the shape of the reference spacelike hypersurface in the non-Hamiltonian extension, where the evolution operators smear over regions of order a universal smallest length $l_0$ . No resolution of either question is known in the literature to date. The problem therefore asks to settle, within the 1950 formalism itself, the hypersurface (and foliation) independence of the stationary states and their energy-momentum eigenvalues in the non-Hamiltonian case, and to establish or refute the physical admissibility of the differential-equation restrictions, including the low-energy correspondence and the convergence of the second-order self-energy for explicit regularized commutation functions.

Background

In the late 1940s the ultraviolet divergences of relativistic quantum field theory appeared to many of its creators to be a structural defect of the Hamiltonian, field-equation formulation itself. Heisenberg had proposed in 1938 to connect the divergences with a universal smallest length $l_0$ (of order the proton radius), below which spacetime localization would cease to be operationally meaningful. In a companion 1950 paper, 'Zur Quantentheorie der Elementarteilchen' (Z. Naturforsch. A 5, 251 (1950)), he therefore proposed a new mathematical scheme: all elementary particles are represented by a single spinor field $\psi$ ; in the interaction representation (the intermediate picture of quantum dynamics in which the field operators evolve freely while the interaction drives the state vector) this field satisfies not the usual singular commutation function $S(x-x')$ (the free Pauli-Jordan function, a singular distribution on the light cone) but an everywhere-regular commutation function $S^R(x-x')=\sum_l c_l S_l(x-x')$ , a finite superposition of free commutation functions for masses $\kappa_l$ with coefficients $c_l$ of both signs; the theory is then completed by a relativistically invariant interaction energy of at least fourth degree in the spinor field. Regularity of $S^R$ is meant to remove all divergent expressions at the root, at the price that the field decomposition involves coefficients of mixed sign, so that the field $\bar\psi$ is no longer the Hermitian adjoint of $\psi$ but involves an independent imaginary unit $j$ ( $j^2=-1$ ), and the interaction density $H(x)$ built from $\psi,\bar\psi$ is in general non-Hermitian.

The paper under discussion here, 'Stationäre Zustände in der relativistischen Quantentheorie der Wellenfelder' (Z. Naturforsch. A 5, 367 (1950)), addresses a gap in that scheme: if only the S-matrix (the scattering matrix, relating the free asymptotic states before and after interaction) is a secure ingredient of a convergent theory, how can the stationary states (bound states, particle masses) that quantum mechanics normally obtains from a time-independent eigenvalue equation be extracted from scattering data? In the Hermitian case the answer is a covariant eigenvalue equation on a spacelike hypersurface $\sigma_0$ transverse to a timelike unit vector $\varepsilon_\mu$ , of the form $\{(J_\mu^0-J_\mu')\varepsilon_\mu+\int_{\sigma_0}d\sigma_\mu\varepsilon_\mu H(x)\}\Phi=0$ , where $J_\mu^0$ is the free energy-momentum operator and $J_\mu'$ the total energy-momentum eigenvalue, and a representation of the stationary state as an operator kernel $O$ built from free-field operators acting on the vacuum. In the non-Hermitian case the naive time-ordered evolution operator $T$ (the $P$ -ordered, i.e. chronologically ordered, exponential of the interaction) is non-unitary, and Heisenberg defines the physical S-matrix by the polar-decomposition ansatz $S=T(T^*T)^{-1/2}$ , which is unitary and invariant under $j\to-j$ ; the stationary-state equation then acquires an extra non-Hamiltonian term $VZV^{-1}$ , where $V$ is built from $T$ and $Z$ from $(V^*V)^{-1/2}$ . He also shows a correspondence limit (for processes with small energy transfer, i.e. energies $E\ll l_0^{-1}$ , the formalism reduces to the ordinary Hamiltonian one) and computes the second-order self-energy (the energy shift of a particle due to the interaction with its own field), where the Hermitian and anti-Hermitian parts combine into $\sum_k H_{kl}H_{lk}/(E_k-E_l)$ , claimed convergent because of $S^R$ .

Two methodological remarks are needed for a modern reader. First, the 1950 restrictions: to obtain a differential (rather than integral) stationary-state equation, the paper works with a discrete, finite free-particle spectrum and interaction densities that are local functions of the fields and finitely many derivatives at a single spacetime point; Heisenberg states that only further development of the theory can decide whether these limitations are acceptable. Second, in the non-Hamiltonian extension the evolution operators $V$ perform spacetime averages over regions of order $l_0$ , so the proof that the kernel $O$ (and the resulting eigenvalues) is independent of the shape of $\sigma_0$ cannot be transferred from the ordinary Hamiltonian case; the paper assumes the hypersurfaces are practically planar on the scale $l_0$ , with radii of curvature much larger than $l_0$ , rather than proving independence. The physical application of the mathematical results to actual elementary particles is explicitly deferred. Related background: the contemporaneous Tomonaga-Schwinger formalism describes states $\Psi[\sigma]$ on spacelike hypersurfaces via local densities $H(x)$ , and Freeman Dyson's 1949 identification of the Feynman rules (the diagrammatic prescription for computing scattering amplitudes) with that formalism and proof of perturbative renormalizability (the property that ultraviolet divergences can be absorbed into finitely many redefined parameters) — as reconstructed in Blum (2017) — made local Lorentz-invariant QFT calculable without a fundamental length; the general question of how bound states are encoded in scattering data was answered in that framework by analyticity of the scattering amplitude (bound states as poles below threshold), as reconstructed historically in Blum (2017). These developments concern the local theory and do not address the two unresolved questions inside Heisenberg's non-Hamiltonian formalism, which no later technical paper took up.

Problem Statement

Within the formalism of Heisenberg's 1950 papers — free field theory with the everywhere-regular commutation function $S^R(x-x')=\sum_l c_l S_l(x-x')$ (with coefficients $c_l$ of both signs and masses $\kappa_l$ satisfying the regularity moment conditions $\sum_l c_l \kappa_l^{\,n}=0$ for $n=0,1,2,3$ ), non-Hermitian interaction density $H(x)$ , the P-ordered (chronologically ordered) evolution operator $T$ , and unitary S-matrix defined by the polar decomposition $S=T(T^*T)^{-1/2}$ — settle the questions the 1950 stationary-states paper explicitly left open:

Hypersurface independence in the non-Hamiltonian case. In the Hermitian case the stationary-state kernel operator $O$ (through which a stationary state is represented on the reference spacelike hypersurface $\sigma_0$ ) is independent of the choice of $\sigma_0$ , so the total energy-momentum eigenvalues $J_\mu'$ are well defined. In the non-Hamiltonian extension the generalized stationary-state equation acquires the term $VZV^{-1}$ (with $V$ built from $T$ and $Z$ from $(V^*V)^{-1/2}$ ), and because the operators $V$ perform spacetime averages over regions of order the smallest length $l_0$ , the Hamiltonian-case proof of $\sigma_0$ -independence no longer transfers. Prove, or disprove by exhibiting a counterexample within the formalism, that in the non-Hamiltonian case the stationary states and their eigenvalues $J_\mu'$ are nevertheless independent of the shape of $\sigma_0$ (and of the foliation vector $\varepsilon_\mu$ , the timelike unit vector along which the family of spacelike hypersurfaces advances), at least in the regime the paper assumed — hypersurfaces practically planar on the scale of $l_0$ , with radii of curvature much larger than $l_0$ — replacing the paper's assumption by a proven statement with controlled error terms, or showing that the eigenvalues genuinely depend on the hypersurface. In the practically-planar regime (radii of curvature much larger than $l_0$ ) the independence claim is quantitative, fixing the regime and the error control explicitly — an explicit lower bound on the radius of curvature in units of $l_0$ , the deviation from planarity measured in a stated norm, and an explicit bound on the resulting remainder — and a counterexample within that same regime refutes it.
Physical admissibility of the differential-equation restrictions. The 1950 derivation of a differential stationary-state equation assumes a discrete, finite free-particle spectrum and interaction densities that are local functions of the fields and finitely many derivatives at a single spacetime point, and it states that only further development of the theory can decide whether these limitations are acceptable. Establish or refute the physical justification of these restrictions inside the formalism by settling each of the following three sub-questions. All three are required. They are independently refutable: a refutation of any one shows the restrictions are not physically justified in that respect while leaving the others open, and establishing all three establishes admissibility in the senses checked.

(a) Convergence of the second-order self-energy. For an explicitly specified regular commutation function $S^R(x-x')=\sum_l c_l S_l(x-x')$ of the kind defined above (the answer must name the coefficients $c_l$ , which are of both signs, and the masses $\kappa_l$ it uses; any one admissible choice suffices), determine whether the second-order self-energy $\sum_k H_{kl}H_{lk}/(E_k-E_l)$ actually converges. Here the index $k$ ranges over the continuum of intermediate states (the free eigenstates of $J_\mu^0$ ), the discrete and finite free-particle spectrum assumed for the differential-equation derivation being suspended for this sub-question; a resolving answer must establish convergence over that domain (with an explicit bound) or exhibit its divergence (the paper's regularity argument is schematic).

(b) Low-energy correspondence. Determine whether the correspondence limit to the ordinary Hamiltonian formalism holds for energies $E\ll l_0^{-1}$ with controlled corrections from the non-Hamiltonian term $VZV^{-1}$ .

(c) $j$ -invariance selection rules. Determine whether the selection rules implied by $j$ -invariance ( $j$ being the independent imaginary unit, $j^2=-1$ , introduced by the mixed-sign field decomposition; no transitions converting an odd number of particles of one sign sector into an odd number of the other) are consistent with an exactly unitary S-matrix and with the stationary states obtained in part 1, whether or not those states are hypersurface-independent.

The questions concern the 1950 formalism; the commutation function, density and S-matrix are those of the 1950 papers.

References

W. Heisenberg, 'Stationäre Zustände in der relativistischen Quantentheorie der Wellenfelder', Zeitschrift für Naturforschung A 5(7), 367-373 (1950). DOI: 10.1515/zna-1950-0702, https://doi.org/10.1515/zna-1950-0702 (the source paper)
W. Heisenberg, 'Zur Quantentheorie der Elementarteilchen', Zeitschrift für Naturforschung A 5(5), 251-259 (1950). DOI: 10.1515/zna-1950-0502, https://doi.org/10.1515/zna-1950-0502 (companion paper defining the regular commutation function, single spinor field, and interaction energy)
W. Heisenberg, 'Zur Frage der Kausalität in der Quantentheorie der Elementarteilchen', Zeitschrift für Naturforschung A 6(6), 281-284 (1951). DOI: 10.1515/zna-1951-0601, https://doi.org/10.1515/zna-1951-0601 (author's own continuation: acausal time-order deviations in the same S-matrix theory)
W. Heisenberg, 'On the mathematical frame of the theory of elementary particles', Communications on Pure and Applied Mathematics 4(1), 15-22 (1951). DOI: 10.1002/cpa.3160040103, https://doi.org/10.1002/cpa.3160040103 (English exposition of the program)
W. Heisenberg, 'Paradoxien des Zeitbegriffs in der Theorie der Elementarteilchen', in 'Festschrift zur Feier des Zweihundertjährigen Bestehens der Akademie der Wissenschaften in Göttingen' (Springer, Berlin, 1951), pp. 50-64. DOI: 10.1007/978-3-642-49913-5_3, https://doi.org/10.1007/978-3-642-49913-5_3 (author's continuation on time-order paradoxes)
S. Watanabe, 'Connection between fusion theory of bosons and nonlocal field theory', Physical Review 91(3), 771-772 (1953). DOI: 10.1103/PhysRev.91.771, https://doi.org/10.1103/PhysRev.91.771 (adjacent nonlocal-field-theory work)
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W. Blum, H.-P. Dürr, H. Rechenberg (eds.), 'Original Scientific Papers / Wissenschaftliche Originalarbeiten' (W. Heisenberg, Collected Works, Springer, 1993). ISBN 978-3-642-70079-8, https://doi.org/10.1007/978-3-642-70079-8 (collected-works reprint volume containing chapters 13-14, which reprint the 1951 papers)
A. S. Blum, 'The state is not abolished, it withers away: How quantum field theory became a theory of scattering', Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 60, 46-80 (2017). DOI: 10.1016/j.shpsb.2017.01.004, https://doi.org/10.1016/j.shpsb.2017.01.004 (historical reconstruction of the transition from stationary-state calculations to scattering-centered QFT)
P. Ruiz de Olano, J. D. Fraser, R. Gaudenzi, A. S. Blum, 'Taking approximations seriously: The cases of the Chew and Nambu-Jona-Lasinio models', Studies in History and Philosophy of Science 93, 82-95 (2022). DOI: 10.1016/j.shpsa.2022.02.013, https://doi.org/10.1016/j.shpsa.2022.02.013 (later historical scholarship)
J. T. Cushing, 'The Importance of Heisenberg's S-Matrix Program for the Theoretical High-Energy Physics of the 1950's', Centaurus 29(2), 110-149 (1986). DOI: 10.1111/j.1600-0498.1986.tb00886.x, https://doi.org/10.1111/j.1600-0498.1986.tb00886.x (standard retrospective historical study of the S-matrix program)
F. J. Dyson, 'The S Matrix in Quantum Electrodynamics', Physical Review 75(11), 1736-1755 (1949). DOI: 10.1103/PhysRev.75.1736, https://doi.org/10.1103/PhysRev.75.1736 (identifies the Feynman rules with the Tomonaga-Schwinger formalism and proves perturbative renormalizability of quantum electrodynamics)
W. Heisenberg, 'Doubts and Hopes in Quantum-Electrodynamics', in 'Scientific Review Papers, Talks, and Books / Wissenschaftliche Übersichtsartikel, Vorträge und Bücher' (W. Heisenberg, Collected Works, Springer, 1984), pp. 509-520. DOI: 10.1007/978-3-642-61742-3_40, https://doi.org/10.1007/978-3-642-61742-3_40 (retrospective essay)