Completing Dirac's Heisenberg-picture reformulation of quantum electrodynamics: rigorous q-number foundations, physical interpretation, gauge-invariant perturbation theory, covariant finiteness, and scope beyond one-electron order $e^2/\hbar c$

Dirac (1965) argued that in quantum electrodynamics (QED) the Heisenberg and Schrödinger pictures are not equivalent — the Hamiltonian that yields sensible Heisenberg field equations admits no Schrödinger-picture solution, not even for the vacuum — and he reformulated QED entirely in the Heisenberg picture using noncommuting 'q-numbers' more general than Hilbert-space operators, thereby eliminating the divergent vacuum-to-vacuum terms of the usual expansion. He recovered the one-electron anomalous magnetic moment and Lamb shift at order $e^2/\hbar c$, but left the program openly incomplete: no mathematical characterization of q-numbers (no limits, differentiation, or integration), only a vague 'intensity' interpretation of the conserved q-number $K$ acting on the time-dependent vacuum ket $|0_t\rangle$, a perturbation expansion built on a non-gauge-invariant small interaction that he said would fail for two-electron problems, mass and charge shifts made finite only by a Lorentz-violating cutoff, and no validation beyond one-electron order $e^2/\hbar c$. The problem is to complete this reformulation into a logically closed theory — or to prove rigorously that no such completion exists. The reformulation remains open: its concerns re-emerge in modern algebraic quantum field theory, whose state-of-the-art review identifies as its remaining open problem whether the nonperturbatively constructed algebras admit physically interpretable states.

Background

Quantum mechanics admits two standard descriptions. In the Schrödinger picture the state of a system is a wave function (a vector in Hilbert space) evolving by Schrödinger's equation $i\hbar\,d\psi/dt=H\psi$ while the observables are fixed operators; in the Heisenberg picture the state is fixed and the observables ('dynamical variables') evolve by $i\hbar\,du/dt=uH-Hu$. For finitely many degrees of freedom the two pictures are connected by a unitary transformation and are equivalent. Quantum electrodynamics (QED) — the quantum theory of electrons, positrons, and the electromagnetic field — appeared to break this equivalence. As Dirac emphasized in 'Quantum Electrodynamics without Dead Wood' (Phys. Rev. 139, B684 (1965), an invited lecture at the January 1965 New York meeting of the American Physical Society), the QED Hamiltonian that gives sensible field equations in the Heisenberg picture admits no solutions of the Schrödinger equation, not even approximate ones and not even for the vacuum: the interaction drives the state vector out of Hilbert space. His conclusion, quoted verbatim: 'The Heisenberg picture is a good picture, the Schrödinger picture is a bad picture, and the two pictures are not equivalent.'

The standard perturbative treatment expands the Schrödinger solution in powers of the coupling; each term corresponds to a Feynman diagram (a graphical code for one term of the expansion). The diagrams describing vacuum-to-vacuum transitions — the vacuum fluctuating into particle pairs and back with no external particles — all carry divergent (infinite) integrals and are simply discarded, and the remaining divergences are absorbed by renormalization, i.e., by redefining the mass and charge parameters so that finite predictions emerge for observables such as the anomalous magnetic moment (the small deviation of the electron's magnetic moment from its Dirac-equation value) and the Lamb shift (the splitting of hydrogen levels absent from the Dirac equation). Dirac regarded discarding the vacuum terms and the renormalization recipe as 'working rules' that replace logical deduction — the 'dead wood' of his title.

His alternative was to work entirely in the Heisenberg picture. The dynamical variables are then 'q-numbers' — Dirac's term for noncommuting quantities — but of a kind more general than operators in Hilbert space, since the kets on which they act lie beyond Hilbert space; chief among these is $|0_t\rangle$, the vacuum at time $t$, defined by the condition that every absorption operator at time $t$ annihilates it. This ket is special to one Lorentz frame, satisfies no Schrödinger equation, and there is no universal vacuum state. In this picture nothing that starts as zero is ever generated, so the vacuum-to-vacuum divergences never appear. A soluble model Hamiltonian, quadratic in an infinite family of fermion emission and absorption operators, exhibits exactly this behavior: the Schrödinger solution is knocked out of Hilbert space, while the Heisenberg equations of motion have unproblematic solutions. Physical states are represented by conserved q-numbers $K$ satisfying

$i\hbar\,\frac{dK}{dt}=i\hbar\,\frac{\partial K}{\partial t}+KH-HK=0,$

and observables at time $t$ are extracted from $K|0_t\rangle$ by normal ordering (arranging all emission operators to the left of all absorption operators, so that only terms without absorption operators survive on $|0_t\rangle$); the squared moduli of the coefficients are postulated to be 'intensities' — Dirac's deliberately weaker substitute for probabilities, since these quantities cannot be normalized. There is 'no determinism' connecting $K|0_t\rangle$ at different times: its value at time $t$ does not determine its value at another time. Perturbation theory is performed on a transformed $K_t$ equation in an interaction-representation-like scheme, and applied — with static external fields treated as c-number potentials inside $H_0$ rather than in the perturbation — to the one-electron anomalous magnetic moment and Lamb shift at order $e^2/\hbar c$ (the fine-structure constant $\alpha$ in Gaussian units), reproducing the usual results. To make the renormalized mass and charge shifts $\delta m$ and $\delta e$ themselves finite and small, a high-energy cutoff must be imposed, sacrificing exact Lorentz invariance — a price Dirac argued is tolerable only because QED neglects other particles (mesons, neutrinos) and is anyway incomplete at high energies.

The paper states its own gaps, which define this problem: the precise mathematical nature of the q-numbers is unknown, and limits, differentiation, and integration cannot be defined for them; the interpretive postulates are 'rather vague'; the perturbation parameter $\int A_\mu j^\mu d^3x$ is not gauge-invariant (gauge invariance being the invariance of physics under the redundancy $A_\mu\to A_\mu+\partial_\mu\chi$ of the electromagnetic potentials), so the scheme 'would not work for two-electron problems in which the Coulomb force is important'; the cutoff breaks Lorentz invariance; and the whole apparatus has been checked only at one-electron order $e^2/\hbar c$.

Haag's theorem (the rigorous result that the interaction picture, and with it the unitary equivalence of free and interacting relativistic field theories, cannot exist) vindicates the letter of Dirac's inequivalence claim; algebraic quantum field theory and its perturbative branch treat observables as elements of operator algebras with states as positive linear functionals — structurally a Heisenberg picture — while the light-front Hamiltonian and gauge-invariant 'dressing' programs inherit specific pieces of his agenda; and interpretive work on the Heisenberg picture has revived in the 2020s foundations literature.

Problem Statement

Complete Dirac's Heisenberg-picture formulation of quantum electrodynamics into a logically closed theory — or prove rigorously that no such completion exists. The starting framework is Dirac (1965): quantum electrodynamics written in the Heisenberg picture, the description in which the dynamical variables rather than the state vector carry the time dependence. Its dynamical variables are q-numbers, noncommuting quantities more general than Hilbert-space operators; a physical state is represented by a q-number $K$ that is a constant of the motion, satisfying $i\hbar\,dK/dt=i\hbar\,\partial K/\partial t+KH-HK=0$ with $H$ the QED Hamiltonian; what is observable at time $t$ is determined by the product $K|0_t\rangle$, where $|0_t\rangle$ is the ket annihilated by every absorption (i.e. particle-annihilation) operator at time $t$ (the vacuum at time $t$ in a given Lorentz frame); and the intensity of an observed configuration is the squared modulus of the coefficient of the corresponding emission (i.e. particle-creation) operators in the expansion of $K|0_t\rangle$ — only emission operators survive on $|0_t\rangle$ — which replaces a probability because these quantities cannot be normalized. The validated content is the one-electron anomalous magnetic moment (the deviation of the electron's magnetic moment from its Dirac-equation value) and Lamb shift (the splitting of hydrogen levels absent from the Dirac equation) at order $e^2/\hbar c$ (the fine-structure constant in Gaussian units), with logarithmic mass and charge shifts $\delta m$ and $\delta e$ made finite only by a high-energy cutoff that breaks Lorentz invariance. The completion must jointly deliver the following five components, which restate the paper's own stated gaps as the single objective of this problem — logical completeness of the theory — rather than five independent problems (each component supplies a premise or a check the others need: the calculus of (1) makes the equations and expansions of (3)-(5) well-defined, the interpretation of (2) is what the calculations of (5) are calculations of, and (3)-(4) are the invariance conditions any physical prediction of the framework must satisfy):

(1) a precise mathematical characterization of the q-numbers, including well-defined limits, differentiation, and integration, rigorous enough to support the field equations and the perturbation expansions;

(2) a physically definite interpretation replacing the provisional intensity postulates — including a definite notion of intensity and the dynamical relation between $K|0_t\rangle$ at different times — that does not presuppose a Schrödinger-picture state vector (the Schrödinger picture being the alternative description in which the state vector evolves and the observables are fixed);

(3) a perturbation theory that is gauge-invariant order by order — invariant under the electromagnetic gauge transformation $A_\mu\to A_\mu+\partial_\mu\chi$ of the four-potential — not based on treating the non-gauge-invariant interaction $\int A_\mu j^\mu d^3x$ ($A_\mu$ the four-potential, $j^\mu$ the charge-current density) as small, and that works at least for two-electron problems in which the Coulomb (electrostatic) force is essential;

(4) finite mass and charge shifts compatible with Lorentz invariance — without an ad hoc non-covariant cutoff — or a specified embedding of the cutoff into a more complete theory;

(5) established calculational control and physical interpretation beyond the validated one-electron order-$e^2/\hbar c$ domain, including at least the two-electron problems in which the Coulomb force is essential that the paper's own analysis places beyond the present scheme.

A rigorous demonstration that no completion of this formulation can exist (for instance, that no state structure over the q-number algebra can satisfy the interpretive postulates together with the field commutation relations and gauge constraints) answers the problem in the negative.

References

1. P. A. M. Dirac, 'Quantum Electrodynamics without Dead Wood', Physical Review 139 (3B), B684-B690 (1965). DOI: 10.1103/PhysRev.139.B684, https://doi.org/10.1103/PhysRev.139.B684
2. P. A. M. Dirac, 'Gauge-Invariant Formulation of Quantum Electrodynamics', Canadian Journal of Physics 33, 650-660 (1955). DOI: 10.1139/p55-081, https://doi.org/10.1139/p55-081
3. A. Katz, 'Is the Heisenberg picture better than the Schrödinger picture?', Il Nuovo Cimento A (Series 10) 44 (4), 871-877 (1966). DOI: 10.1007/BF02719334, https://doi.org/10.1007/BF02719334
4. John Earman and Doreen Fraser, 'Haag's Theorem and its Implications for the Foundations of Quantum Field Theory', Erkenntnis 64, 305-344 (2006). DOI: 10.1007/s10670-005-5814-y, https://doi.org/10.1007/s10670-005-5814-y
5. Thomas Pashby, 'Time and quantum theory: A history and a prospectus', Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 52, 24-38 (2015). DOI: 10.1016/j.shpsb.2015.03.002, https://doi.org/10.1016/j.shpsb.2015.03.002
6. Stanisław D. Głazek, 'Fermion mass mixing and vacuum triviality in the renormalization group procedure for effective particles', Physical Review D 87, 125032 (2013). DOI: 10.1103/PhysRevD.87.125032, https://doi.org/10.1103/PhysRevD.87.125032, preprint arXiv:1305.3702, https://arxiv.org/abs/1305.3702
7. Samuel Kuypers and David Deutsch, 'Everettian relative states in the Heisenberg picture', Proceedings of the Royal Society A 477, 20200783 (2021). DOI: 10.1098/rspa.2020.0783, https://arxiv.org/abs/2008.02328
8. Chiara Marletto, Nicetu Tibau Vidal, and Vlatko Vedral, 'Interference in the Heisenberg picture of quantum field theory, local elements of reality, and fermions', Physical Review D 104, 065013 (2021). DOI: 10.1103/PhysRevD.104.065013, https://doi.org/10.1103/PhysRevD.104.065013
9. Charles Alexandre Bédard, 'Realism and the Inequivalence of the Two Quantum Pictures', arXiv:2510.02138 (2025). arXiv ID: 2510.02138, https://arxiv.org/abs/2510.02138
10. B. J. Hiley and G. Dennis, 'Dirac, Bohm and the Algebraic Approach', arXiv:1901.01979 (2019). arXiv ID: 1901.01979, https://arxiv.org/abs/1901.01979
11. Klaus Fredenhagen and Katarzyna Rejzner, 'Perturbative algebraic quantum field theory', arXiv:1208.1428 (2012). arXiv ID: 1208.1428, https://arxiv.org/abs/1208.1428; published as a book chapter, pp. 17-55 (2015), DOI: 10.1007/978-3-319-09949-1_2, https://doi.org/10.1007/978-3-319-09949-1_2
12. Romeo Brunetti, Klaus Fredenhagen, and Kasia Rejzner, 'Perturbative algebraic quantum field theory and beyond', arXiv:2512.14227 (2025). arXiv ID: 2512.14227, https://arxiv.org/abs/2512.14227; published version in the Encyclopedia of Mathematical Physics (2nd ed.), pp. 464-476 (2025), DOI: 10.1016/b978-0-323-95703-8.00100-2, https://doi.org/10.1016/b978-0-323-95703-8.00100-2
13. Katarzyna Rejzner, 'Remarks on Local Symmetry Invariance in Perturbative Algebraic Quantum Field Theory', Annales Henri Poincaré 16, 205-238 (2015); preprint arXiv:1301.7037 (2013). DOI: 10.1007/s00023-014-0312-x, https://doi.org/10.1007/s00023-014-0312-x, arXiv ID: 1301.7037, https://arxiv.org/abs/1301.7037
14. P. Berghofer and J. François, 'Dressing vs. Fixing: On How to Extract and Interpret Gauge-Invariant Content', Foundations of Physics 54, article 72 (2024). DOI: 10.1007/s10701-024-00809-y, https://doi.org/10.1007/s10701-024-00809-y