The unresolved mathematical foundations of Schwinger's Euclidean quantum electrodynamics: rigorous existence of the interacting theory (canonical eigenstates, indefinite-metric Lorentz-gauge operator formulation, and the functional-integral measure)

Julian Schwinger's paper "Euclidean Quantum Electrodynamics" (Physical Review 115, 721 (1959)) transcribed QED into a four-dimensional Euclidean metric and left five directions explicitly open: the class structure of Lorentz gauges; the existence and Hermiticity of the complementary field operators and their zero-eigenvalue states $|0\rangle$ and $|W\rangle$ for interacting fields; the consistency of indefinite-metric (Lorentz-gauge operator) quantization for interacting QED; the component-counting and internal-symmetry consequences of the real symmetric Euclidean Dirac matrices $\alpha_\mu$ (which force at least $8\times 8$ matrices and exclude the four-component Majorana field); and a mathematically rigorous construction of the functional-integral measure. Later literature substantially settled two directions: Schwinger's own sequel "Euclidean Gauge Transformation" (Physical Review 117, 1407 (1960)) and the general gauge-fixing framework on the gauge-class question, and the 1972-2011 'Euclidean spinors' literature on the fermion component counting. The remaining three directions merge into a single open problem, the constructive-QED existence problem: construct the interacting Maxwell-Dirac theory as a mathematically well-defined Euclidean quantum field theory - via the operator construction with Hermitian complementary fields and null-eigenvalue states, a consistent indefinite-metric Lorentz-gauge operator formulation, or a renormalized Euclidean functional-integral measure satisfying the Osterwalder-Schrader axioms - or prove that no nontrivial continuum limit exists (triviality). This problem remains open.

Background

In the 1950s the archetype of an interacting relativistic quantum field theory was quantum electrodynamics (QED), the theory of the electron (a Dirac spin-1/2 field $\psi$) coupled to the photon (the electromagnetic four-potential $A_\mu$). Julian Schwinger, one of QED's architects, developed in 1958-1959 a new organization of its Green's functions (vacuum expectation values of time-ordered products of fields, from which scattering amplitudes are computed): continue the time coordinate to imaginary values, $t\to -i x_4$, so that all four coordinates enter symmetrically, as in a four-dimensional Euclidean space. This Wick-rotated, Euclidean formulation was announced in Schwinger's note "On the Euclidean Structure of Relativistic Field Theory" (1958) and carried out for QED in "Euclidean Quantum Electrodynamics" (1959).

The 1959 paper proceeds from the radiation gauge (the gauge in which the vector potential is purely transverse, $\nabla\cdot\vec A=0$; physically transparent but with a distinguished time direction) to the Lorentz gauge (the covariant gauge $\partial^\mu A_\mu=0$, with no distinguished time direction) by a gauge transformation of the Green's functions, and then analytically continues to the Euclidean metric. Two structural facts shape the construction. First, in the Euclidean metric the Dirac matrices can be replaced by four matrices $\alpha_\mu$ that are real and symmetric and satisfy the Clifford relation $\{\alpha_\mu,\alpha_\nu\}=2\delta_{\mu\nu}$; Schwinger proved by a representation-counting argument that four such anticommuting real symmetric matrices require matrices of size at least $8\times 8$, so that a Euclidean formulation in this real form excludes the four-component Hermitian (Majorana) spin-1/2 field and forces a doubled, eight-real-component fermion. He noted that this matrix size also permits additional internal symmetry generators, which "appear rather differently in Lorentz or Euclidean form," and deferred that discussion. Second, the Euclidean Green's functions are given an operator construction: next to the Euclidean field operators there are complementary operators (the paper's pairs $A,\psi$ and $B,\phi$, with $[A,B]=i\delta$ and $\{\psi,\phi\}=\delta$), and the generating functional of the Green's functions is the wavefunction, in this canonical representation, of a state $|W\rangle$ built on a state $|0\rangle$; both states are zero-eigenvalue eigenstates of the complementary annihilation-type operators, and exist only if those operators are Hermitian - a property Schwinger established only for free fields. The closed expression additionally invokes a functional (path) integral over field eigenvalues, with an "integration concept" made sense of by summation over arbitrarily small four-dimensional cells: a heuristic, not a constructed measure. Schwinger also observed that imposing the Lorentz-gauge commutation relations on the potential contradicts positivity (the quantity $g_{pp}=-1$ arises where a non-negative one is required), which in an operator formulation forces an indefinite metric (a state space with negative-norm vectors, as in the Gupta-Bleuler covariant quantization of the free electromagnetic field); he declined this route because for interacting fields the consistency of the operator assumptions would have to be established separately. The paper ends by announcing that the discussion of Euclidean Green's functions "will be continued in another publication", and a Note added in proof states that "Lorentz gauge" names a whole class of gauges (for example, the four-dimensionally transverse Lorentz gauge), that radiation-gauge functions constructed from Lorentz-gauge functions are independent of the specific Lorentz gauge chosen, and that this subject "will be discussed further in a later paper."

The Euclidean program became one of the most fruitful ideas of modern field theory. Symanzik (1966) reformulated it with probabilistic (Wiener-integral) methods for a scalar model, and Osterwalder and Schrader (1972-1975) gave it its axiomatic form: a family of Euclidean Green's functions ("Schwinger functions") satisfying certain axioms - most importantly reflection positivity (positivity under reflection in a Euclidean hyperplane), together with Euclidean covariance and regularity - reconstructs a Wightman quantum field theory (the standard axiomatic formulation of a relativistic quantum field theory on physical spacetime) on Minkowski space. This turned the rigorous construction of an interacting QFT into the construction of a measure on fields, Grassmann-valued (i.e., anticommuting) for fermions, satisfying the Osterwalder-Schrader (OS) axioms - the program of constructive quantum field theory. That program has rigorously constructed interacting continuum theories in spacetime dimension at most three, including fermionic Yukawa and Gross-Neveu models, and recently settled the four-dimensional $\varphi^4$ model in the direction of triviality (the continuum limit is a free field), but no interacting four-dimensional continuum theory has been constructed; four-dimensional gauge theories (the Yang-Mills problem, one of the Clay Millennium Prize problems, which asks for a rigorous construction of a four-dimensional non-abelian gauge theory and its mass gap) remain open, as documented in the 2022 review by Dedushenko. For QED specifically, perturbation theory indicates a Landau pole (the running coupling diverges at a finite ultraviolet scale), so the expected answer to the existence question is that continuum QED is trivial - an effective theory valid only below a cutoff; this conjecture is unproven and even contested (Djukanovic, Gegelia and Meißner, 2017). Perturbatively, the local observable algebra of QED has been constructed as formal power series by causal BRST (Becchi-Rouet-Stora-Tyutin, the standard formalism that implements gauge symmetry at each order) perturbation theory (Dütsch and Fredenhagen, 1999), which makes precise exactly what a nonperturbative construction must reproduce.

On the fermion side, Schwinger's eight-component doubling was fully clarified by later work. The standard OS formulation of the Euclidean Dirac theory treats the conjugate field as an independent Grassmann variable (so-called doubling), sacrificing Hermiticity of the Euclidean action; the question was posed as "Is There a Euclidean Field Theory for Fermions" by Fröhlich and Osterwalder (1974). Hermitian formulations of the Euclidean Dirac action with undoubled degrees of freedom were given by Mehta (1990) and derived systematically - as a continuous Wick rotation that also rotates spinor indices, recovering Schwinger's Hermitian action - by van Nieuwenhuizen and Waldron (1996), who credited Schwinger's 1959 observation that no massive Majorana spinors with a reality condition and hermitian action exist in four-dimensional Euclidean space. The residual obstructions for Weyl and Majorana fields, complex structures, and discrete symmetries of the Euclidean formulation were analyzed by Kupsch and Thacker (1990), McKeon and Sherry (2001), and Wetterich (2011), and are summarized in the 2026 review of Bonora and Soldati.

Problem Statement

The problem is the unresolved core of the directions Schwinger left open in "Euclidean Quantum Electrodynamics" (1959): the mathematical existence of interacting four-dimensional QED in the Euclidean formulation. Two of the five source directions have been substantially settled by later work - the class structure of Lorentz gauges and the independence of radiation-gauge functions of the Lorentz-gauge representative (Schwinger's own "Euclidean Gauge Transformation", 1960, and the general gauge-fixing framework), and the fermion component-counting and internal-symmetry questions raised by the real symmetric Euclidean $\alpha_\mu$ matrices (clarified by the Euclidean-spinor literature of 1972-2011). What remains open is a single coherent objective in the three facets the source itself poses for the interacting Maxwell-Dirac theory:

(i) the operator construction: exhibit a Hilbert-space realization in which the Euclidean Green's functional of the full interacting theory is the wavefunction of the state $|W\rangle$ over the zero-eigenvalue state $|0\rangle$ of the complementary annihilation-type field operators, with those operators Hermitian - the properties Schwinger verified only for free fields;

(ii) the indefinite-metric operator formulation: establish, or refute, the mathematical consistency for the interacting theory of Lorentz-gauge operator quantization with an indefinite metric - a state space with negative-norm directions in which the gauge constraints are implemented and the physical-sector Green's functions are well defined;

(iii) the functional-integral construction: construct, on the Euclidean Maxwell-Dirac fields (Grassmann-valued for the fermions), a renormalized measure realizing the paper's formal functional integral, whose Schwinger functions satisfy the Osterwalder-Schrader axioms including reflection positivity, so that a Wightman QFT is reconstructed whose perturbation theory is standard QED.

In modern terms this is the constructive-QED existence problem: either produce a mathematically well-defined nontrivial interacting four-dimensional continuum QED (which would be the first rigorous interacting four-dimensional continuum quantum field theory of any kind), or prove that every continuum limit of the regularized theory is necessarily trivial or nonexistent. The theory at issue is the full interacting theory with the massless photon, including its infrared charged sectors - not a cutoff, massive-vector, or lower-dimensional proxy. The three facets share one object - the existence and consistency of the interacting Euclidean Green's functional itself, which a construction through any one of the three routes establishes.

References

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