In 1976 Wu and Yang exhibited an explicit static sourceless gauge field for the noncompact group SL(2,C) — obtained by converting a complex SU(2) solution — that is everywhere analytic with field strengths decaying as O(1/r2)O(1/r^2), conjectured that static sourceless fields vanishing at spatial infinity exist only for noncompact gauge groups, and asked whether gauge theories with semisimple noncompact groups, whose Hamiltonian is necessarily not positive definite, can have any physical meaning. The compact case of the conjecture was resolved within months for finite-energy configurations by Deser's self-stress no-go theorem (1976), and the absence of compact-group counterexamples is confirmed by the later literature, where all claimed SU(2) solutions turn out to be complex, i.e., real SL(2,C)-valued. What remains open is the conjecture in its literal decay class and the viability question: whether static sourceless gauge fields vanishing at spatial infinity — pointwise, a class formally broader than finite energy — can exist for compact groups, and whether indefinite-energy theories with semisimple noncompact groups admit a unitary, physically sensible quantization, where one-loop no-go results and proposed unitary constructions currently conflict.

Background

Yang–Mills theory is the non-Abelian generalization of Maxwell electromagnetism: the gauge field bμb_\mu takes values in the Lie algebra of a chosen gauge group GG and interacts with itself through the commutators of that algebra. 'Sourceless' means the field is coupled only to itself — there is no external current or matter field — so the field equations are DμFμν=0D_\mu F^{\mu\nu}=0, where FF is the field strength and DD the gauge-covariant derivative. A 'static' solution is time-independent; classically such solutions would represent particle-like lumps held together by their self-interaction, so their existence or absence is a basic question about the classical content of a gauge theory.

For compact gauge groups such as SU(2) (the group of 2×2 unitary matrices with unit determinant) with the standard positive-definite invariant inner product on the Lie algebra (the group metric, essentially the Killing form), the energy (Hamiltonian) is a sum of squares and hence positive definite. Wu and Yang had searched for static sourceless SU(2) solutions since around 1969 and found only singular ones. The search succeeds after complexification: J. P. Hsu found a static sourceless complex solution of the SU(2) equations whose radial profile functions involve hyperbolic functions in the style of the Prasad–Sommerfield monopole solution (the exact 1975 solution of the massless SU(2) monopole equations), and by rules established by Wu and Yang in 1975 a complex SU(2) gauge field is equivalent to a real field for the noncompact group SL(2,C) (the group of 2×2 complex matrices with unit determinant, six real generators), the double cover of the Lorentz group SO(3,1). The 1976 paper exhibits this field ('gauge field A'): it is everywhere analytic (globally regular, in particular nonsingular at the origin) and its field strengths fall off as O(1/r2)O(1/r^2) as r→∞r\to\infty.

The catch is the energy. Contracting group indices in the Lagrangian requires an invariant group metric; for a semisimple Lie algebra (one built as a direct sum of simple, non-Abelian building blocks) the admissible metrics form a cone, and Wu and Yang prove (their Theorem 5) that if any admissible metric is definite the group is compact — so for semisimple noncompact groups like SL(2,C) every invariant metric is indefinite and the Hamiltonian is not positive definite. Consistently, the total Hamiltonian of field A is exactly zero, with positive and negative contributions canceling. This is precisely what evades the no-go machinery: in the same year Deser showed by a self-stress (virial-type) argument that source-free Yang–Mills theory with compact gauge group has no finite-energy non-singular time-independent solutions in 3+1 Minkowski space (the argument uses positivity of the energy; it fails in five dimensions and for indefinite metrics). Coleman and Smarr (1977) strengthened the negative results to non-radiating and geon-like configurations (geons: long-lived lumps of field energy held together by their own self-interaction before eventually radiating away). Complex SU(2) solution families constructed later (Oh; Oh and Teh; Villarroel) are real SL(2,C)-valued in the same sense as field A and provide no compact-group counterexample.

Within this landscape the source paper left two things open. First, the conjecture (Remark C): static sourceless gauge fields vanishing at spatial infinity exist only for noncompact groups. Deser's theorem settles the conjecture for finite-energy configurations, but the conjecture's literal pointwise-decay class is formally broader, and whether slow-decay, infinite-energy compact-group configurations exist is unresolved; on the noncompact side the paper gives the single example SL(2,C), and Sasaki (1978) later produced further sourceless solutions for SO(4,1), the de Sitter group, via instanton-type identifications (instantons: self-dual solutions of the Euclidean gauge-field equations) — the paper itself poses no classification of which noncompact groups admit such solutions. Second, the question (Remark E): whether a gauge theory with semisimple noncompact group — whose energy is necessarily not positive definite — can have any physical meaning. That question received conflicting partial answers: Hsu and Xin (1981) found at one loop that unitarity and local gauge symmetry are incompatible in SL(2,C) Yang–Mills theory (and argued this plausibly holds for any noncompact gauge group), while Cahill and Ozenli (1983) argued that gauging noncompact groups with a dynamical internal metric field preserves unitarity. Modern work on complex gauge theories (complex-group QCD for confinement, SL(2,C) instanton constructions, complex monopoles) continues to build on the complex direction without settling the foundational question.

Problem Statement

Consider pure Yang–Mills theory in (3+1)-dimensional Minkowski space with gauge group GG and no sources: the field equations are DμFμν=0D_\mu F^{\mu\nu}=0.

(1) Existence conjecture. Determine whether static sourceless gauge fields vanishing at spatial infinity exist only for noncompact gauge groups — that is, whether any compact group GG admits a nontrivial static solution that is globally regular (everywhere analytic; in particular nonsingular at r=0r=0) and whose field strengths vanish as r→∞r\to\infty (the boundary/regularity class of the Wu–Yang SL(2,C) solution, whose field strengths are O(1/r2)O(1/r^2)). The finite-energy part of the compact case is settled: no such finite-energy fields exist for compact GG (Deser's self-stress theorem, 1976). The conjecture's literal class — pointwise vanishing of the field strengths — is formally broader, and whether slow-decay, infinite-energy compact-group configurations exist remains open. On the noncompact side, the known examples are the Wu–Yang SL(2,C) field and Sasaki's SO(4,1) solutions; the source paper poses no classification of which noncompact groups admit such solutions, and none is required here.

(2) Physical viability. Determine whether a Yang–Mills theory with semisimple noncompact structure group — whose Hamiltonian is necessarily indefinite (Wu–Yang Theorem 5) — admits a physically sensible, in particular unitary, quantization, where 'unitary' is read operationally as requiring a positive-semidefinite inner product on the physical (gauge-invariant) states: either construct such a quantization or prove a no-go theorem.

The two deliverables correspond to the two separate open questions left by the source paper: the Remark (C) conjecture and the Remark (E) physical-meaning question.

References

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