Dirac's 1962 problem: a genuine unitary representation of the inhomogeneous Lorentz group in an interacting relativistic quantum field theory

In 'The Conditions for a Quantum Field Theory to be Relativistic' (Rev. Mod. Phys. 34, 592 (1962)), P. A. M. Dirac showed that building a relativistic quantum field theory from local energy and momentum densities requires commutation-relation conditions on the energy density (his Eqs. (15), (25), (26) — the condition independently found by Schwinger in 1963), but that integrating the infinitesimal generators to finite transformations meets divergent integrals, so no genuine unitary representation of the inhomogeneous Lorentz (Poincaré) group results; he regarded renormalization as a stopgap and proposed non-local energy densities, non-local commutation relations, or a revision of quantum mechanics as escapes. The subsequent lineage: constructive quantum field theory answered the question positively in two and three spacetime dimensions (interacting theories satisfying the Wightman axioms, including a strongly continuous unitary Poincaré representation), the four-dimensional scalar route is closed by triviality theorems, and the four-dimensional interacting case — the archetype being quantum Yang–Mills theory with a mass gap — remains open (a Clay Mathematics Institute Millennium Prize problem).

Background

Relativistic quantum field theory (QFT) describes elementary particles as operator-valued fields on Minkowski spacetime (the flat spacetime of special relativity). The symmetry group of flat spacetime is the inhomogeneous Lorentz group (IHLG) — also called the Poincaré group — generated by the ten operations of spacetime translation and Lorentz rotation. Since quantum symmetries act on states by unitary (norm-preserving, invertible linear) operators, every quantum system in flat spacetime must carry a unitary representation of the IHLG. Wigner (1939) classified the irreducible unitary representations (those not decomposable into smaller ones) and found that they describe single free particles of various mass and spin, with no interaction between them; Dirac stresses that interaction can only appear once one abandons the view that two representations connected by a unitary transformation are physically equivalent.

Dirac's starting point, developed from his 1949 'instant form' of relativistic dynamics (dynamics referred to states on the hyperplane $x^0=0$, that is, space at a fixed instant of time), is that a QFT should be built from its infinitesimal generators: the ten operators $P_\mu$ (translations) and $M_{\mu\nu}$ (rotations) satisfying the Poincaré commutation relations, expressed through quantities localized in three-dimensional space — a momentum density $K_r$, fixed purely by the geometrical (tensor or spinor) character of the basic fields, and an energy density $U$, with $P_0 = \int U\, d^3x$ and $M_{0r} = \int x_r U\, d^3x$. If the basic fields obey local equal-time commutation relations (commutators that vanish unless the two points coincide, and then involve the Dirac delta distribution $\delta(x-x')$ and its derivatives), the commutator $[U,U']$ of the energy density at two spatial points has a universal delta-supported structure (Dirac's Eq. (15)), and requiring the integrated operators to close the Poincaré algebra imposes algebraic conditions on its coefficients (Eqs. (25), (26)). Schwinger (1963) derived the same condition independently; the resulting equal-time commutation relation of energy densities is now called the Dirac–Schwinger relation, and the anomalous distributional terms that appear in commutators of currents are called Schwinger terms. Dirac discussed the classical Born–Infeld nonlinear electrodynamics (a modification of Maxwell theory designed to bound the field energy) as an illustration.

The problem Dirac identified is that these conditions govern only the infinitesimal algebra. To obtain finite Poincaré transformations one must integrate the infinitesimal operators — mathematically the same task as integrating the equations of motion — and for interacting theories all such attempts led to divergent integrals, so that, in Dirac's words, 'one does not really have a representation of the IHLG at all'. The renormalization method could patch the divergent integrals well enough to compare with experiment, but Dirac judged it 'only a stopgap', since it does not follow logically from established physical laws and does not deliver a genuine representation of the IHLG. This is related to, but distinct from, Haag's theorem — the result that an interacting field cannot be realized in the same Hilbert space as the corresponding free field — the Fock space, the state space built as a direct sum of free multiparticle states — so that the interaction picture of perturbation theory is not literally available.

Dirac's proposed ways out: generalize the construction by departing from the local character of the energy density or from the local commutation relations of the basic fields; and, if all such attempts fail, alter the fundamental laws of quantum mechanics, finding a new principle to replace the use of representations of the IHLG.

Later work reframed rather than answered the question. The Wightman axioms take exactly what Dirac wanted as output — a strongly continuous (continuous in the group parameters) unitary representation of the Poincaré group, together with the spectrum condition (energies nonnegative), local commutativity, and a unique vacuum — as defining axioms, and ask which interactions admit models satisfying them; the Haag–Kastler algebraic approach (1964) axiomatizes the same covariance structure on local algebras of observables. Constructive QFT then rigorously built nontrivial interacting models satisfying the Wightman axioms in two and, later, three spacetime dimensions — superrenormalizable theories (those whose ultraviolet divergences are mild enough to be absorbed into finitely many constants), such as the $P(\varphi)_2$ and $\varphi^4_3$ models — using Euclidean probabilistic constructions and the Osterwalder–Schrader reconstruction (conditions on Euclidean correlation functions, chief among them reflection positivity, that guarantee recovery of a Wightman theory). Wilson's renormalization group (1983) in turn gave renormalization a physical meaning as the scaling behaviour of continuum limits rather than a subtraction trick for divergences.

In four spacetime dimensions, however, the scalar route is closed by triviality theorems — rigorous results showing that the continuum (scaling) limit of the $\varphi^4_4$ model is a free (Gaussian) field with no scattering — and no rigorous interacting construction is known for any four-dimensional model, the archetype being pure Yang–Mills theory (the non-Abelian gauge theory underlying quantum chromodynamics, the theory of the strong nuclear force) for a compact simple gauge group, required to be nontrivial and to have a mass gap (a strictly positive lower bound on excitation energies above the vacuum). This is the substance of the Clay Mathematics Institute Millennium Prize problem 'Yang–Mills existence and mass gap'.

Problem Statement

Does there exist a mathematically consistent, nontrivial (interacting) relativistic quantum field theory that carries a genuine — explicitly constructed, strongly continuous — unitary representation of the inhomogeneous Lorentz (Poincaré) group, with the finite group transformations obtained by an integration of the infinitesimal generators that is free of divergent integrals? Within standard quantum mechanics and locality, the question concerns the physically relevant four spacetime dimensions. The canonical reference archetype is pure quantum Yang–Mills theory for a compact simple gauge group, required in the Clay Mathematics Institute Millennium Prize formulation of that archetype to be nontrivial and to have a mass gap; the existential question above is not restricted to it. Two and three spacetime dimensions are already settled positively by constructive quantum field theory, so the open core of the question is four spacetime dimensions. The question is resolved either by a construction of such a theory or by a proof that none exists in four spacetime dimensions within standard quantum mechanics and locality. If no such theory exists, then Dirac's three alternatives must each be adjudicated, that is, for each one it must be determined whether it yields a consistent, nontrivial (interacting) theory whose transformation law plays the role of, or explicitly replaces, the unitary representation of the inhomogeneous Lorentz group: a departure from the local character of the energy density (only that locality relaxed, the local commutation relations of the basic fields and the unitary representation retained); a departure from the local commutation relations of the basic fields (only local commutativity relaxed, the local energy density and the unitary representation retained); or an alteration of the fundamental laws of quantum mechanics (the unitary representation itself replaced by a new principle), as Dirac feared.

References

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