Scattering content of the non-quasi-free Wightman fields constructed by convex mixing of quasi-free functionals

A 2018 construction by R. Gielerak shows that probability mixtures of stochastically positive Schwinger functionals — in particular of quasi-free functionals of massive free scalar fields — again satisfy all Osterwalder–Schrader axioms, producing in every spacetime dimension large classes of Wightman fields that are non-quasi-free (nonvanishing higher cumulants) yet typically fail cluster decomposition, so their vacuum structure and scattering content are uncontrolled. Whether any model in these classes possesses a nontrivial S-matrix, or whether the entire class is scattering-trivial, was posed as open by the source and remains unresolved. The problem is to develop the scattering-content analysis of the convex-mixed classes — deciding, by Haag–Ruelle scattering theory or an equivalent rigorous formulation, whether any phase in any spacetime dimension (notably d = 4) has nontrivial scattering, or proving that all models in the class are scattering-trivial.

Background

Axiomatic quantum field theory describes a relativistic quantum field through the Wightman axioms: the field is an operator-valued tempered distribution on $d$ -dimensional Minkowski space acting on a Hilbert space carrying a unitary representation of the Poincaré group, with a unique invariant vacuum vector $Ω$ , positivity of the energy spectrum, and commutativity of fields at spacelike separation. The vacuum expectation values $W_n=⟨Ω,φ(f_1)⋯φ(f_n)Ω⟩$ , called Wightman functions, determine the theory; their generating functional on test functions is the Wightman functional.

A standard route to such theories passes through Euclidean signature. The generating functional of the Euclidean correlation functions is called a Schwinger functional. The Osterwalder–Schrader reconstruction theorem states that a Schwinger functional $Γ$ on the Schwartz test-function space satisfying normalization, Hermiticity, a regularity (analyticity) condition, Euclidean invariance, reflection positivity (positivity of the quadratic form $∑ z̄_i z_j Γ(f_i−Rf_j)$ for test functions supported at nonnegative Euclidean times, where $R$ reflects the time coordinate), and stochastic positivity (positivity of $∑ z̄_i z_j Γ(f_i−f_j)$ , which makes $Γ$ the characteristic functional of a probability measure on tempered distributions, so that the Euclidean field is a genuine random distribution) yields, by analytic continuation, a neutral scalar Bose field obeying all the Wightman axioms. A functional is quasi-free (Gaussian) if all its cumulants — the truncated $n$ -point functions obtained from $log Γ$ — of order $n>2$ vanish. Quasi-free functionals reconstruct to free fields or generalized free fields, that is, free-like fields whose two-point function is an arbitrary Källén–Lehmann superposition of massive free two-point functions; all such fields have trivial scattering.

Scattering is formalized by the Haag–Ruelle construction: for a field with a mass gap and an isolated mass shell one builds incoming and outgoing multiparticle states and defines the S-matrix between them; a theory is scattering-trivial if this S-matrix is the identity. Classical triviality criteria (recorded in the standard monographs of the axiomatic framework) imply trivial scattering for generalized free fields and for fields in the Borchers equivalence class of a free field (fields obtainable from a free field by Wick reordering of composite fields, hence physically equivalent to it). Haag–Ruelle theory presupposes the cluster decomposition property, $W_n(x+a,…,y)→W(x)W(y)$ for spacelike $a→∞$ , which is equivalent to uniqueness of the vacuum; a non-clustering Wightman functional represents a mixed phase and can be decomposed into pure (clustering) phases by Choquet-type central decomposition theorems, the convex-analytic result expressing any state as an integral over extremal, indecomposable states.

Gielerak (2018) observed that the class of Schwinger functionals obeying the axioms above is closed under probabilistic mixing: if ${Γ_ω}$ is a uniformly regular, weakly measurable family of such functionals and $dP$ is a probability measure, then $Γ_P(f)=∫dP(ω)Γ_ω(f)$ obeys all the axioms and hence defines a Wightman theory in any dimension $d≥1$ . Mixing the quasi-free functionals $Γ^0_m$ of massive free fields over a measure on the squared mass $m^2$ (supported away from zero) yields functionals that are not quasi-free as soon as the mixing measure has at least two points — for two distinct masses the truncated four-point Schwinger function is explicitly nonzero — while the two-point function is the corresponding Källén–Lehmann mixture. The construction can be iterated (mixing, then replacing by the quasi-free functional with the same two-point function, then mixing again) and applied to the Schwinger functionals of the rigorously constructed interacting scalar theories with polynomial self-interaction — the $P(φ)_2$ models in $d=2$ and $φ^4_3$ in $d=3$ , giving large classes of Wightman fields in any dimension that are non-quasi-free but never clustering unless the mixing measure is a point mass; the author stresses that these mixtures generally do not coincide with the classical decompositions into pure phases.

The physical stakes come from constructive quantum field theory, the programme seeking rigorous existence proofs for interacting relativistic fields. Interacting Wightman theories with nontrivial S-matrix are known in low dimensions ( $P(φ)_2$ , $φ^4_3$ , the sine-Gordon model — a two-dimensional scalar field with periodic self-interaction — and two-dimensional factorizing-S-matrix constructions, in which multiparticle scattering factorizes into two-particle amplitudes), but in four spacetime dimensions no positive-metric, Lorentz-invariant Wightman field with nontrivial scattering is known — the flagship open problem of the field. The nearest four-dimensional results fall short of the Wightman setting: noncommutative (warped-convolution) deformations of the free field produce a model with nontrivial two-body scattering but break Lorentz invariance for $d>2$ , and the covariant stochastic-PDE models of Gielerak and Ługiewicz achieve nontrivial scattering only in indefinite-metric (Krein) spaces. The convex-mixing construction is dimension-independent and preserves all Wightman axioms, including full Lorentz invariance; whether it captures genuine interaction is exactly what is at issue.

Problem Statement

Determine the scattering content of the Wightman models constructed by convex mixing of stochastically positive Schwinger functionals in Gielerak (2018, arXiv:1803.04508): the mixtures $Γ_P(f)=∫dP(m^2)Γ^0_m(f)$ of quasi-free functionals of massive free scalar fields (mixing measures supported away from $m^2=0$ ), their iterations as defined in that paper, and the convex envelopes involving the Schwinger functionals of $P(φ)_2$ ( $d=2$ ) and $φ^4_3$ ( $d=3$ ) models, every mixing measure being taken non-degenerate (support of at least two points) — the degenerate one-atom combinations, which reproduce a single constituent functional, are not counted as members of the class. Because these functionals in general fail cluster decomposition, the question is to decide — by applying Haag–Ruelle scattering theory to the pure (clustering) phases of the mixed functionals, or by an equivalent rigorous formulation of the S-matrix appropriate to these models — whether any model in the class, in any spacetime dimension $d≥1$ (in particular $d=4$ ), has a nontrivial S-matrix; or else to prove that every model in the class is scattering-trivial.

References

Roman Gielerak, "Construction of a class of scalar quantum field theory models in any dimensions", arXiv:1803.04508 (2018), DOI 10.48550/arXiv.1803.04508, https://arxiv.org/abs/1803.04508.
Roman Gielerak, "β-KMS Green functions generating functionals – the convexity based approach", arXiv:1803.09542 (2018), DOI 10.48550/arXiv.1803.09542, https://arxiv.org/abs/1803.09542.
Stephen J. Summers, "A Perspective on Constructive Quantum Field Theory", arXiv:1203.3991 (2012), DOI 10.48550/arXiv.1203.3991, https://arxiv.org/abs/1203.3991.
Harald Grosse, Gandalf Lechner, "Noncommutative Deformations of Wightman Quantum Field Theories", Journal of High Energy Physics 2008(09):131 (2008), DOI 10.1088/1126-6708/2008/09/131, https://arxiv.org/abs/0808.3459.
R. Gielerak, P. Ługiewicz, "4D local quantum field theory models from covariant stochastic partial differential equations I: Generalities", Reviews in Mathematical Physics 13, 335–408 (2001), DOI 10.1142/S0129055X01000685, https://doi.org/10.1142/S0129055X01000685.