The Isham–Salam–Strathdee conjecture: a Wilson operator-product expansion with entire-function coefficients for localizable nonlinearly realized field theories

In a 1973 letter, Isham, Salam and Strathdee conjectured that localizable nonlinearly realized field theories — nonpolynomial theories such as the chiral $SU(2)\times SU(2)$ pion Lagrangian, whose interactions are entire functions of the fields and which carry a dimensional coupling $\varkappa$ with a built-in ultraviolet cutoff of order $1/\varkappa$ — possess a Wilson-type short-distance operator-product expansion whose coefficient functions are entire functions of the dimensionless combination $\varkappa^{2}/x^{2}$, as exemplified by the Wick-theorem identity $T\exp[\varkappa\varphi(x)]\exp[\varkappa\varphi(0)]=\exp[-\varkappa^{2}/(4\pi^{2}x^{2})]$ for a free massless scalar field $\varphi$. The conjecture has never been proven, refuted, or derived axiomatically. The associated analytic-continuation problem — a unique minimally singular continuation of the time-ordered expressions between timelike and spacelike separations, order by order in the dimensionless major coupling $g$ — was carried by the Lehmann–Pohlmeyer program only to fourth order (Pohlmeyer, 1974) before the research program was abandoned. It remains open to establish or refute the entire-function expansion ansatz for this class of theories, to settle the all-orders minimally singular continuation, and to characterize the coefficient functions and dimension numbers; no resolution is known in the literature to date.

Background

The operator-product expansion (OPE) introduced by Kenneth Wilson in 1969 expresses the product of two local quantum-field operators $A(x)B(0)$, in the limit of small space-time separation $x$, as a sum over local operators $C_n$ with singular coefficient functions fixed by the operators' scaling dimensions. In renormalizable four-dimensional theories, whose fields are operator-valued tempered distributions (distributions of at most polynomial growth), the expansion was placed on systematic footing by the configuration-space methods of the period, and its coefficient functions are, up to logarithms, powers of the separation.

In the late 1960s and early 1970s a broad program studied 'nonpolynomial' or 'nonlinearly realized' field theories, in which the interaction depends on the fields through entire functions (functions holomorphic everywhere in each field variable) rather than polynomials. The prototype is the chirally invariant Lagrangian for massless pions, where the symmetry $SU(2)\times SU(2)$ is realized nonlinearly and the field enters through $S=\exp[\varkappa\,\tau\cdot\varphi\,\gamma_5]$, with $\varphi$ the pion triplet, $\tau$ the Pauli matrices acting in isospin, and $\gamma_5$ a Dirac matrix. Two couplings organize the perturbation theory of such theories: a dimensionless 'major' coupling $g$ labeling the order of the expansion, and a dimensional 'minor' coupling $\varkappa$ that provides a built-in ultraviolet cutoff of order $1/\varkappa$, the theory formally tending to a renormalizable one as $\varkappa\to 0$. Volkov constructed the Green's functions of such theories by momentum-space methods and connected the localizability of the theory — the property that its field algebra belongs to the strictly localizable class of Jaffe, i.e. operator-valued distributions acting on non-tempered test-function classes, realized through entire functions of a free field as in Jaffe's construction — with the entire-function character of the nonpolynomial interaction. Epstein, Glaser and Martin proved that localizable theories obey polynomial (Froissart-type) high-energy bounds, which added to the interest of the class.

In a three-page letter of 1973, Isham, Salam and Strathdee proposed how the Wilson expansion should generalize to this class. Because $\varkappa$ is the only new dimensional parameter, any short-distance expansion must involve the dimensionless combination $\varkappa^{2}/x^{2}$, and they conjectured

$$A(x)B(0)=\sum_{n}F_{n}\left(\frac{\varkappa^{2}}{x^{2}}\right)\frac{1}{(x^{2})^{\frac{1}{2}(d_{A}+d_{B}-d_{n})}}\,C_{n}(x),$$

where $d_{A}$, $d_{B}$, $d_{n}$ are numbers (operator dimensions) to be computed, $C_{n}$ are local operators, and the coefficient functions $F_{n}$ are entire functions of $\varkappa^{2}/x^{2}$ of typical exponential form $\exp[\lambda_{n}\,\varkappa^{2}/x^{2}]$ with dimensionless constants $\lambda_{n}$. The explicit evidence offered was the exact evaluation, by Wick's theorem (which reduces time-ordered products of free fields to sums over pairwise contractions), of time-ordered products of exponentials of a free massless scalar field $\varphi$: $T\exp[\varkappa\varphi(x)]\exp[\varkappa\varphi(0)]=\exp[-\varkappa^{2}/(4\pi^{2}x^{2})]$, and the same expression with reversed sign in the exponent when the exponential charges are opposite. Each result is an entire function of $1/x^{2}$, but it has an essential singularity on the light cone $x^{2}=0$: it decays faster than any power when the cone is approached from one side and grows faster than any power from the other. Consequently such expressions are not tempered distributions, and their analytic continuation between timelike and spacelike separations is not uniquely fixed as an ordinary function.

Lehmann and Pohlmeyer had attacked precisely this continuation problem by momentum-space methods: the two-point object (the 'superpropagator', the vacuum expectation value of a time-ordered product of two field exponentials) and its higher-order analogues can be defined, order by order in the major coupling $g$, as continuous linear functionals on suitable test-function spaces by selecting the minimally singular continuation across the light cone — the one that near $x^{2}=0$ grows no faster than the free-field Wick-theorem results already do. By 1973 this program had been carried through third order in $g$: the superpropagator itself by Lehmann and Pohlmeyer (1971) and the third order by Pohlmeyer (1972). What happens at arbitrary order, and whether the entire-function ansatz for the full operator algebra of the interacting theory is correct, was left open — and the broader question of giving the ansatz a first-principles foundation in axiomatic local quantum field theory for strictly localizable fields was explicitly raised by the source as an unresolved issue.

Problem Statement

For the class of localizable nonlinearly realized theories in four space-time dimensions — nonpolynomial theories whose interactions are entire functions of the fields, equipped with a dimensional minor coupling $\varkappa$ (setting a built-in ultraviolet cutoff $1/\varkappa$) and a dimensionless major coupling $g$ labeling perturbative orders, with the chirally invariant massless-pion Lagrangian as prototype — determine the short-distance structure of operator products in the sense proposed by Isham, Salam and Strathdee (1973):

1. The conjecture. Prove or refute the ansatz

$$A(x)B(0)=\sum_{n}F_{n}\left(\frac{\varkappa^{2}}{x^{2}}\right)\frac{1}{(x^{2})^{\frac{1}{2}(d_{A}+d_{B}-d_{n})}}\,C_{n}(x)$$

for local Heisenberg-picture operators $A$, $B$ (operators of the fully interacting theory, not free-field constructions): are the short-distance coefficient functions $F_{n}$ entire functions of the combination $\varkappa^{2}/x^{2}$ (of the typical exponential form $\exp[\lambda_{n}\,\varkappa^{2}/x^{2}]$ with dimensionless constants $\lambda_{n}$)? An affirmative resolution characterizes the entire functions $F_{n}$ and the numbers $d_{A}$, $d_{B}$, $d_{n}$, while a refutation exhibits a theory and operators within the stated class whose coefficient functions fail to be entire in $\varkappa^{2}/x^{2}$.

2. The analytic continuation. Establish or refute, to all orders in the major coupling $g$, the existence of a unique minimally singular analytic continuation of these expressions between timelike and spacelike separations — going beyond the third order available to the source and the fourth order obtained by Pohlmeyer (1974) — or identify the order in $g$ at which uniqueness fails.

3. The foundations. Justify, or refute the possibility of justifying, the ansatz from first principles of local quantum field theory for strictly localizable fields — rather than from free-field Wick-theorem evidence and perturbative heuristics — including the clarification of its relation to the momentum-space constructions of the Green's functions due to Volkov and to Lehmann and Pohlmeyer.

The object of study is the operator algebra of this class of theories, treated in a mathematically definite way. The three parts are independent, each refutable on its own: for part 1, a refutation is an explicit theory and operators within the stated class whose coefficient functions fail to be entire in $\varkappa^{2}/x^{2}$; for part 2, a refutation is an explicit order in the major coupling $g$ at which the minimally singular continuation ceases to be unique; for part 3, a refutation is a demonstration that the ansatz admits no justification from first principles for strictly localizable fields. An affirmative resolution establishes parts 1 and 2 jointly — the conjectured entire-function form, together with the all-orders continuation property and the characterization of $F_{n}$, $d_{A}$, $d_{B}$, $d_{n}$ — while the first-principles foundation of part 3 is posed as a further open question. The extension to matter fields interacting with gravity, mentioned in the source as a further application, is a natural corollary once the above is settled.

References

1. C. J. Isham, Abdus Salam, J. Strathdee, 'Wilson operator products in nonlinearly realized theories', Lettere al Nuovo Cimento 7, 902–904 (1973). DOI: 10.1007/BF02727515, https://doi.org/10.1007/BF02727515 (source paper of the conjecture)
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