Foundations and development of the potential–correlation function duality in statistical mechanics

In 1967 De Dominicis and Englert proved, as a formal identity, a duality in equilibrium statistical mechanics and quantum field theory: the functional relationship expressing the connected n-body Green functions (cumulants) of a system with p-body potentials in terms of those potentials is invariant under the transformation exchanging each potential $v_m$ with $-(-)^m\tilde G_m$ (minus an alternating sign times the connected m-point function), while $\log Z$ changes sign. The duality underlies higher Legendre-transform methods, exact renormalization group reconstructions, and modern divergence-free perturbation schemes. The originating work left the dual formulation without analytic foundations: no existence or convergence conditions, no solvable or truncatable form of the dual equations-of-motion hierarchy (which couples every correlation function to all higher orders), no equal-footing renormalized formulation in observable variables with controlled cancellations in the entropy functional, no diagrammatic classification of those cancellations, and no extension to systems with nonlinear unperturbed effective actions (classical gases, spin or Ising systems). The adjacent higher-Legendre-transform, exact-renormalization-group, and combinatorial-Legendre-transform literatures each develop part of the program formally or perturbatively, and recent sources (Rosten 2012; Jackson–Kempf–Morales 2017) explicitly re-flag the validity and well-posedness gaps. The precise open core survives: turning the formal duality into a well-founded, usable dual formulation.

Background

Equilibrium quantum statistical mechanics describes a gas of identical particles interacting through one-, two-, and higher-body potentials $v_p$ by means of the grand partition function $Z\{v\}$, which generates the n-body thermal Green functions $G_n$ (imaginary-time-ordered expectation values of n particle fields). Each $G_n$ decomposes into connected parts, or cumulants, $\tilde G_n$ — the parts that cannot be factored into products of lower-order functions. In the Feynman path-integral representation the action is linear in the potentials $v_p$, with a Gaussian (free) unperturbed part, so perturbation theory expresses $\log Z$ and every $\tilde G_n$ as formal series of Feynman diagrams with interaction vertices built from the $v_p$ and a free propagator built from the unperturbed (Gaussian, quadratic) part of the action. A program of the 1960s sought to rebuild such expansions around physical quantities — densities and correlation functions — instead of bare potentials, hoping for better convergence and access to nonperturbative effects; De Dominicis contributed a variational formulation of statistical mechanics in terms of observables (1963), and with Balian a generating-function method for eliminating anomalous diagrams (1964). The structural fact beneath this program was isolated by De Dominicis and Englert in 1967: the functional relationship expressing the cumulants $\tilde G_n$ in terms of the potentials is invariant under the transformation exchanging $v_m$ with $-(-)^m\tilde G_m$ everywhere, while $\log Z$ changes sign. In the modern formulation used by Pivovarov: if $W[J]$ is the generating functional of connected Green functions and $I_B[\phi]$ the bare action, then $\exp(I_B[\phi])=\int\mathcal{D}J\,\exp(W[J]-J\phi)$, so the bare action is obtained from $W$ by the same diagrammatic rules with which $W$ is obtained from it — potentials and correlation functions can be exchanged as independent variables. The identity is a symmetric form of the chain of Legendre transforms linking $Z$, the connected functional $W[J]$, the effective action $\Gamma[\phi]$ (the standard Legendre transform of $W$, which generates one-particle-irreducible vertices — diagrams that stay connected when any single internal line is cut), and the higher, n-th Legendre transforms that exchange sources for mean n-point functions; it also reappears inside the exact renormalization group (Wilsonian flow equations that integrate out field modes momentum shell by momentum shell), where the Wilsonian interaction action and the connected correlation functionals form a dual pair. The 1967 note presented these results as formal identities and discussed their content in its conclusion, but the issues that define the present problem are documented across the authors' own program — De Dominicis's 1963 observable formulation, in which the thermodynamical functions become stationary functionals of the correlation functions, and the 1968 companion papers' generalized perturbation theory about an arbitrary unperturbed system characterized by all its Green's functions — and in the modern literature: no conditions of existence or convergence — analytic domains, regularity assumptions, or function spaces — were established; the dual equations of motion (obtained by functional integration by parts of the dual action) form hierarchies coupling each correlation function $C_n$ (the connected functions $\tilde G_n$, now taken as the independent variables of the dual formulation) to all higher-order $C_{n+r}$, whose solvability, truncation, and practical utility remain undeveloped; from a renormalization viewpoint potentials and correlations do not stand on an equal footing (a simple microscopic $v_p$ typically generates infinitely many nonvanishing $C_n$), leaving open which independent observable variables to introduce and how to control the resulting cancellations in the free-energy/entropy functional expressed in correlation variables; the diagrammatic analysis of those cancellations remains open; and the extension to classical, spin, or Ising systems — where the unperturbed effective action is nonlinear — was developed only at the formal level.

Problem Statement

Take the De Dominicis–Englert potential–correlation function duality — the exact formal symmetry exchanging each m-body potential $v_m$ with $-(-)^m\tilde G_m$ (minus an alternating sign times the connected m-point Green function) while flipping the sign of $\log Z$ — and establish it as a well-founded, usable dual formulation of statistical mechanics. Concretely, treating the formal diagrammatic identity itself as settled: (1) Validity. Identify the analytic setting — function spaces for potentials and cumulants, regularity assumptions, and convergence or stability conditions — in which the duality map and its inverse are well-defined, including the sign flip of $\log Z$, and characterize which correlation data define admissible potentials (the reconstruction problem). (2) Dual hierarchy. The dual equations of motion, obtained by functional integration by parts of the dual action, couple every correlation function $C_n$ (the connected functions $\tilde G_n$ in their dual role as the independent variables) to all higher-order $C_{n+r}$; establish the solvability of these hierarchies and truncation or closure schemes with controlled errors. (3) Equal-footing formulation. Construct the renormalized formulation in which potentials and correlation functions enter on an equal footing: specify which independent observable variables replace the microscopic potentials, and control the cancellations in the free-energy/entropy functional expressed in those variables — the stationary thermodynamical functional of De Dominicis's observable formulation (J. Math. Phys. 4, 255, 1963) and the stationarity principle on vacuum fluctuations with respect to stripped correlation functions of Englert–De Dominicis 1968 Part II — so that finitely specified microscopic potentials do not map to unmanageable infinite towers of independent variables. (4) Entropy-functional diagrammatics. Determine which diagrams survive in the entropy functional written in correlation variables, and construct the corresponding irreducible or ladder resummations (closed-form sums of selected infinite families of diagrams, such as ladder chains). (5) Scope. Derive the precise form and consequences of the dual relationships when the unperturbed effective action is nonlinear, as in classical gases and spin (Ising) systems. These five facets are scope descriptors of the single objective — a correlation-based dual formulation with the same standing as the potential-based one.

References

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