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REMARKS ON THE ROLE OF THE THEORY OF CUMULANTS IN THEORETICAL PHYSICS

Philippe Choquard

Published Jan 1, 1961• 0 citations
OpenAlex

Abstract

In his Theory of Observations, T. N. Thiele has introduced the semi- invariants (or cumulaats) into statistical mathematics. These functions, endowed with remarkable properties, have found only few applications in statistics. It is, however, surprising that their interest for theoretical physics has not been recognized from the origin of quantum mechanics. In order to indicate this, a generalized defimtion of cumulants is given which is adaptable to functional analysis, and some problems of theoretical physics where their utilization appears natural are mentioned. It appears that an implicit use of cumulants has been made in modern theories. (tr-auth)

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In his Theory of Observations, T. N. Thiele has introduced the semi- invariants (or cumulaats) into statistical mathematics. These functions, endowed with remarkable properties, have found only few applications in statistics. It is, however, surprising that their interest for theoretical physics has not been recognized from the origin of quantum mechanics. In order to indicate this, a generalized defimtion of cumulants is given which is adaptable to functional analysis, and some problems of theoretical physics where their utilization appears natural are mentioned. It appears that an implicit use of cumulants has been made in modern theories. (tr-auth)
Modern theoretical physics is formulated through several highly successful mathematical frameworks, including Green-function theory, quantum mechanics, quantum field theory, gauge theory and statistical mechanics. Although these theories employ different mathematical objects and were developed independently, they exhibit remarkable mathematical compatibility. Green functions naturally appear in quantum field theory, Hilbert-space methods underlie quantum mechanics, gauge theories are formulated geometrically, and statistical mechanics connects microscopic dynamics with macroscopic thermodynamics. The mathematical origin of this compatibility remains an open conceptual question. This paper investigates whether these apparently different frameworks can be understood as representations of a common underlying mathematical structure. We introduce the Architecture Principle, according to which the primitive object is not a Green function, wave function or Hilbert space, but a Green-Function Architecture describing propagation, interaction, closed transport and structural consistency. Green functions, quantum-mechanical states, gauge transport and statistical ensembles are interpreted as different mathematical representations of the same underlying architecture. The proposed framework does not modify the established equations of physics. Instead, it provides a common structural language that relates several major branches of theoretical physics while preserving their existing mathematical formalisms. The paper develops the basic concepts of the Architecture Principle, introduces an architectural invariant, formulates the Representation Principle and discusses representative applications. Finally, the principal mathematical problems required for a complete representation theory are identified.
Modern theoretical physics is formulated through several highly successful mathematical frameworks. Green functions describe propagation, Hilbert spaces provide the mathematical language of quantum mechanics, gauge theories describe transport and symmetry, quantum field theory combines relativistic fields with quantum mechanics, and statistical mechanics relates microscopic dynamics to thermodynamic observables. Although these theories employ different primitive mathematical structures, they exhibit remarkable mathematical compatibility. The structural origin of this compatibility remains largely unexplained. The purpose of this paper is to investigate whether these apparently distinct frameworks admit a common mathematical description. We introduce the Architecture Principle, according to which the primitive mathematical object is a Green-Function Architecture rather than an isolated Green function, Hilbert space, quantum state, or particle. Green functions, Hilbert-space representations, gauge transport, quantum field theoretic propagators, and statistical ensembles are interpreted as different representations of a common underlying architecture. The associated Architectural Multiplicity provides a structural invariant that organizes these representations within a unified mathematical framework. The present work should be viewed as a structural reconstruction program rather than a replacement for established physical theories. Existing mathematical formalisms remain unchanged; the Architecture Principle instead proposes a common structural foundation that explains their mathematical coherence. The paper develops the basic definitions, introduces the principal architectural invariant, formulates the Representation Principle, discusses applications to several branches of theoretical physics, and identifies the principal mathematical problems that remain open.
The moment generating function of a set of stochastic variables defines the cumulants or the semi-invariants and the cumulant function. It is possible, simply by formal properties of exponential functions, to generaiize to a great extent the concepts of cumulants and cumulant function. The stochastic variables to be considered need not be ordinary c -numbers but they may be q -numbers such as used in quantum mechanics. The exponential function which defines a moment generating function may be any kind of generalized exponential, for example an ordered exponential with a certain prescription for ordering q -number variables. The definition of average may be greatly generalized as far as the condition is fulfilled that the average of unity is unity. After statements of a few basic theorems these generalizations are discussed here with certain examples of application. This generalized cumulant expansion provides us with a point of view from which many existent methods in quantum mechanics and statistical mechanics can be unified.
Recently it has been shown that cumulants significantly simplify the analysis of multipartite weak measurements. Here we consider the mathematical structure that underlies this and find that it can be formulated in terms of what we call the moment algebra. Apart from resulting in simpler proofs, the flexibility of this structure allows generalizations of the original results to a number of weak measurement scenarios, including one where the weakly interacting pointers reach thermal equilibrium with the probed system.