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International Journal of Theoretical Physics

Representation Theorem of Observables on a Quantum System

Oľga Nánásiová, Andrei Yu. Khrennikov

Published Mar 1, 2006• 24 citations
Publisher
DOI
10.1007/s10773-006-9030-6

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This paper interprets Haag’s theorem as more than a technical obstruction to the interaction picture in relativistic quantum field theory. It argues that the theorem reveals a deeper structural limitation of global representation-based ontology: an interacting QFT cannot, under the assumptions of the theorem, be understood as a simple unitary deformation of a free theory within one fixed Fock representation. The paper connects this lesson to algebraic quantum field theory, superselection sectors, locally covariant QFT, Wilsonian renormalization, and effective field theory. It proposes that these developments all point toward a common principle: physical objectivity is not grounded in a privileged representation, but in structural content preserved under admissible transformations between perspectives, representations, sectors, scales, or regions. The paper does not claim to solve Haag’s theorem or construct interacting QFTs. Its aim is interpretive: to show how Haag’s theorem is naturally intelligible within the observer-equivariant framework, where objectivity is understood as invariant or equivariant structure rather than as a view from one absolute representation.
Motivated by models of mixed quantum-classical hybrid theories, we consider the representation of classical observables as Hilbert-space operators based on van Hove's unitary representation of contact transformations. This leads to a reformulation of classical mechanics where the phase space density satisfies the Liouville equation and, more importantly, classical observables are represented by van Hove operators that satisfy a commutation algebra isomorphic to the classical Poisson algebra of functions in phase space. Thus, this Hilbert space approach captures all the essential features of the phase space formulation of classical mechanics in that it reproduces its algebraic structure -- something that is not achieved by other approaches such as the Koopman-von Neumann theory. As an important feature, the definition of appropriate physical states in this formalism imposes certain natural requirements on the phase of the classical wave function, which ensure correct expectation values for the observables. Furthermore, we discuss how to extend the approach to the hybrid van Hove theory of mixed classical-quantum systems and illustrate its application with the example of interacting classical and quantum oscillators. Finally, we focus on the question of how hybrid van Hove theory evades no-go theorems on mixed classical-quantum systems and point to the various assumptions of the no-go theorems that do not apply to our approach. This demonstrates that these no-go theorems are not universally applicable, contrary to their original claims.
In this article, it is argued that the Gibbs-Liouville theorem is a mathematical representation of the statement that closed classical systems evolve deterministically. From the perspective of an observer of the system, whose knowledge about the degrees of freedom of the system is complete, the statement of deterministic evolution is equivalent to the notion that the physical distinctions between the possible states of the system, or, in other words, the information possessed by the observer about the system, is never lost. Thus, it is proposed that the Gibbs-Liouville theorem is a statement about the dynamical evolution of a closed classical system valid in such situations where information about the system is conserved in time. Furthermore, in this article it is shown that the Hamilton equations and the Hamilton principle on phase space follow directly from the differential representation of the Gibbs-Liouville theorem, i.e. that the divergence of the Hamiltonian phase flow velocity vanish. Thus, considering that the Lagrangian and Hamiltonian formulations of classical mechanics are related via the Legendre transformation, it is obtained that these two standard formulations are both logical consequences of the statement of deterministic evolution, or, equivalently, information conservation.