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This paper establishes a unified mathematical framework for density-functional theory in finite-dimensional quantum systems, enabling structural proofs to be applied broadly across various functional theories. By defining a minimal scope based on generalized observables and Hamiltonian components, the work facilitates the analysis of universal functionals, representability, and uniqueness within a rigorous variational context. The systematic formulation connects Lie-algebra observable structures to symplectic geometry, offering a foundational advance for theoretical quantum physics and computational modeling.
Unified Framework for Functional Theories of Quantum Systems
arXiv:2606.06676v1 Announce Type: cross Abstract: We introduce and study a unified framework for density-functional theory and its variants for quantum systems on finite-dimensional Hilbert spaces. These theories seek to reduce the complexity inherent in the many-body quantum problem by describing ground states through reduced variables. The central ingredients of our unified framework are a generalized choice of basic observables, whose expectation values define precisely those reduced variables, and a fixed part of the Hamiltonian characterizing the class of quantum systems under consideration. It is this minimal structure, which we call the scope of a functional theory, that is necessary and sufficient for the formulation of a functional theory. In particular, it allows one to define the universal functionals, establish their convexity and differentiability properties, address representability questions, and prove a Hohenberg-Kohn-type uniqueness result. A purification construction also relates ensemble and weighted-ensemble functionals to the pure-state variant. Particular emphasis is placed on functional theories with Lie-algebra observable structures, connecting the variational framework to symplectic geometry. The result of this work is a systematic mathematical formulation in which structural results can be proved once and applied across a broad class of finite-dimensional functional theories.
Abstract:We introduce and study a unified framework for density-functional theory and its variants for quantum systems on finite-dimensional Hilbert spaces. These theories seek to reduce the complexity inherent in the many-body quantum problem by describing ground states through reduced variables. The central ingredients of our unified framework are a generalized choice of basic observables, whose expectation values define precisely those reduced variables, and a fixed part of the Hamiltonian characterizing the class of quantum systems under consideration. It is this minimal structure, which we call the scope of a functional theory, that is necessary and sufficient for the formulation of a functional theory. In particular, it allows one to define the universal functionals, establish their convexity and differentiability properties, address representability questions, and prove a Hohenberg-Kohn-type uniqueness result. A purification construction also relates ensemble and weighted-ensemble functionals to the pure-state variant. Particular emphasis is placed on functional theories with Lie-algebra observable structures, connecting the variational framework to symplectic geometry. The result of this work is a systematic mathematical formulation in which structural results can be proved once and applied across a broad class of finite-dimensional functional theories.
arXiv:2606.06676 [math-ph]
(or arXiv:2606.06676v1 [math-ph] for this version)
https://doi.org/10.48550/arXiv.2606.06676 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 278 words
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