Transformations of states in operator theory and dynamics

Amil Abraham Thomas

Description TEQF: A Mathematical Framework for Adaptive Perturbation Operators, Normalized Transforms, and Decision-State Dynamics This research monograph presents the mathematical foundations, operator formulations, computational investigations, and proposed extensions of the TEQF (Temporal-Equivariant Quantum Field) framework. The research investigates adaptive perturbation operators acting on mathematical state spaces, with particular emphasis on Hilbert-space formulations, normalized transformations, information pathways, and state-dependent decision mechanisms. The monograph examines: Operator theory: Mathematical definitions, domains, codomains, perturbation mappings, and composition rules. Hilbert-space analysis: Linearity, boundedness, norm preservation, isometries, and associated mathematical conditions. Adaptive decision architecture: State-dependent pathway selection, decision landscapes, and perturbation-based state transitions. Normalized transforms: Mathematical relationships between perturbation operators and subsequent state transformations. Computational validation: Reproducible numerical experiments, randomized testing, mathematical consistency checks, and illustrative counterexamples. Physical interpretation: Exploratory connections to gravitational modelling, wave-state interactions, temporal dynamics, and other potential applications, without asserting experimental confirmation. The accompanying research package includes the consolidated mathematical monograph, historical source documents, original research sketches, supporting diagrams, Python validation scripts, and numerical test results. The computational experiments examine an illustrative adaptive-operator implementation rather than establish the empirical validity of a new physical theory. The work distinguishes established mathematical principles, proposed theoretical constructions, conditional proofs, and unresolved research questions. Research status: Independent mathematical research and computational investigation. The TEQF framework remains under development and has not been independently validated as a physical theory or production-ready computational system. Author: Amil Abraham ThomasResearch affiliation: AIDEN — Independent Mathematical Research and Systems ArchitectureYear: 2026 Keywords: TEQF, Adaptive Operators, Operator Theory, Hilbert Spaces, Perturbation Theory, Nonlinear Operators, Mathematical Transforms, Decision-State Dynamics, Computational Mathematics, Numerical Validation, Mathematical Physics, Applied Mathematics.

Lajos Molnár, Werner Timmermann

Eric Neuscamman, Takeshi Yanai, Garnet Kin‐Lic Chan

Canonical transformation (CT) theory describes dynamic correlation in multireference systems with large active spaces. Here we discuss CT theory's intruder state problem and why our previous approach of overlap matrix truncation becomes infeasible for sufficiently large active spaces. We propose the use of strongly and weakly contracted excitation operators as alternatives for dealing with intruder states in CT theory. The performance of these operators is evaluated for the H(2)O, N(2), and NiO molecules, with comparisons made to complete active space second order perturbation theory and Davidson-corrected multireference configuration interaction theory. Finally, using a combination of strongly contracted CT theory and orbital-optimized density matrix renormalization group theory, we evaluate the singlet-triplet gap of free base porphin using an active space containing all 24 out-of-plane 2p orbitals. Modeling dynamic correlation with an active space of this size is currently only possible using CT theory.

Boris Zelikovich Moroz

Xian Shi

We investigate resource nongenerating operations in both static and dynamical quantum resource theories. For the static scenarios, we derive a sufficient condition for state transformations under resource nongenerating operations. Then we construct a dynamical resource theory where resource nongenerating operations constitute the set of free operations, and we propose an axiomatic approach to quantify the dynamical resource. We further analyze the erasure of the dynamical resources. As applications, we establish bounds on the rate of state conversions under resource nongenerating operations in a generic convex resource theory and obtain capacity bounds for classical communication tasks assisted by dynamical coherence. Our results clarify the key roles of resource nongenerating operations in quantum information processing tasks.