We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous representations in a nested scale of Hilbert spaces. We also construct a couple of examples illustrative of the key features of group representations in rigged Hilbert spaces. Finally, we establish a simple criterion for the integrability of an operator Lie algebra in a rigged Hilbert space.

Inspired by the pioneer work of H.L. Resnikoff, which is described in full detail in the first part of this two-part paper, we give a quantum description of the space [Formula: see text] of perceived colors. We show that [Formula: see text] is the effect space of a rebit, a real quantum qubit, whose state space is isometric to Klein's hyperbolic disk. This chromatic state space of perceived colors can be represented as a Bloch disk of real dimension 2 that coincides with Hering's disk given by the color opponency mechanism. Attributes of perceived colors, hue and saturation, are defined in terms of Von Neumann entropy.

in Hilbert spaces, especially in the study of differential operators and in the mathematical formulation of quantum mechanics. An unbounded operator on The mathematical concept of a Hilbert space generalizes the notion of Euclidean space. It extends the methods of Euclidean geometry and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces of any finite or infinite dimension. A Hilbert space is an abstract vector space, and it has the additional structure of an inner product that allows length and angle to be mea where the functions φn are orthogonal in the sense that ⟨φn, φm⟩ = 0 for all n ≠ m. The individual terms in this series are sometimes referred to as elementary product solutions. However, there are eigenfunction expansions that fail to converge in a suitable sense to a square-integrable function: the missing ingredient, which ensures convergence, is completeness. The second development was the Lebesgue integral, an alternative to the Riemann integral introduced by Henri Lebesgue in 1904. The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that the space L2 of square Lebesgue-integrable functions is a complete metric space. As a consequence of the interplay between geometry and completeness, the 19th century results of Joseph Fourier, Friedrich Bessel and Marc-Antoine Parseval on trigonometric series easily carried over to these more general spaces, resulting in a geometrical and analytical apparatus now usually known as the Riesz–Fischer theorem. Further basic results were proved in the early 20th century. For example, the Riesz representation theorem was independently established by Maurice Fréchet and Frigyes Riesz in 1907. John von Neumann coined the term abstract Hilbert space in his work on unbounded Hermitian operators. Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from a physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them. Von Neumann later used them in his seminal work on the foundations of quantum mechanics, and in his continued work with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups. The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics. In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the…

In the first part of this paper the general perspective of history quantum theories is reviewed. History quantum theories provide a conceptual and mathematical framework for formulating quantum theories without a globally defined Hamiltonian time evolution and for introducing the concept of space time event into quantum theory. On a mathematical level a history quantum theory is characterized by the space of histories, which represent the space time events, and by the space of decoherence functionals which represent the quantum mechanical states in the history approach. The second part of this paper is devoted to the study of the structure of the space of decoherence functionals for some physically reasonable spaces of histories in some detail. The temporal reformulation of standard Hamiltonian quantum theories suggests to consider the case that the space of histories is given by (i) the lattice of projection operators on some Hilbert space or -- slightly more general -- (ii) the set of projection operators in some von Neumann algebra. In the case (i) the conditions are identified under which decoherence functionals can be represented by, respectively, trace class operators, bounded operators or families of trace class operators on the tensor product of the underlying Hilbert space by itself. Moreover we shall discuss the naturally arising representations of decoherence functionals as sesquilinear forms. The paper ends with a discussion of the consequences of the results for

The representation theory of decoherence functionals in history quantum theories In the first part of this paper the general perspective of history quantum theories is reviewed. History quantum theories provide a conceptual and mathematical framework for formulating quantum theories without a globally defined Hamiltonian time evolution and for introducing the concept of space time event into quantum theory. On a mathematical level a history quantum theory is characterized by the space of histories, which represent the space time events, and by the space of decoherence functionals which represent the quantum mechanical states in the history approach. The second part of this paper is devoted to the study of the structure of the space of decoherence functionals for some physically reasonable spaces of histories in some detail. The temporal reformulation of standard Hamiltonian quantum theories suggests to consider the case that the space of histories is given by (i) the lattice of projection operators on some Hilbert space or -- slightly more general -- (ii) the set of projection operators in some von Neumann algebra.

In this paper, we present a homotopical framework for studying invertible gapped phases of matter from the point of view of infinite spin lattice systems, using the framework of algebraic quantum mechanics. We define the notion of quantum state types. These are certain lax-monoidal functors from the category of finite dimensional Hilbert spaces to the category of topological spaces. The universal example takes a finite dimensional Hilbert space to the pure state space of the quasi-local algebra of the quantum spin system with this Hilbert space at each site of a specified lattice. The lax-monoidal structure encodes the tensor product of states, which corresponds to stacking for quantum systems. We then explain how to formally extract parametrized phases of matter from quantum state types, and how they naturally give rise to $\mathscr{E}_\infty$-spaces for an operad we call the "multiplicative" linear isometry operad. We define the notion of invertible quantum state types and explain how the passage to phases for these is related to group completion. We also explain how invertible quantum state types give rise to loop-spectra. Our motivation is to provide a framework for constructing Kitaev's loop-spectrum of bosonic invertible gapped phases of matter. Finally, as a first step towards understanding the homotopy types of the loop-spectra associated to invertible quantum state types, we prove that the pure state space of any UHF algebra is simply connected.

Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical

We establish a rigorous mathematical foundation for Zeckendorf-k-bonacci tensor (ZkT) representations and their embeddings in infinite-dimensional Hilbert spaces. By extending classical Zeckendorf decompositions to k-bonacci sequences with tensor structure, we construct a new class of mathematical objects that serve as the foundation for computational ontology frameworks. Key findings include: (1) ZkT tensors form well-defined mathematical structures with k× dimensions under binary, column-complementary, and no-k constraints; (2) these tensors naturally embed in non-separable Hilbert spaces for all k2, with separability only for finite k=1 systems; (3) the spectral properties of associated evolution operators exhibit kbonacci growth rates with characteristic roots approaching 2 as k→; (4) ZkT structures provide complete orthogonal bases for infinite-dimensional quantum systems; (5) information-theoretic entropy rates scale as log2(rk) where rk is the k-bonacci characteristic root. We prove convergence theorems for ZkT series expansions and establish the mathematical completeness of this framework for representing complex recursive systems. These results provide the essential mathematical infrastructure for observer-based computational theories and quantum tensor applications.