Abstract The focus of this work is a correspondence between the Hilbert space operators on one hand, and doubly periodic generalized functions on the other. The linear map that implements it, referred to as the Q-transform, enables a direct application of the classical Harmonic analysis in a study of quantum systems. In particular, the Q-transform makes it possible to reinterpret the dynamic of a quantum observable as a (typically nonlocal) dynamic of a doubly periodic real function. From this point of view we carry out an analysis of an open quantum system whose dynamics are governed by an asymptotically harmonic Hamiltonian and compact type Lindblad operators. It is established that the initial value problem of the equivalent nonlocal but classical evolution is well posed in the appropriately chosen Sobolev spaces. The second set of results pertains to a generalization of the basic Q-transform and highlights a certain type of asymptotic redundancy. This phenomenon, referred to as the broadband redundancy , is a consequence of a well-known property of the zeros of the Riemann zeta function, namely, the uniform distribution modulo one of their ordinates. Its relevance to the analysis of quantum dynamics is only a special instance of its utility in harmonic analysis in general. It remains to be seen if the phenomenon is significant also in the physical sense, but it appears well-justified—in particular, by the results presented here—to pose such a question.
Abstract The focus of this work is a correspondence between the Hilbert space operators on one hand, and doubly periodic generalized functions on the other. The linear map that implements it, referred to as the Q-transform, enables a direct application of the classical Harmonic analysis in a study of quantum systems. In particular, the Q-transform makes it possible to reinterpret the dynamic of a quantum observable as a (typically nonlocal) dynamic of a doubly periodic real function. From this point of view we carry out an analysis of an open quantum system whose dynamics are governed by an asymptotically harmonic Hamiltonian and compact type Lindblad operators. It is established that the initial value problem of the equivalent nonlocal but classical evolution is well posed in the appropriately chosen Sobolev spaces. The second set of results pertains to a generalization of the basic Q-transform and highlights a certain type of asymptotic redundancy. This phenomenon, referred to as the broadband redundancy , is a consequence of a well-known property of the zeros of the Riemann zeta function, namely, the uniform distribution modulo one of their ordinates. Its relevance to the analysis of quantum dynamics is only a special instance of its utility in harmonic analysis in general. It remains to be seen if the phenomenon is significant also in the physical sense, but it appears well-justified—in particular, by the results presented here—to pose such a question.
The focus of this work is a correspondence between the Hilbert space operators on one hand, and doubly periodic generalized functions on the other. The linear map that implements it, referred to as the Q-transform, enables a direct application of the classical Harmonic Analysis in a study of quantum systems. In particular, the Q-transform makes it possible to reinterpret the dynamic of a quantum observable as a (typically nonlocal) dynamic of a classical observable. From this point of view we carry out an analysis of an open quantum system whose dynamics are governed by an asymptotically harmonic Hamiltonian and compact type Lindblad operators. It is established that the initial value problem of the equivalent nonlocal but classical evolution is well posed in the appropriately chosen Sobolev spaces. The second set of results pertains to a generalization of the basic Q-transform and highlights a certain type of asymptotic redundancy. This phenomenon, referred to as the broadband redundancy, is a consequence of a well-known property of the zeros of the Riemann zeta function, namely, the uniform distribution modulo one of their ordinates. Its relevance to the analysis of quantum dynamics is only a special instance of its utility in harmonic analysis in general. It remains to be seen if the phenomenon is significant also in the physical sense, but it appears well-justified---in particular, by the results presented here---to pose such a question.
A correspondence is established between measure-preserving, ergodic dynamics of a classical harmonic oscillator and a quantum mechanical gauge theory on two-dimensional Minkowski space. This correspondence is realized through an isometric embedding of the L2(μ) space on the circle associated with the oscillator’s invariant measure, μ, into a Hilbert space H of sections of a C-line bundle over Minkowski space. This bundle is equipped with a covariant derivative induced from an SO+(1, 1) gauge field on the corresponding inertial frame bundle, satisfying the Yang–Mills equations. Under this embedding, the Hamiltonian operator of a Lorentz-invariant quantum system, constructed as a geometrical Laplace-type operator on bundle sections, pulls back to the generator of the unitary group of Koopman operators governing the evolution of classical observables of the harmonic oscillator, with Koopman eigenfunctions of zero, positive, and negative eigenfrequencies corresponding to quantum eigenstates of zero (“vacuum”), positive (“matter”), and negative (“antimatter”) energies. The embedding also induces a pair of operators acting on classical observables of the harmonic oscillator, exhibiting canonical position–momentum commutation relationships. These operators have the structure of order-1/2 fractional derivatives and therefore display a form of non-locality. In a second part of this work, we study a quantum mechanical representation of the classical harmonic oscillator using a one-parameter family of reproducing kernel Hilbert spaces, K^τ, associated with the time-τ transition kernel of a fractional diffusion on the circle. As shown in recent work, these spaces are unital Banach *-algebras of functions. It is found that the evolution of classical observables in these spaces takes place via a strongly continuous, unitary Koopman evolution group, which exhibits a stronger form of classical–quantum consistency than the L2(μ) case. Specifically, for every real-valued classical observable in K^τ, there exists a quantum mechanical observable, whose expectation value is consistent with classical function evaluation. This allows for a description of classical state space dynamics, classical statistics, and quantum statistics of the harmonic oscillator within a unified framework.
The Quantum Harmonic Oscillator is one of the most important models in Quantum Mechanics. Analogous to the classical mass vibrating back and forth on a spring, the quantum oscillator system has attracted substantial attention over the years because of its importance in many advanced and difficult quantum problems. This dissertation deals with solving generalized models of the time-dependent Schrodinger equation which are called generalized quantum harmonic oscillators, and these are characterized by an arbitrary quadratic Hamiltonian of linear momentum and position operators. The primary challenge in this work is that most quantum models with timedependence are not solvable explicitly, yet this challenge became the driving motivation for this work.
In this dissertation, the methods used to solve the time-dependent Schrodinger equation are the fundamental singularity (or Green's function) and the Fourier (eigenfunction expansion) methods. Certain Riccati- and Ermakov-type systems arise, and these systems are highlighted and investigated. The overall aims of this dissertation are to show that quadratic Hamiltonian systems are completely integrable systems, and to provide explicit approaches to solving the time-dependent SchrDodinger equation governed by an arbitrary quadratic Hamiltonian operator. The methods and results established in the dissertation are not yet well recognized in the literature, yet hold for high promise for further future research.
Finally, the most recent results in the dissertation correspond to the harmonic oscillator group and its symmetries. A simple derivation of the maximum kinematical invariance groups of the free particle and quantum harmonic oscillator is constructed from the view point of the Riccati- and Ermakov-type systems, which shows an alternative to the traditional Lie Algebra approach. To conclude, a missing class of solutions of the time-dependent Schrodinger equation for the simple harmonic oscillator in one dimension is constructed. Probability distributions of the particle linear position and momentum, are emphasized with Mathematica animations. The eigenfunctions qualitatively differ from the traditional standing waves of the one-dimensional Schrodinger equation. The physical relevance of these dynamic states is still questionable, and in order to investigate their physical meaning, animations could also be created for the squeezed coherent states. This will be addressed in future work.
The quantum dynamics of arbitrary N-level systems, including dissipative systems, are modeled exactly here by the dynamics of classical coupled oscillators. A one-to-one correspondence is established between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. Although the well-known equivalence [SU(2), SO(3) homomorphism] of two-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary N-level systems, the representation of quantum dynamics by a system of coupled harmonic oscillators in one physical dimension is general for any N. The time evolution of an N-level system [generated by a complex element of the SU(N) group], is first represented as the rotation of a real state vector in (unphysical) hyperspace, as previously known for density matrix states and also extended here to include Schr\"odinger states. The resulting rotor in n Euclidean dimensions [the rotation group SO(n)] is then mapped directly to n oscillators in one physical dimension, which significantly reduces the level of abstraction required to visualize quantum dynamics compared to vector models or generalized Bloch spheres in higher dimensions. The number of such oscillators needed to represent N-level systems scales as N2 for the density matrix formalism but increases only linearly with N for Schr\"odinger states. Values for the classical coupling constants are readily derived from the system Hamiltonian, allowing construction of classical mechanical systems that provide insight into the dynamics of abstract quantum systems (new dynamical invariants) as well as a metric for characterizing the interface between quantum and classical mechanics. A distinctive attribute of the quantum-classical connection as presented here is the necessity for both positive and negative couplings and, in the case of dissipative systems, antisymmetric couplings.
In this work, we are in the position to view a measurement of a physical observable as an experiment in the sense of probability theory. To every physical observable, a sample space called the spectrum of the observable is therefore available. We have investigated the Wasserstein spaces over the spectrums of a quantum observables, i.e. the space of all probability measures defined on the suitable σ -algebra of subsets of the spectrum equipped with certain metric. It has been shown here the existence of one-to-one correspondence between the set of all quantum states and the set of all distribution functions defined on the spectrum of an observable. Thanks to the correspondence, every quantum state can be represented by a probability measure associated with the corresponding distribution function. It has been shown that the Wasserstein space over the spectrum of an observable is homeomorphic to the Wasserstein space over the spectrum of another observable. Such homeomorphism is referred to as generalized Fourier transform. By using the work of von Renesse, it has been shown that the Newtonian dynamics of the system can be formulated in the Wasserstein space over the spectrum of each observable which is equivalent to the Schroedinger dynamics, starting from the solution of Newton equation of motion in the Wasserstein space over the spectrum of a certain observable.
Functions of positive type on locally compact abelian groups can be defined as positive functionals on group algebras, and play a remarkable role in probability theory and in classical statistical mechanics. By Bochner's celebrated theorem, indeed, they are Fourier-Stieltjes transforms of finite positive measures. Hence, a properly normalized nonzero function of positive type on (the group of translations on) phase space provides a realization of a classical state, so it may be called a function of classical positive type. A similar result holds in the quantum setting as well, where a generalized kind of functions of positive type on phase space — the so-called functions of quantum positive type — are related, via the Fourier-Plancherel transform, to the Wigner quasi-probability distributions. In this paper, we will argue that, as in the classical setting, the notion of function of quantum positive type is of a group-theoretical nature. Exploiting an interesting interplay between functions of classical and quantum positive type, we will then provide an interesting characterization of a class of semi-groups of operators that describe the evolution of certain open quantum systems which are of interest in quantum information science. Finally, a suitable extension of this framework to generalized phase spaces that are relevant for current applications will be proposed.