Hilbert Space Of States Research Articles

The Riesz basis property of discrete operators and application to a Euler-Bernoulli beam equation with boundary linear feedback control

In this paper, we give an abstract condition of Riesz basis generation for discrete operators in Hilbert spaces, from which we show that the generalized eigenfunctions of a Euler–Bernoulli beam equation with boundary linear feedback control form a Riesz basis for the state Hilbert space. As an consequence, the asymptotic expression of eigenvalues together with exponential stability are readily presented.

Riesz Basis Approach to the Stabilization of a Flexible Beam with a Tip Mass

Using an abstract condition of Riesz basis generation of discrete operators in the Hilbert spaces, we show, in this paper, that a sequence of generalized eigenfunctions of an Euler--Bernoulli beam equation with a tip mass under boundary linear feedback control forms a Riesz basis for the state Hilbert space. In the meanwhile, an asymptotic expression of eigenvalues and the exponential stability are readily obtained. The main results of [ SIAM J. Control Optim., 36 (1998), pp. 1962--1986] are concluded as a special case, and the additional conditions imposed there are removed.

Canonical and quantum FRW cosmological solutions inM-theory

We present a canonical and quantum cosmological investigation of aspatially flat, four-dimensional Friedmann-Robertson-Walker(FRW) model that is derived from the M-theory effective action obtainedoriginally by Billyard, Coley, Lidsey and Nilsson (BCLN). The analysismakes use of two sets of canonical variables, the Shanmugadhasan gauge-invariantcanonical variables and the `hybrid' variables which diagonalize the Hamiltonian. Wefind the observables and discuss in detail the phase space of the classical theory.In particular, a region of phase space exists that describes a four-dimensionalFRW spacetime first contracting from a strong-coupling regime and then expanding toa weak-coupling regime, while the internal space alwayscontracts. We find the quantum solutions of the model andobtain the positive norm Hilbert space of states. Finally, thecorrespondence between wavefunctions and classical solutionsis outlined.

On the coherent states, displacement operators and quasidistributions associated with deformed quantum oscillators

The Wigner (W), Husimi-Kano (Q) and Glauber-Sudarshan (P) quasidistributions are generalized to f-deformed quasidistributions which extend the parametric family of s-ordered quasidistributions of Cahill and Glauber. The deformation procedure is obtained via a canonical nonisometric transform of the displacement operators which preserves the form of the standard creation-annihilation commutation relation, hence the Heisenberg-Weyl algebra, but changes the scalar product in the Hilbert space of the oscillator states. A whole class of new resolutions of the identity is introduced. The time evolution equation for the new generalized quasidistributions is derived.

Hodge decomposition theorem for Abelian two-form gauge theory

We show that the BRST/anti-BRST invariant 3+1 dimensional 2-form gauge theory has further nilpotent symmetries (dual BRST /anti-dual BRST) that leave the gauge fixing term invariant. The generator for the dual BRST symmetry is analogous to the co-exterior derivative of differential geometry. There exists a bosonic symmetry which keeps the ghost terms invariant and it turns out to be the analogue of the Laplacian operator. The Hodge duality operation is shown to correspond to a discrete symmetry in the theory. The generators of all these continuous symmetries are shown to obey the algebra of the de Rham cohomology operators of differential geometry. We derive the extended BRST algebra constituted by six conserved charges and discuss the Hodge decomposition theorem in the quantum Hilbert space of states.

Functional integral formulation of the Thirring model with two fermion species

A two-dimensional Abelian model with quartic interaction among two species of Fermi fields is analyzed through a functional integral formulation. We consider the bosonization of the generating functional of the model in order to establish the fermion-boson mapping in the Hilbert space of states. The grand partition function of the statistical mechanical system associated with the effective bosonized theory is obtained and the corresponding exact equation of state exhibits a critical line characterizing a Kosterlitz-Thouless phase transition.

Structural aspects of functional integral bosonization

We discuss structural aspects of functional integral bosonization of two-dimensional models. We show that the use of auxiliary vector fields enlarges the Hilbert space by the introduction of an external field algebra that should not be considered as an element of the intrinsic algebraic structure defining the model. These aspects are discussed in a model with well known and established results in the literature, by considering the Abelian reduction of the Wess-Zumino-Witten theory to reconstruct in the Hilbert space of states Coleman's proof of the fermion-boson mapping between the massive Thirring and sine-Gordon theories. We show that the factorization of the partition function will generally lead to incorrect conclusions concerning the physical content of the model, such as the existence of infinitely delocalized states and the violation of the asymptotic factorization property of the Wightman functions. In order to exert control on the effect of the redundant Bose fields and obtain the fermion-boson mapping in the Hilbert space of states, the functional integral bosonization must be performed on the generating functional.

Quantum gravity as a dissipative deterministic system

It is argued that the so-called holographic principle will obstruct attempts to produce physically realistic models for the unification of general relativity (GR) with quantum mechanics, unless determinism in the latter is restored. The notion of time in GR is so different from the usual one in elementary particle physics that we believe that certain versions of hidden-variable theories can - and must - be revived. A completely natural procedure is proposed, in which the dissipation of information plays an essential role. Unlike earlier attempts, it allows us to use strictly continuous and differentiable classical field theories as a starting point (although discrete variables, leading to fermionic degrees of freedom, are also welcome), and we show how an effective Hilbert space of quantum states emerges naturally when one attempts to describe the solutions statistically. Our theory removes some of the mysteries of the holographic principle; apparently non-local features are to be expected when the quantum degrees of freedom of the world are projected onto a lower-dimensional black hole horizon. Various examples and models illustrate the points we wish to make, notably a model showing that massless, non-interacting neutrinos are deterministic.

On the relation between Schrödinger and von Neumann equations

The relation between the density matrix obeying the von Neumann equation and the wave function obeying the Schrodinger equation is discussed in connection with the superposition principle of quantum states. The definition of the ray-addition law is given, and its relation to the addition law of vectors in the Hilbert space of states and the role of a constant phase factor of the wave function is elucidated. The superposition law of density matrices, Wigner functions, and tomographic probabilities describing quantum states in the probability representation of quantum mechanics is studied. Examples of spin-1/2 and Schrodinger-cat states of the harmonic oscillator are discussed. The connection of the addition law with the entanglement problem is considered.

Riesz basis generation of discrete operators and application to a linear thermoelastic system

In this paper, we give a new result of Riesz basis generation property for discrete operators in Hilbert spaces, from which, we show that the generalized eigenfunctions of a one-dimensional linear thermoelastic system with Dirichlet-Dirichlet boundary condition form a Riesz basis in the state Hilbert space. This develops a parallel result which is already known for the same systems with Dirichlet-Neumann or Neumann-Dirichlet boundary conditions [S.W.Hansen, J.Math.Anal.Appl., 167(1992), 429-442].

Derivative state analysis of intramolecular vibrational energy redistribution of acetylene

The intramolecular vibrational energy redistribution (IVR) of acetylene in the electronically ground state is analyzed by a novel method called derivative state analysis [K. Someda, Bull. Chem. Soc. Jpn. 69, 3037 (1996)]. The derivative state analysis clarifies how the Hilbert space of quantum states are explored in IVR, and we are thereby able to classify the quantum states and to know their genealogy. The Hamiltonian presented by Temsamani and Herman [M. A. Temsamani and M. Herman, J. Chem. Phys. 102, 6371 (1995)] is adopted in the analysis. The IVR from three different types of initial vibrational states are analyzed: (1) The initial states excited in the trans-bending (ν4) mode undergo a rapid and instantaneous IVR when v4⩾10. (2) The IVR from the states with the CH antisymmetric stretch (ν3) excitation has an onset at v3=6 and is sequential. The vibrational energy is transmitted from the ν3 mode to the CC stretch (ν2) in the first stage and then to the trans- and cis-bending. (3) Simultaneous excitation in ν2 and ν4 modes also gives rise to sequential IVR, in which the energy flows via cis-bending (ν5) to symmetric CH stretch (ν1) and ν2.

Darboux transformation of the coherent states of a nonrelativistic free particle

Various systems of soliton-potential coherent states have been derived. Measures are found that realize the expansion of a unit operator in terms of projectors on these states. Two different realizations of Hilbert state space are constructed as functions holomorphic throughout the complex plane and with a certain constraint on their increase.

Bulk versus boundary dynamics in anti–de Sitter spacetime

We investigate the details of the bulk-boundary correspondence in Lorentzian signature anti--de Sitter space. Operators in the boundary theory couple to sources identified with the boundary values of non-normalizable bulk modes. Such modes do not fluctuate and provide classical backgrounds on which bulk excitations propagate. Normalizable modes in the bulk arise as a set of saddlepoints of the action for a fixed boundary condition. They fluctuate and describe the Hilbert space of physical states. We provide an explicit, complete set of both types of modes for free scalar fields in global and Poincar\'e coordinates. For ${\mathrm{AdS}}_{3},$ the normalizable and non-normalizable modes originate in the possible representations of the isometry group $\mathrm{SL}(2,R{)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{SL}(2,R{)}_{R}$ for a field of given mass. We discuss the group properties of mode solutions in both global and Poincar\'e coordinates and their relation to different expansions of operators on the cylinder and on the plane. Finally, we discuss the extent to which the boundary theory is a useful description of the bulk spacetime.

The quantum tetrahedron in 3 and 4 dimensions

Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries of the tetrahedron in R^3, we use geometric quantization to obtain a Hilbert space of states. This Hilbert space has a basis of states labeled by the areas of the faces of the tetrahedron together with one more quantum number, e.g. the area of one of the parallelograms formed by midpoints of the tetrahedron's edges. Repeating the procedure for the tetrahedron in R^4, we obtain a Hilbert space with a basis labelled solely by the areas of the tetrahedron's faces. An analysis of this result yields a geometrical explanation of the otherwise puzzling fact that the quantum tetrahedron has more degrees of freedom in 3 dimensions than in 4 dimensions.

Quantum black hole entropy

We derive an exact formula for the dimensionality of the Hilbert space of the boundary states of SU(2) Chern-Simons theory, which, according to the recent work of Ashtekar et al., leads to the Bekenstein-Hawking entropy of a four dimensional Schwarzschild black hole. Our result stems from the relation between the (boundary) Hilbert space of the Chern-Simons theory with the space of conformal blocks of the Wess-Zumino model on the boundary 2-sphere.

Diffusive Realisations of Fractional Integrodifferential Operators: Structural Analysis Under Approximation

Fractional integrals and derivatives are usually presented as non-standard convolutions, but they are also linked to classical diffusive operators on a Hilbert state space of infinite dimension. A variety of equivalent realisations are proposed, which can naturally be approximated by truncation in finite dimension. This process gives rise to convergent approximation algorithms from which some numerical simulations are derived.

Quantum spin dynamics (QSD): III. Quantum constraint algebra and physical scalar product in quantum general relativity

This paper deals with several technical issues of non-perturbative four-dimensional Lorentzian canonical quantum gravity in the continuum that arose in connection with the recently constructed Wheeler-DeWitt quantum constraint operator. The Wheeler-DeWitt constraint of quantum general relativity mixes the diffeomorphism superselection sectors for diffeomorphism-invariant theories of connections that were previously discussed in the literature. From it one can construct diffeomorphism-invariant operators which do not necessarily commute with the Hamiltonian constraint but which still mix those sectors and which, at the diffeomorphism-invariant level, encode physical information. Thus, if one adopts, as before in the literature, the strategy to solve the diffeomorphism constraint before the Hamiltonian constraint then those sectors become spurious. The inner product for diffeomorphism-invariant states can be fixed by requiring that diffeomorphism group averaging is a partial isometry. The established non-anomalous constraint algebra is clarified by computing commutators of duals of constraint operators. The full classical constraint algebra is faithfully implemented on the diffeomorphism-invariant Hilbert space in an appropriate sense. The Hilbert space of diffeomorphism-invariant states can be made separable if a natural new superselection principle is satisfied. We propose a natural physical scalar product for quantum general relativity by extending the group-average approach to the case of non-self-adjoint constraint operators like the Wheeler-DeWitt constraint. Equipped with this inner product, the construction of physical observables is straightforward.

Unification of Quantum Theory and Relativity

A underlying dynamical structure for bothrelativity and quantum theory —“superrelativity” — has been proposedin order to overcome the well-known incompatibilitybetween these theories. The relationship between curvature of spacetime (gravity) andcurvature of the projective Hilbert space of purequantum states is established as well.

On the Global Dynamics of a Controlled Viscous Burgers' Equation

In this paper we consider a boundary control problem for a forced Burgers' equation in a Hilbert state space consisting of square integrable functions on a finite interval. Our first main result consists in proving the global in time existence of solutions of the closed loop boundary control system for arbitrary L2 initial data and quite general forcing terms (disturbances). For the unforced problem, our main interest is in stability of equilibria, while for the forced problem we are interested in steady state behavior of solutions. We note that the uncontrolled (open loop), unforced problem is not asymptotically stable on L2. However, for positive gains we show that the unforced closed loop system is globally Lyapunov stable and locally exponentially stable. In addition, for nonzero stationary forcing, we show that there is a local absorbing ball and that the corresponding nonlinear semigroup is compact for all t > 0. Using this fact, we are able to deduce several consequences regarding the existence and properties of local attractors. Our method of proof is based on a systematic investigation of the smoothing properties of the controlled dynamics. In particular we establish various regularity results for the dynamics of the closed loop system.

On diagonalization of a composite-system observable. Separability

We develop a general algebraic method for investigating the diagonalizability of a composite-system observable in the, so-called, noncorrelated basis of the Hilbert state space of a composite ("two-body") quantum system. The method consists in rewriting a composite-system obsrvable in linear form, for which one directly and uniquely obtains an answer concerning the diagonalizability. The method has been applied to the analysis of the concept of separability of the interaction part of the Hamiltonian of the composite system, "quantum system plus environment (apparatus)". The application leads to a new (more precise) operational definition of separability, which represents a necessary condition concerning the occurrence of decoherence in the, so-called, orthodox approach to the theory of decoherence.