Hilbert Space Of States Research Articles

Ground-State Properties of Extended Two-Channel Kondo Model

Ground-state properties are examined for an extended two-channel Kondo model where the Hilbert space of the localized states is extended to include a singlet state in addition to the doublet states. By means of zero-th order variational wavefunctions with different symmetries, which are associated with the non-Fermi-liquid and the Fermi-liquid ground states, we demonstrate that the channel exchange coupling via the localized singlet state stabilizes the Fermi-liquid wavefunction. The ground-state phase diagrams, which are in qualitative agreement with the previous study performed by Koga and Shiba, are obtained. The comparison to the structure of the resultant wavefunctions suggests that a unique non-Fermi-liquid (Fermi-liquid) fixed point exists, irrespective of the localized ground state.

A combinatorial approach to diffeomorphism invariant quantum gauge theories

Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs. Then a family of graphs embedded in the space manifold (satisfying certain properties) induces a representation of the algebra of physical observables. We construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold, and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes. Even though the two models are different at the kinematical level, they provide unitarily equivalent representations of the algebra of physical observables in separable Hilbert spaces of physical states (their s-knot basis is countable). Hence, the combinatorial framework is compatible with the usual interpretation of quantum field theory.

Quanta Without Quantization

The dimensional properties of fields in classical general relativity lead to a tangent tower structure which gives rise directly to quantum mechanical and quantum field theory structures without quantization. We derive all the fundamental elements of quantum mechanics from the tangent tower structure, including fundamental commutation relations, a Hilbert space of pure and mixed states, measurable expectation values, Schrödinger time evolution, "collapse" of a state and the probability interpretation. The most central elements of string theory also follow, including an operator valued mode expansion like that in string theory as well as the Virasoro algebra with central charges.

Canonical quantization of the Gowdy model

The family of Gowdy universes with the spatial topology of a three-torus is studied both classically and quantum mechanically. Starting with the Ashtekar formulation of Lorentzian general relativity, we introduce a gauge fixing procedure to remove almost all of the non-physical degrees of freedom. In this way, we arrive at a reduced model that is subject only to one homogeneous constraint. The phase space of this model is described by means of a canonical set of elementary variables. These are two real, homogeneous variables and the Fourier coefficients for four real fields that are periodic in the angular coordinate which does not correspond to a Killing field of the Gowdy spacetimes. We also obtain the explicit expressions for the line element and reduced Hamiltonian. We then proceed to quantize the system by representing the elementary variables as linear operators acting on a vector space of analytic functionals. The inner product on that space is selected by imposing Lorentzian reality conditions. We find the quantum states annihilated by the operator that represents the homogeneous constraint of the model and construct with them the Hilbert space of physical states. Finally, we derive the general form of the quantum observables of the model.

Superrelativity as an element of a final theory

A underlying dynamical structure for both relativity and quantum theory-``superrelativity'' has been proposed in order to overcome the well known incompatibility between these theories. The relationship between curvature of spacetime (gravity) and curvature of the projective Hilbert space of pure quantum states is established as well.

Projection operator approach to constrained systems

Recently, within the context of the phase space coherent state path integral quantisation of constrained systems, John Klauder introduced a reproducing kernel for gauge invariant physical states, which involves a projection operator onto the reduced Hilbert space of physical states, avoids any gauge fixing conditions, and leads to a specific measure for the integration over Lagrange multipliers. Here, it is pointed out that this approach is also devoid of any Gribov problems and always provides for an effectively admissible integration over all gauge orbits of gauge invariant systems. This important aspect of Klauder's proposal is explicitly confirmed by two simple examples.

Quantum theory of geometry: I. Area operators

A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces are introduced and shown to be self-adjoint on the underlying (kinematical) Hilbert space of states. It is shown that their spectra are purely discrete, indicating that the underlying quantum geometry is far from what the continuum picture might suggest. Indeed, the fundamental excitations of quantum geometry are one dimensional, rather like polymers, and the three-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite-dimensional subspaces which can be interpreted as the spaces of states of spin systems. Using this property, the complete spectrum of the area operators is evaluated. The general framework constructed here will be used in a subsequent paper to discuss three-dimensional geometric operators, e.g. the ones corresponding to volumes of regions.

Derivative-State Analysis of a Sequential Intramolecular Vibrational Energy Re-distribution

Abstract The Mori expansion adapted to pure-state dynamics is shown to be useful for analyzing a sequential intramolecular vibrational energy re-distribution (IVR). The expansion produces non-stationary basis states, “derivative states”, which form a chain of mutually interacting quantum states beginning at a given initial state. An analysis based on this chain clearly demonstrates how the Hilbert space of quantum states is explored in sequential IVR. The derivative states can be classified into several “generations”, and, thereby, a genealogy of quantum states can be discussed.

Family of geometric phases.

A family of geometric phases associated with the evolution of a quantum system is introduced. These phases share the property of depending on the curve determined by the Schr\"odinger equation as well as on the choice of an orthonormal basis in the Hilbert space of states. Their role in the expansion of the wave function is examined, and a comparison with previous results about Berry phases is made. \textcopyright{} 1996 The American Physical Society.

HAMILTONIAN FORMALISM FOR BLACK HOLES AND QUANTIZATION II

We quantize by the Dirac-Wheeler-DeWitt method the canonical formulation of the Schwarzschild black hole developed in a previous paper. We investigate the properties of the operators that generate rigid symmetries of the Hamiltonian, establish the form of the invariant measure under the rigid transformations, and determine the gauge fixed Hilbert space of states. We also prove that the reduced quantization method leads to the same Hilbert space for a suitable gauge fixing.

On the nonnegative self-adjoint solutions of the operator Riccati equation for infinite dimensional systems

The nonnegative self-adjoint solutions of the operator Riccati equation (ORE) are studied for stabilizable semigroup Hilbert state space systems with bounded sensing and control. Basic properties of the maximal solution of the ORE are investigated: stability of the corresponding closed loop system, structure of the kernel, Hilbert-Schmidt property. Similar properties are obtained for the nonnegative self-adjoint solutions of the ORE. The analysis leads to a complete classification of all nonnegative self-adjoint solutions, which is based on a bijection between these solutions and finite dimensional semigroup invariant subspaces contained in the antistable unobservable subspace.

Scattering states of plektons (particles with braid group statistics) in 2+1 dimensional quantum field theory

A Haag-Ruelle scattering theory for particles with braid group statistics is developed, and the arising structure of the Hilbert space of multiparticle states is analyzed.

First quantized noncritical relativistic Polyakov string

The first quantization of the relativistic Brink-DiVecchia-Howe-Polyakov (BDHP) string in the range 1 < d < 25 is considered. It is shown that using the Polyakov sum over bordered surfaces in the Feynman path integral quantization scheme one gets a consistent quantum mechanics of relativistic one-dimensional extended objects in the range 1 < d < 25. In particular the BDHP string propagator is exactly calculated for arbitrary initial and final string configurations and the Hilbert space of physical states of noncritical BDHP string is explicitly constructed. The resulting theory is equivalent to the Fairlie-Chodos-Thorn massive string model. In contrast to the conventional conformal field theory approach to noncritical string and random surfaces in the Euclidean target space the path integral formulation of the Fairlie-Chodos-Thorn string obtained in this paper does not rely on the principle of conformal invariance. Some consequences of this feature for constructing a consistent relativistic string theory based on the “splitting-joining” interaction are discussed.

Wormholes as a basis for the Hilbert space in Lorentzian gravity

We carry out to completion the quantization of a Friedmann-Robertson-Walker model provided with a conformal scalar field, and of a Kantowski-Sachs spacetime minimally coupled to a massless scalar field. We prove that the Hilbert space determined by the reality conditions that correspond to Lorentzian gravity admits a basis of wormhole wave functions. This result implies that the vector space spanned by the quantum wormholes can be equipped with an unique inner product by demanding an adequate set of Lorentzian reality conditions, and that the Hilbert space of wormholes obtained in this way can be identified with the whole Hilbert space of physical states for Lorentzian gravity. In particular, all the normalizable quantum states can then be interpreted as superpositions of wormholes. For each of the models considered here, we finally show that the physical Hilbert space is separable by constructing a discrete orthonormal basis of wormhole solutions.

Universal Propagator for Group-Related Coherent States

In earlier work universal propagators were introduced for the Heisenberg —Weyl group, the affine group, and the rotation group. By generalizing these constructions we show here that it is possible to introduce a universal propagator for a rather general unitary Lie group. In the context of coherent-state representations, the universal propagator is a single function independent of any particular choice of fiducial vector, which nonetheless, propagates all coherent state Hilbert space representatives correctly. PACS numbers: 02.20.Qs, 03.65.Ca

Reality conditions in non-perturbative quantum cosmology

We carry out the non-perturbative canonical quantization of several types of cosmological models that have already been studied in the geometrodynamic formulation using the complex path-integral approach. We establish a relation between the choices of complex contours in the path integral and the sets of reality conditions for which the metric representation is well defined, proving that the ambiguity in the selection of complex contours disappears when one imposes suitable reality conditions. In most of the cases, the wavefunctions defined by means of the path integral turn out to be non-normalizable and cannot be accepted as proper quantum states. Moreover, the wavefunctions of the Universe picked out in quantum cosmology by the no-boundary condition and the tunneling proposals do not belong, in general, to the Hilbert space of quantum states. Finally, we show that different sets of reality conditions can lead to equivalent quantum theories. This fact enables us to extract physical predictions corresponding to Lorentzian gravity from quantum theories constructed with other than Lorentzian reality conditions.

Noncritical W-strings and minimal models

We perform a BRST analysis of the physical states described by a general noncriticalW-string. A crucial feature of our analysis is that we introduce a special basis in the Hilbert space of physical states in which the BRST operator splits into a nested sum of nilpotent BRST operators. We argue that the cohomology of each nilpotent BRST operator occurring in the “nested” sum is closely related to a specificW mimimal model. We discuss in detail the special case of the noncriticalW 3-string.

Construction of field algebras with quantum symmetry from local observables

It has been discussed earlier that ( weak quasi-) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra $\A$ of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey local braid relation.

Local non-abelian gauge symmetry of the Hubbard model

It is shown that the field operators of an electron system on a lattice can be decomposed into direct products of two kinds of operators acting in two separate Hilbert spaces. The Hilbert space of electron states thus becomes a direct product of two Hilbert spaces. By this fact a certain class of electron systems exhibits a formal separation of charge and spin degrees of freedom into two kinds of elementary excitations. A typical example of such a system is given by the Hubbard model. The separation of charge and spin resulting from the new representation of the field operators can be considered as a rigorous realization and generalization of an idea expressed by Anderson concerning the separation of spin and charge degrees of freedom in strongly correlated electron systems. The new representation of electron field operators implies the existence of a localU(2) gauge symmetry in the theory. The theory of superconductivity based on the Hubbard model is then represented by a non-abelian gauge field theory.

Inequalities on the time evolution of probabilities

The author considers probabilities pM(t) for quantum mechanical systems to be found in a subspace M of the Hilbert space of states and derive an upper bound on the change in time of PM(t) which contains only the Hamiltonian and the projector PM onto M but not the density matrix.