Vectors In Hilbert Space Research Articles

Hypersensitivity to perturbation in the quantum kicked top.

For the quantum kicked top we study numerically the distribution of Hilbert-space vectors evolving in the presence of a small random perturbation. For an initial coherent state centered in a chaotic region of the classical dynamics, the evolved perturbed vectors are distributed essentially like random vectors in Hilbert space. In contrast, for an initial coherent state centered near an elliptic (regular) fixed point of the classical dynamics, the evolved perturbed vectors remain close together, explore only a few dimensions of Hilbert space, and do not explore them randomly. These results support and extend the results of earlier studies, thereby providing additional support for a characterization of quantum chaos that uses concepts from information theory.

Quantum Kinematic Approach to the Geometric Phase. I. General Formalism

A new approach to the theory of the geometric phase in quantum mechanics, based entirely on kinematic ideas, is developed. It is shown that a gauge and reparametrization invariant phase can be naturally associated with any smooth open curve of unit vectors in Hilbert space. Expression of this phase in terms of total and dynamical parts, rewriting it exclusively in terms of the density matrix, the special roles of geodesics and the Bargmann invariants all emerge very naturally. Application of the formalism to standard quantum evolution, adiabatic or otherwise, gives back familiar results and also answers questions relating to effects of unitary transformations and symmetries. An application to the case of light beams propagating along an arbitrary fibre, in the geometrical optic limit, is worked out in detail. Connections to differential geometric approaches, and extension to nonunitary evolution, are also worked out.

Hypersensitivity to perturbations in the quantum baker’s map

We analyze a randomly perturbed quantum version of the baker’s transformation, a prototype of an area-conserving chaotic map. By numerically simulating the perturbed evolution, we estimate the information needed to follow a perturbed Hilbert-space vector in time. We find that the Landauer erasure cost associated with this information grows very rapidly and becomes much larger than the maximum statistical entropy given by the logarithm of the dimension of Hilbert space. The quantum baker’s map thus displays a hypersensitivity to perturbations that is analogous to behavior found earlier in the classical case. This hypersensitivity characterizes “quantum chaos” in a way that is directly relevant to statistical physics.

The hilbert space formulation of the TLM method

AbstractThe Hilbert space representation of the TLM method for time‐domain computation of electromagnetic fields and the algebraic computation of the discrete Green's function are investigated. The complete field state is represented by a Hilbert space vector. The space and time evolution of the field state vector is governed by operator equations in Hilbert space. The discrete Green's functions may be represented by a Neumann series in space‐ and time‐shift operators. The Hilbert space representation allows the description of the geometric structures by projection operators, too. The system of difference equations governing the time evolution of the electromagnetic field in configuration space is derived from the operator equation for the field state vector in the Hilbert space.

Removal of singularities from collective-variable theory by incorporating relativistic invariance.

The equations of motion which govern the evolution of collective variables introduced into a nonlinear Klein-Gordon system are obtained by projecting the nonlinear field equation onto vectors in Hilbert space. For systems in the zero-kink sector, which admit breather and kink-antikink dynamics, the magnitude of some vectors onto which the field equation is projected become zero at particular instants in time. At these instants in time, the projections are not defined, and consequently the collective-variable equations of motion themselves are not defined: the transformation from the original field variables to the collective variables becomes singular. Although the singularity problem has already appeared in various contexts in the literature, it has not been resolved in the context of a collective-variable theory which contains collective variables in addition to that of the center of mass. For such cases, we show the singularities are removed by correctly taking into account the relativistic invariance of the underlying theory and give an example Ansatz which admits a well-defined collective-variable theory which would otherwise have been singular. We comment on the increased complexity of the present Ansatz relative to the utility of collective-variable theory.

Antinormally ordering some multimode exponential operators by virtue of the IWOP technique

The author shows that the technique of integration within an ordered product (IWOP) provides one with a very simple approach to deriving the normal product form of some multimode exponential operators, which greatly simplifies the calculations of normalising some state vectors in Hilbert space.

Time in a quantum mechanical world

In quantum mechanics it is usual to represent physical reality as a vector in Hilbert space at a particular time, with evolution being governed by Schrodinger's equation which involves an externally imposed time parameter. This leads to difficulties if one wishes to regard the universe as a quantum mechanical system, because there should no time external to such a system. The approach in this paper is to represent the totality of reality as a vector in Hilbert space. The author shows how time evolution follows, where time now is defined in terms of the states of a quantum mechanical clock which is part of the system. Rather than the correlation between the clock states and the states of the rest of the system arising because both are governed by an imposed law involving an external time parameter, it is seen that this correlation is of the separation-independent Einstein-Podolsky-Rosen type. The total reality vector, which incorporates the whole history of the system, is shown to be a zero-energy eigenstate of the system Hamiltonian. He discusses systems of finite and infinite lifetime, and is able to answer the question: what was the state before the initial state? He concludes that the quantum mechanical system of this paper is a reasonable representation of the observed universe.

Normally ordering some multimode exponential operators by virtue of the IWOP technique

The author shows that the technique of integration within an ordered product (IWOP) provides one with a very simple approach to deriving the normal product form of some multimode exponential operators, which greatly simplifies the calculations of normalising some state vectors in Hilbert space.

On the Monotonicity of Exact Upper Sequences for Sums of Independent Random Vectors in Hilbert Space

On the Monotonicity of Exact Upper Sequences for Sums of Independent Random Vectors in Hilbert Space

General setting for Berry's phase.

It is shown that Berry's phase appears in a more general context than realized so far. The evolution of the quantum system need be neither unitary nor cyclic and may be interrupted by quantum measurements. A key ingredient in this generalization is the use of some ideas introduced by Pancharatnam in his study of the interference of polarized light, which, when carried over to quantum mechanics, allow a meaningful comparison of the phase between any two nonorthogonal vectors in Hilbert space.

Parallelism between quantum mechanics and theory of stochastic processes

It is shown that the state vector in many-particle Hilbert space corresponds to the probability density of a stationary continuous Markov process where those expansion coefficients of the state vector which vary slowly in time and their complex conjugated quantities play the role of the random variables.

Covariant Quantization of Spinor Fields in a Given Gravitational Field

AbstractIn coupling gravity with the quantum field theory, unitary transformations, depending on space‐time‐points, were considered and derivatives were introduced, which imply a nonintegrable parallel transport of the state vectors of Hilbert space [1]. The Dirac equation, built with these generalized derivatives, is quantized in a prescribed classical gravitational field. The quantization can be performed in complete analogy to the usual procedure in Minkowski space, but the quantum state vector becomes path dependent. In carrying out the quantization, two two‐component classical spinor fields necessarily occur, which obey Weyl's equation. The considered quantized Dirac equations are also picture‐covariant, that is they have the same from in each physical picture, especially in the Heisenberg picture and the Schrödinger picture.

A Method of Calculating Bound States in a Unified Field Model

Abstract In a model in which the usual elementary particles (leptons, quarks, photons, weak bosons, gluons, and so on) are bound states of truly elementary fermions we present a method for the calculation of the masses of these bound states. The kinetic energy of this model contains derivatives of second order so that the four-fermion interaction becomes renormalizable. The method uses explicit representations for the Hilbert space vectors of bound states.

A new semiclassical approach to the molecular dynamics: Label variable classical mechanics

A new semiclassical approach to molecular collision dynamics is developed. In this approach the state of a system is described by a unit vector in Hilbert space. By choosing a reference unit vector and a continuous mapping, a correspondence between the vector in Hilbert space and the point in a label space is established, i.e., the vector in Hilbert space is parametrized or labeled by complex variables. It is shown that the label variables formally obey classical mechanics. Thus, the time evolution of the vectors in Hilbert space, i.e., the evolution of the states of the system, can be determined by calculating the evolution of the label variables classically. To illustrate this idea, the formalism for calculating the vibrational transition probability in the collinear collision A+BC is presented, and a demonstrative calculation for the collinear Secrest–Johnson model of He+H2 vibrationally inelastic scattering has been carried out. The comparison with the exact quantum results shows that the agreement is encouragingly good.

Zur theorie rotierender und schwingender schaufelkränze in einer unterschallstromung durch einen ringkanal

AbstractIn this paper the three‐dimensional perturbation flow induced by a rotating and oscillating blade row which operates in a subsonic flow in axial direction of an annular channel is studied. The velocity potential is reduced to the infinite Hilbert space vector of Fourier coefficients of an eigen‐function expansion with respect to vanishing normal derivatives on both cylinder walls. These coefficients satisfy an infinite set of ordinary differential equations of second order after an application of a one‐dimensional Fourier transform in axial direction.Several canonical two‐part mixed boundary value problems are then investigated by reduction to “infinite two‐by‐two‐Wiener‐Hopf functional systems”. In case of strong factorizability of certain matrix‐operator‐valued functions on the line these systems may be solved explicitely. Criteria for the factorization are not given here.

Erdös Conjecture on Sums of Distinct Numbers

A conjecture of Erdös that a set of n distinct numbers having the most linear combinations with coefficients 0,1 all equal are n integers of smallest magnitude is here proven. The result follows from a theorem of Stanley that implies that the integers from n have the most such linear combinations having k distinct values for every k. The same result is shown to hold for complex numbers and vectors in Hilbert space. It is shown that the number of linear combinations taking on k distinct values is maximized by the same configuration, for every k. Generalization to the case in which irregular distinctness restrictions are imposed is also given.

Generalization of the von Neumann theorems to irreproducible measurements within quantum field theory. IV

It is shown that the theorem of the first part of this cycle of papers necessarily, by virtue of only the fundamental principles of quantum theory, implies the existence of two fundamentally different types of evolution of isolated quantum macrosystems, i.e., the S and PS evolutions of pure states. As for isolated quantum microsystems, they can only S-evolve. The paper considers the fundamental specific singularities of PS evolution, which follow strictly from the theorem itself, as well as its corollary, proved in the first part [1] of this cycle of papers, also only on the basis of the fundamental principles of quantum theory. These singularities consist in the fact that, regardless of the commutativity of any observable\(\mathop \xi \limits^ \wedge\) of a system described by a vector in physical Hilbert space with the S-operator of this system, the probability distribution of this observable\(\mathop \xi \limits^ \wedge\) during measurement (with an appropriate instrument Ξ), providing complete information about the ξ characteristic of the system, is not conserved during the process of PS evolution. Nor is even the expectation of the result of measurement of the observable\(\mathop \xi \limits^ \wedge\) conserved with time.

Quantum axiomatics, representation theorem, and communicability of observations in physics

We show how a fundamental assumption in the Dirac formulation of quantum mechanics, namely that the states of a physical system at a particular time are mathematically represented by unit vectors in Hilbert space, can be deduced from certain aspects of our experimental procedures and of the observed outcome of quantum mechanical experiments. Our assumptions have clear empirical meaning and the results hold true for any dimensionality of the system, without anomalies in low dimensions which exist in the two well-known axiomatic approaches to quantum mechanics. The propositional logic approach of Birkhoff and von Neumann does not work for quantum systems of dimension less than four and requires an assumption which does not have an empirical basis. Jordan algebra axioms, on the other hand, also lead to anomalies in low dimensions and, moreover, are formal and cannot be directly physically interpreted. In our work it was possible to avoid these shortcomings.

Smoothing operators for field domains

We consider sharp-time fields and write down, in terms of the fields, simple and explicit expressions for operators with very strong smoothing properties. When applied to any vector in Hilbert space, the resulting vector is in the domain of any power of the space-smeared fields, and it even is entire for the fields. It is shown that in this way one obtains a common dense domain of definition on which the field operators are essentially self-adjoint. Attention is focused on the space S of rapidly decreasing C∞ functions as smearing functions for the fields; here the smoothing operators are simply products of exponentials of the field smeared with Hermite functions.

Atmospheric oscillations—V. The propagator matrix and the transmission of an electrostatic potential along the geomagnetic field lines

An account is given of a new mathematical formulation of the process of transmission of an electrostatic field along magnetic field lines, assuming the latter to be equipotentials of the electrostatic field. The theory is presented in terms of a dipole magnetic field, but is capable of extension to higher multipoles.By means of vector and matrix representations in Hilbert space, corresponding to the description of the field variables in terms of their spherical harmonic components, a propagator matrix is constructed which, given the harmonic expansion of the potential over a sphere of radius r o, enables the corresponding expansion at a general radius to be determined. Proposed areas of application of the technique include three-dimensional ionospheric dynamo theory, and tidal motions in the thermosphere which are excited by electric fields transmitted upwards from the dynamo region; in both cases an analytical rather than a geometrical or numerical specification of the potential is essential.