The Structure of the Algebra of Observables in the Intermediate Situation of the ∈-Model

We look at the action of the spin-1/2 operatorsof quantum mechanics on the state of an entity in aphysical way, and use this as a guideline to define theoperators of the intermediate situations of a general spin-1/2 measurement model called the-model. Then we test the possible linearity ofthe operators so constructed.     

Relativistic quantum mechanics of spin-zero and spin-half particles

We consider the quantum mechanics of directly interacting relativistic particles of spin-zero and spin-half. We introduce a scalar product in the vector space of physical states which is finite, positive definite and relativistically invariant and keeps orthogonal eigenstates of total four momentum belonging to different eigenvalues. This allows us to show that the vector space of physical states is, in fact, a Hilbert space. The case of two particles is explicitly considered and the Cauchy problem of physical wave function illustrated. The problem of a spin-1/2 particle interacting with a spin-zero particle is considered and a new equation is proposed for two spin-1/2 particles interacting via the most general form of interaction possible. The restrictions due to Hermiticity, space inversion and time reversal invariance are also considered.     

Quantum mechanics of ‘free’ spin-1/2 particles in an expanding universe

This paper deals with the gravi-quantum mechanical interaction on the level of the first quantisation and in the framework of a metric theory of gravitation (no field quantisation). The interaction is introduced by embedding the quantum mechanics of the otherwise unaffected (i.e. free) spin-1/2 particle in the given curved space-time of the 3-flat expanding Robertson-Walker universe. The metric acts thereby as an external field. The corresponding Hilbert space formalism is established in interpreting the generally covariant theory of the Dirac field in the Riemann space in question as the Dirac representation of the spin-1/2 particle in the Schrödinger picture. The evolution operator is then extracted out of the general relativistic Dirac equation, while contractions of the symmetric energy momentum tensor with the tetrad vectors of the reference system lead to the operators of energy, linear momentum and total angular momentum. The temporal behaviour of the corresponding expectation values is calculated.     

Using quaternions to design composite pulses for spin-1 NQR

The fictitious spin-1/2 operators are well known to describe the evolution of a pure nuclear quadrupole resonance (NQR) system; particularly, the application of a radio-frequency pulse at one of the NQR transition frequencies is equivalent to a three-dimensional rotation in a space defined by the corresponding fictitious spin-1/2 operators. We demonstrate, theoretically and experimentally, that consecutive noncommuting rotations applied at the same transition frequency are well described by a single rotation given by quaternion parameterization of the rotations in ficitious spin-1/2 operator space. This new route could greatly save computing time and efforts. We extend this approach to design composite pulses that compensate for the effects of the radio-frequency field inhomogeneity for a powder sample of spin-1 nuclei.     

Exploring the entanglement of free spin-(1/2), spin-1 and spin-2 fields

In this study, we explore the entanglement of free spin-(1/2), spin-1, and spin-2 fields. We start with an example involving Majorana fields in 1+1 and 2+1 dimensions. Subsequently, we perform the Bogoliubov transformation and express the vacuum state with a particle pair state in the configuration space, which is used to calculate the entropy. This clearly demonstrates that the entanglement entropy originates from the particles across the boundary.Finally, we generalize this method to free spin-1 and spin-2 fields. These higher free massless spin fields have wellknown complications owing to gauge redundancy. We deal with the redundancy by gauge-fixing in the light-cone gauge. We show that this gauge provides a natural tensor product structure in the Hilbert space, while surrendering explicit Lorentz invariance. We also use the Bogoliubov transformation to calculate the entropy. The area law emerges naturally by this method.     

More about Operators Generated by Partial Isometries

It has been suggested by several authors [1, 2] that quantum mechanical canonical transformations may be generalized by admitting partially isometric operators instead of unitary transformations used so far [3, 4]. It is known that it is possible to transform a Heisenberg couple into a corresponding one in a different Hilbert space. We shall show that the operators Q = VqV⁺ and P = VpV⁺ obtained in this way — which are unitarily equivalent to EqE and EpE, respectively, in the initial domain M of V onto which E projects — though symmetric in general will not be selfadjoint, and also present an example of this. Although it does not seem to be possible to settle the question of the existence of self-adjoint extensions definitely in the general case, the example of operators generated from the Schrödinger couple q and p shows the existence of such extensions having the spectrum of angle and z-component of angular momentum. Transducing the argument further we shall show that by choosing a different subspace N ? M ?? H it is well possible to generate the action and phase operator of the quantum mechanical harmonic oscillator with the correct spectrum.     

Integrable models for confined fermions: applications to metallic grains

We study integrable models for electrons in metals when the single particle spectrum is discrete. The electron–electron interactions are BCS-like pairing, Coulomb repulsion, and spin-exchange coupling. These couplings are, in general, nonuniform in the sense that they depend on the levels occupied by the interacting electrons. By using the realization of spin-1/2 operators in terms of electrons the models describe spin-1/2 models with nonuniform long range interactions and external magnetic field. The integrability and the exact solution arise since the model Hamiltonians can be constructed in terms of Gaudin models. Uniform pairing and the resulting orthodox model correspond to an isotropic limit of the Gaudin Hamiltonians. We discuss possible applications of this model to a single grain and to a system of few interacting grains.     

Generalized Effect Algebras of Positive Operators Densely Defined on Hilbert Spaces

Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely, we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.     

Realization of the three-dimensional quantum Euclidean space by differential operators

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by q-deformation. Simultaneously, angular momentum is deformed to , it acts on the q-Euclidean space that becomes a -module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on functions on . On a factorspace of a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a q-lattice. Received: 27 June 2000 / Published online: 9 August 2000     

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations kα + μα^? of a boson annihilation operator α and a boson creation operator α^? is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ ? μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ? 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α^? to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α^? are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ ? μ/k as a parameter.     

Stabilization of a steady state in oscillators coupled by a digital delayed connection

We propose the mapping of polynomial of degree 2S constructed as a linear combination of powers of spin-S (for simplicity, we called as spin-S polynomial) onto spin-crossover state. The spin-S polynomial in general can be projected onto non-symmetric degenerated spin up (high-spin) and spin down (low-spin) momenta. The total number of mapping for each general spin-S is given by 2(2^2S ? 1). As an application of this mapping, we consider a general non-bilinear spin-S Ising model which can be transformed onto spin-crossover described by Wajnflasz model. Using a further transformation we obtain the partition function of the effective spin-1/2 Ising model, making a suitable mapping the non-symmetric contribution leads us to a spin-1/2 Ising model with a fixed external magnetic field, which in general cannot be solved exactly. However, for a particular case of non-bilinear spin-S Ising model could become equivalent to an exactly solvable Ising model. The transformed Ising model exhibits a residual entropy, then it should be understood also as a frustrated spin model, due to competing parameters coupling of the non-bilinear spin-S Ising model.     

Dynamical simulation of integrable and nonintegrable models in the Heisenberg picture

The numerical simulation of quantum many-body dynamics is typically limited by the linear growth of entanglement with time. Recently numerical studies have shown that for 1D Bethe-integrable models the simulation of local operators in the Heisenberg picture can be efficient. Using the spin-1/2 XX chain as generic example of an integrable model that can be mapped to free fermions, we provide a simple explanation for this. We show furthermore that the same reduction of complexity applies to operators that have a high-temperature autocorrelation function which decays slower than exponential, i.e., with a power law. Thus efficient simulability may already be implied by a single conservation law as we will illustrate numerically for the spin-1 XXZ model.     

N.M.R. spectra of molecules containing quadrupolar nuclei

The general expression giving the N.M.R. band-shape of a spin-1/2 nucleus coupled to a spin-1 nucleus is considered in detail for the ‘fast-exchange’ limit, where the quadrupolar-induced relaxation rates of the spin-1 spin states are much greater than the coupling constant between the nuclei. The problem of obtaining the coupling constant from measurements on the band-shape of the spin-1/2 resonance is discussed, and it is concluded that this is only possible without further information when departures from lorentzian shape are observed. Possible sources of extra data are mentioned. The case of 3,4,5-trichloro-2,6-difluoropyridine is examined. The (N, F) coupling constant is obtained from the ¹⁵N satellites in ¹⁹F resonance. Hence values of the spin-lattice relaxation time for ¹⁴N are derived as a function of temperature.     

Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras

Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms.While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract *-algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory.     

A Generalized Quantum Theory

In quantum mechanics, the selfadjoint Hilbert space operators play a triple role as observables, generators of the dynamical groups and statistical operators defining the mixed states. One might expect that this is typical of Hilbert space quantum mechanics, but it is not. The same triple role occurs for the elements of a certain ordered Banach space in a much more general theory based upon quantum logics and a conditional probability calculus (which is a quantum logical model of the Lüders-von Neumann measurement process). It is shown how positive groups, automorphism groups, Lie algebras and statistical operators emerge from one major postulate—the non-existence of third-order interference [third-order interference and its impossibility in quantum mechanics were discovered by Sorkin (Mod Phys Lett A 9:3119–3127, 1994)]. This again underlines the power of the combination of the conditional probability calculus with the postulate that there is no third-order interference. In two earlier papers, its impact on contextuality and nonlocality had already been revealed.     

Entropy increase for a class of dynamical maps

The dynamical evolution of a quantum system is described by a one parameter family of linear transformations of the space of self-adjoint trace class operators (on the Hilbert space of the system) into itself, which map statistical operators to statistical operators. We call such transformations dynamical maps. We give a sufficient condition for a dynamical map A not to decrease the entropy of a statistical operator. In the special case of an N-level system, this condition is also necessary and it is equivalent to the property that A preserves the central state.     

Asymptotic properties of the solutions of linear and nonlinear spin field equations in Minkowski space

In this paper I will first derive, based on energy estimations and geometric invariance, the asymptotic behavior of solutions of linear spin field equations in Minkowski space. It generalizes the result in [3] where it was proved for the spin-1 and spin-2 cases. The techniques are then applied to Yang-Mills equations, the result improves the previous one in [1] by allowing the initial data to have charge, dipole and quadrupole moments. The Lie derivative operator for spinors and some properties will be also discussed; they can be used to simplify some algebraic calculations of [4].This research is partially supported by a grant from NSF under DMS-8610730     

Quantum phenomena and the zeropoint radiation field

The stationary solutions for a bound electron immersed in the random zeropoint radiation field of stochastic electrodynamics are studied, under the assumption that the characteristic Fourier frequencies of these solutions are not random. Under this assumption, the response of the particle to the field is linear and does not mix frequencies, irrespectively of the form of the binding force; the fluctuations of the random field fix the scale of the response. The effective radiation field that supports the stationary states of motion is no longer the free vacuum field, but a modified form of it with new statistical properties. The theory is expressed naturally in terms of matrices (or operators), and it leads to the Heisenberg equations and the Hilbert space formalism of quantum mechanics in the radiationless approximation. The connection with the poissonian formulation of stochastic electrodynamics is also established. At the end we briefly discuss a few important aspects of quantum mechanics which the present theory helps to clarify.On leave of absence at Mathematics Department, University College London. Gower Street, London WC1, U.K.     

Open multistate Majorana model

The Majorana model in the presence of dissipation and dephasing is considered. First, it is proven that increasing the Hilbert space dimension the system becomes more and more fragile to quantum noise, whether dephasing or dissipation are mainly present. Second, it is shown that, contrary to its ideal counterpart, the dynamics related to the open Majorana model cannot be considered as the combined dynamics of a set of independent spin-1/2 models.