Lightlike simultaneity,comoving observers and distances in general relativity

We state a condition for an observer to be comoving with another observer in general relativity, based on the concept of lightlike simultaneity. Taking into account this condition, we study relative velocities, Doppler effect and light aberration. We obtain that comoving observers observe the same light ray with the same frequency and direction, and so gravitational redshift effect is a particular case of Doppler effect. We also define a distance between an observer and the events that it observes, called lightlike distance, obtaining geometrical properties. We show that lightlike distance is a particular case of radar distance in the Minkowski space-time and generalizes the proper radial distance in the Schwarzschild space-time. Finally, we show that lightlike distance gives us a new concept of distance in Robertson–Walker space-times, according to Hubble law.     

An Operational Characterization for Optimal Observation of Potential Properties in Quantum Theory and Signal Analysis

Quantum logic introduced a paradigm shift in the axiomatization of quantum theory by looking directly at the structural relations between the closed subspaces of the Hilbert space of a system. The one dimensional closed subspaces correspond to testable properties of the system, forging an operational link between theory and experiment. Thus a property is called actual, if the corresponding test yields “yes” with certainty. We argue a truly operational definition should include a quantitative criterion that tells us when we ought to be satisfied that the test yields “yes” with certainty. This question becomes particularly pressing when we inquire how the usual definition can be extended to cover potential, rather than actual properties. We present a statistically operational candidate for such an extension and show that its representation automatically captures some essential Hilbert space structure. If it is the nature of observation that is responsible for the Hilbert space structure, then we should be able to give examples of theories with scope outside the domain of quantum theory, that employ its basic structure, and that describe the optimal extraction of information. We argue signal analysis is such an example. This work was supported by the Flemish Fund for Scientific Research (FWO) project G.0362.03N.     

Effect Algebras Are Not Adequate Models for Quantum Mechanics

We show that an effect algebra E possess an order-determining set of states if and only if E is semiclassical; that is, E is essentially a classical effect algebra. We also show that if E possesses at least one state, then E admits hidden variables in the sense that E is homomorphic to an MV-algebra that reproduces the states of E. Both of these results indicate that we cannot distinguish between a quantum mechanical effect algebra and a classical one. Hereditary properties of sharpness and coexistence are discussed and the existence of {0,1} and dispersion-free states are considered. We then discuss a stronger structure called a sequential effect algebra (SEA) that we believe overcomes some of the inadequacies of an effect algebra. We show that a SEA is semiclassical if and only if it possesses an order-determining set of dispersion-free states.     

Positive Commutators in Non-Equilibrium Quantum Statistical Mechanics

The method of positive commutators, developed for zero temperature problems over the last twenty years, has been an essential tool in the spectral analysis of Hamiltonians in quantum mechanics. We extend this method to positive temperatures, i.e. to non-equilibrium quantum statistical mechanics. We use the positive commutator technique to give an alternative proof of a fundamental property of a certain class of large quantum systems, called Return to Equilibrium. This property says that equilibrium states are (asymptotically) stable: if a system is slightly perturbed from its equilibrium state, then it converges back to that equilibrium state as time goes to infinity. Received: 27 December 2000 / Accepted: 21 June 2001     

State observer for stochastic resonance in bistable system

We investigate numerically stochastic resonance variation in bistable system. Putting signals with different intensities into a bistable system, or adjusting system structural parameters for an input, will produce different motion trajectories. These trajectories contain both chaos and non-chaos parts. We classify these states according to the state of resonance and chaos, and propose a state observer. In this way, bistable system becomes an observable nonlinear system. State observer can provide us with a new perspective that is not available through conventional observation methods. These investigations contribute to nonlinear system optimization, and can also be applied to other interesting noise-induced phenomena.     

Insufficiency of the quantum state for deducing observational probabilities

It is usually assumed that the quantum state is sufficient for deducing all probabilities for a system. This may be true when there is a single observer, but it is not true in a universe large enough that there are many copies of an observer. Then the probability of an observation cannot be deduced simply from the quantum state (say as the expectation value of the projection operator for the observation, as in traditional quantum theory). One needs additional rules to get the probabilities. What these rules are is not logically deducible from the quantum state, so the quantum state itself is insufficient for deducing observational probabilities. This is the measure problem of cosmology.     

Application of discrete-time variable structure control in the vibration reduction of a flexible structure

This paper studies the application of using the discrete-time variable structure control method to reduce the vibration of the flexible structure. The structure is subjected to arbitrary, unmeasurable disturbance forces. The concept of independent modal space control is adopted, and the system is studied by the discrete-time model. Here, discrete sensors and actuators are used. We choose the modal filters as the state estimator to obtain the modal co-ordinates and modal velocities for the modal space control. A discrete-time variable structure controller with a disturbance force observer is adopted due to its distinguished robustness property of insensitiveness to parameter uncertainties and external disturbances. The included disturbance force observer can observe the unknown disturbance modal forces, which are used in the discrete-time variable structure control law to cancel out the excitations. The upperbound limitations of the unknown disturbances in the variable structure control, therefore, are no longer needed. The switching surface, in the discrete-time variable structure control system, is designed in an optimal sense. That is, along the switching surface, the cost function of the states is minimized. The investigation of this research focuses on the optimal switching surface design and the control performances of the discrete-time variable structure controller. The performance of estimating the disturbance modal forces and the robustness property of the control law are also discussed.     

Positive Commutator Method in Nonequilibrium Statistical Mechanics

We extend the method of positive commutators, which was very successfully applied to zero temperature problems, to positive temperatures, i.e. to nonequilibrium quantum statistical mechanics. Using this technique, we give another proof of a fundamental property of large quantum systems, called Return to Equilibrium. This property says that if a system is slightly perturbed from its equilibrium state, then it converges back to it as time goes to infinity.     

Symmetry and Topology in Quantum Logic

A test space is a collection of non-empty sets, usually construed as the catalogue of (discrete) outcome sets associated with a family of experiments. Subject to a simple combinatorial condition called algebraicity, a test space gives rise to a “quantum logic”—that is, an orthoalgebra. Conversely, all orthoalgebras arise naturally from algebraic test spaces. In non-relativistic quantum mechanics, the relevant test space is the set ℱ F(H) of frames (unordered orthonormal bases) of a Hilbert space H. The corresponding logic is the usual one, i.e., the projection lattice L(H) of H. The test space ℱ F(H) has a strong symmetry property with respect to the unitary group of H, namely, that any bijection between two frames lifts to a unitary operator. In this paper, we consider test spaces enjoying the same symmetry property relative to an action by a compact topological group. We show that such a test space, if algebraic, gives rise to a compact, atomistic topological orthoalgebra. We also present a construction that generates such a test space from purely group-theoretic data, and obtain a simple criterion for this test space to be algebraic. PACS: 02.10.Ab; 02.20.Bb; 03.65.Ta.     

Hilbert's 17th problem and the quantumness of states

A state of a quantum system can be regarded as classical (quantum) with respect to measurements of a set of canonical observables if and only if there exists (does not exist) a well defined, positive phase-space distribution, the so called Glauber-Sudarshan P representation. We derive a family of classicality criteria that requires that the averages of positive functions calculated using P representation must be positive. For polynomial functions, these criteria are related to Hilbert's 17th problem, and have physical meaning of generalized squeezing conditions; alternatively, they may be interpreted as nonclassicality witnesses. We show that every generic nonclassical state can be detected by a polynomial that is a sum-of-squares of other polynomials. We introduce a very natural hierarchy of states regarding their degree of quantumness, which we relate to the minimal degree of a sum-of-squares polynomial that detects them.     

Substituting a qubit for an arbitrarily large number of classical bits

We show that a qubit can be used to substitute for a classical analog system requiring an arbitrarily large number of classical bits to represent digitally. Let a physical system S interact locally with a classical field varphi(x) as S travels directly from point A to point B. Our task is to use S to answer a simple yes/no question about varphi(x). If S is a qubit, the task can be done perfectly. We show that any classical system S must encode an arbitrarily large number of classical bits to solve the same task. This result implies a large quantum advantage in the memory size necessary for some computations. We also show that no finite amount of one-way classical communication can perfectly simulate the effect of quantum entanglement.     

On the Quantum Boltzmann Equation

We give a nonrigorous derivation of the nonlinear Boltzmann equation from the Schrödinger evolution of interacting fermions. The argument is based mainly on the assumption that a quasifree initial state satisfies a property called restricted quasifreenessin the weak coupling limit at any later time. By definition, a state is called restricted quasifree if the four-point and the eight-point functions of the state factorize in the same manner as in a quasifree state.     

INVESTIGATION OF NON-LINFAR SCHRODINGER EQUATION SOLITON SOLUTION

Packet-like soliton solution of the non-linear Schrödinger's equation has been in-vestigated.We found that a state of a free particle can possibly be described by thissoliton solution,with the quantum property and the classical property of a particleas its limiting cases.A set of biorthogonal eigen-functions of non-hermitian opera-tor,which can be used in the pertubation expansion,has been found.We discoveredthat the discrete eigenvalue mode corresponds to the“classical”motion of a particle,and the continuous eigenvalue modes correspond to the“quantum”motion.Wesuggested that the parameter u describing the state of a system can be used to identifywhether the system is“quantum”or“classical”.     

State Property Systems and Closure Spaces: A Study of Categorical Equivalence

We show that the natural mathematical structureto describe a physical entity by means of its states andits properties within the Geneva–Brussels approachis that of a state property system. We prove that the category of state property systems (andmorphisms) SP is equivalent to the category ofclosure spaces (and continuous maps) Cls. We showthe equivalence of the state determinationaxiom for state property systems with the T₀separation axiom for closure spaces. We alsoprove that the category SP ₀ ofstate-determined state property systems is equivalent tothe category L ₀ of based completelattices. In this sense the equivalence of SP andCls generalizes the equivalence ofCls ₀ (T₀ closure spaces)and L ₀ proven by Erne(1984).     

State Property Systems and Orthogonality

The structure of a state property system was introduced to formalize in a complete way the operational content of the Geneva–Brussels approach to the foundations of quantum mechanics (Aerts, D. International Journal of Theoretical Physics, 38, 289–358, 1999; Aerts, D. in Quantum Mechanics and the Nature of Reality, Kluwer Academic; Aerts, D., Colebunders, E., van der Voorde, A., and van Steirteghem, B. International Journal of Theoretical Physics, 38, 359–385, 1999), and the category of state property systems was proven to be equivalent to the category of closure spaces (Aerts, D., Colebunders, E., van der Voorde, A., and van Steirteghem, B., International Journal of Theoretical Physics, 38, 359–385, 1999; Aerts, D., Colebunders, E., van der Voorde, A., and van Steirteghem, B., The construct of closure spaces as the amnestic modification of the physical theory of state property systems, Applied Categorical Structures, in press). The first axioms of standard quantum axiomatics (state determination and atomisticity) have been shown to be equivalent to the T₀ and T₁ axioms of closure spaces (van Steirteghem, B., International Journal of Theoretical Physics, 39, 955, 2000; van der Voorde, A., International Journal of Theoretical Physics, 39, 947–953, 2000; van der Voorde, A., Separation Axioms in Extension Theory for Closure Spaces and Their Relevance to State Property Systems, Doctoral Thesis, Brussels Free University, 2001), and classical properties to correspond to clopen sets, leading to a decomposition theorem into classical and purely nonclassical components for a general state property system (Aerts, D., van der Voorde, A., and Deses, D., Journal of Electrical Engineering, 52, 18–21, 2001; Aerts, D., van der Voorde, A., and Deses, D. International Journal of Theoretical Physics; Aerts, D. and Deses, D., Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Computation, and Axiomatics, World Scientific, Singapore, 2002). The concept of orthogonality, very important for quantum axiomatics, had however not yet been introduced within the formal scheme of the state property system. In this paper we introduce orthogonality in an operational way, and define ortho state property systems. Birkhoff's well known biorthogonal construction gives rise to an orthoclosure and we study the relation between this orthoclosure and the operational orthogonality that we introduced.     

A Coaccelerated Observer

We analyze the situation of an observer coaccelerated relative to a linearly accelerated charge, in order to find whether he can observe the radiation emitted from the accelerated charge. It is found that the seemingly special situation of the coaccelerated observer relative to any other observer, is deduced from a wrong use of the retarded coordinate system, when such a system is inadmissible. It is also found that the coaccelerated observer has no special position other than any other observer, and hence, he can observe any physical events as any other observer.     

Perfect Transfer of Many-Particle Quantum State via High-Dimensional Systems with Spectrum-Matched Symmetry

The quantum state transmission through the medium of high-dimensional many-particle system (boson or spinless fermion) is generally studied with a symmetry analysis. We discover that, if the spectrum of a Hamiltonian matches the symmetry of a fermion or boson system in a certain fashion, a perfect quantum state transfer can be implemented without any operation on the medium with pre-engineered nearest neighbor (NN). We also study a simple but realistic near half-filled tight-binding fermion system with uniform NN hopping integral. We show that an arbitrary many-particle state near the fermi surface can be perfectly transferred to its translational counterpart.     

Perfect Transfer of Many-Particle Quantum State via High-Dimensional Systems with Spectrum-Matched Symmetry

The quantum state transmission through the medium of high-dimensional many-particle system （boson or spinless fermion） is generally studied with a symmetry analysis. We discover that, if the spectrum of a Hamiltonian matches the symmetry of a fermion or boson system in a certain fashion, a perfect quantum state transfer can be implemented without any operation on the medium with pre-engineered nearest neighbor （NN）. We also study a simple but realistic near half-filled tight-bindlng fermion system wlth uniform NN hopping integral. We show that an arbitrary many-particle state near the fermi surface can be perfectly transferred to its translational counterpart.