Logics in realization and modeling from data

Systemrealization is the construction of a state-space model given input-output data of a system. One approach, briefly summarized here, is the subspace method. In the deterministic realization problem, the data are used in alinear fashion, whereas the stochastic realization problem usesquadratic forms in the data. This dichotomy is related to the basic assumptions of repeatability or nonrepeatability of the input-output experiments performed on the system. In particular, the logic of the system is constructed, closely following the axiomatic foundations of physics. It is shown that this logic is Boolean in the deterministic and quantal in the stochastic case. The system dynamics is obtained from the data-induced measures one can define on the lattice.     

On the quantization of dissipative systems

A recent paper of Dekker on the quantization of dissipative systems is examined in some detail. It is argued that one can construct a large number of classical equivalent Hamiltonians for damped systems. These can be formally quantized according to Dirac's method, and the resulting equations are mathematically consistent, but yield different eigenfunctions for the same classical system. However, this procedure should be rejected on physical grounds. That is in quantum mechanics, unlike classical dynamics, the definition of the time derivative of a dynamical variable is unique, and is given by the commutator of the proper Hamiltonian (or the energy operator) and that variable. If the proper Hamiltonian is used for the quantization of a damped system, then the quantal equations are inconsistent for the cases where the rate of energy dissipation depends on the velocity of the particle. As an alternative approach to the quantal theory of dissipative phenomena, a generalization of the Hamilton-Jacobi formalism is considered, where the equation for the principle functionS, depends not only on the space and time derivatives ofS, but onS itself. This leads to a new class of damped systems in classical mechanics. The original Schrödinger method of quantization via the Hamilton-Jacobi equation has been applied to this class of dissipative systems, with the result that the wave equation in this case is a solution of a non-linear Schrödinger-Langevin equation. This formulation has no analogue in the Hamiltonian approach, since in the latter, the resulting wave equation is always linear.Supported in part by a grant from the National Research Council of Canada.     

Quantal Noether Identities and Their Applications

Based on the phase-space generating functional of the Green function for a system with a regular/singular Lagrangian, the quantal canonical Noether identities (NI) under the local and non-local transformation in extended phase have been derived, respectively. The result holds true whether the Jacobian of the transformation is equal to unity or not. Based on the configuration-space generating functional of the gauge-invariant system obtained by using Faddeev-Popov (FP) trick, the quantal NI under the local and non-local transformation in configuration space have been also deduced. It is showed that for a system with a singular Lagriangian one must use the effective action in the quantal NI instead of the classical action in corresponding classical NI. It is pointed out that in certain cases, the quantal NI may be converted into the quantal (weak) conservation laws by using the quantal equations of motion. This algorithm to derive the quantal conservation laws differs from the quantal first Noether theorem. The preliminary applications of this formulation to Yang-Mills (YM) fields and non-Abelian Chern-Simons (CS) theories are given. The quantal conserved quantities for non-local transformation in YM fields are obtained. The conserved BRS and PBRS quantities at the quantum level in non-Abelian CS theories are also found. The property of fractional spin in CS theories is discussed. PACS no11.10. Ef; 11.30.−j 11.15. −q.     

Quantum action principle in curved space

Schwinger's action principle is formulated for the quantum system which corresponds to the classical system described by the LagrangianL _c( , x)=(M/2)g_ij(x) ^{i j}–v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider onlyc-number variations of coordinates and time. The Euler-Lagrange equation, the canonical commutation relations, and the canonical equations of motion are derived from this principle in a consistent manner. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant. The judicious choice of the quantal Lagrangian is essential in our formulation. A quantum mechanical analog of Noether's theorem, which relates the invariance of the quantal action with a conservation law, is established. The ambiguities in the quantal Lagrangian are also discussed and it is pointed out that the requirement of invariance is not sufficient to determine uniquely the quantal Lagrangian and the Hamiltonian.     

Quantal Poincaré-Cartan Integral Invariant for Field Theory

On the basis of the phase-space generating function of Green function for a system with a regular/singular Lagrangian, the quantal Poincaré-Cartan integral invariant (PCII) for field theory is derived. This PCII is equivalent to the quantal canonical equations. For this case in which the Jacobian of the transformation does not equalto unity, the quantal PCII can still be derived. This case is different from the quantal first Noether theorem. The quantal PCII connected with canonical equations and canonical transformation is also discussed.     

On the Mössbauer hyperfine spectra of paramagnetic clusters

The hyperfine interaction of a system containing two weakly coupled paramagnetic ions is investigated. It is shown that even a very weak interaction cannot be broken by a large magnetic field, if the Zeeman interaction of the two paramagnetic centers is identical. Small differences in theg-values, however, yield immediately a quite different hyperfine pattern which corresponds to two isolated paramagnetic centers in the case of a large external magnetic field. Mössbauer experiments on reduced bacterial ferredoxin where one molecule contains two iron clusters withS=1/2 each are used to analyze the interaction of two paramagnetic centers.     

The quantal Poincaré-Cartan integral invariantfor singular higher-order Lagrangian in field theories

Based on the phase-space generating functional of the Green function for a system with a regular/singular higher-order Lagrangian, the quantal Poincaré-Cartan integral invariant (QPCII) for the higher-order Lagrangian in field theories is derived. It is shown that this QPCII is equivalent to the quantal canonical equations. For the case in which the Jacobian of the transformation may not be equal to unity, the QPCII can still be derived. This case is different from the quantal first Noether theorem. The relations between QPCII and a canonical transformation and those between QPCII and the Hamilton-Jacobi equation at the quantum level are also discussed.Received: 26 May 2004, Revised: 2 December 2004, Published online: 16 March 2005     

Study of a fast collective mode in deep inelastic reactions: Quantal fluctuations

We have measured for the 430 MeV⁸⁶Kr+^92.98Mo systems the triple differential cross sectiond ³ σ/dEdAdZ (E=kinetic energy,A=mass andZ=atomic number) of the reaction products in the angular range (θ=25, 45°). We focussed attention on a fast collective mode in deep inelastic reactions: the neutron excess or charge equilibration. This mode is shown to relax very quickly to equilibrium within a time scale of the order of 10^?22s. It is shown that this collective degree of freedom exhibit a quantal behavior which can be seen in the observation of quantal fluctuations. The experimental results are discussed in terms of a simple equilibrium model. The giant dipole resonance of the composite system could be closely connected to this neutron excess mode.     

Fluctuation,relaxation, and extensivity of macrovariables in nonequilibrium systems

The extensive property of a macrovariable is proved for a quantal system whose Hamiltonian depends on time and for a stochastic system whose temporal evolution operator depends on time. These generalized situations are concerned with bulk-contact open systems. The extensive property, fluctuation, and nonlinear relaxation are investigated explicitly by calculating rigorously generating functions in exactly soluble models such as the linear stochastic model and linearXY model. The relation between the nonlinear critical slowing down and linear critical slowing down is also discussed.     

Eikonal description of quantal scattering

Elastic and inelastic quantal scattering is described by a theory in which the contribution of a range of impact parameters to the scattering amplitude is determined by a phase integral (“eikonal”) which is integrated along a real curved “quantal” trajectory. This amplitude reduces to the Glauber expression in the high-energy, forward-angle limit, and to the usual semiclassical amplitude in the classical limit. The formulation can be applied to the study of heavy-ion scattering. The quantal trajectories are investigated analytically for the case of Coulomb scattering. A numerical analysis of elastic ¹⁶O¹⁶O scattering is carried out. The results show appreciable improvement as compared with the Glauber approximation.     

Universality in the length spectrum of integrable systems

The length spectrum of periodic orbits in integrable hamiltonian systems can be expressed in terms of the set of winding numbers {M ₁,…,M _f} on thef-tori. Using the Poisson summation formula, one can thus express the density, Σδ(T−T _M), as a sum of a smooth average part and fluctuations about it. Working with homogeneous separable potentials, we explicitly show that the fluctuations are due to quantal energies. Further, their statistical properties are universal and typical of a Poisson process as in the corresponding quantal energy eigenvalues. It is interesting to note however that even though long periodic orbits in chaotic billiards have similar statistical properties, the form of the fluctuations are indeed very different.     

A unified quantum theory of mechanics and thermodynamics. Part III. Irreducible quantal dispersions

This part of the paper concludes the presentation of the unified theory. It is shown that the theory requires the existence of, and applies only to, irreducible quantal dispersions associated with pure or mixed states. Two experimental procedures are given for the operational verification of such dispersions. Because the existence of irreducible dispersions associated with mixed states is required by Postulate 4 of the theory, and because Postulate 4 expresses the basic implications of the second law of classical thermodynamics, it is concluded that the second law is a manifestation of phenomena characteristic of irreducible quantal dispersions associated with the elementary constituents of matter.Parts I, IIa, and IIb of this paper appeared inFound. Phys. 6, 15, 127, 439 (1976), respectively. The numbering of the sections, equations, and references in this part continues from the previous parts.     

Quantum decoherence from adiabatic entanglement with external one or a few degrees of freedom

Based on the Born-Oppenhemer approximation, the concept of adiabatic quantum entanglement is introduced to account for quantum decoherence of a quantum system due to its interaction with a large system of one or a few degrees of freedom. In the adiabatic limit, it is shown that the wave function of the total system formed by the quantum system plus the large system can be factorized as an entangled state with correlation between adiabatic quantum states and quasi-classical motion configurations of the large system. In association with a novel viewpoint about quantum measurement, which has been directly verified by most recent experiments [e.g., S. Durr et al., Nature 33, 359 (1998)], it is shown that the adiabatic entanglement is indeed responsible for the quantum decoherence and thus can be regarded as a “clean” quantum measurement when the large system behaves as a classical object. By taking the large system respectively to be a macroscopically distinguishable spatial variable, a high spin system and a harmonic oscillator with a coherent initial state, three illustrations are presented with their explicit solutions in this paper. Received 26 February 2000 and Received in final form 14 July 2000     

非定域量子Noether恒等式及应用

基于高阶微商奇异拉氏量系统的相空间生成泛函,导出了定域和非定域变换下的量子正则Noether恒等式;对高阶微商规范不变系统,导出了位形空间中定域和非定域变换下的量子Noether恒等式.指出在某些情形下,由量子Noether恒等式可导致系统的量子守恒律.这种求守恒律的程式与量子Noether(第一)定理不同.用于高阶微商非AbelChern-Simons(CS)理论,求出某些非定域等变换下的量子守恒量.     

Spin waves and static susceptibility in the weakly doped t-J model

We investigate the spin dynamics in weakly doped high-temperature superconductors. The system is described by the two-dimensional t-J model. Our focus is on the interaction between mobile holes and spin waves. The calculations are based on a recently introduced cumulant method for computing the ground state energy of correlated electronic systems. Contrary to previous works using dynamical quantities like correlation functions or spectral densities our approach contains a static view to the system. This new method treats spin and hole dynamics on the same basis and allows for the calculation of static and dynamical quantities. We present results for spin-wave energies and transverse static susceptibilities for small hole concentrations and various values of t/J. We find a strong renor-malization of the spin-wave energies due to the spin-hole interaction. In agreement with neutron scattering experiments the spin-wave velocity vanishes at a critical hole density of a few percent which is equivalent to the instability of the antiferromagnetic order.     

Propagation and interaction of ion-acoustic solitary waves in a quantum electron-positron-ion plasma

This paper discusses the existence of ion-acoustic solitary waves and their interaction in a dense quantum electron-positron-ion plasma by using the quantum hydrodynamic equations.The extended Poincar’e-Lighthill-Kuo perturbation method is used to derive the Korteweg-de Vries equations for quantum ion-acoustic solitary waves in this plasma.The effects of the ratio of positrons to ions unperturbation number density p and the quantum diffraction parameter H e (H p) on the newly formed wave during interaction,and the phase shift of the colliding solitary waves are studied.It is found that the interaction between two solitary waves fits linear superposition principle and these plasma parameters have significantly influence on the newly formed wave and phase shift of the colliding solitary waves.The investigations should be useful for understanding the propagation and interaction of ion-acoustic solitary waves in dense astrophysical plasmas (such as white dwarfs) as well as in intense laser-solid matter interaction experiments.     

Subleading long-range interactions and violations of finite size scaling

We study the behavior of systems in which the interaction contains a long-range component that does not dominate the critical behavior. Such a component is exemplified by the van der Waals force between molecules in a simple liquid-vapor system. In the context of the mean spherical model with periodic boundary conditions we are able to identify, for temperatures close above T _c, finite-size contributions due to the subleading term in the interaction that are dominant in this region decaying algebraically as a function of L. This mechanism goes beyond the standard formulation of the finite-size scaling but is to be expected in real physical systems. We also discuss other ways in which critical point behavior is modified that are of relevance for analysis of Monte Carlo simulations of such systems. Received 21 November 2000 and Received in final form 28 February 2001     

Rare earth paramagnetism,as studied by μSR

Some recent result of muon spin relaxation measurements in rare earth metals and intermetallic compounds are reviewed. Special emphasis is put on measurements that relate to the properties of correlated regions of spins existing relatively far above the ordering temperature in the rare earth ions. As far as comparable data from paramagnetic neutron scattering exist, they will be discussed in the same framework. For each temperature the correlated regions (or short-lived magnetic clusters) are characterized by their size, possible anisotropy with respect to the crystalline axes and their lifetime. The actual form of the interaction between the rare earth spins themselves and with the crystal fields determine the temperature dependence of these properties; a strong dipole interaction can, for instance, be expected to change the critical behaviour nearT _c . Much of the time will be devoted to experiments on Gd-metal where there are experimental indications that several interesting phenomena occur: (1) a strong effect of a cross-over from a non-conserved dynamics (dipolar) regime to a conserved (exchange dominated) regime some 10 K aboveT _c , (2) an anisotropy of the magnetic clusters with respect to the hexagonalc-axis, and (3), a persistence of spin correlations far aboveT _c . Some attempts to correlate the rare earth spin relaxation times measured in this region with cluster lifetimes deduced from neutron scattering will be reviewed, as well as a model for understanding these lifetimes in terms of temperature dependent cluster wall motion, which is determined by exchange and magnetic anisotropy parameters. Effects of possible quantum correlations originating from the “spin system+bath” interaction will be mentioned.     

The Vlasov equation for Coulomb systems and the Husimi picture

We explore the validity of the Vlasov equation as a semi-classical approximation of time-dependent Hartree-Fock and time-dependent LDA theories. We discuss the h → 0 limit for the propagation of quantal wavefunctions in terms of classical densities. The h → 0 limit is studied formally by means of its Wigner and Husimi phase-space representations. We consider an application to the valence electron cloud of metal clusters and show a comparison between quantal and Vlasov dynamics in this case.