Renormalization group study of quantum fluctuations near classical critical points of Hamiltonian systems

A general framework is considered for treating quantum corrections to the classical limit in the Wigner function formalism. We discuss the quantal effect on the classical phenomena such as period doubling and the breakup of KAM tori. By using an exact renormalization group method, the scaling factor for Planck's constant is derived as an eigenvalue of the linearized renormalization transformation.     

Light metal ions in water: Quantal and classical simulations for Li+

Path integral Monte Carlo computations have been done to study the local structure of water molecules around an isolated lithium ion (⁷Li⁺) at T=298K. The solute was treated as a quantal particle and the water solvent was treated classically. The water-water interaction was modelled by the MCY pair-potential and the solute-water interaction by the Kistenmacher et al. pair-potential. Purely classical simulations, at the same conditions and using the same model potentials, where also performed for comparison.

Significant changes are observed on the results of the quantal simulations when compared with the results of the classical simulation. The major difference is the coordination number that increases from 5, the result of the classical simulation, to 6, for the quantal simulation. In addition, structural analysis of the generated configurations showed that the local structure of the water molecules surrounding the ion is also clearly different in the two simulations. The reliability of the results is discussed.     

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.     

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.     

高阶微商规范不变系统的整体对称性和量子守恒律

分别从Faddeev–Popov(FP)和Faddeev–Senjanovic(FS)路径积分量子化方法对高阶微商规范不变系统导致的位形空间和相空间生成泛函出发,导出规范系统在量子水平下的守恒律,用于高阶Maxwell非AbelChern–Simons(CS)理论.得到了高阶Maxwell非AbelCS理论与标量场耦合系统的量子BRS守恒荷和量子守恒角动量,无论从位形空间或相空间的生成泛函出发,其结果是相同的.并对CS理论中的分数自旋性质给予了讨论.     

Topological Chern—Simons vortices in the O (3) σ -model

Based on the phase-space path integral (functional integral) for a system with a regular or singular Lagrangian, the generalized Ward identities for phase space generating functional under the global transformation in phase space are derived respectively. The canonical Noether theorem at the quantum level is also established. It is pointed out that the connection between the symmetries and conservation laws in classical theories, in general,is no longer preserved in quantum theories. The advantage of our formulation is that we do not need to carry out the integration over the canonical momenta as usually performed. Applying the present formulation to Yang-Mills theory, the quantal BRS conserved quantity and Ward-Takahashi identity for BRS tranformation are derived; the Ward identities for gaugeghost proper vertices and new quantal conserved quantity are also found. In comparison of quantal conservation laws with those one deriving from configuration-space path integral using the Faddeev-Popov(F-P) trick is discussed. A precise study of path-integral quantisation for a nonlinear sigma model with Hopf and Chern-Simons (CS) terms is reexamined. It has been shown that the angular momentum at the quantum level is equal to classical (Noether ) one. Applying our formulation to non-Abelian CS theory, the quantal conserved angular momentum of this system is obtained which differs from classical one in that one needs to take into account the contribution of angular momenta of ghost fields.     

Quantum chaos of atoms in a resonant cavity

A system of atoms interacting with a radiation field in a resonant cavity is studied under conditions when the dynamics in the classical limit is stochastic. This situation is called quantum chaos. Equations of motion are obtained for the quantum-mechanical expectation values which take into account the quantum correlation functions. It is shown that in a situation corresponding to quantum chaos, the quantum corrections grow exponentially, making the evolution of the system essentially quantal after a certain time tau( variant Planck's over 2pi ) has elapsed. Analytical and numerical analysis show that in this regime the time tau( variant Planck's over 2pi ) obeys the logarithmic law tau( variant Planck's over 2pi ) approximately ln N (N is the number of atoms), and not the law tau( variant Planck's over 2pi ) approximately N(alpha) (alpha is a certain constant of order unity), as would be the case in the absence of chaos.     

Zero field isotropic Muonium Kubo — Toyabe relaxation functions

Static zero field Gaussian Kubo — Toyabe relaxation functions for muons in isotropic muonium atoms are presented. That is, as with diamagnetic muons, an average of the spin dynamics of a muon in an isolated isotropic ground state muonium atom is taken over an isotropic Gaussian continuous classical local random magnetic field distribution. This motion approximates the exact quantal spin dynamics generated by the dipole-dipole interactions between the muonium atom and the surrounding nuclear spins associated with the site at which the muonium atom has stopped. Expressions are derived for triplet muonium only since, in general, singlet muonium is not observed. For normal nuclear spins and ground state muonium, the resulting relaxation functions are identical to the standard diamagnetic function (except for a shift in the time scale).     

非Abel Chern-Simons理论中量子水平的分数自旋性质

与经典水平下的研究不同，研究了（2＋1）维含非Abel Chern-Simons 项的非线性σ模 型量子水平的分数自旋性质.根据约束Hamilton系统的Faddeev-Senjanovic(FS)路径积分量 子化方案,对该系统进行量子化，由量子Noether定理给出了量子守恒角动量，说明了在量子 水平上该系统仍具有分数自旋的性质. 关键词： 约束Hamilton系统 分数自旋 O(3)非线性σ模型     

Semiclassical calculations with nuclear model potentials in the ħ-resummation approach

The single-particle densityρ(r) of a system of fermions can be calculated in a tractable way as the Laplace inverse of the Bloch density describing the system. The complex integrals involved can be solved very easily by the saddle-point method. The semiclassical nature of this approach is illustrated in the simple example of the single-particle level density of a harmonic oscillator potential. It is then applied to calculate the total energy of particles in different mean field potentials. The exact Bloch density being generally unknown, different approximate forms are used in our calculations which correspond to a partial resummation of the Wigner-Kirkwood?-expansion. The resulting local densities reproduce the exact density distributions on the average, without quantal oscillations. They are well defined everywhere, even beyond the classical turning point, in contrast to the original Wigner-Kirkwood approach.     

Semiclassical stochastic trajectory approach to molecule-solid surface scattering

A semiclassical stochastic trajectory (SST) approach to the sudy of collision induced transitions in gas molecule-solid surface scattering is presented. The time-dependent Schrödinger equation provides the time-evolution of the transition amplitudes for the molecular internal states. Classical mechanics is used to describe the molecule's center of mass motion as well as the surface atoms' motion — the latter through the generalized Langevin equation (GLE) method which allows the treatment of non-rigid surfaces (i.e. surface temperature effects). These quantum and classical equations of motion are coupled through the use of a time-dependent interaction potential in the Schrödinger equation and the use of the expectation value of the interaction potential in the classical equations of motion. Advantages of the SST approach include: (1) flexibility in the choice of quantum versus classical coordinates; (c) strict energy conservation for non-dissipative system; and (3) realistic treatment of surface many-body effects within the GLE. The SST technique is applied to the study of vibrational and rotational inelasticity in a model H₂Pt(111) system. As an initial test, results obtained assuming a rigid, smooth surface with an exponentially repulsive potential are compared to exact quantal and quasi-classical trajectory values to determine the accuracy and utility of the SST approach. A limited practical application is presented for the same H₂Pt(111) system but for a non-rigid surface. These results, calculated at low gas kinetic energies, indicate that surface energy transfer and surface temperature effects should be minimal for this type of system, even though the energy gaps are quite similar for rotational and phonon degrees of freedom.     

Propositional systems and measurements. III. Quasitensorproducts of certain orthomodular lattices

Continuing our investigations on propositional systems without assumption of the covering law, we introduce a quasi-tensor-product of a complete atomic orthomodular lattice with a complete atomic Boolean lattice. This product has a universal property with respect to postulates on propositional systems of coupled physical systems. We use it to describe measurements on a purely quantal object by a purely classical apparatus and find no nontrivial proposition of the object to be commensurable with its quantal negation. If the object is not purely quantal, the central propositions are commensurable. By this, it is shown directly that useful apparatuses must have a quantal microstructure.Dedicated to the 60th birthday of Professor G. Ludwig     

Delinearization of quantum logic

The algebraic structure of the set of elementary observables of a delinearized quantal theory is described. As the delinearization procedure provides a kind of classical representation for any quantal theory, its relation to the traditional hypothesis of hidden variables is discussed.     

Calculations of Stationary Fission Rate by the Partition Function Method

The stationary nuclear fission rate including both thermodynamic and quantal fluctuations are studied by using the partition function method, which is related to the imaginary part of the free energy by R_f = (2/h)k_BT_cIm In Z (Z is the partition function). We calculate the classical fission rate with a coordinatedependent mass within a onedimensional model. The contributions of'additional degrees of fieedom, angular momentum and viscosity to the quantal fission rate at finite temperature are also discussed. Differences between the two- or three-dimensional and the onedimensional cases are analyzed by using the {c, h, a} parametrization.     

Semiclassical scattering theory with complex trajectories. I. Elastic waves

The WKB approximation to the one-particle Schrödinger equation is used to obtain the wave function at a given point as a sum of semiclassical terms, each of them corresponding to a different classical trajectory ending up at the same point. Besides the usual, real trajectories, also possible complex solutions of the classical equations of motion are considered. The simplicity of the method makes its use easy in practical cases and allows realistic calculations. The general solution of the one-dimensional WKB equations for an arbitrary number of complex turning points is given, and the solution is applied to calculate the position of the Regge poles of the scattering amplitude. The solution of the WKB equations in three dimensions for a central analytical potential is also obtained in a way that can be easily generalized to N-dimensions, provided the problem is separable. A multiple reflection series is derived, leading to a separation of the scattering amplitude into a smooth “background” term (single reflection approximation) that can be treated using classical but complex trajectories and a second resonating term that can be treated using the Sommerfeld-Watson transformation. The physical interpretation of the complex solutions of the classical equations of motion is given: they describe diffractive effects such as Fresnel, Fraunhofer diffraction, or the penetration of the quantal wave into shadow regions of caustics. They arise also in the scattering by a complex potential in an absorptive medium. The comparison with exact quantal calculations shows an astonishingly good agreement, and establishes the complex semiclassical approximation as a quantitative tool even in cases where the potential varies rapidly within a fraction of a wavelength. An approximate property of classical paths is discussed. The general pattern of the trajectories depends only on the product ? = EΘ, and not on energy and angle separately. This property is confirmed by experiments and besides the signature it gives for the semiclassical behavior, it simplifies considerably the search for all trajectories scattering through the same angle. Finally, a general classification of the different types of elastic heavy ion cross sections is given.     

Fundamental Physics

A survey is given of the elegant physics of N-particle systems, both classical and quantal, non-relativistic (NR) and relativistic, non-gravitational (SR) and gravitational (GR). Chapter 1 deals exclusively with NR systems; the correspondence between classical and quantal systems is highlighted and summarized in two tables of Sec. 1.3. Chapter 2 generalizes Chapter 1 to the relativistic regime, including Maxwell’s theory of electromagnetism. Chapter 3 follows Einstein in allowing gravity to curve the spacetime arena; its Sec. 3.2 is devoted to the yet missing theory of elementary particles, which should determine their properties and interactions. If completed, it would replace QFT; promising is the ‘metron’ approach.     

奇异拉氏量系统的整体量子正则对称性质

从奇异拉氏量系统相空间路径积分的量子化形式出发，导出了系统在增广相空间整体变换下的广义正则Ｗａｒｄ恒等式和量子水平的守恒荷，一般这些守恒荷有别于经典Ｎｏｅｔｈｅｒ荷．给出了在杨-Ｍｉｌｓ场论中的应用，找到了新守恒荷     

Exact asymptotics for one-dimensional diffusion with mobile traps

We consider a diffusing particle, with diffusion constant D', moving in one dimension in an infinite sea of noninteracting mobile traps with diffusion constant D and density rho. We show that the asymptotic behavior of the survival probability, P(t), satisfies lim([-ln(P(t)]/sqrt[rho2Dt]=4/sqrt[pi], independent of D'. The result comes from obtaining upper and lower bounds on P(t), and showing that they coincide asymptotically. We also obtain exact results for P(t) to first order in D(')/D for an arbitrary finite number of traps.