A Variational Principle for Markov Processes

In this note, we first present a result concerning a variational principle for general Markov processes. Then we apply it to spin particle systems to obtain a full form of a variational principle characterizing the stationary Markov laws of the systems. A related extreme decomposition for any stationary distribution of such Markov systems is also given.     

Variational Principle and Almost Quasilocality for Renormalized Measures

We study the variational principle for some non-Gibbsian measures. We give a necessary and sufficient condition for the validity of the implication zero relative entropy density implies common version of conditional probabilities (so-called second part of the variational principle). Applying this to noisy decimations of the low-temperature phases of the Ising model, we obtain almost sure quasilocality for these measures and the second part of the variational principle. For the projection of low temperature Ising phases on a one-dimensional layer, we also obtain the second part of the variational principle.     

Variational Principle for Some Renormalized Measures

We show that some measures suffering from the so-called renormalization group pathologies satisfy a variational principle and that the corresponding limit of the pressure, with boundary conditions in a set of measure 1, is equal to the pressure of the Ising model modulo a scale factor.     

Difference Discrete Variational Principle in Discrete Mechanics and Symplectic Algorithm

We propose the difference discrete variational principle in discrete mechanics and symplectic algorithmwith variable step-length of time in finite duration based upon a noncommutative differential calculus established inthis paper. This approach keeps both symplecticity and energy conservation discretely. We show that there exists thediscrete version of the Euler-Lagrange cohomology in these discrete systems. We also discuss the solution existencein finite time-length and its site density in continuous limit, and apply our approach to the pendulum with periodicperturbation. The numerical results are satisfactory.     

Difference Discrete Variational Principle in Discrete Mechanics and Symplectic Algorithm

We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This approach keeps both symplecticity and energy conservation discretely. We show that there exists the discrete version of the Euler-Lagrange cohomology in these discrete systems. We also discuss the solution existence in finite time-length and its site density in continuous limit, and apply our approach to the pendulum with periodic perturbation. The numerical results are satisfactory.     

球形平均场加四极-四极和对力模型在单j壳内的量子相交叉行为(英文)

介绍了单j壳的球形平均场加几何四极-四极和标准对力模型的量子相交叉行为。在单j=15/2的壳内,计算了随模型控制参数变化的多个物理量如低激发能级、激发态间重叠积分、低激发态间的B（E2）比值和电四极矩比值。结果显示,在类转动到对激发相的演化中,多个物理量在交叉区存在非常明显的变化,如B（E2;4₁ → 2₁）/B（E2;2₁ → 0_g）,B（E2;4₂ → 2₁）/B（E2;2₁ → 0_g）等,并且这些变化在核子数达到半满壳时尤为显著。此外,尽管当j较小时,由几何四极-四极相互作用得到的低激发能级不满足转动谱规律,但当j值足够大时,这些低激发能级满足转动谱规律。The analysis of the quantum phase crossover behavior in the spherical shell model mean-field plus the geometric quadrupole-quadrupole (Q·Q) and standard pairing model within a single-j shell is reported. Several quantities, such as low-lying excitation energies, the overlaps of excited states, ratios of some B(E2) and electric quadrupole moments of some low-lying states as functions of the control parameter of the model in a j=15/2 shell are calculated. The results show that there are noticeable changes in the crossover region of the rotational-like to the pair-excitation phase transition, such as B(E2;4₁ → 2₁)/B(E2;2₁ → 0_g) and B(E2;4₂ → 2₁)/B(E2;2₁ → 0_g), especially in the half-filling case. Though the low-lying excitation energies generated from the geometric quadrupole-quadrupole interaction not satisfy the pattern of a rotational spectrum when j is small, these energies follow the pattern of a rotational spectrum when j is sufficiently large.     

基于变分原理的自适应方法在统一坐标系中的应用

在统一坐标系的基础上引入变分方法,并从自适应网格的正交性、光滑性、疏密程度等角度考虑,获取速度系数h的椭圆方程,从而在边界上可以自由控制h的取值,以适应不同物理问题的需要.算例证明,变分法在统一坐标系中的应用是可行的,在边界上可以满足物理要求.     

Variational principle and dynamical equations of discrete nonconservative holonomic systems

By analogue with the methods and processes in continuous mechanics, a Lagrangian formulation and a Hamiltonian formulation of discrete mechanics are obtained. The dynamical equations including Euler--Lagrange equations and Hamilton's canonical equations of the discrete nonconservative holonomic systems are derived on a discrete variational principle. Some illustrative examples are also given.     

用独立元格等效作用量研究三维U(1)格点规范理论的相结构

在格点规范理论中,应用变分累积展开方法,引入独立元格等效作用量,研究了三维U(1)格点规范理论的元格内能,得到连续的内能曲线,与Monte Carlo(MC)数据相符,表明理论是禁闭的.     

Quadratic Actions in Dependent Fields and the Action Principle

General field theories are considered, within the functional differential formalism of quantum field theory, with interaction Lagrangian densities L _I (x;λ), with λ a generic coupling constant, such that the following expression ∂ L _I (x;λ)/∂ λ may be expressed as quadratic functions in dependent fields but may, in general, be arbitrary functions of independent fields. These necessarily include, as special cases, present renormalizable gauge theories. It is shown, in a unified manner, that the vacuum-to-vacuum transition amplitude (the generating functional) may be explicitly derived in functional differential form which, in general, leads to modifications to computational rules by including such factors as Faddeev–Popov ones and modifications thereof which are explicitly obtained. The derivation is given in the presence of external sources and does not rely on any symmetry and invariance arguments as is often done in gauge theories and no appeal is made to path integrals.     

Effective Action for a Statistical System with a Field Dependent Wave Function

We compute the first order correction in ħ to the field dependent wave function in Statistical Field Theory. These corrections are evaluated by several usual methods. We limit ourselves to a one dimensional model in order to avoid the usual difficulties with the UV divergences that are not relevant for our purposes. The main result of the paper is that the various methods yield different corrections to the wave function. Moreover, we give arguments to show that the perturbative integration of the exact renormalization group provides the right result.     

高压静电吸附除尘新型板擦的研制

利用高压静电对粉尘有良好絮凝、吸附作用的特点,首先设计静电除尘的模拟吸附装置进行了粉尘吸附的实验,然后设计并组装了具有能够降低小粒径粉尘密度、吸附收集粉尘、防止扬尘等特点的静电吸附除尘新型板擦。     

Variational Principle for a Schrdinger Equation with Non-Hermitian Hamiltonian and Position-Dependent Mass

A classical field theory for a Schrodinger equation with a non-Hermitian Hamiltonian describing a particle with position-dependent mass has been recently advanced by Nobre and Rego-Monteiro(NR)[Phys.Rev.A 88(2013)032105].This field theory is based on a variational principle involving the wavefunction Ψ(x,t) and an auxiliary fieldΦ{x,t).It is here shown that the relation between the dynamics of the auxiliary field Φ(x,t) and that of the original wavefunction Ψ(x,t) is deeper than suggested by the NR approach.Indeed,we formulate a variational principle for the aforementioned Schrodinger equation which is based solely on the wavefunction Ψ(x,t).A continuity equation for an appropriately defined probability density,and the concomitant preservation of the norm,follows from this variational principle via Noether's theorem.Moreover,the norm-conservation law obtained by NR is reinterpreted as tie preservation of the inner product between pairs of solutions of the variable mass Schrodinger equation.     

The Principle of Covariance and the Hamiltonian Formulation of General Relativity

The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.     

The Variational Principle for a Class of Asymptotically Abelian C*-Algebras

Let (A,α) be a C^*-dynamical system. We introduce the notion of pressure P _α(H) of the automorphism α at a self-adjoint operator H∈A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense α-invariant *-subalgebra ? of A such that for all pairs a,b∈? the C^*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ℕ such that [α ^j (a),b]= 0 for |j|≥p. For systems in this class we prove the variational principle, i.e. show that P _α(H) is the supremum of the quantities h _φ(α) −φ(H), where h _φ(α) is the Connes–Narnhofer–Thirring dynamical entropy of α with respect to the α-invariant state φ. If H∈A, and P _α(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h _φ(α), and any state of finite maximal entropy is a trace. Received: 19 April 2000 / Accepted: 14 June 2000     

Spherical gravitational collapse with photon emission and a generalized Schwarzchild interior solution

The general dynamical equations for perfect fluid filled spheres with an outward flux of photons are derived. The vital role played by the energy density of the free gravitational field in accelerating photon production has been emphasized. It is pointed out that even when the material energy density is finite, the energy density of the free gravitational field can take infinitely large values resulting in vanishing surface area of the star. A generalized Schwarzschild interior solution with conformally flat geometry but with photon emission has been obtained. It is pointed out that the interior conformal coordinate system bears a strong resemblance to the exterior Krushkal coordinates. It is shown that for spherical star the invariant velocity of the fluid particles, falling towards the centre, is proportional to its radius suggesting that the outer envelopes collapse at a faster rate than the core part. It is shown that the interior radiating solution can be matched with generalized Schwarzchild exterior solution.     

Effective Action for Noncommutative U（1） Gauge Theory with Higher Dimensional Terms

In this paper we apply the assumption of our recent work in noncommutative scalar models to the noncommutative U（1） gauge theories. This assumption is that the noneommutative effects start to be visible continuously from a scale ANC and that below this scale the theory is a commutative one. Based on this assumption and using background field method and loop calculations, an effective action is derived for noncommutative U（1） gauge theory. It will be shown that the corresponding low energy effective theory is asymptotically free and that under this condition the noncommutative quadratic IR divergences will not appear. The effective theory contains higher dimensional terms, which become more important at high energies. These terms predict an elastic photon-photon scattering due to the noncommutativity of space. The coefficients of these higher dimensional terms also satisfy a positivity constraint indicating that in this theory the related diseases of superluminal signal propagating and bad analytic properties of S-matrix do not exist. In the last section, we will apply our method to the noncommutative extra dimension theories.     

Variational quantum algorithms for trace norms and their applications

The trace norm of matrices plays an important role in quantum information and quantum computing. How to quantify it in today's noisy intermediate scale quantum(NISQ) devices is a crucial task for information processing. In this paper, we present three variational quantum algorithms on NISQ devices to estimate the trace norms corresponding to different situations.Compared with the previous methods, our means greatly reduce the requirement for quantum resources. Numerical experiments are provided to illustrate the effectiveness of our algorithms.     

Effective Action for Noncommutative U(1) Gauge Theory with Higher Dimensional Terms

In this paper we apply the assumption of our recent work in noncommutative scalar models to the noncommutative U(1) gauge theories. This assumption is that the noncommutative effects start to be visible continuously from a scale λ_NC and that below this scale the theory is a commutative one. Based on thisassumption and using background field method and loop calculations, an effective action is derived for noncommutative U(1) gauge theory. It will be shown that the corresponding low energy effective theory is asymptotically free and that under this condition the noncommutative quadratic IR divergences will not appear. The effective theory contains higher dimensional terms, which become more important at high energies. These terms predict an elastic photon-photon scattering due to the noncommutativity of space. Thecoefficients of these higher dimensional terms also satisfy a positivity constraint indicating that in this theory the related diseases of superluminal signal propagating and bad analytic properties of S-matrix do not exist. In the last section, we will apply our method to the noncommutative extra dimension theories.