From quantum trajectories to classical orbits

The evolution of open quantum systems can be “unraveled” into individual “trajectories” in a variety of ways. In the mesoscopic regime, quantum jump (QJ) trajectories approach a diffusive limit similar to quantum state diffusion (QSD). In the classical limit, both unravelings show the rise of classical orbits for both regular and chaotic systems.     

从量子谱到经典轨道：矩形腔中的弹子球

用体系的本征值和本征波函数定义一种新的量子谱函数．这种量子谱函数的傅里叶变换包含了体系从一个给定点到另一个给定点的许多经典轨道的信息．以二维矩形腔中的弹子球运动体系为例的初步研究验证了这一结论．　　 关键词： 经典量子对应 半经典物理     

Magnetic monopoles

Summary Theoretical foundations for the existence of magnetic monopoles are given. Consequences extracted from cosmological and astrophysical considerations on the abundance of magnetic monopoles today are reviewed. The latest experimental results in the search for classical monopoles as well as for grand unified monopoles are summarized.

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Riassunto Si danno i fondamenti teorici per l'esistenza di monopoli magnétici. Si esaminano le conseguenze tratte da considerazioni cosmologiche e astrofisiche sull'abbondanza di monopoli magnetici attuale. Si riassumono i piú recenti risultati sperimentali sulla ricerca di monopoli classici e di quelli della grande unificazione.

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Резюме Приводятся теоретические обоснования сушствования магнитных монополей. Анализируются следствия из космологических и астрофизических рассмотрений, касающиеся распространенноси магнитных мононолей в настоящее время. Приводятся последние экспериментальные результаты по поискы классических монополей и монополей Великого Объединения.

     

Magnetic monopoles

These are notes of the first part of the lectures given at the JINR-ISU Baikal Summer School on Physics of Elementary Particles and Astrophysics (July 2010). I review classical monopole solutions of the SU(2) Yang-Mills-Higgs theory providing a pedagogical introduction into to the theory of non-Abelian monopoles both in the BPS limit and beyond of it. I briefly discuss monopole dynamics, the idea of the moduli space and some of the basic properties which are connected with the field theoretical aspects of these classical solutions.     

From quantum mechanics to classical statistical physics: Generalized Rokhsar-Kivelson Hamiltonians and the “Stochastic Matrix Form” decomposition

Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar-Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with stochastic classical systems described by a Master equation of the matrix type, hence their name. It then follows that the equilibrium partition function of the stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.     

Two-dimensional supersymmetry: From SUSY quantum mechanics to integrable classical models

Two known two-dimensional SUSY quantum mechanical constructions—the direct generalization of SUSY with first-order supercharges and higher-order SUSY with second-order supercharges—are combined for a class of 2-dim quantum models, which are not amenable to separation of variables. The appropriate classical limit of quantum systems allows us to construct SUSY-extensions of original classical scalar Hamiltonians. Special emphasis is placed on the symmetry properties of the models thus obtained—the explicit expressions of quantum symmetry operators and of classical integrals of motion are given for all (scalar and matrix) components of SUSY-extensions. Using Grassmanian variables, the symmetry operators and classical integrals of motion are written in a unique form for the whole Superhamiltonian. The links of the approach to the classical Hamilton-Jacobi method for related “flipped” potentials are established.     

Nonstandard analysis in classical physics and quantum formal scattering

After a rigorous introduction to hyperreal numbers, we give in terms of non standard analysis, (1) a Lagrangian statement of classical physics, and (2) a statement of formal quantum scattering.     

Outline of a classical theory of quantum physics and gravitation

It is argued that in the manner in which the Galilean-Newtonian physics may be said to have explained the Ptolemaic-Copernican theories in terms which have since been called classical, so also Milner's theories of the structure of matter may be said to explain present day quantum and relativistic theory. In both cases the former employ the concept of force and the latter, by contrast, are geometrical theories. Milner envisaged space as being stressed, whereas Einstein thought of it as strained. Development of Milner's theory from criticisms and suggestions made by Kilmister has taken it further into the realms of quantum and gravitational physics, where it is found to give a more physically comprehensible explanation of the phenomena. Further, it shows why present day quantum theory is cast in a statistical form. The theory is supported by many predictions such as the ratio of Planck's constant to the mass of the electron, the value of the fine structure constant and reason for apparent variations in past measurements, the magnetic moment of the electron and proton of the stable particles such as the neutron Λ and Σ together with the kaon, and a relation between the universal gravitational constant and Hubble's constant—all within published experimental accuracy. The latest results to be accounted for by the theory are the masses of the newly discovered ψ particles and confirmation of the value of the decay of Newton's gravitational constant obtained from lunar measurements. While this paper is being typed, new particles are rapidly being discovered—the latest being a neutral ψ particle. A short Appendix discusses the significance of these.     

The concept of indistinguishable particles in classical and quantum physics

The consequences of the following definition of indistinguishability are analyzed. Indistinguishable classical or quantum particles are identical classical or quantum particles in a state characterized by a probability measure, a statistical operator respectively, which is invariant under any permutation of the particles. According to this definition the particles of classical Maxwell-Boltzmann statistics are indistinguishable.     

Magnetic monopoles, alive

We review recent developments in understanding the physics of the magnetic monopoles in unbroken non-Abelian gauge theories. Since numerical data on the monopoles are accumulated in lattice simulations, the continuum theory is understood as the limiting case of the lattice formulation. We emphasize physical effects related to the monopoles. In particular, we discuss the monopole-antimonopole potential at short and larger distances as well as a dual formulation of the gluodynamics, relevant to the physics of the confinement.     

Magnetic monopoles with no strings

Previous treatments of the charge-monopole system have used either a lagrangian singular along the Dira string or a multiple valued action integral. Here we formulate an action principle for this system which involves no strings and is single. This is achieved by writing the lagrangian directly in terms of the variables of a suitable fibre bundle. Canonical quantization is carried out in a simple way and known results are recovered.     

Magnetic monopoles in gauge theories

Magnetic monopoles in gauge theories are investigated. Let G be the gauge group and H the group of symmetries which are not spontaneously broken. The existence of magnetic monopoles is proved in the case when the group G has a compact covering group but the covering group of H is non-compact.