Space-like symmetrization operator and local commutativity axiom

The correct definition of the space-like symmetrization operator is given. Using this operator, an extension of the Fock space is constructed. In this space an operator function, which satisfies the local commutativity axiom in a more complete sense than the free field, is defined. It is shown that this operator function satisfies all the other axioms of the field theory, except the spectral one.     

Particle dynamics on Snyder space

We examine Hamiltonian formalism on Euclidean Snyder space. The latter corresponds to a lattice in the quantum theory. For any given dynamical system, it may not be possible to identify time with a real number parametrizing the evolution in the quantum theory. The alternative requires the introduction of a dynamical time operator. We obtain the dynamical time operator for the relativistic (nonrelativistic) particle, and use it to construct the generators of Poincaré (Galilei) group on Snyder space.     

Bosonization and quantum hydrodynamics

It is shown that it is possible to bosonize fermions in any number of dimensions using the hydrodynamic variables, namely the velocity potential and density. The slow part of the Fermi field is defined irrespective of dimensionality and the commutators of this field with currents and densities are exponentiated using the velocity potential as conjugate to the density. An action in terms of these canonical bosonic variables is proposed that reproduces the correct current and density correlations. This formalism in one dimension is shown to be equivalent to the Tomonaga-Luttinger approach as it leads to the same propagator and exponents. We compute the one-particle properties of a spinless homogeneous Fermi system in two spatial dimensions with long-range gauge interactions and highlight the metal-insulator transition in the system. A general formula for the generating function of density correlations is derived that is valid beyond the random phase approximation. Finally, we write down a formula for the annihilation operator in momentum space directly in terms of number conserving products of Fermi fields.     

Diffusion in inhomogeneous media

The problem is to establish the correct diffusion equation in a medium that is inhomogeneous and whose temperature also varies in space. As a special model we study particles whose phase space distribution obeys Kramers' equation with a generalized collision operator. In the usual limit of strong collisions a diffusion equation is obtained. This equation contains additional drift terms, which depend on the form of the collision operator. They cannot be expressed as a mobility and a diffusion coefficient, unless the decay law of the velocity happens to be linear. Conclusion: no universal form of the diffusion equation exists, but each system has to be studied individually.Dedicated to Professor Harry Thomas on the occasion of his 60th birthday     

On the derivation of the Schroedinger equation in a Riemannian manifold

Under certain conditions it is shown that the kinetic part of the dynamical operator of a quantum mechanical system with a Riemannian manifold as configuration space is the Laplace-Beltrami operator.     

A stochastic approach to the Euler-Poincaré number of the loop space of a developable orbifold

We define a regularised version of the de Rham operator over the free loop space. We perform a semi-classical approximation of it, such that the index of the limit operator is equal to the “orbit Euler characteristic” of physicists.     

Theory of quantum error correction for general noise

A measure of quality of an error-correcting code is the maximum number of errors that it is able to correct. We show that a suitable notion of "number of errors" e makes sense for any quantum or classical system in the presence of arbitrary interactions. Thus, e-error-correcting codes protect information without requiring the usual assumptions of independence. We prove the existence of large codes for both quantum and classical information. By viewing error-correcting codes as subsystems, we relate codes to irreducible representations of operator algebras and show that noiseless subsystems are infinite-distance error-correcting codes.     

Adaptive multigrid algorithm for the lattice Wilson-Dirac operator

We present an adaptive multigrid solver for application to the non-Hermitian Wilson-Dirac system of QCD. The key components leading to the success of our proposed algorithm are the use of an adaptive projection onto coarse grids that preserves the near null space of the system matrix together with a simplified form of the correction based on the so-called γ5-Hermitian symmetry of the Dirac operator. We demonstrate that the algorithm nearly eliminates critical slowing down in the chiral limit and that it has weak dependence on the lattice volume.     

Quantum Brachistochrone problem and the geometry of the state space in pseudo-Hermitian quantum mechanics

A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining the inner product of physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT-symmetric or other non-Hermitian Hamiltonians than those given by Hermitian Hamiltonians.     

A horseshoe for the doubling operator: Topological dynamics for metric universality

The doubling operator, properly defined on the space of smooth maps on the interval at the boundary of chaos, yields a dynamical system in this function space. Even if one restricts oneself to the space of real analytic maps, there is evidence that the dynamics of the doubling operator contains a horseshoe whose symbolic dynamics is described by the one-sided shift on two symbols. We indicate also how some of the global aspects of this dynamics could be recognized in a physical experiment on the transition to chaos.     

Entropy increase for a class of dynamical maps

The dynamical evolution of a quantum system is described by a one parameter family of linear transformations of the space of self-adjoint trace class operators (on the Hilbert space of the system) into itself, which map statistical operators to statistical operators. We call such transformations dynamical maps. We give a sufficient condition for a dynamical map A not to decrease the entropy of a statistical operator. In the special case of an N-level system, this condition is also necessary and it is equivalent to the property that A preserves the central state.     

Density Matrix and Squeezed State for Two Harmonic Oscillators with a Kinetic Coupling

We study two harmonic oscillators with a kinetic coupling system. By taking a unitary transformation approach, we turn the system into the Fock space of two independent harmonic oscillators and derive the density matrix for it. The corresponding unitary operator U is characteristic of including frequency-jump squeezing transformation. By virtue of the technique of the integration within an ordered product of operators, we manifestly show that the ground state of the system is a squeezed state.     

Multiple-quantum operator algebra spaces and description for unitary time evolution of multilevel spin systems

A general operator algebra formalism is proposed for describing the unitary time evolution of multilevel spin systems. The time-evolutional propagator of a multilevel spin system is decomposed completely into a product of a series of elementary propagators. Then the unitary time evolution of the system can be determined exactly through the decomposed propagator. This decomposition may be simplified with the help of the properties of the finite dimensional Liouville operator space and of its three operator subspaces, and the operator algebra structure of spin Hamiltonian of the system. The Liouville operator space contains the even-order multiple-quantum, the zero-quantum, and the longitudinal magnetization and spin order operator subspace, and moreover, each former subspace contains its following subspaces. The propagator can be decomposed readily and completely for a spin system whose Hamiltonian is a member of the longitudinal magnetization and spin order operator subspace. If the Hamiltonian of a spin system is a zero-quantum operator this decomposition may be implemented by making a zero-quantum unitary transformation on the Hamiltonian to convert it into the diagonalized Hamiltonian, while if the Hamiltonian is an even-order multiple-quantum operator the decomposition may be carried out by diagonalizing the Hamiltonian with an even-order multiple-quantum unitary transformation. When the Hamiltonian is a member of the Liouville operator space but not any element of its three subspaces the decomposition may be achieved first by making an odd-order multiple-quantum and then an even-order multiple-quantum unitary transformation to convert it into the diagonalized Hamiltonian. Parameter equations to determine the unknown parameters in the decomposed propagator are derived for the general case and approaches to solve the equations are proposed.     

Bounded energy approximation to an unstable quantum system

Any state can be approximated by the so-called bounded energy states; this gives a possibility to construct approximations with clear physical meaning to the state space, reduced evolution operator and other characteristics of an unstable system. We discuss it in detail, especially in connection with the Weisskopf-Wigner (semigroup) condition.     

一般阻尼随机微分方程的拟局部振荡算法

提出了一种模拟随机微分方程的拟局部振荡算法,即利用算符劈裂方法和势函数的泰勒展开,对噪声作用下耗散粒子的时间演化算符进行分解,得到了对应涨落行为的扩散算符和对应确定轨迹的漂移算符.其中局部简谐势场的涨落过程可获得解析解,而剩余的确定项则利用简单的Euler算法积分.应用到几个算例并与常用的两种算法相比较,结果表明:本算法随时间步长最稳定,可使用较大的时间步长.     

Steady-state currents through nanodevices: a scattering-states numerical renormalization-group approach to open quantum systems

We propose a numerical renormalization group (NRG) approach to steady-state currents through nanodevices. A discretization of the scattering-states continuum ensures the correct boundary condition for an open quantum system. We introduce two degenerate Wilson chains for current carrying left- and right-moving electrons reflecting time-reversal symmetry in the absence of a finite bias V. We employ the time-dependent NRG to evolve the known steady-state density operator for a noninteracting junction into the density operator of the fully interacting nanodevice at finite bias. We calculate the differential conductance as function of V, T, and the external magnetic field.     

Vertex Operators in 4D Quantum Gravity Formulated as CFT

We study vertex operators in 4D conformal field theory derived from quantized gravity, whose dynamics is governed by the Wess-Zumino action by Riegert and the Weyl action. Conformal symmetry is equal to diffeomorphism symmetry in the ultraviolet limit, which mixes positive-metric and negative-metric modes of the gravitational field and thus these modes cannot be treated separately in physical operators. In this paper, we construct gravitational vertex operators such as the Ricci scalar, defined as space-time volume integrals of them are invariant under conformal transformations. Short distance singularities of these operator products are computed and it is shown that their coefficients have physically correct signs. Furthermore, we show that conformal algebra holds even in the system perturbed by the cosmological constant vertex operator as in the case of the Liouville theory shown by Curtright and Thorn.     

The Schrödinger and Pauli-Dirac Oscillators in Noncommutative Phase Space

We investigate the non-relativistic Schrödinger and Pauli-Dirac oscillators in noncommutative phase space using the five-dimensional Galilean covariant framework. The Schrödinger oscillator presented the correct energy spectrum whose non isotropy is caused by the noncommutativity with an expected similarity between this system and the particle in a magnetic field. A general Hamiltonian for the 3-dimensional Galilean covariant Pauli-Dirac oscillator was obtained and it presents the usual terms that appears in commutative space, like Zeeman effect and spin-orbit terms. We find that the Hamiltonian also possesses terms involving the noncommutative parameters that are related to a type of magnetic moment and an electric dipole moment.     

Discreteness of area and volume in quantum gravity

We study the operator that corresponds to the measurement of volume, in non-perturbative quantum gravity, and we compute its spectrum. The operator is constructed in the loop representation, via a regularization procedure; it is finite, background independent, and diffeomorphism-invariant, and therefore well defined on the space of diffeomorphism invariant states (knot states). We find that the spectrum of the volume of any physical region is discrete. A family of eigenstates are in one to one correspondence with the spin networks, which were introduced by Penrose in a different context. We compute the corresponding component of the spectrum, and exhibit the eigenvalues explicitly. The other eigenstates are related to a generalization of the spin networks, and their eigenvalues can be computed by diagonalizing finite dimensional matrices. Furthermore, we show that the eigenstates of the volume diagonalize also the area operator. We argue that the spectra of volume and area determined here can be considered as predictions of the loop-representation formulation of quantum gravity on the outcomes of (hypothetical) Planck-scale sensitive measurements of the geometry of space.     

On vector radiative transfer equation in curvilinear coordinate systems

The differential operator of polarized radiative transfer equation is examined in case of homogeneous medium in Euclidean three-dimensional space with arbitrary curvilinear coordinate system defined in it. This study shows that an apparent rotation of polarization plane along the light ray with respect to the chosen reference plane for Stokes parameters generally takes place, due to purely geometric reasons. Analytic expressions for the differential operator of transfer equation dependent on the components of metric tensor and their derivatives are found, and the derivation of differential operator of polarized radiative transfer equation has been made a standard procedure. Considerable simplifications take place if the coordinate system is orthogonal.