Table of contents

The 'little group' for massless particles (namely, the Lorentz transformations Λ that leave a null vector invariant) is isomorphic to the Euclidean group E2: translations and rotations in a plane. We show how to obtain explicitly the rotation angle of E2 as a function of Λ and we relate that angle to Berry's topological phase. Some particles admit both signs of helicity, and it is then possible to define a reduced density matrix for their polarization. However, that density matrix is physically meaningless because it has no transformation law under the Lorentz group, even under ordinary rotations.

We present and apply a generalized coarse-graining method of reducing the Becker–Döring model; originally formulated to describe the stepwise aggregation and fragmentation of clusters during nucleation. Previous formulations of the coarse-graining procedure have allowed a temporal rescaling of the coarse-grained reaction rates; this is generalized to allow the rescaling to depend on cluster size. The form of this factor is derived for general reaction rates and general mesh function so that the steady-state solution is preserved; in the case of an even mesh function the kinetics can also be accurately reproduced. With a size-dependent mesh function the equilibrium solution and the form of convergence to this state are matched for a specific example. Finally we consider reaction rates relevant to the classical nucleation theory of spherical cluster growth, and numerically compare solutions of the full system to the generalized coarse-grained system in both constant monomer and constant mass formulations, demonstrating the accuracy of the method.

For many-particle systems defined on lattices we investigate the global structure of effective Hamiltonians and observables obtained by means of a suitable basis transformation. We study transformations which lead to effective Hamiltonians conserving the number of excitations. The same transformation must be used to obtain effective observables. The analysis of the structure shows that effective operators give rise to a simple and intuitive perspective on the initial problem. The systematic calculation of n-particle irreducible quantities becomes possible constituting a significant progress. Details on how to implement the approach perturbatively for a large class of systems are presented.

We revisit the relationship between the Nernst theorem and the Kelvin–Planck statement of the second law. We propose that the exchange of entropy uniformly vanishes as the temperature goes to zero. The analysis of this assumption shows that is equivalent to the fact that the compensation of a Carnot engine scales with the absorbed heat so that the Nernst theorem should be embedded in the statement of the second law.

We study the Schrödinger equation of the hydrogen atom in the (arbitrarily) strong magnetic field in two dimensions, which is an integrable and separable system. The energy spectrum is very interesting as it has infinitely many accumulation points located at the values of the Landau energy levels of a free electron in the uniform magnetic field. In the polar coordinates, the canonical (not kinetic!) angular momentum has a precise eigenvalue and we have the one-dimensional radial Schrödinger equation, which is an ordinary second-order differential equation whose analytical exact solution is unknown. We describe the qualitative properties of the energy spectrum, and we propose a semi-analytical method to numerically calculate the eigenenergies (the representation matrix of the Hamiltonian in the Landau basis is analytically calculated and is exactly known). Also, we use a number of useful analytical approximation methods, such as semiclassical approximations, the perturbation method, the variational method and the Taylor power expansion of the potential around the minimum to estimate the ground-state energy and the higher levels.

The fringed tunnelling, which can be observed in strongly coupled 1.5-dimensional barrier systems as well as in autonomous two-dimensional barrier systems, is a manifestation of intrinsic multi-dimensional effects in the tunnelling process. In this paper, we investigate such an intrinsic multi-dimensional effect on the tunnelling by means of classical dynamical theory and semiclassical theory, which are extended to the complex domain. In particular, we clarify the underlying classical mechanism which enables multiple tunnelling trajectories to simultaneously contribute to the wavefunction, thereby resulting in the formation of the remarkable interference fringe on it. Theoretical analyses are carried out in the low-frequency regime based upon a complexified adiabatic tunnelling solution, together with the Melnikov method extended to the complex domain. These analyses reveal that the fringed tunnelling is a result of a heteroclinic-like entanglement between the complexified stable manifold of the barrier-top unstable periodic orbit and the incident beam set. Tunnelling particles reach the real phase plane very promptly, guided by the complexified stable manifold, which gives quite a different picture of the tunnelling from the ordinary instanton mechanism. More fundamentally, the entanglement is attributed to a divergent movement of movable singularities of the classical trajectory, namely, to a singular dependence of singularities on its initial condition.

Reversible cellular automata are invertible discrete dynamical systems which have been widely studied both for analysing interesting theoretical questions and for obtaining relevant practical applications, for instance, simulating invertible natural systems or implementing data coding devices. An important problem in the theory of reversible automata is to know how the local behaviour which is not invertible is able to yield a reversible global one. In this sense, symbolic dynamics plays an important role for obtaining an adequate representation of a reversible cellular automaton. In this paper we prove the equivalence between a reversible automaton where the ancestors only differ at one side (technically with one of the two Welch indices equal to 1) and a full shift. We represent any reversible automaton by a de Bruijn diagram, and we characterize the way in which the diagram produces an evolution formed by undefined repetitions of two states. By means of amalgamations, we prove that there is always a way of transforming a de Bruijn diagram into the full shift. Finally, we provide an example illustrating the previous results.

An integral equation describing the fuel distribution necessary to maintain a flat flux in a nuclear reactor in two group transport theory is reduced to the solution of a singular integral equation. The formalism developed enables the physical aspects of the problem to be better understood and its relationship with the corresponding diffusion theory model is highlighted. The integral equation is solved by reducing it to a non-singular Fredholm equation which is then evaluated numerically.

We construct one-parameter deformation of the Dorfman Hamiltonian operator for the Riemann hierarchy using the quasi-Miura transformation from topological field theory. In this way, one can get the rational approximate symmetries of the KdV equation and then investigate its bi-Hamiltonian structure.

We construct a Poisson map between manifolds with linear Poisson brackets corresponding to the Lie algebras e(3) and so(4). Using this map we establish a connection between the deformed Kowalevski top on e(3) proposed by Sokolov and the Kowalevski top on so(4). The connection between these systems leads to the separation of variables for the deformed system on e(3) and yields the natural 5 × 5 Lax pair for the Kowalevski top on so(4).

We study the existence of a natural 'linearization' process for generalized connections on an affine bundle. It is shown that this leads to an affine generalized connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalized connections on such bundles.

We deal with a family of generalized coherent states obtained by means of operators of an unitary irreducible representation of the group of affine transformations of the real line. We prove that the ranges of the corresponding coherent state transforms coincide with spaces of bound states of the Landau Hamiltonian in the hyperbolic plane. This provides us with a new characterization of hyperbolic Landau states.

We derive a new integrable system of evolutions describing the dynamics of the transverse strain waves propagating in a bulk crystal with ion impurities possessing the spin-1/2. This system describes the evolution of the picosecond acoustic pulses corresponding to 'a few cycle pulses'. The Lax representation for this system and its particular cases are presented. Soliton solutions corresponding to the common and particular cases are found.

We derive the wavefunction and energy level formula for two charged particles with magnetic interaction, i.e., the Hamiltonian includes both two-body Coulomb interaction and a kinetic coupling. It is by virtue of the EPR entangled state representation we can conveniently derive the exact result.

We present a quantum parameter estimation theory for a generalized Pauli channel Γ_θ : (^d) → (^d), where the parameter θ is regarded as a coordinate system of the probability simplex ^d2−1. We show that for each degree n of extension (id ⊗ Γ_θ)^⊗n : ((^d ⊗ ^d)^⊗n) → ((^d ⊗ ^d)^⊗n), the SLD Fisher information matrix for the output states takes the maximum when the input state is an n-tensor product of a maximally entangled state τ^ME ∊ (^d ⊗ ^d). We further prove that for the corresponding quantum Cramér–Rao inequality, there is an efficient estimator if and only if the parameter θ is ∇^m-affine in ^d2−1. These results rely on the fact that the family {id ⊗ Γ_θ(τ^ME)}_θ of output states can be identified with ^d2−1 in the sense of quantum information geometry. This fact further allows us to investigate submodels of generalized Pauli channels in a unified manner.

An algebraic method of constructing potentials for which the Schrödinger equation with position dependent mass can be solved exactly is presented. A general form of the generators of su(1,1) algebra has been employed with a unified approach to the problem. Our systematic approach reproduces a number of earlier results and also leads to some novelties. We show that the solutions of the Schrödinger equation with position dependent mass are free from the choice of parameters for position dependent mass. Two classes of potentials are constructed that include almost all exactly solvable potentials.

We show how to convert a quantum stabilizer code to a one- or two-way entanglement distillation protocol. The proposed conversion method is a generalization of those of Shor–Preskill and Nielsen–Chuang. The recurrence protocol and the quantum privacy amplification protocol are equivalent to the protocols converted from [[2, 1]] stabilizer codes. We also give an example of a two-way protocol converted from a stabilizer better than the recurrence protocol and the quantum privacy amplification protocol. The distillable entanglement by the class of one-way protocols converted from stabilizer codes for a certain class of states is equal to or greater than the achievable rate of stabilizer codes over the channel corresponding to the distilled state, and they can distill asymptotically more entanglement from a very noisy Werner state than the hashing protocol.

We study the quantization problem of relativistic scalar and spinning particles interacting with a radiation electromagnetic field by using the path integral and the external source method. The spin degrees of freedom are described in terms of Grassmann variables and the Feynman kernel is obtained through functional integration on both Bose and Fermi variables. We provide rigorous proof that the Feynman amplitudes are only determined by the classical contribution and we explicitly evaluate the propagators.

This paper discusses skyrmions on the 3-sphere coupled to fermions. The resulting Dirac equation commutes with a generalized angular momentum G. For G = 0 the Dirac equation can be solved explicitly for a constant Skyrme configuration and also for a SO(4) symmetric hedgehog configuration. We discuss how the spectrum changes due to the presence of a non-trivial winding number, and also consider more general Skyrme configurations numerically.