Table of contents

Based on the idea of balance between the internal and external potentials, it is shown that for a two-component Bose–Einstein condensate (BEC) confined in an optical lattice the presence of a spacetime periodic laser field can induce a family of exact Floquet states (EFSs). The atomic-number densities of the EFSs are illustrated and the atomic current, phase blowing-up and the quantum reflection are investigated. The balance conditions and blowing-up region on parameter space are found, and the influences of phase blowing-up to the velocity fields, flow densities are revealed. It is demonstrated that the BEC motions can be controlled by adjusting the laser frequency, wave vector and amplitude.

We study the dynamics of a kink in a one-lane asymmetric simple exclusion process with detachment and attachment of particles at arbitrary sites. For a system with one site of detachment and attachment we find that the kink is trapped by the site, and the probability distribution of the kink position is described by the overdumped Fokker–Planck equation with a V-shaped potential. When the detachment and attachment take place at every site, we confirm that the kink motion is equivalent to the diffusion in a harmonic potential. We compare our results with the Monte Carlo simulation, and check the quantitative validity of our theoretical prediction of the potential form.

Recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices. Here we show how these results can be extended to other classes of graphs, significantly increasing the number and variety of solved problems. Any graph that can be decomposed into a certain arrangement of triangles, which we call self-dual, gives a class of lattices whose percolation thresholds can be found exactly by a recently introduced triangle–triangle transformation. We use this method to generalize Wierman's solution of the bow-tie lattice to yield several new solutions. We also give another example of a self-dual arrangement of triangles that leads to a further class of solvable problems. There are certainly many more such classes.

We compute the mixed correlation function in a way which involves only the orthogonal polynomials with degrees close to n (in some sense like the Christoffel–Darboux theorem for non-mixed correlation functions). We also derive new representations for the differential systems satisfied by the bi-orthogonal polynomials, and we find new formulae for the spectral curve. In particular we prove the conjecture of M Bertola, claiming that the spectral curve is the same curve which appears in the loop equations.

A new algorithm for linearization of a third-order ordinary differential equation is presented. The algorithm consists of composition of two operations: reducing order of an ordinary differential equation and using the Lie linearization test for the obtained second-order ordinary differential equation. The application of the algorithm to several ordinary differential equations is given.

We study the derivative of the Legendre function of the first kind, P_ν(z), with respect to its degree ν. At first, we provide two contour integral representations for ∂P_ν(z)/∂ν. Then, we proceed to investigate the case of [∂P_ν(z)/∂ν]_ν=n, with n being an integer; this case is met in some physical and engineering problems. Since it holds that , we focus on the sub-case of n being a non-negative integer. We show that where R_n(z) is a polynomial in z of degree n. We present alternative derivations of several known explicit expressions for R_n(z) and also add some new. A generating function for R_n(z) is also constructed. Properties of the polynomials V_n(z) = [R_n(z) + (−1)ⁿR_n(−z)]/2 and W_n−1(z) = −[R_n(z) − (−1)ⁿR_n(−z)]/2 are also investigated. It is found that W_n−1(z) is the Christoffel polynomial, well known from the theory of the Legendre function of the second kind, Q_n(z). As examples of applications of the results obtained, we present non-standard derivations of some representations of Q_n(z), sum to closed forms some Legendre series, evaluate some definite integrals involving Legendre polynomials and also derive an explicit representation of the indefinite integral of the Legendre polynomial squared.

This paper treats the dynamics and scattering of a model of coupled oscillating systems, a finite dimensional one and a wave field on the half line. The coupling is realized producing the family of self-adjoint extensions of the suitably restricted self-adjoint operator describing the uncoupled dynamics. The spectral theory of the family is studied and the associated quadratic forms constructed. The dynamics turns out to be Hamiltonian and the Hamiltonian is described, including the case in which the finite-dimensional systems comprise nonlinear oscillators; in this case, the dynamics is shown to exist as well. In the linear case, the system is equivalent, on a dense subspace, to a wave equation on the half line with higher order boundary conditions, described by a differential polynomial p(∂_x) explicitly related to the model parameters. In terms of such structure, the Lax–Phillips scattering of the system is studied. In particular, we determine the scattering operator, which turns out to be unitarily equivalent to the multiplication operator given by the rational function −p(iκ)*/p(iκ), the incoming and outgoing translation representations and the Lax–Phillips semigroup, which describes the evolution of the states which are neither incoming in the past nor outgoing in the future.

Preparation of the pseudopure states in homonuclear systems of dipolar coupling spins is closely examined. An extremely important role of the non-diagonal part of zeroth-order coherence in the construction of the pseudopure state has been shown. Simulations of the preparation process of pseudopure states with the real molecular structures (a rectangular (1-chloro-4-nitrobenzene molecule), a chain (hydroxyapatite molecule), a ring (benzene molecule) and a double ring (cyclopentane molecule)) open the way to experimental NMR testing of the obtained results. The proposed method could be considered as a fruitful technology for quantum computation.

We discuss the hyperfine shifts of the positronium levels in a relativistic framework, starting from a two-fermion wave equation where, in addition to the Coulomb potential, the magnetic interaction between spins is described by a Breit term. We write the system of four first-order differential equations describing this model. We discuss its mathematical features, mainly in relation to possible singularities that may appear at finite values of the radial coordinate. We solve the boundary value problems both in the singular and non-singular cases and we develop a perturbation scheme, well suited for numerical computations, that allows us to calculate the hyperfine shifts for any level, according to well-established physical arguments that the Breit term must be treated at the first perturbative order. We discuss our results, comparing them with the corresponding values obtained from semi-classical expansions.

We construct a generalized concurrence for general multipartite states based on local W-class and GHZ-class operators. We explicitly construct the corresponding concurrence for three-partite states. The construction of the concurrence is interesting since it is based on local operators and therefore gives some hints on the classification of multipartite states.

A rotation number in the case of one-dimensional maps is introduced. As is shown, this rotation number is equivalent to the already known rotation number in the case of two-dimensional maps. The definition of the rotation number is given in two steps. First, it is defined for periodic orbits inside a window of organized motion (WOM). We show that in this case our definition coincides with the definition of the over-rotation number. Then, our definition is further generalized for chaotic orbits outside the WOMs. Thus, we obtain a unified definition of the rotation number for the whole area of the chaotic zone of the bifurcation diagram, having a number of useful applications. Namely, it can be used as a tool to distinguish whether an orbit is contained within a WOM or not, as a tool of numerical location of the bifurcation points, of the band mergings, as well as of the boundary points of a WOM. Finally, a method of numerical calculation of the percentage of the cumulative width of the WOMs in every particular segment (chaotic band) of the chaotic zone is given.

In this paper, we present explicit results for the fusion of irreducible and higher rank representations in two logarithmically conformal models, the augmented c_2,3 = 0 model as well as the augmented Yang–Lee model at c_2,5 = −22/5. We analyse their spectrum of representations which is consistent with the symmetry and associativity of the fusion algebra. We also describe the first few higher rank representations in detail. In particular, we present the first examples of consistent rank 3 indecomposable representations and describe their embedding structure. Knowing these two generic models we also conjecture the general representation content and fusion rules for general augmented c_p,q models.

We determine for each of the simple, simply connected and compact Lie groups SU(n), Spin(4n + 2) and E₆ with a complex fundamental representation that particular region inside the unit disc in the complex plane which is filled by their mean eigenvalues. We give analytical parameterizations for the boundary curves of these so-called trace figures. The area enclosed by a trace figure turns out to be a rational multiple of π in each case. We calculate also the length of the boundary curve and determine the radius of the largest circle that fits into a trace figure. The discrete centre of the corresponding compact Lie group shows up prominently in the form of cusp points of the trace figure placed symmetrically on the unit circle. For the exceptional Lie groups G₂, F₄ and E₈ with trivial centre we determine the (negative) lower bounds on their mean eigenvalues lying within the real interval [− 1, 1]. We find the rational boundary values −2/7, − 3/13 and −1/31 for G₂, F₄ and E₈, respectively.

Inspired by a recent work that proposes using coherent states to evaluate the Feynman kernel in noncommutative space, we provide an independent formulation of the path-integral approach for quantum mechanics on the Moyal plane, with the transition amplitude defined between two coherent states of mean position coordinates. In our approach, we invoke solely a representation of the noncommutative algebra in terms of commutative variables. The kernel expression for a general Hamiltonian was found to contain Gaussian-like damping terms, and it is non-perturbative in the sense that it does not reduce to the commutative theory in the limit of vanishing θ—the noncommutative parameter. As an example, we studied the free particle's propagator which turned out to be oscillating with period being the product of its mass and θ. Further, it satisfies the Pauli equation for a charged particle with its spin aligned to a constant, orthogonal B field in the ordinary Landau problem, thus providing an interesting evidence of how noncommutativity can induce spin-like effects at the quantum mechanical level.

The main focus of the second, enlarged edition of the book Mathematica for Theoretical Physics is on computational examples using the computer program Mathematica in various areas in physics. It is a notebook rather than a textbook. Indeed, the bookis just a printout of the Mathematica notebooks included on the CD. The second edition is divided into two volumes, the first covering classical mechanics and nonlinear dynamics, the second dealing with examples in electrodynamics, quantum mechanics, general relativity and fractal geometry.

The second volume is not suited for newcomers because basic and simple physical ideas which lead to complex formulas are not explained in detail. Instead, the computer technology makes it possible to write down and manipulate formulas of practically any length.

For researchers with experience in computing, the book contains a lot of interesting and non-trivial examples. Most of the examples discussed are standard textbook problems, but the power of Mathematica opens the path to more sophisticated solutions. For example, the exact solution for the perihelion shift of Mercury within general relativity is worked out in detail using elliptic functions.

The virial equation of state for molecules' interaction with Lennard-Jones-like potentials is discussed, including both classical and quantum corrections to the second virial coefficient. Interestingly, closed solutions become available using sophisticated computing methods within Mathematica. In my opinion, the textbook should not show formulas in detail which cover three or more pages—these technical data should just be contained on the CD. Instead, the textbook should focus on more detailed explanation of the physical concepts behind the technicalities. The discussion of the virial equation would benefit much from replacing 15 pages of Mathematica output with 15 pages of further explanation and motivation. In this combination, the power of computing merged with physical intuition would be of benefit even for newcomers.

In summary, this book shows in a convincing manner how classical problems in physics can be attacked with modern computing technology. The second volume is interesting for experienced users of Mathematica. For students, the textbook can be very useful in combination with a seminar.

We have spent more than twenty years applying supersymmetry (SUSY) to elementary particle physics and attempting to find an experimental manifestation of this symmetry. Terning's monograph demonstrates the strong influence of SUSY on theoretical elaborations in the field of elementary particles. It gives both an overview of modern supersymmetry in elementary particle physics and calculation techniques.

The author, trying to be closer to applications of SUSY in the real world of elementary particles, is also anticipating the importance of supersymmetry for rigorous study of nonperturbative phenomena in quantum field theory. In particular, he presents the `exact' SUSY β function using instanton methods, phenomena of anomalies and dualities.

Supersymmetry algebra is introduced by adding two anticommuting spinor generators to Poincaré algebra and by presenting massive and massless supermultiplets of its representations. The author prefers to use mostly the component description of field contents of the theories in question rather than the superfield formalism. Such a style makes the account closer to physical chartacteristics.

Relations required by SUSY among β functions of the gauge, Yukawa and quartic interactions are checked by direct calculations as well as to all orders in perturbation theory, thus demonstrating that SUSY survives quantization.

A discussion is included of the hierarchy problem of different scales of weak and strong interactions and its possible solution by the minimal supersymmetric standard model. Different SUSY breaking mechanisms are presented corresponding to a realistic phenomenology.

The monograph can also be considered as a guide to `duality' relations connecting different SUSY gauge theories, supergravities and superstrings. This is demonstrated referring to the particular properties and characteristics of these theories (field contents, scaling dimensions of appropriate operators etc). In particular, the last chapter deals with the AdS/CFT correspondence.

The author explains clearly most of the arguments in discussions and refers for further details to original papers (with corresponding arXiv numbers), selected lists of which appear at the end of each chapter (there are more than 300 references in the book).

Considered as a whole the book covers primers on quantum fields, Feynman diagrams, renormalization procedure and renormalization groups, as well as the representation theory of classical linear Lie algebras. Some necessary information on irreducible representations of su(N), so(N) and sp(2N) is given in an appendix.

There are in the text short historical and biographical notes concerning those scientists who made important contributions to the subject of the monograph: S Coleman, Yu Golfand, E Witten and others.

Most of the seventeen chapters contain a few exercises to check the reader's understanding of the corresponding material.

This monograph will be useful for graduate students and researchers in the field of elementary particles.