Table of contents

There will be two important changes in the structure of this journal. From the first issue in January 2007 the name of the journal will be Journal of Physics A: Mathematical and Theoretical. Also, there will be a new section entitled `Fast Track Communications: Short Innovative Papers'. Letters to the Editor will no longer be published in the journal.

Change of journal name. The new name, Journal of Physics A: Mathematical and Theoretical, is an accurate description of the content, nature and spirit of the journal as described in the scope statement on the inside front cover:

`A major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.'

The word `General' is no longer appropriate for the scope of this journal. This name change will have no effect on the subject material of the papers the journal publishes.

Fast Track Communications: Short Innovative Papers. Beginning with issue 1 of the 2007 volume, the journal will publish a new article type, Fast Track Communications. Fast Track Communications (FTCs) are defined as outstanding short papers reporting new and timely developments in mathematical and theoretical physics. These high-quality articles will be of importance to readers of Journal of Physics A: Mathematical and Theoretical but are not expected to meet any criterion of `general interest'. Conjectural articles are welcomed. FTCs will be processed quickly and are available free to readers in the electronic journal.

To facilitate a fast review process we urge authors of FTCs to restrict the length of their article to eight journal pages (5000 words). The key criteria for acceptance will be quality, novelty and timeliness. FTCs will be published at the front of the journal and will be free to readers through IOPSelect. Submissions are welcomed from 1 October 2006. For details of how to submit an FTC please visit the journal homepage, www.iop.org/journals/jphysa.

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We discuss a nonlinear stochastic master equation that governs the time evolution of the estimated quantum state. Its differential evolution corresponds to the infinitesimal updates that depend on the time-continuous measurement of the true quantum state. The new stochastic master equation couples to the two standard stochastic differential equations of time-continuous quantum measurement. For the first time, we can prove that the calculated estimate almost always converges to the true state, also at low-efficiency measurements. We show that our single-state theory can be adapted to weak continuous ensemble measurements as well.

We establish a link between the two functional approaches: a mesoscopic field theory developed recently by Ciach and Stell (2000 J. Mol. Liq.87 253) for the study of ionic models and an exact statistical field theory based on the method of collective variables.

Random walks in random environments and their diffusion analogues have been a source of surprising phenomena and challenging problems, especially in the non-reversible situation, since they began to be studied in the 1970s. We review the model, available results and techniques, and point out several gaps in the understanding of these processes.

From an experimental-mathematical perspective we analyse 'Ising-class' integrals. These are structurally related n-dimensional integrals we call C_n, D_n, E_n, where D_n is a magnetic susceptibility integral central to the Ising theory of solid-state physics. We first analyse We had conjectured—on the basis of extreme-precision numerical quadrature—that C_n has a finite large-n limit, namely C_∞ = 2 e^−2γ, with γ being the Euler constant. On such a numerological clue we are able to prove the conjecture. We then show that integrals D_n and E_n both decay exponentially with n, in a certain rigorous sense. While C_n, D_n remain unresolved for n ⩾ 5, we were able to conjecture a closed form for E₅. Our experimental results involved extreme-precision, multidimensional quadrature on intricate integrands; thus, a highly parallel computation was required.

We construct the explicit Q-operator incorporated with the sl₂-loop-algebra symmetry of the six-vertex model at roots of unity. The functional relations involving the Q-operator, the six-vertex transfer matrix and fusion matrices are derived from the Bethe equation, parallel to the Onsager-algebra-symmetry discussion in the superintegrable N-state chiral Potts model. We show that the whole set of functional equations is valid for the Q-operator. Direct calculations in certain cases are also given here for clearer illustration about the nature of the Q-operator in the symmetry study of root-of-unity six-vertex model from the functional-relation aspect.

The class of norm-dependent random matrix ensembles is studied in the presence of an external field. The probability density of the ensemble depends on the trace of the squared random matrices, but is otherwise arbitrary. An exact mapping to superspace is performed. A transformation formula is derived which gives the probability density in superspace as a single integral over the probability density in ordinary space. This is done for orthogonal, unitary and symplectic symmetries. In the case of unitary symmetry, some explicit results for the correlation functions are derived.

The transmission of a single soliton is investigated numerically across an interface between two Toda lattices, which are connected by a harmonic lattice. The soliton transmission coefficient is used as a measure of transmission. When the spring constant (κ) of the harmonic spring is small and the number of harmonic springs is greater than or equal to 2, a delay in the transmission of the soliton is found for proper κ. It is shown that the delay in the soliton transmission is due to the existence of the quasi-localization of the wave in the harmonic lattice and the agreement of the time scale of the motion between the two springs.

We study a physical model of the O(3)-invariant coupled integrable dispersionless equations that describes the dynamic of a focused system within the background of a plane gravitational field. The investigation is carried out both numerically and analytically, and realized beneath some assumptions superseding the structure constant with the structure function implemented in Lie algebra and quasigroup theory, respectively. The energy density and topological structures such as loop soliton are examined.

Using a realization of the q-exponential function as an infinite multiplicative series of the ordinary exponential functions we obtain new nonlinear connection formulae of the q-orthogonal polynomials such as q-Hermite, q-Laguerre and q-Gegenbauer polynomials in terms of their respective classical analogues.

We consider those Gaussian unitary ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painlevé IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable.

Transformation coefficients between standard bases for irreducible representations of the symmetric group S_n and split bases adapted to the subgroup (n₁ + n₂ = n) are considered. We first provide a selection rule and an identity rule for the subduction coefficients which allow us to decrease the number of unknowns and equations arising from the linear method by Pan and Chen. Then, using the reduced subduction graph approach, we may look at higher multiplicity instances. As a significant example, an orthonormalized solution for the first multiplicity-three case, which occurs in the decomposition of the irreducible representation [4, 3, 2, 1] of S₁₀ into [3, 2, 1] ⊗ [3, 1] of S₆ × S₄, is presented and discussed.

We show that given a nonvanishing particular solution of the equation the corresponding differential operator can be factorized into a product of two first-order operators. The factorization allows us to reduce the above equation to a first-order equation which in a two-dimensional case is a Vekua equation of a special form. Under quite general conditions on the coefficients p and q, we obtain an algorithm which allows us to construct in explicit form positive formal powers (solutions of the Vekua equation generalizing the usual powers (z − z₀)ⁿ, n = 0, 1, ...). This result means that under quite general conditions one can construct an infinite system of exact solutions of the above equation explicitly and, moreover, at least when p and q are real valued this system will be complete in ker(divp grad + q) in the sense that any solution of the above equation in a simply connected domain Ω can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of Ω. Finally, we give a similar factorization of the operator (divp grad + q) in a multidimensional case and obtain a natural generalization of the Vekua equation which is related to second-order operators in a similar way as its two-dimensional prototype does.

The main difficulty in utilizing the O(4) symmetry of the hydrogen atom in practical calculations is the dependence of the Fock stereographic projection on energy. This is due to the fact that the wavefunctions of the states with different energies are proportional to the hyperspherical harmonics (HSH) corresponding to different points on the hypersphere. Thus, the calculation of the matrix elements reduces to the problem of re-expanding HSH in terms of HSH depending on different points on the hypersphere. We solve this problem by applying the technique of multipole expansions for four-dimensional HSH. As a result, we obtain the multipole expansions whose coefficients are the matrix elements of the boost operator taken between hydrogen wavefunctions (i.e., hydrogen form factors). The explicit expressions for those coefficients are derived. It is shown that the hydrogen matrix elements can be presented as derivatives of an elementary function. Such an operator representation is convenient for the derivation of recurrence relations connecting matrix elements between states corresponding to different values of the quantum numbers n and l.

We study associative multiplications in semi-simple associative algebras over compatible with the usual one. An interesting class of such multiplications is related to the affine Dynkin diagrams of , and -type. In this paper we investigate in detail the multiplications of the -type and integrable matrix ODEs and PDEs generated by them.

The Schwinger representation and the Marumori–Yamamura–Tokunaga boson expansion are used to describe the Lipkin model in terms of generalized coherent states. The groundstate, first excited state and RPA energies are obtained within several variant types of coherent states. It has been found that generalized coherent states defined in consonance with the parity symmetry of the model describe particularly well the transition from weak to strong coupling, providing a remarkable improvement of the mean-field description of the transition zone.

A one-particle non-relativistic quantum mechanical solvable model in two-dimensional space is given. The Hamiltonian is the sum of kinetic and interaction parts. Interactions are separable and can be centred at n arbitrary points of the plane. Conditions for the existence and for the number of bound states in finite linear chains are formulated in terms of the parameters of the interactions and intercentre distances. Scattering problems are also considered. Finally, when the interactions are centred in a single centre, it is shown that the model remains solvable in the presence of a uniform magnetic field of arbitrary intensity.

It is shown that for an adaptive quantum estimation scheme based on locally unbiased measurements, the sequence of maximum likelihood estimators is strongly consistent and asymptotically efficient.

The polarization of a multi-component vector soliton (of the Manakov type) can be thought of as a state vector of a system of qubits (a register of quantum information). A change of this state on demand via colliding the register pulse with other solitons is shown to be possible with arbitrary accuracy. The parameters (polarization, pulse width, velocity) of the register-switching solitons corresponding to the computationally universal set of quantum gates are found. Physical realizations of information processing using effects of the self-focusing of optical media or of the self-induced transparency are considered.

The tomographic approach to quantum mechanics is revisited as a direct tool to investigate the violation of Bell-like inequalities. Since quantum tomograms are well defined probability distributions, the tomographic approach is emphasized to be the most natural one to compare the predictions of classical and quantum theory. Examples of inequalities for two qubits and two qutrits are considered in the tomographic probability representation of spin states.

We analyse some compositeness effects and their relation with entanglement. We show that the purity of a composite system increases, in the sense of the expectation values of the deviation operators, with large values of the entanglement between the components of the system. We also study the validity of Pauli's principle in composite systems. It is valid within the limits of application of the approach presented here. We also present an example of two identical fermions, one of them entangled with a distinguishable particle, where the exclusion principle cannot be applied. This result can be important in the description of open systems.

An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) is presented. The method that was used was to replace the Weyl–Heisenberg algebra by other dynamical algebra of low dimension that admits infinite-dimensional irreducible representations with naturally defined generalized coherent states. We have selected the Lie algebra , because this algebra possesses the necessary characteristics to realize the hypercomputation and also because such algebra has been identified as the dynamical algebra associated with many relatively simple quantum systems. In addition to an algebraic adaptation of KHQA over the algebra , we presented an adaptation of KHQA over some concrete physical referents: the infinite square well, the infinite cylindrical well, the perturbed infinite cylindrical well, the Pöschl–Teller potentials, the Holstein–Primakoff system and the Laguerre oscillator. We conclude that it is possible to have many physical systems within condensed matter and quantum optics in which it is possible to consider an implementation of KHQA.

In this paper we give a brief outline of the applications of the generalized Heun equation (GHE) in the context of quantum field theory in curved spacetimes. In particular, we relate the separated radial part of a massive Dirac equation in the Kerr–Newman metric to the static perturbations for the non-extremal Reissner–Nordström solution to a GHE.

In standard quantum mechanics, it is not possible to directly extend the Schrödinger equation to spinors, so the Pauli equation must be derived from the Dirac equation by taking its non-relativistic limit. Hence, it predicts the existence of an intrinsic magnetic moment for the electron and gives its correct value. In the scale relativity framework, the Schrödinger, Klein–Gordon and Dirac equations have been derived from first principles as geodesics equations of a non-differentiable and continuous spacetime. Since such a generalized geometry implies the occurrence of new discrete symmetry breakings, this has led us to write Dirac bi-spinors in the form of bi-quaternions (complex quaternions). In the present work, we show that, in scale relativity also, the correct Pauli equation can only be obtained from a non-relativistic limit of the relativistic geodesics equation (which, after integration, becomes the Dirac equation) and not from the non-relativistic formalism (that involves symmetry breakings in a fractal 3-space). The same degeneracy procedure, when it is applied to the bi-quaternionic 4-velocity used to derive the Dirac equation, naturally yields a Pauli-type quaternionic 3-velocity. It therefore corroborates the relevance of the scale relativity approach for the building from first principles of the quantum postulates and the quantum tools. This also reinforces the relativistic and fundamentally quantum nature of spin, which we attribute in scale relativity to the non-differentiability of the quantum spacetime geometry (and not only of the quantum space). We conclude by performing numerical simulations of spinor geodesics, that allow one to gain a physical geometric picture of the nature of spin.

We present a time evolving path-integral method for solving the Landau–Fokker–Planck equation to compute kinetic transport coefficients in a fully ionized plasma. The electron distribution function is advanced in time by means of the conservative short-time propagators, which we previously obtained. The validated integral operator takes into account both electron–electron and electron–ion collisions without linearizing the original Fokker–Planck collisional operator. The resulting integral formulation in velocity space is applied here to evaluate the local transport coefficients if inhomogeneities in configuration space appear. We define an effective source term through a flux particle balance in a thin slab of plasma, which leads to a nonhomogeneous Fokker–Planck equation. Hence, this new term locally models the so-called Vlasov term appearing in the general kinetic equation. Arbitrary departures from Maxwellian equilibrium can be dealt with this effective source term that preserves the positiveness of the electron distribution function, even in the runaway limit. For small perturbations of the equilibrium, the classical Spitzer and Harm transport coefficients are recovered, while a very strong reduction of the heat flux takes place for large temperature gradients, as predicted by some authors in different theories.