Table of contents

Under the name prime decomposition (PD), a unique decomposition of an arbitrary N-dimensional density matrix into a sum of separable density matrices with dimensions determined by the coprime factors of N is introduced. For a class of density matrices a complete tensor product factorization is achieved. The construction is based on the Chinese remainder theorem, and the projective unitary representation of by the discrete Heisenberg group . The PD isomorphism is unitarily implemented and it is shown to be co-associative and to act on as comultiplication. Density matrices with complete PD are interpreted as group-like elements of . To quantify the distance of from its PD a trace-norm correlation index is introduced and its invariance groups are determined.

The commutation relations for a linear system are given directly in terms of the classical energy and equations of motion, without the need for a normal mode expansion. The formula is illustrated in the case of the free electromagnetic field.

We study complex eigenvalues of large symmetric random matrices of the form , where both and are real symmetric, is a random Gaussian and is such that when . When the model can be used to describe the universal statistics of S-matrix poles (resonances) in the complex energy plane. We derive the ensuing distribution of the resonance widths which generalizes the well known distribution to the case of overlapping resonances. We also consider a different class of `almost real' matrices when is random and uncorrelated with .

We consider the p-spin spherical spin-glass model in the presence of an external magnetic field as a general example of a mean-field system where a one-step replica symmetry breaking (1-RSB) occurs. In this context we compute the complexity of the Thouless-Anderson-Palmer states, performing a quenched computation. We find what the general connection is between this method and the standard static 1-RSB one, formulating a clear mapping between the parameters used in the two different calculations. A dynamical analysis of the model confirms the validity of our results.

We apply periodic orbit theory to a quantum billiard on a torus with a variable number (N) of randomly distributed circular scatterers. Provided the scatterers are much smaller than the wavelength they may be regarded as sources of pure s-wave diffraction. The relevant part of the spectral determinant is due only to diffractive periodic orbits. We formulate this diffractive zeta function in terms of an transfer matrix, which is transformed to real form. The determinant is shown to reproduce the full density of states for generic configurations if . The zeros of the determinant are computed numerically. We study the statistics exhibited by these spectra. The numerical results suggest that the spectra tend to GOE statistics as the number of scatterers increases for typical members of the ensemble. A peculiar situation arises for configurations with four scatterers and kR tuned to , where the statistics appears to be perfectly Poissonian.

The Gaussian wavefunctional approach is developed in thermofield dynamics. We construct the thermal vacuum wavefunctional, its creation as well as annihilation operators, and accordingly the thermo-particle excited states. For a (D + 1)-dimensional scalar field system with an arbitrary potential whose Fourier representation exists in the sense of tempered distributions, we calculate the finite-temperature Gaussian effective potential (FTGEP), one- and two-thermo-particle-state energies. The zero-temperature limit of each of them is just the corresponding result in quantum field theory, and the FTGEP can lead to the same one for some concrete models as calculated by the imaginary-time Green function.

We give a criterion for the simultaneous existence or non-existence of two long-range orders for two observables, at finite temperatures, for quantum lattice many body systems. Our analysis extends previous results of Tian (1997 30 841) limited to the ground state of similar models. The proof involves an inequality of Dyson-Lieb-Simon (1978 J. Stat. Phys. 18 335) which connects the Duhamel two-point function to the usual correlation function. An application to the special case of the Holstein model is discussed.

We study two-dimensional classically integrable field theory with independent boundary conditions at each end, and obtain three possible generating functions for integrals of motion when this model is an ultralocal one. Classically integrable boundary conditions can be found when solving boundary equations. In the quantum case, we also find that the unitarity condition of the quantum R-matrix is sufficient to construct commutative quantities with boundaries, and its reflection equations are obtained.

Exact characteristic trajectories are specified for the time-propagating Wigner phase-space distribution function. They are especially simple - indeed, classical - for the quantized simple harmonic oscillator, which serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase space. Applications to duality transformations in field theory are discussed.

A number of new coagulation models depending on a parameter is derived. The dependence is considered in two different ways. If the parameter takes its maximal value then in the first case we obtain a new discrete kinetic equation. We demonstrate that its continuous version is simply the Oort-Hulst's coagulation model. In the second case, the maximum of the corresponding parameter yields the Smoluchowski coagulation equation. At the minimal values of both parameters we arrive at another new kinetic equation. These three models form a `triangle' connecting the two known coagulation equations `situated' in its vertexes (Smoluchowski and Oort-Hulst equations) via an alteration of the parameters. Also, a comparative analysis of these three models is presented. As an advantage of the Oort-Hulst approach we compute the coagulation front and establish a connection between the infringement of the mass conservation law and convergence of the coagulation front to infinity.

Quasi-spherical harmonics, are derived and presented for half-odd-integer values of and m. The form of the factor is identical to that in the case of integer and m: . However, the domain of these functions in the half-odd-integer case is rather than the domain in the case of integer and m (the true spherical harmonics). The form of the factor, (an associated Legendre function) is (as in the integer case) the factor multiplied by a polynomial in of degree (an associated Legendre polynomial). A substantial difference between the spherical (integer and m) and quasi-spherical (half-odd-integer and m) Legendre functions is that the latter have an irrational factor of whereas the factor of the truly spherical functions is an integer power of . The domain of both the true and quasi-spherical associated Legendre functions is the same: . A table of the associated Legendre functions is presented for both integer and half-odd-integer values of and m, for , and for . The table displays the similarity between the functions for integer and m (which are well known) and those for half-odd-integer and m (which have not been recognized previously).

The graded contractions of pseudo-Euclidean Lie algebra are studied. The non-equivalent gradings of of type and a are extended to the entire Lie algebra , using the action of on the Abelian ideal (the translations.). The graded contractions embed into a large family of six-dimensional Lie algebras. The family includes solvable, nilpotent and nonsolvable Lie algebras, both decomposable and indecomposable ones. The distinction between graded and Inönü-Wigner contractions is analysed. The physically most interesting Lie algebras obtained by the contractions are the inhomogeneous Galilei and pseudo-Galilei.

A universal version of the Hamilton-Jacobi equation on arises from the Liouville-Arnol'd theorem for a completely integrable system on a finite-dimensional manifold M. We give necessary and sufficient conditions for such complete integrability to imply a canonical separability of both this universal Hamilton-Jacobi equation and its traditional counterpart. The geodesic case is particularly interesting. We show that these conditions also apply for systems of second-order ordinary differential equations (contact flows) which are not Euler-Lagrange. The Kerr metric, the Toda lattice and a completely integrable contact flow are given as examples.

A model of a 4D open string with non-Grassmann spinning variables is considered. The nonlinear gauge, which is invariant under both Poincaré and scale transformations of spacetime, is used for subsequent studies. It is shown that the reduction of the canonical Poisson structure from the original phase space to the surface of constraints and gauge conditions gives the degenerated Poisson brackets. Moreover it is shown that such a reduction is non-unique. The concept of adjunct phase space is introduced. The consequences for subsequent relativistic invariant quantization are discussed. Deduced dependence of spin J from the square of mass of the string generalizes the `Regge spectrum' for conventional theory.

We consider the general problem of plane motion of a charged particle under the action of potential forces in the same plane and a magnetic field orthogonal to it. Four new integrable cases are pointed out in which the gyroscopic (zero-potential) forces play an essential role. Physical interpretation is given for some of the results in terms of motion under potential and Lorentz' forces in the rotating plane.