We discuss the split between system and measuring apparatus, i.e. non-unitary nature, of the conventional quantum mechanics to motivate a causal unitary description of nature. We then describe causal quantum mechanics of de Broglie and Bohm. We conclude by presenting a version of recently proposed causal quantum mechanics which treats position and momentum variable symmetrically.

A generalized Schrödinger formalism has been presented which is obtained as a Hilbert space representation of a Liouville equation generalized to include the action as a dynamical variable, in addition to the positions and the momenta. This formalism applied to a classical mechanical system had been shown to yield a similar set of Schrödinger like equations for the classical dynamical system of charged particles in a magnetic field. The novel quantum-like predictions for this classical mechanical system have been experimentally demonstrated and the results are presented.

The kinematic approach to the theory of the geometric phase is outlined. This phase is shown to be the simplest invariant under natural groups of transformations on curves in Hilbert space. The connection to the Bargmann invariant is brought out, and the case of group representations described.

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable. In this lecture I review the theoretical formulation of supersymmetric quantum mechanics and discuss many of its applications. I show that the well-known exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials and shape invariance. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Further, it is pointed out that the connection between the solutions of the Dirac equation and the Schrödinger equation is exactly same as between the solutions of the MKdV and the KdV equations.

It is shown that a violation of signal locality and unitarity occur in a particular merger of quantum mechanics and general relativity.

We report some progress on the quark confinement problem in 2 + 1 dim. pure Yang-Mills theory, using Euclidean instanton methods. The instantons are regularized Wu-Yang ‘monopoles’, whose long range Coulomb field is screened by collective effects. Such configurations are stable to small perturbations unlike the case of singular, undressed monopoles. Using exact non-perturbative results for the 3-dim. Coulomb gas, where Debye screening holds for arbitrarily low temperatures, we show in a self-consistent way that a mass gap is dynamically generated in the gauge theory. The mass gap also determines the size of the monopoles. We also identify the disorder operator of the model in terms of the Sine-Gordon field of the Coulomb gas and hence obtain a dual representation whose symmetry is the centre ofSU(2).

The concept of discrete symmetries in classical and quantum physics is reviewed. An account is given ofCP violation observed in theK-meson system and of other experiments whereCP symmetry has been tested. The present theoretical ideas onCP violation within the standard model, and problems needing extension of the model are described. Finally, ideas and experimental approaches toCP violation beyond the standard model are reviewed in brief.

Starting with the very definition of chaos, we demonstrate that the study of chaos is not an abstract one but can lead to some useful practical applications. With the advent of some powerful mathematical techniques and with the availability of fast computers, it is now possible to study the fascinating phenomena of chaos — the subject which is truly interdisciplinary. The essential role played by fractals, strange attractors, Poincare maps, etc., in the study of chaotic dynamics, is briefly discussed. Phenomena of self-organization, coherence in chaos and control of chaos in plasmas is highlighted.

A review of the generic features as well as the exact analytical solutions of coupled scalar field equations governing nonlinear wave modulations in plasmas is presented. Coupled sets of equations like the Zakharov system, the Schrödinger-Boussinesq system and the Schrödinger-KDV system are considered. For stationary solutions, the latter two systems yield a generic system of a pair of coupled, ordinary differential equations with many free parameters. Different classes of exact analytical solutions of the generic system which are valid in different regions of the parameter space are obtained. The generic system is shown to generalize the Hénon-Heiles equations in the field of nonlinear dynamics to include a case when the kinetic energy in the corresponding Hamiltonian is not positive definite. The relevance of the generic system to other equations like the self-dual Yang-Mills equations, the complex KDV equation and the complexified classical dynamical equations is also pointed out.

The description of gravitational clustering is essentially statistical but its origin is dynamical. Hence both aspects of clustering: dynamical and statistical, must be understood in order to arrive at a proper appreciation of the subject of gravitational instability and the formation and evolution of the large scale structure of the Universe. Key dynamical aspects of gravitational clustering such as the Zeldovich approximation and its extension — the adhesion model are reviewed. Statistical indicators of clustering such as the correlation function and percolation theory, as applied to the large scale structure of the Universe have also been focussed on.

The rotation curves of galaxies are modelled using very special properties of an hydrodynamically turbulent fluid possessing helicity fluctuations. The development of correlations among these fluctuations leads to the formation of organized structures characterized by a new flat branch of the spatial energy spectrum in addition to the well known Kolmogorov spectrum. It is proposed that the flat nature of the rotation curves of galaxies may be a result of the energy cascading processes occuring in turbulent galactic atmospheres. Thus, in this model, there is no need of invoking dark matter to account for the flat rotation curves of galaxies.

When the dynamics of a system partitions the phase space of configurations into very many disjoint sectors, we are faced with an assignment problem: Given a configuration, how can we tell which sector it belongs to? We study this problem in connection with the dynamics of deposition and evaporation ofk particles at a time, from a lattice substrate. Fork ≥ 3, the system shows complex behaviour: (a) The number of disjoint sectors in phase space grows exponentially with the size. (b) The asymptotic time dependence of the autocorrelation function shows slow decays, with power laws which depend on the sector. Both (a) and (b) are explained in terms of a nonlocal construct known as the irreducible string (IS), formed from a particle configuration by applying a deletion algorithm. The IS provides a label for sectors; the multiplicity of possible IS’s accounts for (a), and let us determine sector numbers and sizes. The elements of the IS are conserved; thus their motion is responsible for the slow modes of the system, and accounts for (b) as well.