Background Independent Quantum Mechanics, Metric of Quantum States, and Gravity: A Comprehensive Perspective

This paper presents a comprehensive perspective of the metric of quantum states with a focus on the background independent metric structures. We also explore the possibilities of geometrical formulations of quantum mechanics beyond the quantum state space and Kahler manifold. The metric of quantum states in the classical configuration space with the pseudo-Riemannian signature and its possible applications are explored. On contrary to the common perception that a metric for quantum state can yield a natural metric in the configuration space with the limit when Planck constant vanishes, we obtain the metric of quantum states in the configuration space without imposing this limiting condition. Here, Planck constant is absorbed in the quantity like Bohr radii. While exploring the metric structure associated with Hydrogen like atom, we witness another interesting finding that the invariant lengths appear in the multiple of Bohr radii.Comment: 25 Pages;journal reference added:Published in- Int. J. Theor. Phys. 46 (2007) 3216-3229. References revise

Self Replication and Signalling

It is known that if one could clone an arbitrary quantum state one could send signal faster than the speed of light. However it remains interesting to see that if one can perfectly self replicate an arbitrary quantum state, does it violate the no signalling principle? Here we see that perfect self replication would also lead to superluminal signalling.Comment: Modified version of quant-ph/0510221, Accepted in International Journal of Theoretical Physic

Elementary Particles and Spin Representations

We emphasize that the group-theoretical considerations leading to SO(10) unification of electro-weak and strong matter field components naturally extend to space-time components, providing a truly unified description of all generation degrees of freedoms in terms of a single chiral spin representation of one of the groups SO(13,1), SO(9,5), SO(7,7) or SO(3,11). The realization of these groups as higher dimensional space-time symmetries produces unification of all fundamental fermions is a single space-time spinor.Comment: 4 page

Three lepton decay modes of the proton

We consider the three lepton decay modes of the proton within the proton decay interpretation of the atmospheric neutrino anomaly. We construct higher dimensional operators in the framework of the standard model. The operators which allow the particularly interesting decay mode are of dimension 10 involving $SU(2)_L$ non-singlet higgs. We show how these operators can be comparable to the dimension 9 operators. We then present a simple left-right symmetric model which can give rise to the desired proton decay modes of the right order of magnitude.Comment: 14 pages, UTPT-93-13, Latex file with one postscript figure to be cut from the end of the Latex fil

Geometric Phases for Mixed States during Cyclic Evolutions

The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical one-form is defined whose line integral gives the geometric phase which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment [Phys. Rev. Lett. \textbf{85}, 2845 (2000)] for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.Comment: 9 pages, 1 figur

Fluctuation, time-correlation function and geometric Phase

We establish a fluctuation-correlation theorem by relating the quantum fluctuations in the generator of the parameter change to the time integral of the quantum correlation function between the projection operator and force operator of the ``fast'' system. By taking a cue from linear response theory we relate the quantum fluctuation in the generator to the generalised susceptibility. Relation between the open-path geometric phase, diagonal elements of the quantum metric tensor and the force-force correlation function is provided and the classical limit of the fluctuation-correlation theorem is also discussed.Comment: Latex, 12 pages, no figures, submitted to J. Phys. A: Math & Ge

Quantum confinement effects on the ordering of the lowest-lying excited states in conjugated chains

The symmetrized density matrix renormalization group approach is applied within the extended Hubbard-Peierls model (with parameters U/t, V/t, and bond alternation \delta) to study the ordering of the lowest one-photon (1^{1}B^{-}_u) and two-photon (2^{1}A^{+}_g) states in one- dimensional conjugated systems with chain lengths, N, up to N=80 sites. Three different types of crossovers are studied, as a function of U/t, \delta, and N. The U-crossover emphasizes the larger ionic character of the 2A_g state compared to the lowest triplet excitation. The \delta crossover shows strong dependence on both N and U/t. The N-crossover illustrates the more localized nature of the 2A_g excitation relative to the 1B_u excitation at intermediate correlation strengths.Comment: Latex file; figures available upon request. Submitted to PR