Operator Coproduct-Realization of Quantum Group Transformations in Two Dimensional Gravity, I.

A simple connection between the universal $R$ matrix of $U_q(sl(2))$ (for spins \demi and $J$) and the required form of the co-product action of the Hilbert space generators of the quantum group symmetry is put forward. This gives an explicit operator realization of the co-product action on the covariant operators. It allows us to derive the quantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of $U_q(sl(2))$ realized by (what we call) fixed point commutation relations. This is explained by showing that the link between the algebra of field transformations and that of the co-product generators is much weaker than previously thought. The central charges of our extended $U_q(sl(2))$ algebra, which includes the Liouville zero-mode momentum in a nontrivial way are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry U_q(sl(2))\odot U_{\qhat}(sl(2)) related to the presence of both of the screening charges of 2D gravity.Comment: 33 pages, latex, no figure

Quantum Group Structure and Local Fields in the Algebraic Approach to 2D Gravity

This review contains a summary of work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables -the Liouville exponentials and the Liouville field itself - and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.Comment: 38 pages, LaTex file. Proceedings of the Vth International Conference on Mathematical Physics, Strings and Quantum gravity, Alushta, Ukraine 199

The bicomplex quantum Coulomb potential problem

Generalizations of the complex number system underlying the mathematical formulation of quantum mechanics have been known for some time, but the use of the commutative ring of bicomplex numbers for that purpose is relatively new. This paper provides an analytical solution of the quantum Coulomb potential problem formulated in terms of bicomplex numbers. We define the problem by introducing a bicomplex hamiltonian operator and extending the canonical commutation relations to the form [X_i,P_k] = i_1 hbar xi delta_{ik}, where xi is a bicomplex number. Following Pauli's algebraic method, we find the eigenvalues of the bicomplex hamiltonian. These eigenvalues are also obtained, along with appropriate eigenfunctions, by solving the extension of Schrodinger's time-independent differential equation. Examples of solutions are displayed. There is an orthonormal system of solutions that belongs to a bicomplex Hilbert space.Comment: Clarifications; some figures removed; version to appear in Can. J. Phy

Reactor power systems for manned earth orbital applications

Design requirements for reactor power system of manned earth orbital space statio

Light-Cone Quantization of the Liouville Model

We present the quantization of the Liouville model defined in light-cone coordinates in (1,1) signature space. We take advantage of the representation of the Liouville field by the free field of the Backl\"{u}nd transformation and adapt the approch by Braaten, Curtright and Thorn. Quantum operators of the Liouville field $\partial_{+}\phi$, $\partial_{-}\phi$, $e^{g\phi}$, $e^{2g\phi}$ are constructed consistently in terms of the free field. The Liouville model field theory space is found to be restricted to the sector with field momentum $P_{+}=-P_{-}$, $P_{+}> 0$ , which is a closed subspace for the Liouville theory operator algebra.Comment: 16 p, EFI-92-6

Abelian Current Algebra and the Virasoro Algebra on the Lattice

We describe how a natural lattice analogue of the abelian current algebra combined with free discrete time dynamics gives rise to the lattice Virasoro algebra and corresponding hierarchy of conservation laws.Comment: 11 pages, LATEX, HU-TFT-93-2

Quantum and Classical Gauge Symmetries in a Modified Quantization Scheme

The use of the mass term as a gauge fixing term has been studied by Zwanziger, Parrinello and Jona-Lasinio, which is related to the non-linear gauge $A_{\mu}^{2}=\lambda$ of Dirac and Nambu in the large mass limit. We have recently shown that this modified quantization scheme is in fact identical to the conventional {\em local} Faddeev-Popov formula {\em without} taking the large mass limit, if one takes into account the variation of the gauge field along the entire gauge orbit and if the Gribov complications can be ignored. This suggests that the classical massive vector theory, for example, is interpreted in a more flexible manner either as a gauge invariant theory with a gauge fixing term added, or as a conventional massive non-gauge theory. As for massive gauge particles, the Higgs mechanics, where the mass term is gauge invariant, has a more intrinsic meaning. It is suggested to extend the notion of quantum gauge symmetry (BRST symmetry) not only to classical gauge theory but also to a wider class of theories whose gauge symmetry is broken by some extra terms in the classical action. We comment on the implications of this extended notion of quantum gauge symmetry.Comment: 14 pages. Substantially revised and enlarged including the change of the title. To appear in International Journal of Modern Physics