2,336 research outputs found
Operator Coproduct-Realization of Quantum Group Transformations in Two Dimensional Gravity, I.
A simple connection between the universal matrix of (for
spins \demi and ) 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
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 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
Scattering Mechanism in Modulation-Doped Shallow Two-Dimensional Electron Gases
We report on a systematic investigation of the dominant scattering mechanism
in shallow two-dimensional electron gases (2DEGs) formed in modulation-doped
GaAs/Al_{x}Ga_{1-x}As heterostructures. The power-law exponent of the electron
mobility versus density, mu \propto n^{alpha}, is extracted as a function of
the 2DEG's depth. When shallower than 130 nm from the surface, the power-law
exponent of the 2DEG, as well as the mobility, drops from alpha \simeq 1.65
(130 nm deep) to alpha \simeq 1.3 (60 nm deep). Our results for shallow 2DEGs
are consistent with theoretical expectations for scattering by remote dopants,
in contrast to the mobility-limiting background charged impurities of deeper
heterostructures.Comment: 4 pages, 3 figures, modified version as accepted in AP
Domain Walls in a FRW Universe
We solve the equations of motion for a scalar field with domain wall boundary
conditions in a Friedmann-Robertson-Walker (FRW) spacetime. We find (in
agreement with Basu and Vilenkin) that no domain wall solutions exist in de
Sitter spacetime for h = H/m >= 1/2, where H is the Hubble parameter and m is
the scalar mass. In the general FRW case we develop a systematic perturbative
expansion in h to arrive at an approximate solution to the field equations. We
calculate the energy momentum tensor of the domain wall configuration, and show
that the energy density can become negative at the core of the defect for some
values of the non-minimal coupling parameter xi. We develop a translationally
invariant theory for fluctuations of the wall, obtain the effective Lagrangian
for these fluctuations, and quantize them using the Bunch-Davies vacuum in the
de Sitter case. Unlike previous analyses, we find that the fluctuations act as
zero-mass (as opposed to tachyonic) modes. This allows us to calculate the
distortion and the normal-normal correlators for the surface. The normal-normal
correlator decreases logarithmically with the distance between points for large
times and distances, indicating that the interface becomes rougher than in
Minkowski spacetime.Comment: 23 pages, LaTeX, 7 figures using epsf.tex. Now auto-generates P
Free fields via canonical transformations of matter-coupled 2D dilaton gravity models
It is shown that the 1+1-dimensional matter-coupled Jackiw-Teitelboim model
and the model with an exponential potential can be converted by means of
appropriate canonical transformations into a bosonic string theory propagating
on a flat target space with an indefinite signature. This makes it possible to
consistently quantize these models in the functional Schroedinger
representation thus generalizing recent results on CGHS theory.Comment: 15 pages, Late
A Note on Background (In)dependence
In general quantum systems there are two kinds of spacetime modes, those that
fluctuate and those that do not. Fluctuating modes have normalizable
wavefunctions. In the context of 2D gravity and ``non-critical'' string theory
these are called macroscopic states. The theory is independent of the initial
Euclidean background values of these modes. Non-fluctuating modes have
non-normalizable wavefunctions and correspond to microscopic states. The theory
depends on the background value of these non-fluctuating modes, at least to all
orders in perturbation theory. They are superselection parameters and should
not be minimized over. Such superselection parameters are well known in field
theory. Examples in string theory include the couplings (including the
cosmological constant) in the matrix models and the mass of the two-dimensional
Euclidean black hole. We use our analysis to argue for the finiteness of the
string perturbation expansion around these backgrounds.Comment: 16 page
Soliton quantization and internal symmetry
We apply the method of collective coordinate quantization to a model of
solitons in two spacetime dimensions with a global symmetry. In
particular we consider the dynamics of the charged states associated with
rotational excitations of the soliton in the internal space and their
interactions with the quanta of the background field (mesons). By solving a
system of coupled saddle-point equations we effectively sum all tree-graphs
contributing to the one-point Green's function of the meson field in the
background of a rotating soliton. We find that the resulting one-point function
evaluated between soliton states of definite charge exhibits a pole on
the meson mass shell and we extract the corresponding S-matrix element for the
decay of an excited state via the emission of a single meson using the standard
LSZ reduction formula. This S-matrix element has a natural interpretation in
terms of an effective Lagrangian for the charged soliton states with an
explicit Yukawa coupling to the meson field. We calculate the leading-order
semi-classical decay width of the excited soliton states discuss the
consequences of these results for the hadronic decay of the resonance
in the Skyrme model.Comment: 23 pages, LA-UR-93-299
Classical Scattering in Dimensional String Theory
We find the general solution to Polchinski's classical scattering equations
for dimensional string theory. This allows efficient computation of
scattering amplitudes in the standard Liouville background.
Moreover, the solution leads to a mapping from a large class of time-dependent
collective field theory backgrounds to corresponding nonlinear sigma models.
Finally, we derive recursion relations between tachyon amplitudes. These may be
summarized by an infinite set of nonlinear PDE's for the partition function in
an arbitrary time-dependent background.Comment: 15 p
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