2,424 research outputs found

### 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

### 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

### The Quantum Group Structure of 2D Gravity and Minimal Models II: The Genus-Zero Chiral Bootstrap

The F and B matrices associated with Virasoro null vectors are derived in
closed form by making use of the operator-approach suggested by the Liouville
theory, where the quantum-group symmetry is explicit. It is found that the
entries of the fusing and braiding matrices are not simply equal to
quantum-group symbols, but involve additional coupling constants whose
derivation is one aim of the present work. Our explicit formulae are new, to
our knowledge, in spite of the numerous studies of this problem. The
relationship between the quantum-group-invariant (of IRF type) and
quantum-group-covariant (of vertex type) chiral operator-algebras is fully
clarified, and connected with the transition to the shadow world for
quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce
to the simpler transformation of Babelon and one of the author (J.-L. G.) in a
suitable infinite limit defined by analytic continuation. The above two types
of operators are found to coincide when applied to states with Liouville
momenta going to $\infty$ in a suitable way. The introduction of
quantum-group-covariant operators in the three dimensional picture gives a
generalisation of the quantum-group version of discrete three-dimensional
gravity that includes tetrahedra associated with 3-j symbols and universal
R-matrix elements. Altogether the present work gives the concrete realization
of Moore and Seiberg's scheme that describes the chiral operator-algebra of
two-dimensional gravity and minimal models.Comment: 56 pages, 22 figures. Technical problem only, due to the use of an
old version of uuencode that produces blank characters some times suppressed
by the mailer. Same content

### 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

### 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

### 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

### Lotka--Volterra Type Equations and their Explicit Integration

In the present note we give an explicit integration of some
two--dimensionalised Lotka--Volterra type equations associated with simple Lie
algebras, other than the familiar $A_n$ case, possessing a representation
without branching. This allows us, in particular, to treat the first
fundamental representations of $A_r$, $B_r$, $C_r$, and $G_2$ on the same
footing.Comment: 3 pages LATEX fil

### 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

### Photon Production from a Quark-Gluon-Plasma at Finite Baryon Chemical Potential

We compute the photon production of a QCD plasma at leading order in the
strong coupling with a finite baryon chemical potential. Our approach starts
from the real time formalism of finite temperature field theory. We identify
the class of diagrams contributing at leading order when a finite chemical
potential is added and resum them to perform a full treatment of the LPM effect
similar to the one performed by Arnold, Moore, and Yaffe at zero chemical
potential. Our results show that the contribution of $2\mapsto 3$ and $3\mapsto
2$ processes grows as the chemical potential grows.Comment: 28 pages, 14 figure

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