922 research outputs found
Resonance equals reducibility for A-hypergeometric systems
Classical theorems of Gel'fand et al., and recent results of Beukers, show
that non-confluent Cohen-Macaulay A-hypergeometric systems have reducible
monodromy representation if and only if the continuous parameter is A-resonant.
We remove both the confluence and Cohen-Macaulayness conditions while
simplifying the proof.Comment: 9 pages, final versio
The super algebra and its associated generalized KdV hierarchies
We construct the super algebra as a certain reduction of the
second Gel'fand-Dikii bracket on the dual of the Lie superalgebra of
super pseudo-differential operators. The algebra is put in manifestly
supersymmetric form in terms of three superfields , with
being the energy momentum tensor and and being
conformal spin and superfields respectively. A search for integrable
hierarchies of the generalized KdV variety with this algebra as Hamiltonian
structure gives three solutions, exactly the same number as for the
(super KdV) and (super Boussinesq) cases.Comment: 16 pages, LaTeX, UTAS-PHYS-92-3
A possible mathematics for the unification of quantum mechanics and general relativity
This paper summarizes and generalizes a recently proposed mathematical
framework that unifies the standard formalisms of special relativity and
quantum mechanics. The framework is based on Hilbert spaces H of functions of
four space-time variables x,t, furnished with an additional indefinite inner
product invariant under Poincar\'e transformations, and isomorphisms of these
spaces that preserve the indefinite metric. The indefinite metric is
responsible for breaking the symmetry between space and time variables and for
selecting a family of Hilbert subspaces that are preserved under Galileo
transformations. Within these subspaces the usual quantum mechanics with
Schr\"odinger evolution and t as the evolution parameter is derived.
Simultaneously, the Minkowski space-time is isometrically embedded into H,
Poincar\'e transformations have unique extensions to isomorphisms of H and the
embedding commutes with Poincar\'e transformations. The main new result is a
proof that the framework accommodates arbitrary pseudo-Riemannian space-times
furnished with the action of the diffeomorphism group
Invariant tensors and Casimir operators for simple compact Lie groups
The Casimir operators of a Lie algebra are in one-to-one correspondence with
the symmetric invariant tensors of the algebra. There is an infinite family of
Casimir operators whose members are expressible in terms of a number of
primitive Casimirs equal to the rank of the underlying group. A systematic
derivation is presented of a complete set of identities expressing
non-primitive symmetric tensors in terms of primitive tensors. Several examples
are given including an application to an exceptional Lie algebra.Comment: 11 pages, LaTeX, minor changes, version in J. Math. Phy
Translation-finite sets, and weakly compact derivations from \lp{1}(\Z_+) to its dual
We characterize those derivations from the convolution algebra
to its dual which are weakly compact. In particular, we
provide examples which are weakly compact but not compact. The characterization
is combinatorial, in terms of "translation-finite" subsets of ,
and we investigate how this notion relates to other notions of "smallness" for
infinite subsets of . In particular, we show that a set of
strictly positive Banach density cannot be translation-finite; the proof has a
Ramsey-theoretic flavour.Comment: v1: 14 pages LaTeX (preliminary). v2: 13 pages LaTeX, submitted. Some
streamlining, renumbering and minor corrections. v3: appendix removed. v4:
Modified appendix reinstated; 14 pages LaTeX. To appear in Bull. London Math.
Soc
One-dimensional Chern-Simons theory and the genus
We construct a Chern-Simons gauge theory for dg Lie and L-infinity algebras
on any one-dimensional manifold and quantize this theory using the
Batalin-Vilkovisky formalism and Costello's renormalization techniques. Koszul
duality and derived geometry allow us to encode topological quantum mechanics,
a nonlinear sigma model of maps from a 1-manifold into a cotangent bundle T*X,
as such a Chern-Simons theory. Our main result is that the partition function
of this theory is naturally identified with the A-genus of X. From the
perspective of derived geometry, our quantization construct a volume form on
the derived loop space which can be identified with the A-class.Comment: 61 pages, figures, final versio
coherent state operators and invariant correlation functions and their quantum group counterparts
Coherent state operators (CSO) are defined as operator valued functions on
G=SL(n,C), homogeneous with respect to right multiplication by lower triangular
matrices. They act on a model space containing all holomorphic finite
dimensional representations of G with multiplicity 1. CSO provide an analytic
tool for studying G invariant 2- and 3-point functions, which are written down
in the case of . The quantum group deformation of the construction gives
rise to a non-commutative coset space. We introduce a "standard" polynomial
basis in this space (related to but not identical with the Lusztig canonical
basis) which is appropriate for writing down invariant 2-point
functions for representaions of the type and .
General invariant 2-point functions are written down in a mixed
Poincar\'e-Birkhoff-Witt type basis.Comment: 33 pages, LATEX, preprint IPNO/TH 94-0
Ladder operators for isospectral oscillators
We present, for the isospectral family of oscillator Hamiltonians, a
systematic procedure for constructing raising and lowering operators satisfying
any prescribed `distorted' Heisenberg algebra (including the
-generalization). This is done by means of an operator transformation
implemented by a shift operator. The latter is obtained by solving an
appropriate partial isometry condition in the Hilbert space. Formal
representations of the non-local operators concerned are given in terms of
pseudo-differential operators. Using the new annihilation operators, new
classes of coherent states are constructed for isospectral oscillator
Hamiltonians. The corresponding Fock-Bargmann representations are also
considered, with specific reference to the order of the entire function family
in each case.Comment: 13 page
Simple derivation of general Fierz-type identities
General Fierz-type identities are examined and their well known connection
with completeness relations in matrix vector spaces is shown. In particular, I
derive the chiral Fierz identities in a simple and systematic way by using a
chiral basis for the complex matrices. Other completeness relations
for the fundamental representations of SU(N) algebras can be extracted using
the same reasoning.Comment: 9pages. Few sentences modified in introduction and in conclusion.
Typos corrected. An example added in introduction. Title modifie
New global stability estimates for the Calder\'on problem in two dimensions
We prove a new global stability estimate for the Gel'fand-Calder\'on inverse
problem on a two-dimensional bounded domain or, more precisely, the inverse
boundary value problem for the equation on ,
where is a smooth real-valued potential of conductivity type defined on a
bounded planar domain . The principal feature of this estimate is that it
shows that the more a potential is smooth, the more its reconstruction is
stable, and the stability varies exponentially with respect to the smoothness
(in a sense to be made precise). As a corollary we obtain a similar estimate
for the Calder\'on problem for the electrical impedance tomography.Comment: 18 page
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