3,588 research outputs found
On the quantization of polygon spaces
Moduli spaces of polygons have been studied since the nineties for their
topological and symplectic properties. Under generic assumptions, these are
symplectic manifolds with natural global action-angle coordinates. This paper
is concerned with the quantization of these manifolds and of their action
coordinates. Applying the geometric quantization procedure, one is lead to
consider invariant subspaces of a tensor product of irreducible representations
of SU(2). These quantum spaces admit natural sets of commuting observables. We
prove that these operators form a semi-classical integrable system, in the
sense that they are Toeplitz operators with principal symbol the square of the
action coordinates. As a consequence, the quantum spaces admit bases whose
vectors concentrate on the Lagrangian submanifolds of constant action. The
coefficients of the change of basis matrices can be estimated in terms of
geometric quantities. We recover this way the already known asymptotics of the
classical 6j-symbols
Confusability graphs for symmetric sets of quantum states
For a set of quantum states generated by the action of a group, we consider
the graph obtained by considering two group elements adjacent whenever the
corresponding states are non-orthogonal. We analyze the structure of the
connected components of the graph and show two applications to the optimal
estimation of an unknown group action and to the search for decoherence free
subspaces of quantum channels with symmetry.Comment: 7 pages, no figures, contribution to the Proceedings of the XXIX
International Colloquium on Group-Theoretical Methods in Physics, August
22-26, Chern Institute of Mathematics, Tianjin, Chin
Gauge Field Theory Coherent States (GCS) : I. General Properties
In this article we outline a rather general construction of diffeomorphism
covariant coherent states for quantum gauge theories.
By this we mean states , labelled by a point (A,E) in the
classical phase space, consisting of canonically conjugate pairs of connections
A and electric fields E respectively, such that (a) they are eigenstates of a
corresponding annihilation operator which is a generalization of A-iE smeared
in a suitable way, (b) normal ordered polynomials of generalized annihilation
and creation operators have the correct expectation value, (c) they saturate
the Heisenberg uncertainty bound for the fluctuations of and
(d) they do not use any background structure for their definition, that is,
they are diffeomorphism covariant.
This is the first paper in a series of articles entitled ``Gauge Field Theory
Coherent States (GCS)'' which aim at connecting non-perturbative quantum
general relativity with the low energy physics of the standard model. In
particular, coherent states enable us for the first time to take into account
quantum metrics which are excited {\it everywhere} in an asymptotically flat
spacetime manifold. The formalism introduced in this paper is immediately
applicable also to lattice gauge theory in the presence of a (Minkowski)
background structure on a possibly {\it infinite lattice}.Comment: 40 pages, LATEX, no figure
Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths
We study the dependence of the quantum graph Hamiltonian, its resolvent, and
its spectrum on the vertex conditions and graph edge lengths. In particular,
several results on the interlacing (bracketing) of the spectra of graphs with
different vertex conditions are obtained and their applications are discussed.Comment: 19 pages, 1 figur
Index theorems for quantum graphs
In geometric analysis, an index theorem relates the difference of the numbers
of solutions of two differential equations to the topological structure of the
manifold or bundle concerned, sometimes using the heat kernels of two
higher-order differential operators as an intermediary. In this paper, the case
of quantum graphs is addressed. A quantum graph is a graph considered as a
(singular) one-dimensional variety and equipped with a second-order
differential Hamiltonian H (a "Laplacian") with suitable conditions at
vertices. For the case of scale-invariant vertex conditions (i.e., conditions
that do not mix the values of functions and of their derivatives), the constant
term of the heat-kernel expansion is shown to be proportional to the trace of
the internal scattering matrix of the graph. This observation is placed into
the index-theory context by factoring the Laplacian into two first-order
operators, H =A*A, and relating the constant term to the index of A. An
independent consideration provides an index formula for any differential
operator on a finite quantum graph in terms of the vertex conditions. It is
found also that the algebraic multiplicity of 0 as a root of the secular
determinant of H is the sum of the nullities of A and A*.Comment: 19 pages, Institute of Physics LaTe
3nj Morphogenesis and Semiclassical Disentangling
Recoupling coefficients (3nj symbols) are unitary transformations between
binary coupled eigenstates of N=(n+1) mutually commuting SU(2) angular momentum
operators. They have been used in a variety of applications in spectroscopy,
quantum chemistry and nuclear physics and quite recently also in quantum
gravity and quantum computing. These coefficients, naturally associated to
cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and
analytical features that make them fashinating objects to be studied on their
own. In this paper we develop a bottom--up, systematic procedure for the
generation of 3nj from 3(n-1)j diagrams by resorting to diagrammatical and
algebraic methods. We provide also a novel approach to the problem of
classifying various regimes of semiclassical expansions of 3nj coefficients
(asymptotic disentangling of 3nj diagrams) for n > 2 by means of combinatorial,
analytical and numerical tools
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