804 research outputs found
Quantisations of piecewise affine maps on the torus and their quantum limits
For general quantum systems the semiclassical behaviour of eigenfunctions in
relation to the ergodic properties of the underlying classical system is quite
difficult to understand. The Wignerfunctions of eigenstates converge weakly to
invariant measures of the classical system, the so called quantum limits, and
one would like to understand which invariant measures can occur that way,
thereby classifying the semiclassical behaviour of eigenfunctions. We introduce
a class of maps on the torus for whose quantisations we can understand the set
of quantum limits in great detail. In particular we can construct examples of
ergodic maps which have singular ergodic measures as quantum limits, and
examples of non-ergodic maps where arbitrary convex combinations of absolutely
continuous ergodic measures can occur as quantum limits. The maps we quantise
are obtained by cutting and stacking
An algebraic interpretation of the Wheeler-DeWitt equation
We make a direct connection between the construction of three dimensional
topological state sums from tensor categories and three dimensional quantum
gravity by noting that the discrete version of the Wheeler-DeWitt equation is
exactly the pentagon for the associator of the tensor category, the
Biedenharn-Elliott identity. A crucial role is played by an asymptotic formula
relating 6j-symbols to rotation matrices given by Edmonds.Comment: 10 pages, amstex, uses epsf.tex. New version has improved
presentatio
The Hamiltonian Formulation of Higher Order Dynamical Systems
Using Dirac's approach to constrained dynamics, the Hamiltonian formulation
of regular higher order Lagrangians is developed. The conventional description
of such systems due to Ostrogradsky is recovered. However, unlike the latter,
the present analysis yields in a transparent manner the local structure of the
associated phase space and its local sympletic geometry, and is of direct
application to {\em constrained\/} higher order Lagrangian systems which are
beyond the scope of Ostrogradsky's approach.Comment: 17 pages. Revised: references adde
Quantum curves and topological recursion
This is a survey article describing the relationship between quantum curves
and topological recursion. A quantum curve is a Schr\"odinger operator-like
noncommutative analogue of a plane curve which encodes (quantum) enumerative
invariants in a new and interesting way. The Schr\"odinger operator annihilates
a wave function which can be constructed using the WKB method, and
conjecturally constructed in a rather different way via topological recursion.Comment: This article arose out of the Banff workshop Quantum Curves and
Quantum Knot Invariants. Comments welcome. 20 pages, 1 figur
Correlation functions of twist fields from Ward identities in the massive Dirac theory
We derive non-linear differential equations for correlation functions of U(1)
twist fields in the two-dimensional massive Dirac theory. Primary U(1) twist
fields correspond to exponential fields in the sine-Gordon model at the
free-fermion point, and it is well-known that their vacuum two-point functions
are determined by integrable differential equations. We extend part of this
result to more general quantum states (pure or mixed) and to certain
descendents, showing that some two-point functions are determined by the
sinh-Gordon differential equations whenever there is translation and parity
invariance, and the density matrix is the exponential of a bilinear expression
in fermions. We use methods involving Ward identities associated to the
copy-rotation symmetry in a model with two independent, anti-commuting copies.
Such methods were used in the context of the thermally perturbed Ising quantum
field theory model. We show that they are applicable to the Dirac theory as
well, and we suggest that they are likely to have a much wider applicability to
free fermion models in general. Finally, we note that our form-factor study of
descendents twist fields combined with a CFT analysis provides a new way of
evaluating vacuum expectation values of primary U(1) twist fields: by deriving
and solving a recursion relation.Comment: 31 page
Open problems, questions, and challenges in finite-dimensional integrable systems
The paper surveys open problems and questions related to different aspects
of integrable systems with finitely many degrees of freedom. Many of the open
problems were suggested by the participants of the conference “Finite-dimensional
Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version
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