1,384 research outputs found
Tensor Rank, Invariants, Inequalities, and Applications
Though algebraic geometry over is often used to describe the
closure of the tensors of a given size and complex rank, this variety includes
tensors of both smaller and larger rank. Here we focus on the tensors of rank over , which has as a dense subset the orbit
of a single tensor under a natural group action. We construct polynomial
invariants under this group action whose non-vanishing distinguishes this orbit
from points only in its closure. Together with an explicit subset of the
defining polynomials of the variety, this gives a semialgebraic description of
the tensors of rank and multilinear rank . The polynomials we
construct coincide with Cayley's hyperdeterminant in the case , and thus
generalize it. Though our construction is direct and explicit, we also recast
our functions in the language of representation theory for additional insights.
We give three applications in different directions: First, we develop basic
topological understanding of how the real tensors of complex rank and
multilinear rank form a collection of path-connected subsets, one of
which contains tensors of real rank . Second, we use the invariants to
develop a semialgebraic description of the set of probability distributions
that can arise from a simple stochastic model with a hidden variable, a model
that is important in phylogenetics and other fields. Third, we construct simple
examples of tensors of rank which lie in the closure of those of rank
.Comment: 31 pages, 1 figur
Classical and Quantum Interaction of the Dipole
A unified and fully relativistic treatment of the interaction of the electric
and magnetic dipole moments of a particle with the electromagnetic field is
given. New forces on the particle due to the combined effect of electric and
magnetic dipoles are obtained. Four new experiments are proposed, three of
which would observe topological phase shifts.Comment: 10 pages, Latex/Revtex. Some minor errors have been correcte
Topology, Locality, and Aharonov-Bohm Effect with Neutrons
Recent neutron interferometry experiments have been interpreted as
demonstrating a new topological phenomenon similar in principle to the usual
Aharonov-Bohm (AB) effect, but with the neutron's magnetic moment replacing the
electron's charge. We show that the new phenomenon, called Scalar AB (SAB)
effect, follows from an ordinary local interaction, contrary to the usual AB
effect, and we argue that the SAB effect is not a topological effect by any
useful definition. We find that SAB actually measures an apparently novel spin
autocorrelation whose operator equations of motion contain the local torque in
the magnetic field. We note that the same remarks apply to the Aharonov-Casher
effect.Comment: 9 page
Adventures in Invariant Theory
We provide an introduction to enumerating and constructing invariants of
group representations via character methods. The problem is contextualised via
two case studies arising from our recent work: entanglement measures, for
characterising the structure of state spaces for composite quantum systems; and
Markov invariants, a robust alternative to parameter-estimation intensive
methods of statistical inference in molecular phylogenetics.Comment: 12 pp, includes supplementary discussion of example
Quench-induced spontaneous currents in rings of ultracold fermionic atoms
We have measured the rate of spontaneous current formation in ring-shaped
ensembles of fermionic Li atoms, following a thermal quench through the BCS
superfluid phase transition. For the fastest quenches, the mean square winding
number follows a scaling law with exponent = 0.24(2), in line with
predictions of the Kibble-Zurek (KZ) model for mean-field BCS theory. We use a
hybrid quench protocol involving simultaneous evaporation and interaction
ramps, with a long system lifetime allowing characterization of a different
rate of spontaneous current formation in the slow-quench regime, where
finite-size effects are important. Comparing our observations to a quasi-1D
stochastic Ginzburg-Landau model, we find quantitative agreement for fast
quenches, but only qualitative agreement for slow quenches.Comment: 6 pages, 4 figure
Noncyclic Pancharatnam phase for mixed state SU(2) evolution in neutron polarimetry
We have measured the Pancharatnam relative phase for spin-1/2 states. In a
neutron polarimetry experiment the minima and maxima of intensity modulations,
giving the Pancharatnam phase, were determined. We have also considered general
SU(2) evolution for mixed states. The results are in good agreement with
theory.Comment: 5 pages, 4 figures, to be published in Phys.Lett.
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