1,210 research outputs found
More on complexity of operators in quantum field theory
Recently it has been shown that the complexity of SU() operator is
determined by the geodesic length in a bi-invariant Finsler geometry, which is
constrained by some symmetries of quantum field theory. It is based on three
axioms and one assumption regarding the complexity in continuous systems. By
relaxing one axiom and an assumption, we find that the complexity formula is
naturally generalized to the Schatten -norm type. We also clarify the
relation between our complexity and other works. First, we show that our
results in a bi-invariant geometry are consistent with the ones in a
right-invariant geometry such as -local geometry. Here, a careful analysis
of the sectional curvature is crucial. Second, we show that our complexity can
concretely realize the conjectured pattern of the time-evolution of the
complexity: the linear growth up to saturation time. The saturation time can be
estimated by the relation between the topology and curvature of SU() groups.Comment: Modified the Sec. 4.1, where we offered a powerful proof: if (1) the
ket vector and bra vector in quantum mechanics contain same physics, or (2)
adding divergent terms to a Lagrangian will not change underlying physics,
then complexity in quantum mechanics must be bi-invariant
Entropies from coarse-graining: convex polytopes vs. ellipsoids
We examine the Boltzmann/Gibbs/Shannon and the
non-additive Havrda-Charv\'{a}t / Dar\'{o}czy/Cressie-Read/Tsallis \
\ and the Kaniadakis -entropy \ \
from the viewpoint of coarse-graining, symplectic capacities and convexity. We
argue that the functional form of such entropies can be ascribed to a
discordance in phase-space coarse-graining between two generally different
approaches: the Euclidean/Riemannian metric one that reflects independence and
picks cubes as the fundamental cells and the symplectic/canonical one that
picks spheres/ellipsoids for this role. Our discussion is motivated by and
confined to the behaviour of Hamiltonian systems of many degrees of freedom. We
see that Dvoretzky's theorem provides asymptotic estimates for the minimal
dimension beyond which these two approaches are close to each other. We state
and speculate about the role that dualities may play in this viewpoint.Comment: 63 pages. No figures. Standard LaTe
The excitation spectrum for weakly interacting bosons in a trap
We investigate the low-energy excitation spectrum of a Bose gas confined in a
trap, with weak long-range repulsive interactions. In particular, we prove that
the spectrum can be described in terms of the eigenvalues of an effective
one-particle operator, as predicted by the Bogoliubov approximation.Comment: LaTeX, 32 page
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