5,425 research outputs found
The topological Bloch-Floquet transform and some applications
We investigate the relation between the symmetries of a Schr\"odinger
operator and the related topological quantum numbers. We show that, under
suitable assumptions on the symmetry algebra, a generalization of the
Bloch-Floquet transform induces a direct integral decomposition of the algebra
of observables. More relevantly, we prove that the generalized transform
selects uniquely the set of "continuous sections" in the direct integral
decomposition, thus yielding a Hilbert bundle. The proof is constructive and
provides an explicit description of the fibers. The emerging geometric
structure is a rigorous framework for a subsequent analysis of some topological
invariants of the operator, to be developed elsewhere. Two running examples
provide an Ariadne's thread through the paper. For the sake of completeness, we
begin by reviewing two related classical theorems by von Neumann and Maurin.Comment: 34 pages, 1 figure. Key words: topological quantum numbers, spectral
decomposition, Bloch-Floquet transform, Hilbert bundle. V3: a subsection has
been added; V4: some proofs have been simplified; V5: final version to be
published (with a new title
Symmetry and History Quantum Theory: An analogue of Wigner's Theorem
The basic ingredients of the `consistent histories' approach to quantum
theory are a space \UP of `history propositions' and a space \D of
`decoherence functionals'. In this article we consider such history quantum
theories in the case where \UP is given by the set of projectors \P(\V) on
some Hilbert space \V. We define the notion of a `physical symmetry of a
history quantum theory' (PSHQT) and specify such objects exhaustively with the
aid of an analogue of Wigner's theorem. In order to prove this theorem we
investigate the structure of \D, define the notion of an `elementary
decoherence functional' and show that each decoherence functional can be
expanded as a certain combination of these functionals. We call two history
quantum theories that are related by a PSHQT `physically equivalent' and show
explicitly, in the case of history quantum mechanics, how this notion is
compatible with one that has appeared previously.Comment: To appear in Jour.Math.Phys.; 25 pages; Latex-documen
A Covariant Information-Density Cutoff in Curved Space-Time
In information theory, the link between continuous information and discrete
information is established through well-known sampling theorems. Sampling
theory explains, for example, how frequency-filtered music signals are
reconstructible perfectly from discrete samples. In this Letter, sampling
theory is generalized to pseudo-Riemannian manifolds. This provides a new set
of mathematical tools for the study of space-time at the Planck scale: theories
formulated on a differentiable space-time manifold can be completely equivalent
to lattice theories. There is a close connection to generalized uncertainty
relations which have appeared in string theory and other studies of quantum
gravity.Comment: 4 pages, RevTe
Polynomial functors and combinatorial Dyson-Schwinger equations
We present a general abstract framework for combinatorial Dyson-Schwinger
equations, in which combinatorial identities are lifted to explicit bijections
of sets, and more generally equivalences of groupoids. Key features of
combinatorial Dyson-Schwinger equations are revealed to follow from general
categorical constructions and universal properties. Rather than beginning with
an equation inside a given Hopf algebra and referring to given Hochschild
-cocycles, our starting point is an abstract fixpoint equation in groupoids,
shown canonically to generate all the algebraic structure. Precisely, for any
finitary polynomial endofunctor defined over groupoids, the system of
combinatorial Dyson-Schwinger equations has a universal solution,
namely the groupoid of -trees. The isoclasses of -trees generate
naturally a Connes-Kreimer-like bialgebra, in which the abstract
Dyson-Schwinger equation can be internalised in terms of canonical
-operators. The solution to this equation is a series (the Green function)
which always enjoys a Fa\`a di Bruno formula, and hence generates a
sub-bialgebra isomorphic to the Fa\`a di Bruno bialgebra. Varying yields
different bialgebras, and cartesian natural transformations between various
yield bialgebra homomorphisms and sub-bialgebras, corresponding for example to
truncation of Dyson-Schwinger equations. Finally, all constructions can be
pushed inside the classical Connes-Kreimer Hopf algebra of trees by the
operation of taking core of -trees. A byproduct of the theory is an
interpretation of combinatorial Green functions as inductive data types in the
sense of Martin-L\"of Type Theory (expounded elsewhere).Comment: v4: minor adjustments, 49pp, final version to appear in J. Math. Phy
Gallavotti-Cohen-Type symmetry related to cycle decompositions for Markov chains and biochemical applications
We slightly extend the fluctuation theorem obtained in \cite{LS} for sums of
generators, considering continuous-time Markov chains on a finite state space
whose underlying graph has multiple edges and no loop. This extended frame is
suited when analyzing chemical systems. As simple corollary we derive in a
different method the fluctuation theorem of D. Andrieux and P. Gaspard for the
fluxes along the chords associated to a fundamental set of oriented cycles
\cite{AG2}.
We associate to each random trajectory an oriented cycle on the graph and we
decompose it in terms of a basis of oriented cycles. We prove a fluctuation
theorem for the coefficients in this decomposition. The resulting fluctuation
theorem involves the cycle affinities, which in many real systems correspond to
the macroscopic forces. In addition, the above decomposition is useful when
analyzing the large deviations of additive functionals of the Markov chain. As
example of application, in a very general context we derive a fluctuation
relation for the mechanical and chemical currents of a molecular motor moving
along a periodic filament.Comment: 23 pages, 5 figures. Correction
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