1,401 research outputs found
A categorical framework for the quantum harmonic oscillator
This paper describes how the structure of the state space of the quantum
harmonic oscillator can be described by an adjunction of categories, that
encodes the raising and lowering operators into a commutative comonoid. The
formulation is an entirely general one in which Hilbert spaces play no special
role. Generalised coherent states arise through the hom-set isomorphisms
defining the adjunction, and we prove that they are eigenstates of the lowering
operators. Surprisingly, generalised exponentials also emerge naturally in this
setting, and we demonstrate that coherent states are produced by the
exponential of a raising morphism acting on the zero-particle state. Finally,
we examine all of these constructions in a suitable category of Hilbert spaces,
and find that they reproduce the conventional mathematical structures.Comment: 44 pages, many figure
Solutions of Word Equations over Partially Commutative Structures
We give NSPACE(n log n) algorithms solving the following decision problems.
Satisfiability: Is the given equation over a free partially commutative monoid
with involution (resp. a free partially commutative group) solvable?
Finiteness: Are there only finitely many solutions of such an equation? PSPACE
algorithms with worse complexities for the first problem are known, but so far,
a PSPACE algorithm for the second problem was out of reach. Our results are
much stronger: Given such an equation, its solutions form an EDT0L language
effectively representable in NSPACE(n log n). In particular, we give an
effective description of the set of all solutions for equations with
constraints in free partially commutative monoids and groups
AQFT from n-functorial QFT
There are essentially two different approaches to the axiomatization of
quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and
functorial QFT, going back to Atiyah and Segal. More recently, based on ideas
by Baez and Dolan, the latter is being refined to "extended" functorial QFT by
Freed, Hopkins, Lurie and others. The first approach uses local nets of
operator algebras which assign to each patch an algebra "of observables", the
latter uses n-functors which assign to each patch a "propagator of states".
In this note we present an observation about how these two axiom systems are
naturally related: we demonstrate under mild assumptions that every
2-dimensional extended Minkowskian QFT 2-functor ("parallel surface transport")
naturally yields a local net. This is obtained by postcomposing the propagation
2-functor with an operation that mimics the passage from the Schroedinger
picture to the Heisenberg picture in quantum mechanics.
The argument has a straightforward generalization to general
pseudo-Riemannian structure and higher dimensions.Comment: 39 pages; further examples added: Hopf spin chains and asymptotic
inclusion of subfactors; references adde
Duality for convex monoids
Every C*-algebra gives rise to an effect module and a convex space of states,
which are connected via Kadison duality. We explore this duality in several
examples, where the C*-algebra is equipped with the structure of a
finite-dimensional Hopf algebra. When the Hopf algebra is the function algebra
or group algebra of a finite group, the resulting state spaces form convex
monoids. We will prove that both these convex monoids can be obtained from the
other one by taking a coproduct of density matrices on the irreducible
representations. We will also show that the same holds for a tensor product of
a group and a function algebra.Comment: 13 page
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