30,239 research outputs found
Scaling Algebras and Renormalization Group in Algebraic Quantum Field Theory
For any given algebra of local observables in Minkowski space an associated
scaling algebra is constructed on which renormalization group (scaling)
transformations act in a canonical manner. The method can be carried over to
arbitrary spacetime manifolds and provides a framework for the systematic
analysis of the short distance properties of local quantum field theories. It
is shown that every theory has a (possibly non-unique) scaling limit which can
be classified according to its classical or quantum nature. Dilation invariant
theories are stable under the action of the renormalization group. Within this
framework the problem of wedge (Bisognano-Wichmann) duality in the scaling
limit is discussed and some of its physical implications are outlined.Comment: 47 pages, no figures, ams-late
Internal labelling operators and contractions of Lie algebras
We analyze under which conditions the missing label problem associated to a
reduction chain of (simple) Lie algebras
can be completely solved by means of an In\"on\"u-Wigner contraction
naturally related to the embedding. This provides a new interpretation of the
missing label operators in terms of the Casimir operators of the contracted
algebra, and shows that the available labeling operators are not completely
equivalent. Further, the procedure is used to obtain upper bounds for the
number of invariants of affine Lie algebras arising as contractions of
semisimple algebras.Comment: 20 pages, 2 table
Singular dimensions of the N=2 superconformal algebras. I
Verma modules of superconfomal algebras can have singular vector spaces with
dimensions greater than 1. Following a method developed for the Virasoro
algebra by Kent, we introduce the concept of adapted orderings on
superconformal algebras. We prove several general results on the ordering
kernels associated to the adapted orderings and show that the size of an
ordering kernel implies an upper limit for the dimension of a singular vector
space. We apply this method to the topological N=2 algebra and obtain the
maximal dimensions of the singular vector spaces in the topological Verma
modules: 0, 1, 2 or 3 depending on the type of Verma module and the type of
singular vector. As a consequence we prove the conjecture of Gato-Rivera and
Rosado on the possible existing types of topological singular vectors (4 in
chiral Verma modules and 29 in complete Verma modules). Interestingly, we have
found two-dimensional spaces of singular vectors at level 1. Finally, by using
the topological twists and the spectral flows, we also obtain the maximal
dimensions of the singular vector spaces for the Neveu-Schwarz N=2 algebra (0,
1 or 2) and for the Ramond N=2 algebra (0, 1, 2 or 3).Comment: Latex, 37 page
Entanglement entropy for a Maxwell field: Numerical calculation on a two dimensional lattice
We study entanglement entropy (EE) for a Maxwell field in 2+1 dimensions. We
do numerical calculations in two dimensional lattices. This gives a concrete
example of the general results of our recent work on entropy for lattice gauge
fields using an algebraic approach. To evaluate the entropies we extend the
standard calculation methods for the entropy of Gaussian states in canonical
commutation algebras to the more general case of algebras with center and
arbitrary numerical commutators. We find that while the entropy depends on the
details of the algebra choice, mutual information has a well defined continuum
limit. We study several universal terms for the entropy of the Maxwell field
and compare with the case of a massless scalar field. We find some interesting
new phenomena: An "evanescent" logarithmically divergent term in the entropy
with topological coefficient which does not have any correspondence with
ultraviolet entanglement in the universal quantities, and a non standard way in
which strong subadditivity is realized. Based on the results of our
calculations we propose a generalization of strong subadditivity for the
entropy on some algebras that are not in tensor product.Comment: 27 pages, 15 figure
Quantum Error Correction of Observables
A formalism for quantum error correction based on operator algebras was
introduced in [1] via consideration of the Heisenberg picture for quantum
dynamics. The resulting theory allows for the correction of hybrid
quantum-classical information and does not require an encoded state to be
entirely in one of the corresponding subspaces or subsystems. Here, we provide
detailed proofs for the results of [1], derive a number of new results, and we
elucidate key points with expanded discussions. We also present several
examples and indicate how the theory can be extended to operator spaces and
general positive operator-valued measures.Comment: 22 pages, 1 figure, preprint versio
The Douglas Lemma for von Neumann Algebras and Some Applications
In this article, we discuss the well-known Douglas lemma on the relationship
between majorization and factorization of operators, in the context of von
Neumann algebras. We give a proof of the Douglas lemma for von Neumann algebras
which is essential for some of our applications. We discuss several
applications of the Douglas lemma and prove some new results about left (or,
one-sided) ideals of von Neumann algebras. A result by Loebl-Paulsen
characterizes -convex subsets of as those
subsets which contain -segments generated from operators in the subset
( denotes the set of bounded operators on a complex
Hilbert space .) We define the notion of pseudo -convexity
for a subset of a Hilbert -bimodule over a -algebra with the
aspiration of it being a practical technical tool in establishing
-convexity of a subset. For a von Neumann algebra , we prove
the equivalence of the notions of -convexity and pseudo -convexity in
Hilbert -bimodules. This generalizes the aforementioned
Loebl-Paulsen result which may be formulated in a straightforward manner in the
setting of -convexity in Hilbert -bimodules.Comment: 16 page
The characteristic polynomial of the Adams operators on graded connected Hopf algebras
The Adams operators on a Hopf algebra are the convolution powers
of the identity of . We study the Adams operators when is graded
connected. They are also called Hopf powers or Sweedler powers. The main result
is a complete description of the characteristic polynomial (both eigenvalues
and their multiplicities) for the action of the operator on each
homogeneous component of . The eigenvalues are powers of . The
multiplicities are independent of , and in fact only depend on the dimension
sequence of . These results apply in particular to the antipode of (the
case ). We obtain closed forms for the generating function of the
sequence of traces of the Adams operators. In the case of the antipode, the
generating function bears a particularly simple relationship to the one for the
dimension sequence. In case H is cofree, we give an alternative description for
the characteristic polynomial and the trace of the antipode in terms of certain
palindromic words. We discuss parallel results that hold for Hopf monoids in
species and -Hopf algebras.Comment: 36 pages; two appendice
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