24,170 research outputs found
Towards a -function in 4D quantum gravity
We develop a generally applicable method for constructing functions, ,
which have properties similar to Zamolodchikov's -function, and are
geometrically natural objects related to the theory space explored by
non-perturbative functional renormalization group (RG) equations. Employing the
Euclidean framework of the Effective Average Action (EAA), we propose a
-function which can be defined for arbitrary systems of gravitational,
Yang-Mills, ghost, and bosonic matter fields, and in any number of spacetime
dimensions. It becomes stationary both at critical points and in classical
regimes, and decreases monotonically along RG trajectories provided the
breaking of the split-symmetry which relates background and quantum fields is
sufficiently weak. Within the Asymptotic Safety approach we test the proposal
for Quantum Einstein Gravity in dimensions, performing detailed numerical
investigations in . We find that the bi-metric Einstein-Hilbert truncation
of theory space introduced recently is general enough to yield perfect
monotonicity along the RG trajectories, while its more familiar single-metric
analog fails to achieve this behavior which we expect on general grounds.
Investigating generalized crossover trajectories connecting a fixed point in
the ultraviolet to a classical regime with positive cosmological constant in
the infrared, the -function is shown to depend on the choice of the
gravitational instanton which constitutes the background spacetime. For de
Sitter space in 4 dimensions, the Bekenstein-Hawking entropy is found to play a
role analogous to the central charge in conformal field theory. We also comment
on the idea of a `- connection' and the `-bound' discussed
earlier.Comment: 15 figures; additional comment
Covariance and Fisher information in quantum mechanics
Variance and Fisher information are ingredients of the Cramer-Rao inequality.
We regard Fisher information as a Riemannian metric on a quantum statistical
manifold and choose monotonicity under coarse graining as the fundamental
property of variance and Fisher information. In this approach we show that
there is a kind of dual one-to-one correspondence between the candidates of the
two concepts. We emphasis that Fisher informations are obtained from relative
entropies as contrast functions on the state space and argue that the scalar
curvature might be interpreted as an uncertainty density on a statistical
manifold.Comment: LATE
Intensional and Extensional Semantics of Bounded and Unbounded Nondeterminism
We give extensional and intensional characterizations of nondeterministic
functional programs: as structure preserving functions between biorders, and as
nondeterministic sequential algorithms on ordered concrete data structures
which compute them. A fundamental result establishes that the extensional and
intensional representations of non-deterministic programs are equivalent, by
showing how to construct a unique sequential algorithm which computes a given
monotone and stable function, and describing the conditions on sequential
algorithms which correspond to continuity with respect to each order.
We illustrate by defining may and must-testing denotational semantics for a
sequential functional language with bounded and unbounded choice operators. We
prove that these are computationally adequate, despite the non-continuity of
the must-testing semantics of unbounded nondeterminism. In the bounded case, we
prove that our continuous models are fully abstract with respect to may and
must-testing by identifying a simple universal type, which may also form the
basis for models of the untyped lambda-calculus. In the unbounded case we
observe that our model contains computable functions which are not denoted by
terms, by identifying a further "weak continuity" property of the definable
elements, and use this to establish that it is not fully abstract
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