28,342 research outputs found
Depth, Highness and DNR degrees
We study Bennett deep sequences in the context of recursion theory; in
particular we investigate the notions of O(1)-deepK, O(1)-deepC , order-deep K
and order-deep C sequences. Our main results are that Martin-Loef random sets
are not order-deepC , that every many-one degree contains a set which is not
O(1)-deepC , that O(1)-deepC sets and order-deepK sets have high or DNR Turing
degree and that no K-trival set is O(1)-deepK.Comment: journal version, dmtc
Anomalies and Fermion Content of Grand Unified Theories in Extra Dimensions
The restrictions imposed by anomaly cancellation on the chiral fermion
content of nonsupersymmetric gauge theories based on various groups are studied
in spacetime dimension D=6, 8, and 10. In particular, we show that the only
mathematically consistent chiral SU(5) theory in D=6 contains three
nonidentical generations.Comment: 15 pages, revtex. v2: references added to match published versio
Self-consistent Green's functions formalism with three-body interactions
We extend the self-consistent Green's functions formalism to take into
account three-body interactions. We analyze the perturbative expansion in terms
of Feynman diagrams and define effective one- and two-body interactions, which
allows for a substantial reduction of the number of diagrams. The procedure can
be taken as a generalization of the normal ordering of the Hamiltonian to fully
correlated density matrices. We give examples up to third order in perturbation
theory. To define nonperturbative approximations, we extend the equation of
motion method in the presence of three-body interactions. We propose schemes
that can provide nonperturbative resummation of three-body interactions. We
also discuss two different extensions of the Koltun sum rule to compute the
ground state of a many-body system.Comment: 26 pages, 19 figure
One-particle irreducible functional approach - a new route to diagrammatic extensions of DMFT
We present an approach which is based on the one-particle irreducible (1PI)
generating functional formalism and includes electronic correlations on all
length-scales beyond the local correlations of dynamical mean field theory
(DMFT). This formalism allows us to unify aspects of the dynamical vertex
approximation (D\GammaA) and the dual fermion (DF) scheme, yielding a
consistent formulation of non-local correlations at the one- and two-particle
level beyond DMFT within the functional integral formalism. In particular, the
considered approach includes one-particle reducible contributions from the
three- and more-particle vertices in the dual fermion approach, as well as some
diagrams not included in the ladder version of D\GammaA. To demonstrate the
applicability and physical content of the 1PI approach, we compare the
diagrammatics of 1PI, DF and D\GammaA, as well as the numerical results of
these approaches for the half-filled Hubbard model in two dimensions.Comment: 36 pages, 12 figures, updated versio
A simple way to generate high order vacuum graphs
We describe an efficient practical procedure for enumerating and regrouping
vacuum Feynman graphs of a given order in perturbation theory. The method is
based on a combination of Schwinger-Dyson equations and the
two-particle-irreducible ("skeleton") expansion. The regrouping leads to
skeletons containing only free propagators, together with "ring diagrams"
containing all the self-energy insertions. As a consequence, relatively few
diagrams need to be drawn and integrations carried out at any single stage of
the computation and, in low dimensions, overlapping ultraviolet/infrared
subdivergences can be cleanly isolated. As an illustration we enumerate the
graphs contributing to the 4-loop free energy in QCD, explicitly in a continuum
and more compactly in a lattice regularization.Comment: 19 pages. Reference added. To appear in Phys.Rev.
On the Solvability of the Mind-Body Problem
The mind-body problem is analyzed in a physicalist perspective. By combining the concepts of emergence and algorithmic information theory in a thought experiment employing a basic nonlinear process, it is shown that epistemically strongly emergent properties may develop in a physical system. Turning to the significantly more complex neural network of the brain it is subsequently argued that consciousness is epistemically emergent. Thus reductionist understanding of consciousness appears not possible; the mind-body problem does not have a reductionist solution. The ontologically emergent character of consciousness is then identified from a combinatorial analysis relating to universal limits set by quantum mechanics, implying that consciousness is fundamentally irreducible to low-level phenomena
The Value of Help Bits in Randomized and Average-Case Complexity
"Help bits" are some limited trusted information about an instance or
instances of a computational problem that may reduce the computational
complexity of solving that instance or instances. In this paper, we study the
value of help bits in the settings of randomized and average-case complexity.
Amir, Beigel, and Gasarch (1990) show that for constant , if instances
of a decision problem can be efficiently solved using less than bits of
help, then the problem is in P/poly. We extend this result to the setting of
randomized computation: We show that the decision problem is in P/poly if using
help bits, instances of the problem can be efficiently solved with
probability greater than . The same result holds if using less than
help bits (where is the binary entropy function),
we can efficiently solve fraction of the instances correctly with
non-vanishing probability. We also extend these two results to non-constant but
logarithmic . In this case however, instead of showing that the problem is
in P/poly we show that it satisfies "-membership comparability," a notion
known to be related to solving instances using less than bits of help.
Next we consider the setting of average-case complexity: Assume that we can
solve instances of a decision problem using some help bits whose entropy is
less than when the instances are drawn independently from a particular
distribution. Then we can efficiently solve an instance drawn from that
distribution with probability better than .
Finally, we show that in the case where is super-logarithmic, assuming
-membership comparability of a decision problem, one cannot prove that the
problem is in P/poly by a "black-box proof.
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