38 research outputs found
Exotic Smoothness and Quantum Gravity
Since the first work on exotic smoothness in physics, it was folklore to
assume a direct influence of exotic smoothness to quantum gravity. Thus, the
negative result of Duston (arXiv:0911.4068) was a surprise. A closer look into
the semi-classical approach uncovered the implicit assumption of a close
connection between geometry and smoothness structure. But both structures,
geometry and smoothness, are independent of each other. In this paper we
calculate the "smoothness structure" part of the path integral in quantum
gravity assuming that the "sum over geometries" is already given. For that
purpose we use the knot surgery of Fintushel and Stern applied to the class
E(n) of elliptic surfaces. We mainly focus our attention to the K3 surfaces
E(2). Then we assume that every exotic smoothness structure of the K3 surface
can be generated by knot or link surgery a la Fintushel and Stern. The results
are applied to the calculation of expectation values. Here we discuss the two
observables, volume and Wilson loop, for the construction of an exotic
4-manifold using the knot and the Whitehead link . By using Mostow
rigidity, we obtain a topological contribution to the expectation value of the
volume. Furthermore we obtain a justification of area quantization.Comment: 16 pages, 1 Figure, 1 Table subm. Class. Quant. Grav
Exotic R^4 and quantum field theory
Recent work on exotic smooth R^4's, i.e. topological R^4 with exotic
differential structure, shows the connection of 4-exotics with the
codimension-1 foliations of , SU(2) WZW models and twisted K-theory
, . These results made it possible
to explicate some physical effects of exotic 4-smoothness. Here we present a
relation between exotic smooth R^4 and operator algebras. The correspondence
uses the leaf space of the codimension-1 foliation of S^3 inducing a von
Neumann algebra as description. This algebra is a type III_1 factor
lying at the heart of any observable algebra of QFT. By using the relation to
factor II, we showed that the algebra can be interpreted as
Drinfeld-Turaev deformation quantization of the space of flat SL(2,\mathbb{C})
connections (or holonomies). Thus, we obtain a natural relation to quantum
field theory. Finally we discuss the appearance of concrete action functionals
for fermions or gauge fields and its connection to quantum-field-theoretical
models like the Tree QFT of Rivasseau.Comment: 15 pages, 2 figures, Based on the talk presented at Quantum Theory
and Symmetries 7, Prague, August 7-13, 2011, JPconf styl
Absolute spacetime: the twentieth century ether
All gauge theories need ``something fixed'' even as ``something changes.''
Underlying the implementation of these ideas all major physical theories make
indispensable use of an elaborately designed spacetime model as the ``something
fixed,'' i.e., absolute. This model must provide at least the following
sequence of structures: point set, topological space, smooth manifold,
geometric manifold, base for various bundles. The ``fine structure'' of
spacetime inherent in this sequence is of course empirically unobservable
directly, certainly when quantum mechanics is taken into account. This issue is
at the basis of the difficulties in quantizing general relativity and has been
approached in many different ways. Here we review an approach taking into
account the non-Boolean properties of quantum logic when forming a spacetime
model. Finally, we recall how the fundamental gauge of diffeomorphisms (the
issue of general covariance vs coordinate conditions) raised deep conceptual
problems for Einstein in his early development of general relativity. This is
clearly illustrated in the notorious ``hole'' argument. This scenario, which
does not seem to be widely known to practicing relativists, is nevertheless
still interesting in terms of its impact for fundamental gauge issues.Comment: Contribution to Proceedings of Mexico Meeting on Gauge Theories of
Gravity in honor of Friedrich Heh
Quantum D-branes and exotic smooth R^4
In this paper, we present the idea that the formalism of string theory is
connected with the dimension 4 in a new way, not covered by phenomenological or
model-building approaches. The main connection is given by structures induced
by small exotic smooth R^4's having intrinsic meaning for physics in dimension
4. We extend the notion of stable quantum D-branes in a separable
noncommutative C* algebras over convolution algebras corresponding to the
codimension-1 foliations of S^3 which are mainly connected to small exotic R^4.
The tools of topological K-homology and K-theory as well KK-theory describe
stable quantum branes in the C* algebras when naturally extended to algebras.
In case of convolution algebras, small exotic smooth R^4's embedded in exotic
R^4 correspond to a generalized quantum branes on the algebras. These results
extend the correspondence between exotic R^4 and classical D and NS branes from
our previous work.Comment: 16 pages, no figure, see arXiv/1101.3169 for Part 1 This is part 2 of
the work based on the talk "Small exotic smooth and string
theory" given at the International Congress of Mathematicians, ICM2010,
19-28.08.2010, Hyderabad, Indi
Analytical and numerical investigations of Evolutionary Algorithms in continuous spaces
. We investigate the biologically motivated selfreproduction strategies, by numerical and analytical calculations. In the analytical part we show that each of these strategies can be reduced to a eigenvalue problem of Sturm-Liouville-type. The properties of the landscape and the dynamics of the optimization are encoded in the spectrum of the Hamiltonian, which is different in both cases. We discuss some model cases with exact solutions and compare it with simulations. 1 Introduction The optimization problem appears in several fields of physics and mathematics. It is known from mathematics that every local minimum of a convex function defined over a convex set is also the global minimum of the function. But the main problem is to find this optimum. From the physical point of view every dynamical process can be considered in terms of finding the optimum of the action functional. The best example is the trajectory of the free point mass in mechanics which follows the shortest way between..