384 research outputs found
A note on the Zassenhaus product formula
We provide a simple method for the calculation of the terms c_n in the
Zassenhaus product for
non-commuting a and b. This method has been implemented in a computer program.
Furthermore, we formulate a conjecture on how to translate these results into
nested commutators. This conjecture was checked up to order n=17 using a
computer
A constructive algorithm for the Cartan decomposition of SU(2^N)
We present an explicit numerical method to obtain the Cartan-Khaneja-Glaser
decomposition of a general element G of SU(2^N) in terms of its `Cartan' and
`non-Cartan' components. This effectively factors G in terms of group elements
that belong in SU(2^n) with n<N, a procedure that can be iterated down to n=2.
We show that every step reduces to solving the zeros of a matrix polynomial,
obtained by truncation of the Baker-Campbell-Hausdorff formula, numerically.
All computational tasks involved are straightforward and the overall truncation
errors are well under control.Comment: 15 pages, no figures, matlab file at
http://cam.qubit.org/users/jiannis
Manifolds with large isotropy groups
We classify all simply connected Riemannian manifolds whose isotropy groups
act with cohomogeneity less than or equal to two.Comment: 21 page
Exact solutions in Einstein-Yang-Mills-Dirac systems
We present exact solutions in Einstein-Yang-Mills-Dirac theories with gauge
groups SU(2) and SU(4) in Robertson-Walker space-time , which
are symmetric under the action of the group SO(4) of spatial rotations. Our
approach is based on the dimensional reduction method for gauge and
gravitational fields and relates symmetric solutions in EYMD theory to certain
solutions of an effective dynamical system.
We interpret our solutions as cosmological solutions with an oscillating
Yang-Mills field passing between topologically distinct vacua. The explicit
form of the solution for spinor field shows that its energy changes the sign
during the evolution of the Yang-Mills field from one vacuum to the other,
which can be considered as production or annihilation of fermions.
Among the obtained solutions there is also a static sphaleron-like solution,
which is a cosmological analogue of the first Bartnik-McKinnon solution in the
presence of fermions.Comment: 18 pages, LaTeX 2
Quantum simulations under translational symmetry
We investigate the power of quantum systems for the simulation of Hamiltonian
time evolutions on a cubic lattice under the constraint of translational
invariance. Given a set of translationally invariant local Hamiltonians and
short range interactions we determine time evolutions which can and those that
can not be simulated. Whereas for general spin systems no finite universal set
of generating interactions is shown to exist, universality turns out to be
generic for quadratic bosonic and fermionic nearest-neighbor interactions when
supplemented by all translationally invariant on-site Hamiltonians.Comment: 9 pages, 2 figures, references added, minor change
Centralizers of maximal regular subgroups in simple Lie groups and relative congruence classes of representations
In the paper we present a new, uniform and comprehensive description of
centralizers of the maximal regular subgroups in compact simple Lie groups of
all types and ranks. The centralizer is either a direct product of finite
cyclic groups, a continuous group of rank 1, or a product, not necessarily
direct, of a continuous group of rank 1 with a finite cyclic group. Explicit
formulas for the action of such centralizers on irreducible representations of
the simple Lie algebras are given.Comment: 27 page
Brownian Motions on Metric Graphs
Brownian motions on a metric graph are defined. Their generators are
characterized as Laplace operators subject to Wentzell boundary at every
vertex. Conversely, given a set of Wentzell boundary conditions at the vertices
of a metric graph, a Brownian motion is constructed pathwise on this graph so
that its generator satisfies the given boundary conditions.Comment: 43 pages, 7 figures. 2nd revision of our article 1102.4937: The
introduction has been modified, several references were added. This article
will appear in the special issue of Journal of Mathematical Physics
celebrating Elliott Lieb's 80th birthda
Oxidation = group theory
Dimensional reduction of theories involving (super-)gravity gives rise to
sigma models on coset spaces of the form G/H, with G a non-compact group, and H
its maximal compact subgroup. The reverse process, called oxidation, is the
reconstruction of the possible higher dimensional theories, given the lower
dimensional theory. In 3 dimensions, all degrees of freedom can be dualized to
scalars. Given the group G for a 3 dimensional sigma model on the coset G/H, we
demonstrate an efficient method for recovering the higher dimensional theories,
essentially by decomposition into subgroups. The equations of motion, Bianchi
identities, Kaluza-Klein modifications and Chern-Simons terms are easily
extracted from the root lattice of the group G. We briefly discuss some aspects
of oxidation from the E_{8(8)}/SO(16) coset, and demonstrate that our formalism
reproduces the Chern-Simons term of 11-d supergravity, knows about the
T-duality of IIA and IIB theory, and easily deals with self-dual tensors, like
the 5-tensor of IIB supergravity.Comment: LaTeX, 8 pages, uses IOP style files; Talk given at the RTN workshop
``The quantum structure of spacetime and the geometric nature of fundamental
interactions'', Leuven, September 200
Classification of Singular Fibres on Rational Elliptic Surfaces in Characteristic Three
We determine and list all possible configurations of singular fibres on
rational elliptic surfaces in characteristic three. In total, we find that 267
distinct configurations exist. This result complements Miranda and Persson's
classification in characteristic zero, and Lang's classification in
characteristic two.Comment: 40 Pages. Minor typos correcte
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