515 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
A simple method for finite range decomposition of quadratic forms and Gaussian fields
We present a simple method to decompose the Green forms corresponding to a
large class of interesting symmetric Dirichlet forms into integrals over
symmetric positive semi-definite and finite range (properly supported) forms
that are smoother than the original Green form. This result gives rise to
multiscale decompositions of the associated Gaussian free fields into sums of
independent smoother Gaussian fields with spatially localized correlations. Our
method makes use of the finite propagation speed of the wave equation and
Chebyshev polynomials. It improves several existing results and also gives
simpler proofs.Comment: minor correction for t<
SU(5) grand unification on a domain-wall brane from an E_6-invariant action
An SU(5) grand unification scheme for effective 3+1-dimensional fields
dynamically localised on a domain-wall brane is constructed. This is achieved
through the confluence of the clash-of-symmetries mechanism for symmetry
breaking through domain-wall formation, and the Dvali-Shifman gauge-boson
localisation idea. It requires an E_6 gauge-invariant action, yielding a
domain-wall solution that has E_6 broken to differently embedded SO(10) x U(1)
subgroups in the two bulk regions on opposite sides of the wall. On the wall
itself, the unbroken symmetry is the intersection of the two bulk subgroups,
and contains SU(5). A 4+1-dimensional fermion family in the 27 of E_6 gives
rise to localised left-handed zero-modes in the 5^* + 10 + 1 + 1 representation
of SU(5). The remaining ten fermion components of the 27 are delocalised exotic
states, not appearing in the effective 3+1-dimensional theory on the
domain-wall brane. The scheme is compatible with the type-2 Randall-Sundrum
mechanism for graviton localisation; the single extra dimension is infinite.Comment: 21 pages, 9 figures. Minor changes to text and references. To appear
in Phys. Rev.
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
Characteristic Dynkin diagrams and W-algebras
We present a classification of characteristic Dynkin diagrams for the ,
, and algebras. This classification is related to the
classification of \cw(\cg,\ck) algebras arising from non-Abelian Toda models,
and we argue that it can give new insight on the structure of algebras.Comment: 20 page
On the spatial Markov property of soups of unoriented and oriented loops
We describe simple properties of some soups of unoriented Markov loops and of
some soups of oriented Markov loops that can be interpreted as a spatial Markov
property of these loop-soups. This property of the latter soup is related to
well-known features of the uniform spanning trees (such as Wilson's algorithm)
while the Markov property of the former soup is related to the Gaussian Free
Field and to identities used in the foundational papers of Symanzik, Nelson,
and of Brydges, Fr\"ohlich and Spencer or Dynkin, or more recently by Le Jan
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