295 research outputs found
Exotic Differential Operators on Complex Minimal Nilpotent Orbits
Let O be the minimal nilpotent adjoint orbit in a classical complex
semisimple Lie algebra g. O is a smooth quasi-affine variety stable under the
Euler dilation action on g. The algebra of differential operators on O is
D(O)=D(Cl(O)) where the closure Cl(O) is a singular cone in g. See \cite{jos}
and \cite{bkHam} for some results on the geometry and quantization of O.
We construct an explicit subspace of commuting
differential operators which are Euler homogeneous of degree -1. The space
is finite-dimensional, g-stable and carries the adjoint
representation. consists of (for ) non-obvious order
4 differential operators obtained by quantizing symbols we obtained previously.
These operators are "exotic" in that there is (apparently) no geometric or
algebraic theory which explains them. The algebra generated by is a
maximal commutative subalgebra A of D(X). We find a G-equivariant algebra
isomorphism R(O) to A, , such that the formula defines a positive-definite Hermitian inner product on
R(O).
We will use these operators to quantize O in a subsequent paper.Comment: 34 pages, corrected some typos, changed conten
Minimal Unitary Realizations of Exceptional U-duality Groups and Their Subgroups as Quasiconformal Groups
We study the minimal unitary representations of noncompact exceptional groups
that arise as U-duality groups in extended supergravity theories. First we give
the unitary realizations of the exceptional group E_{8(-24)} in SU*(8) as well
as SU(6,2) covariant bases. E_{8(-24)} has E_7 X SU(2) as its maximal compact
subgroup and is the U-duality group of the exceptional supergravity theory in
d=3. For the corresponding U-duality group E_{8(8)} of the maximal supergravity
theory the minimal realization was given in hep-th/0109005. The minimal unitary
realizations of all the lower rank noncompact exceptional groups can be
obtained by truncation of those of E_{8(-24)} and E_{8(8)}. By further
truncation one can obtain the minimal unitary realizations of all the groups of
the "Magic Triangle". We give explicitly the minimal unitary realizations of
the exceptional subgroups of E_{8(-24)} as well as other physically interesting
subgroups. These minimal unitary realizations correspond, in general, to the
quantization of their geometric actions as quasi-conformal groups as defined in
hep-th/0008063.Comment: 28 pages. Latex commands removed from the abstract for the arXiv. No
changes in the manuscrip
Globally controlled universal quantum computation with arbitrary subsystem dimension
We introduce a scheme to perform universal quantum computation in quantum
cellular automata (QCA) fashion in arbitrary subsystem dimension (not
necessarily finite). The scheme is developed over a one spatial dimension
-element array, requiring only mirror symmetric logical encoding and global
pulses. A mechanism using ancillary degrees of freedom for subsystem specific
measurement is also presented.Comment: 7 pages, 1 figur
Minimum orbit dimension for local unitary action on n-qubit pure states
The group of local unitary transformations partitions the space of n-qubit
quantum states into orbits, each of which is a differentiable manifold of some
dimension. We prove that all orbits of the n-qubit quantum state space have
dimension greater than or equal to 3n/2 for n even and greater than or equal to
(3n + 1)/2 for n odd. This lower bound on orbit dimension is sharp, since
n-qubit states composed of products of singlets achieve these lowest orbit
dimensions.Comment: 19 page
Muon capture on light nuclei
This work investigates the muon capture reactions 2H(\mu^-,\nu_\mu)nn and
3He(\mu^-,\nu_\mu)3H and the contribution to their total capture rates arising
from the axial two-body currents obtained imposing the
partially-conserved-axial-current (PCAC) hypothesis. The initial and final A=2
and 3 nuclear wave functions are obtained from the Argonne v_{18} two-nucleon
potential, in combination with the Urbana IX three-nucleon potential in the
case of A=3. The weak current consists of vector and axial components derived
in chiral effective field theory. The low-energy constant entering the vector
(axial) component is determined by reproducting the isovector combination of
the trinucleon magnetic moment (Gamow-Teller matrix element of tritium
beta-decay). The total capture rates are 393.1(8) s^{-1} for A=2 and 1488(9)
s^{-1} for A=3, where the uncertainties arise from the adopted fitting
procedure.Comment: 6 pages, submitted to Few-Body Sys
Non-Abelian Tensor Multiplet Equations from Twistor Space
We establish a Penrose-Ward transform yielding a bijection between
holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual
tensor fields on six-dimensional flat space-time. Extending the twistor space
to supertwistor space, we derive sets of manifestly N=(1,0) and N=(2,0)
supersymmetric non-Abelian constraint equations containing the tensor
multiplet. We also demonstrate how this construction leads to constraint
equations for non-Abelian supersymmetric self-dual strings.Comment: v3: 23 pages, revised version published in Commun. Math. Phy
Yang-Mills theory for bundle gerbes
Given a bundle gerbe with connection on an oriented Riemannian manifold of
dimension at least equal to 3, we formulate and study the associated Yang-Mills
equations. When the Riemannian manifold is compact and oriented, we prove the
existence of instanton solutions to the equations and also determine the moduli
space of instantons, thus giving a complete analysis in this case. We also
discuss duality in this context.Comment: Latex2e, 7 pages, some typos corrected, to appear in J. Phys. A:
Math. and Ge
Valence bond solid formalism for d-level one-way quantum computation
The d-level or qudit one-way quantum computer (d1WQC) is described using the
valence bond solid formalism and the generalised Pauli group. This formalism
provides a transparent means of deriving measurement patterns for the
implementation of quantum gates in the computational model. We introduce a new
universal set of qudit gates and use it to give a constructive proof of the
universality of d1WQC. We characterise the set of gates that can be performed
in one parallel time step in this model.Comment: 26 pages, 9 figures. Published in Journal of Physics A: Mathematical
and Genera
A Coboundary Morphism For The Grothendieck Spectral Sequence
Given an abelian category with enough injectives we show that a
short exact sequence of chain complexes of objects in gives rise
to a short exact sequence of Cartan-Eilenberg resolutions. Using this we
construct coboundary morphisms between Grothendieck spectral sequences
associated to objects in a short exact sequence. We show that the coboundary
preserves the filtrations associated with the spectral sequences and give an
application of these result to filtrations in sheaf cohomology.Comment: 18 page
Quantum circuits with uniformly controlled one-qubit gates
Uniformly controlled one-qubit gates are quantum gates which can be
represented as direct sums of two-dimensional unitary operators acting on a
single qubit. We present a quantum gate array which implements any n-qubit gate
of this type using at most 2^{n-1} - 1 controlled-NOT gates, 2^{n-1} one-qubit
gates and a single diagonal n-qubit gate. The circuit is based on the so-called
quantum multiplexor, for which we provide a modified construction. We
illustrate the versatility of these gates by applying them to the decomposition
of a general n-qubit gate and a local state preparation procedure. Moreover, we
study their implementation using only nearest-neighbor gates. We give upper
bounds for the one-qubit and controlled-NOT gate counts for all the
aforementioned applications. In all four cases, the proposed circuit topologies
either improve on or achieve the previously reported upper bounds for the gate
counts. Thus, they provide the most efficient method for general gate
decompositions currently known.Comment: 8 pages, 10 figures. v2 has simpler notation and sharpens some
result
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