101,286 research outputs found
Separation of noncommutative differential calculus on quantum Minkowski space
Noncommutative differential calculus on quantum Minkowski space is not
separated with respect to the standard generators, in the sense that partial
derivatives of functions of a single generator can depend on all other
generators. It is shown that this problem can be overcome by a separation of
variables. We study the action of the universal L-matrix, appearing in the
coproduct of partial derivatives, on generators. Powers of he resulting quantum
Minkowski algebra valued matrices are calculated. This leads to a nonlinear
coordinate transformation which essentially separates the calculus. A compact
formula for general derivatives is obtained in form of a chain rule with
partial Jackson derivatives. It is applied to the massive quantum Klein-Gordon
equation by reducing it to an ordinary q-difference equation. The rest state
solution can be expressed in terms of a product of q-exponential functions in
the separated variables.Comment: 33 page
Derivatives of Multilinear Functions of Matrices
Perturbation or error bounds of functions have been of great interest for a
long time. If the functions are differentiable, then the mean value theorem and
Taylor's theorem come handy for this purpose. While the former is useful in
estimating in terms of and requires the norms of the
first derivative of the function, the latter is useful in computing higher
order perturbation bounds and needs norms of the higher order derivatives of
the function.
In the study of matrices, determinant is an important function. Other scalar
valued functions like eigenvalues and coefficients of characteristic polynomial
are also well studied. Another interesting function of this category is the
permanent, which is an analogue of the determinant in matrix theory. More
generally, there are operator valued functions like tensor powers,
antisymmetric tensor powers and symmetric tensor powers which have gained
importance in the past. In this article, we give a survey of the recent work on
the higher order derivatives of these functions and their norms. Using Taylor's
theorem, higher order perturbation bounds are obtained. Some of these results
are very recent and their detailed proofs will appear elsewhere.Comment: 17 page
Massive Nonplanar Two-Loop Maximal Unitarity
We explore maximal unitarity for nonplanar two-loop integrals with up to four
massive external legs. In this framework, the amplitude is reduced to a basis
of master integrals whose coefficients are extracted from maximal cuts. The
hepta-cut of the nonplanar double box defines a nodal algebraic curve
associated with a multiply pinched genus-3 Riemann surface. All possible
configurations of external masses are covered by two distinct topological
pictures in which the curve decomposes into either six or eight Riemann
spheres. The procedure relies on consistency equations based on vanishing of
integrals of total derivatives and Levi-Civita contractions. Our analysis
indicates that these constraints are governed by the global structure of the
maximal cut. Lastly, we present an algorithm for computing generalized cuts of
massive integrals with higher powers of propagators based on the Bezoutian
matrix method.Comment: 54 pages, 9 figures, v2: journal versio
Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
We derive a set of first-order differential equations obeyed by the S-matrix of planar maximally supersymmetric Yang-Mills theory. The equations, based on the Yangian symmetry of the theory, involve only finite and regulator-independent quantities and uniquely determine the all-loop S-matrix. When expanded in powers of the coupling they give derivatives of amplitudes as single integrals over lower-loop, higher-point amplitudes/Wilson loops. We outline a derivation for the equations using the Operator Product Expansion for Wilson loops. We apply them on a few examples at two- and three-loops, reproducing a recent result on the two-loop NMHV hexagon and fixing previously undermined coefficients in a recent Ansatz for the three-loop MHV hexagon. In addition, we consider amplitudes restricted to a two-dimensional subspace of Minkowski space and derive a particularly simple closed set of equations in that case
Large N_c in chiral perturbation theory
The construction of the effective Lagrangian relevant for the mesonic sector
of QCD in the large N_c limit meets with a few rather subtle problems. We
thoroughly examine these and show that, if the variables of the effective
theory are chosen suitably, the known large N_c counting rules of QCD can
unambiguously be translated into corresponding counting rules for the effective
coupling constants. As an application, we demonstrate that the Kaplan-Manohar
transformation is in conflict with these rules and is suppressed to all orders
in 1/N_c. The anomalous dimension of the axial singlet current generates an
additional complication: The corresponding external field undergoes
nonmultiplicative renormalization. As a consequence, the Wess-Zumino-Witten
term, which accounts for the U(3)_R x U(3)_L anomalies in the framework of the
effective theory, contains pieces that depend on the running scale of QCD. The
effect only shows up at nonleading order in 1/N_c, but requires specific
unnatural parity contributions in the effective Lagrangian that restore
renormalization group invariance.Comment: 56 page
Application of asymptotic expansions of maximum likelihood estimators errors to gravitational waves from binary mergers: the single interferometer case
In this paper we describe a new methodology to calculate analytically the
error for a maximum likelihood estimate (MLE) for physical parameters from
Gravitational wave signals. All the existing litterature focuses on the usage
of the Cramer Rao Lower bounds (CRLB) as a mean to approximate the errors for
large signal to noise ratios. We show here how the variance and the bias of a
MLE estimate can be expressed instead in inverse powers of the signal to noise
ratios where the first order in the variance expansion is the CRLB. As an
application we compute the second order of the variance and bias for MLE of
physical parameters from the inspiral phase of binary mergers and for noises of
gravitational wave interferometers . We also compare the improved error
estimate with existing numerical estimates. The value of the second order of
the variance expansions allows to get error predictions closer to what is
observed in numerical simulations. It also predicts correctly the necessary SNR
to approximate the error with the CRLB and provides new insight on the
relationship between waveform properties SNR and estimation errors. For example
the timing match filtering becomes optimal only if the SNR is larger than the
kurtosis of the gravitational wave spectrum
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