Rotational invariance

Summary

In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. Mathematics Functions For example, the function :f(x,y) = x^2 + y^2 is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any angle θ :x' = x \cos \theta - y \sin \theta :y' = x \sin \theta + y \cos \theta the function, after some cancellation of terms, takes exactly the same form :f(x',y') = {x}^2 + {y}^2 The rotation of coordinates can be expressed using matrix form using the rotation matrix, :\begin{bmatrix} x' \ y' \ \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \ \end{bmatrix}\begin{bmatrix} x \ y \ \end{bmatrix}. or symbolically x′ = Rx. Symbolically, the rotation invariance of a real-valued fu

Official source

https://en.wikipedia.org/wiki/Rotational_invariance

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Asymptotic freedom, Haldane gap and edge states of SU(N) spin chains

Samuel Gozel

In this thesis we study various one-dimensional quantum spin systems with SU(2) and SU(N) symmetry. We investigate the short-distance behavior of the SU(2) Heisenberg model in the limit of large spin and show that there exists an extended regime where perturbation theory, in the form of spin-wave theory, can be successfully applied. The reason for this perturbative regime stems from the asymptotic freedom of the nonlinear sigma model onto which the Heisenberg model can be mapped. When considering observables which respect the rotational invariance of the model, we observe a cancellation of infrared divergences in the perturbative expansions, leading to a meaningful description of correlation functions. We then turn to the study of SU(N) models. Building on the representation theory of the SU(N) group and on the matrix product state (MPS) formalism, we introduce a generic method to construct Affleck-Kennedy-Lieb-Tasaki (AKLT) models having edge states described by any self-conjugate irreducible representation (irrep) of SU(N). A simple example is given by a spin-1 AKLT model having spin-1 edge states. The phase transition between this model and the original AKLT model is shown to be continuous and to correspond to a topological phase transition described by the SU(2) level-1 Wess-Zumino-Witten conformal field theory universality class. In addition we study an SU(3) AKLT model for the 3-box symmetric irrep at each site. We demonstrate that the edge states are adjoint edge irreps, we extract its correlation length and provide a useful construction as an optimal MPS. Finally, we develop a density matrix renormalization group algorithm based on standard Young tableaus and subduction coefficients to make full use of the non-abelian symmetry and to investigate the SU(3) Heisenberg model with 3-box symmetric irrep at each site. We show that the model has a finite gap above the singlet ground state, in agreement with an extension of the Haldane conjecture to SU(3) chains in the fully symmetric irreps. We also argue that there are five branches of elementary excitations living in four different irreps, each of which is gapped.

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Hybrid inflation in the complex plane

Valerie Fiona Domcke, Kohei Kamada

Supersymmetric hybrid inflation is an exquisite framework to connect inflationary cosmology to particle physics at the scale of grand unification. Ending in a phase transition associated with spontaneous symmetry breaking, it can naturally explain the generation of entropy, matter and dark matter. Coupling F-term hybrid inflation to soft supersymmetry breaking distorts the rotational invariance in the complex inflaton plane - an important fact, which has been neglected in all previous studies. Based on the delta N formalism, we analyze the cosmological perturbations for the first time in the full two-field model, also taking into account the fast-roll dynamics at and after the end of inflation. As a consequence of the two-field nature of hybrid inflation, the predictions for the primordial fluctuations depend not only on the parameters of the Lagrangian, but are eventually fixed by the choice of the inflationary trajectory. Recognizing hybrid inflation as a two-field model resolves two shortcomings often times attributed to it: the fine-tuning problem of the initial conditions is greatly relaxed and a spectral index in accordance with the PLANCK data can be achieved in a large part of the parameter space without the aid of supergravity corrections. Our analysis can be easily generalized to other (including large-field) scenarios of inflation in which soft supersymmetry breaking transforms an initially single-field model into a multi-field model.

Iop Publishing Ltd2014

We address the problem of reconstructing scalar and vector functions from non-uniform data. The reconstruction problem is formulated as a minimization problem where the cost is a weighted sum of two terms. The first data term is the quadratic measure of goodness of fit, whereas the second regularization term is a smoothness functional. We concentrate on the case where the later is a semi-norm involving differential operators. We are interested in a solution that is invariant with respect to scaling and rotation of the input data. We first show that this is achieved whenever the smoothness functional is both scale- and rotation-invariant. In the first part of the thesis, we address the scalar problem. An elegant solution having the above mentioned invariant properties is provided by Duchon's method of thin-plate splines. Unfortunately, the solution involves radial basis functions that are poorly conditioned and becomes impractical when the number of samples is large. We propose a computationally efficient alternative where the minimization is carried out within the space of uniform B-splines. We show how the B-spline coefficients of the solution can be obtained by solving a well-conditioned, sparse linear system of equations. By taking advantage of the refinable nature of B-splines, we devise a fast multiresolution-multigrid algorithm. We demonstrate the effectiveness of this method in the context of image processing. Next, we consider the reconstruction of vector functions from projected samples, meaning that the input data do not contain the full vector values, but only some directional components. We first define the rotational invariance and the scale invariance of a vector smoothness functional, and then characterize the complete family of such functionals. We show that such a functional is composed of a weighted sum of two sub-functionals: (i) Duchon's scalar semi-norm applied on the divergence field; (ii) and the same applied to each component of the rotational field. This forms a three-parameter family, where the first two are the Duchon's order of the above sub-functionals, and the third is their relative weight. Our family is general enough to include all vector spline formulations that have been proposed so far. We provide the analytical solution for this minimization problem and show that the solution can be expressed as a weighted sum of vector basis functions, which we call the generalized vector splines. We construct the linear system of equations that yields the required weights. As in the scalar case, we also provide an alternative B-spline solution for this problem, and propose a fast multigrid algorithm. Finally, we apply our vector field reconstruction method to cardiac motion field recovery from ultrasound pulsed wave Doppler data, and demonstrate its clinical potential.

EPFL2005

Asymptotic freedom, Haldane gap and edge states of SU(N) spin chains

Samuel Gozel

EPFL2020

EPFL2005