Marcel Novaes
Integration of polynomials over the classical groups of unitary, orthogonal and symplectic matrices can be reduced to basic building blocks known as Weingarten functions. We present an elementary derivation of these functions.
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Alignment in products of positive matrices and Birkhoff's contraction formula: an elementary direct derivation
本文通过直接分析给出正矩阵乘积的尖锐错位界，从而初等地导出对齐性质和Birkhoff收缩公式。
We give an elementary derivation of the Montgomery phase formula for the motion of an Euler top, using only basic facts about the Euler equation and parallel transport on the 2-sphere (whose holonomy is seen to be responsible for the geometric phase). We also give an approximate geometric interpretation of the geometric phase for motions starting close to an unstable equilibrium point.
We review several techniques and ideas initiated by a remarkable work by Spencer [26], used and further developed in numerous subsequent researches. We also describe a relatively short and elementary derivation of the spectral and strong dynamical Anderson localization from the fixed-energy analysis of the Green functions, obtained either by the Multi-Scale Analysis (MSA) or by the Fractional-Moment Method (FMM). This derivation goes in the same direction as the Simon--Wolf criterion [28], but provides quantitative estimates, applies also to multi-particle models and, combined with a simplifie…
A multi-parafermion basis of states for the Z_k parafermionic models is derived. Its generating function is constructed by elementary steps. It corresponds to the Andrews multiple-sum which enumerates partitions whose parts separated by the distance k-1 differ by at least 2. Two analogous bases are derived for graded parafermions; one of these entails a new expression for their fermionic characters.
We give a derivation of the dispersion law $ε(p)=\hbar^2p^2/2m +\widetilde V(p)-\widetilde V(0)$, where $\widetilde V(p)$ is the Fourier transform of the pair interaction potential $V(r)$. (The interaction between particles $x_i$ and $x_j$ is $V(|x_i-x_j|)$.)
We develop a powerful framework to calculate expectation values of polynomials and moments on compact Lie groups based on elementary representation-theoretic arguments and an integration by parts formula. In the setting of lattice gauge theory, we generalize expectation value formulas for products of Wilson loops by Chatterjee and Jafarov to arbitrary compact Lie groups, and study explicit examples for many classical compact Lie groups and the exceptional Lie group $G_2$. Extending classical results by Collins and Lévy, we use our framework to derive expectation value formulas of polynomials o…
Current quantum theories of an elementary free particle assume unitary space inversion and anti-unitary time reversal operators. In so doing robust classes of possible theories are discarded. The present work shows that consistent theories can be derived through a strictly deductive development from the principle of relativistic invariance and position covariance, also with anti-unitary space inversion and unitary time reversal operators. In doing so the class of possible consistent theories is extended for positive but also zero mass particles. In particular, consistent theories for a Klein-G…
An elementary 'quantum-mechanical' derivation of the conditions for a system of functions to form a Reisz basis of a Hilbert space on a finite interval is presented.
Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the framework of biorthogonal systems, to obtain an alternate derivation. The central factorial numbers play key roles in both derivations.
We provide an elementary and self-contained derivation of formulae for products and ratios of characteristic polynomials from classical groups using classical results due to Weyl and Littlewood.
In this work we provide an elementary derivation of the indefinite spin groups in low-dimensions. Our approach relies on the isomorphism of Cl(p+1, q+1) to the algebra 2x2 matrices with entries in Cl(p,q), simple properties of Kronecker products, characterization of when an even dimensional real (resp. complex) matrix represents a complex (resp. quaternionic) linear transformation, and basic aspects of the isomorphism between M(4, R) and the quaternion tensor product. Of these the last is arguably the most vital. Among other things it yields naturally, a surprisingly ubiquitous role for an alt…
In this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue of a $(N\times N)$ random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE) and by suitably adapting a method of orthogonal polynomials developed by Gross and Matytsin in the context of Yang-Mills theory in two dimensions, we provide a rather simple derivation of the Tracy-Widom law for GUE. Our derivation is based on the elementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations. As an added bonus, this method also all…
The purpose of this note is to give an elementary derivation of a lower bound on the relativistic Thomas-Fermi-Weizsäcker-Dirac functional of Thomas-Fermi type and to apply it to get an upper bound on the excess charge of this model.
The usual action integral of classical electrodynamics is derived starting from Lanczos's electrodynamics -- a pure field theory in which charged particles are identified with singularities of the homogeneous Maxwell's equations interpreted as a generalization of the Cauchy-Riemann regularity conditions from complex to biquaternion functions of four complex variables. It is shown that contrary to the usual theory based on the inhomogeneous Maxwell's equations, in which charged particles are identified with the sources, there is no divergence in the self-interaction so that the mass is finite, …
Tetrahedral $L$-operators, tensor Schur polynomials and $q$-deformed loop elementary symmetric functions
本文基于四面体L-算子研究三维分区函数，提出可表示为张量Schur多项式类乘积的分区函数，推导Schur多项式洗牌公式及多个恒等式的统一，并展示其在多物种完全非对称排斥过程稳态中的应用，同时确定q-变形环初等对称函数的显式形式。
We discuss some fundamental properties of the XXZ spin chain, which are important in the algebraic Bethe-ansatz derivation for the multiple-integral representations of the spin-s XXZ correlation function with an arbitrary product of elementary matrices. For instance, we construct Hermitian conjugate vectors in the massless regime and introduce the spin-s Hermitian elementary matrices.
Orthogonal signed-distance coordinates and vector calculus near evolving curves and surfaces
本文基于符号距离函数推导了演化曲线与曲面边界层的正交坐标系，整理了向量微积分恒等式，并提供代码以在广泛物理系统的边界层渐近推导中一致地考虑几何效应。
Averages of products of characteristic polynomials and the law of real eigenvalues for the real Ginibre ensemble
本文利用特征多项式乘积的平均，基础推导了刻画大尺寸极限下实Ginibre系综实特征值律的Borodin-Sinclair-Forrester-Nagao Pfaffian点过程，揭示了实特征值统计的隐藏辛对称性及K点相关函数的渐近精确积分表示。
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