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Register a new userQuaternionic eigenvalue problem
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Authors
S. De Leo
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We discuss the (right) eigenvalue equation for $mathbb{H}$, $mathbb{C}$ and $mathbb{R}$ linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows to translate the quaternionic problem into an {em equivalent} real or complex counterpart. Interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics.
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An eigenvalue problem relevant for non-linear sigma model with singular metric is considered. We prove the existence of a non-degenerate pure point spectrum for all finite values of the size R of the system. In the infrared (IR) regime (large R) the eigenvalues admit a power series expansion around IR critical point Rtoinfty. We compute high order coefficients and prove that the series converges for all finite values of R. In the ultraviolet (UV) limit the spectrum condenses into a continuum spectrum with a set of residual bound states. The spectrum agrees nicely with the central charge computed by the Thermodynamic Bethe Ansatz method
The quaternionic Hopf surface, HL, is associated with a non-compact moduli space, ML, of stable holomorphic SL(2,C) bundles. ML is open in MLc, the corresponding compact moduli space of holomorphic SL(2,C) bundles, and naturally fibers over an open set of the quaternionic projective line HP^1. We pull back to ML natural locally conformal kaehler and hyperkaehler structures from MLc, and lift natural sub-pseudoriemannian and optical structures from HP^1. Unexpectedly, the holomorphic maps connecting these structures solve the the classical Dirac-Higgs equations of the unbroken Standard Model. These equations include: all observed fermionic and bosonic fields of all three generations with the correct color, weak isospin, and hypercharge values; a Higgs field coupling left and right fermion fields; and a pp-wave gravitational metric. We hypothesize that physics is essentially the geometry of ML, both algebraic (quantum) and differential (classical). We further show that the Yang-Mills equations with fermionic currents also naturally emerge, along with an induced action on the ML structure sheaf equivalent to the time-evolution operator of the associated quantum field theory.
Eigenvalue problems for linear differential equations, such as time-independent Schrodinger equations, can be generalized to eigenvalue problems for nonlinear differential equations. In the nonlinear context a separatrix plays the role of an eigenfunction and the initial conditions that give rise to the separatrix play the role of eigenvalues. Previously studied examples of nonlinear differential equations that possess discrete eigenvalue spectra are the first-order equation $y(x)=cos[pi xy(x)]$ and the first, second, and fourth Painleve transcendents. It is shown here that the differential equations for the first and second Painleve transcendents can be generalized to large classes of nonlinear differential equations, all of which have discrete eigenvalue spectra. The large-eigenvalue behavior is studied in detail, both analytically and numerically, and remarkable new features, such as hyperfine splitting of eigenvalues, are described quantitatively.
We consider the single eigenvalue fluctuations of random matrices of general Wigner-type, under a one-cut assumption on the density of states. For eigenvalues in the bulk, we prove that the asymptotic fluctuations of a single eigenvalue around its classical location are Gaussian with a universal variance which agrees with the GOE and GUE cases. Our method is based on a dynamical approach to mesoscopic linear spectral statistics which reduces their behavior on short scales to that on larger scales. We prove a central limit theorem for linear spectral statistics on larger scales via resolvent techniques and show that for certain classes of test functions, the leading order contribution to the variance is universal, agreeing with the GOE/GUE cases.
We show that Tsirelsons problem concerning the set of quantum correlations and Connes embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchbergs QWEP conjecture) are essentially equivalent. Specifically, Tsirelsons problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the set of correlations between commuting C*-algebras. Connes embedding problem asks whether any separable II$_1$ factor is a subfactor of the ultrapower of the hyperfinite II$_1$ factor. We show that an affirmative answer to Connes question implies a positive answer to Tsirelsons. Conversely, a positve answer to a matrix valued version of Tsirelsons problem implies a positive one to Connes problem.
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