# Several integrals of quaternionic field on hyperbolic matrix space

Some integrals of matrix spaces over a quaternionic field have been calculated in this work. The associated volume of hyperbolic matrix spaces over a quaternionic field has also been calculated by making use of these integrals, and it is of great significance in calculating related kernel functions of these spaces.

**Comments:**21 pages

## Similar Publications

Totally Asymmetric Simple Exclusion Process (TASEP) on $\mathbb{Z}$ is one of the classical exactly solvable models in the KPZ universality class. We study the "slow bond" model, where TASEP on $\mathbb{Z}$ is imputed with a slow bond at the origin. The slow bond increases the particle density immediately to its left and decreases the particle density immediately to its right. Read More

The Kundu-Eckhaus equation is a nonlinear partial differential equation which seems in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. In spite of its importance, exact solution to this nonlinear equation are rarely found in literature. In this work, we solve this equation and present a new approach to obtain the solution by means of the combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). Read More

Using diagrammatic techniques, we provide an explicit proof of the single ring theorem, including the recent extension for the correlation function built out of left and right eigenvectors of a non-Hermitian matrix. We present the operational formalism allowing to map mutually the two distinct areas of free random variables: Hermitian positive definite operators and non-normal R-diagonal operators, realized as the large size limit of biunitarily invariant random matrices. Read More

A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other not. A complete Lie symmetry classification, including a number of the cases characterised being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications tumour growth). Read More

In this note we aim to characterize the cylindrical Wigner measures associated to regular quantum states in the Weyl C*-algebra of canonical commutation relations. In particular, we provide conditions at the quantum level sufficient to prove the concentration of all the corresponding cylindrical Wigner measures as Radon measures on suitable topological vector spaces. The analysis is motivated by variational and dynamical problems in the semiclassical study of bosonic quantum field theories. Read More

We prove that the ground space projections of a subspace of energy operators in a matrix *-algebra are the greatest projections of the algebra under certain operator cone constraints. The lattice of ground space projections being coatomistic, we also discuss its maximal elements as building blocks. We demonstrate the results with (commutative) two-local three-bit Hamiltonians. Read More

We discuss the possibility of protecting the state of a quantum system that goes through noise by measurements and operations before and after the noise process. We extend our previous result on nonexistence of "truly quantum" protocols that protect an unknown qubit state against the depolarizing noise better than "classical" ones [Phys. Rev. Read More

Time-independent gauge transformations are implemented in the canonical formalism by the Gauss law which is not covariant. The covariant form of Gauss law is conceptually important for studying asymptotic properties of the gauge fields. For QED in $3+1$ dimensions, we have developed a formalism for treating the equations of motion (EOM) themselves as constraints, that is, constraints on states using Peierls' quantization. Read More

In this paper we extend the approach of orthogonal polynomials for extreme value calculations of Hermitian random matrices, developed by Nadal and Majumdar [1102.0738], to normal random matrices and 2D Coulomb gases in general. Firstly, we show that this approach provides an alternative derivation of results in the literature. Read More

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter space dS$_4$ and construct a smooth and spatially homogeneous magnetic solution to the Yang-Mills equations. Slicing dS$_4$ as ${\mathbb R}\times S^3$, via an SU(2)-equivariant ansatz we reduce the Yang-Mills equations to ordinary matrix differential equations and further to Newtonian dynamics in a double-well potential. Its local maximum yields a Yang-Mills solution whose color-magnetic field at time $\tau\in{\mathbb R}$ is given by $\tilde{B}_a=-\frac12 I_a/(R^2\cosh^2\!\tau)$, where $I_a$ for $a=1,2,3$ are the SU(2) generators and $R$ is the de Sitter radius. Read More