Symplectic Matrix Research Articles

Commutators of skew-involutions

Let $\mathrm{SL}_n(\mathbb{F})$ be the group of all $n\times n$ matrices over a field $\mathbb{F}$ with determinant $1$. Denote by $I$ ($I_n$) the ($n\times n$) identity matrix. A matrix $A$ is called skew-involution if $A^2=-I$. It is proved that every matrix in $\mathrm{SL}_{2n}(\mathbb{F})$ is a product of at most three commutators of skew-involutions if $\mathbb{F}\ne\mathbb{Z}_3$ and $\mathrm{SL}_{2n}(\mathbb{F})\ne\mathrm{SL}_{2}(\mathbb{Z}_2)$, and at most four commutators of skew-involutions if $\mathbb{F}=\mathbb{Z}_3$ and $n>1$. Every complex symplectic matrix is a product of two commutators of complex symplectic skew-involutions, and every real symplectic matrix is a product of not more than four commutators of real symplectic skew-involutions.

Revealing symmetries in quantum computing for many-body systems

Abstract We develop a method to deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan–Wigner form in which they can act on multi-qubit states. Symmetries, such as point-group symmetries in molecules, are apparent in the standard second quantized form of the Hamiltonian. They are, however, masked when the Hamiltonian is translated into a Pauli matrix representation required for its operation on qubits. To reveal these symmetries we prove a general theorem that provides a straightforward method to calculate the transformation of Pauli tensor strings under symmetry operations. They are a subgroup of the Clifford group transformations and induce a corresponding group representation inside the symplectic matrices. We finally give a simplified derivation of an affine qubit encoding scheme which allows for the removal of qubits due to Boolean symmetries and thus reduces effort in quantum computations for many-body systems.

Dynamic continualization of mechanical metamaterials with quasi-periodic microstructure.

This article focuses on characterizing a class of quasi-periodic metamaterials created through the repeated arrangement of an elementary cell in a fixed direction. The elementary cell consists of two building blocks made of elastic materials and arranged according to the generalized Fibonacci sequence, giving rise to a quasi-periodic finite microstructure, also called Fibonacci generation. By exploiting the transfer matrix method, the frequency band structure of selected periodic approximants associated with the Fibonacci superlattice, i.e. the layered quasi-periodic metamaterial, is determined. The self-similarity of the frequency band structure is analysed by means of the invariants of the symplectic transfer matrix as well as the transmission coefficients of the finite clusters of Fibonacci generations. A high-frequency continualization scheme is then proposed to identify integral-type or gradient-type non-local continua. The frequency band structures obtained from the continualization scheme are compared with those derived from the Floquet-Bloch theory to validate the proposed scheme. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1).'

A Maslov-Type Index in Dimension 2

In this article, we define an index of the Maslov type for paths of 2×2 orthogonal symplectic matrices. The starting point is an arbitrary 2×2 orthogonal symplectic matrix rather than the identity matrix. We use this index to explain the geometric intersection number of a pair of Lagrangian paths and compare it with the Cappell–Lee–Miller index.

Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson

In this tutorial, we summarize four important matrix decompositions commonly used in quantum optics, namely the Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson decompositions. The first two of these decompositions are specialized versions of the singular-value decomposition when applied to symmetric or symplectic matrices. The third factors any symplectic matrix in a unique way in terms of matrices that belong to different subgroups of the symplectic group. The last one instead gives the symplectic diagonalization of real, positive definite matrices of even size. While proofs of the existence of these decompositions exist in the literature, we review explicit constructions to implement these decompositions using standard linear algebra packages and functionalities such as singular-value, polar, Schur, and QR decompositions, and matrix square roots and inverses.

Almost Commuting Scheme of Symplectic Matrices and Quantum Hamiltonian Reduction

Losev introduced the scheme X of almost commuting elements (i.e., elements commuting upto a rank one element) of g=sp(V)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}=\\mathfrak {sp}(V)$$\\end{document} for a symplectic vector space V and discussed its algebro-geometric properties. We construct a Lagrangian subscheme Xnil\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{nil}$$\\end{document} of X and show that it is a complete intersection of dimension dim(g)+12dim(V)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {dim}(\\mathfrak {g})+\\frac{1}{2}\ ext {dim}(V)$$\\end{document} and compute its irreducible onents. We also study the quantum Hamiltonian reduction of the algebra D(g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {D}(\\mathfrak {g})$$\\end{document} of differential operators on the Lie algebra g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}$$\\end{document} tensored with the Weyl algebra with respect to the action of the symplectic group, and show that it is isomorphic to the spherical subalgebra of a certain rational Cherednik algebra of Type C. We contruct a category Cc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}_c$$\\end{document} of D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {D}$$\\end{document}-modules whose characteristic variety is contained in Xnil\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{nil}$$\\end{document} and construct an exact functor from this category to the category O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}$$\\end{document} of the above rational Cherednik algebra. Simple objects of the category Cc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}_c$$\\end{document} are mirabolic analogs of Lusztig’s character sheaves.

Orthogonal-symplectic matrices and their parametric representation

The method of computing the parametric representation of an even orthogonal symplectic matrix is considered. The dimension of the family of such matrices is calculated. The general structure of matrices of small even dimensions up to 8 is discussed in detail. Theorem on the structure of a skew symmetric matrix generating a generic orthogonal symplectic matrix is proven. The problem of constructing an orthogonal symplectic matrix of dimension 4 by a given vector is solved. The application of this transformation to the study of families of periodic solutions to an autonomous Hamiltonian system with two degrees of freedom is discussed.

Wigner Analysis of Operators. Part II: Schrödinger Equations

We study the phase-space concentration of the so-called generalized metaplectic operators whose main examples are Schrödinger equations with bounded perturbations. To reach this goal, we perform a so-called A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document}-Wigner analysis of the previous equations, as started in Part I, cf. Cordero and Rodino (Appl Comput Harmon Anal 58:85–123, 2022). Namely, the classical Wigner distribution is extended by considering a class of time–frequency representations constructed as images of metaplectic operators acting on symplectic matrices A∈Sp(2d,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}\\in Sp(2d,\\mathbb {R})$$\\end{document}. Sub-classes of these representations, related to covariant symplectic matrices, reveal to be particularly suited for the time–frequency study of the Schrödinger evolution. This testifies the effectiveness of this approach for such equations, highlighted by the development of a related wave front set. We first study the properties of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document}-Wigner representations and related pseudodifferential operators needed for our goal. This approach paves the way to new quantization procedures. As a byproduct, we introduce new quasi-algebras of generalized metaplectic operators containing Schrödinger equations with more general potentials, extending the results contained in the previous works (Cordero et al. in J Math Pures Appl 99(2):219–233, 2013, J Math Phys 55(8):081506, 2014).

Characteristic polynomials of orthogonal and symplectic random matrices, Jacobi ensembles & L-functions

Starting from Montgomery’s conjecture, there has been a substantial interest on the connections of random matrix theory and the theory of L-functions. In particular, moments of characteristic polynomials of random matrices have been considered in various works to estimate the asymptotics of moments of L-function families. In this paper, we first consider joint moments of the characteristic polynomial of a symplectic random matrix and its second derivative. We obtain the asymptotics, along with a representation of the leading order coefficient in terms of the solution of a Painlevé equation. This gives us the conjectural asymptotics of the corresponding joint moments over families of Dirichlet L-functions. In doing so, we compute the asymptotics of a certain additive Jacobi statistic, which could be of independent interest in random matrix theory. Finally, we consider a slightly different type of joint moment that is the analogue of an average considered over [Formula: see text] in various works before. We obtain the asymptotics and the leading order coefficient explicitly.

Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices

We investigate the joint moments of derivatives of characteristic polynomials over the unitary symplectic group Sp(2N) and the orthogonal ensembles SO(2N) and O−(2N) . We prove asymptotic formulae for the joint moments of the n 1th and n 2th derivatives of the characteristic polynomials for all three matrix ensembles. Our results give two explicit formulae for each of the leading order coefficients, one in terms of determinants of hypergeometric functions and the other as combinatorial sums over partitions. We use our results to put forward conjectures on the joint moments of derivatives of L-functions with symplectic and orthogonal symmetry.

A metaplectic perspective of uncertainty principles in the linear canonical transform domain

We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.

Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics.

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton's equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton's equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton's and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton's, variational, and Schrödinger equations are described. Employing the Hénon-Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton's equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes.

Excursus on modulation spaces via metaplectic operators and related time-frequency representations

We provide a comprehensive overview of the theoretical framework surrounding modulation spaces and their characterizations, particularly focusing on the role of metaplectic operators and time-frequency representations. We highlight the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. In particular, this work provides new characterizations via the submanifold of shift-invertible symplectic matrices. Similar results hold for the Wiener amalgam spaces.

First order structure-preserving perturbation theory for eigenvalues of symplectic matrices: Part II

The first order eigenvalue perturbation theory for structure-preserving perturbations of real or complex symplectic matrices started in [21] is continued and completed. While in [21] all eigenvalues were considered that qualitatively behave similarly under structure-preserving and structure-ignoring perturbations, the focus of this paper is on one particular case when the behavior under structure-preserving perturbations differs significantly from the one under structure-ignoring perturbations. This case occurs for the exceptional eigenvalues ±1 and is caused by restrictions in the possible Jordan structure of symplectic matrices. The main result gives asymptotic expansions of specific perturbed eigenvalues that are not yet covered by the results from [21]. While the main theorem requires that the rank of the perturbation is equal to the rank of its first order term, it is also discussed what can be said if this hypothesis is not satisfied.

Random Matrix Statistics in Propagating Correlation Fronts of Fermions.

We theoretically study propagating correlation fronts in noninteracting fermions on a one-dimensional lattice starting from an alternating state, where the fermions occupy every other site. We find that, in the long-time asymptotic regime, all the moments of dynamical fluctuations around the correlation fronts are described by the universal correlation functions of Gaussian orthogonal and symplectic random matrices at the soft edge. Our finding here sheds light on a hitherto unknown connection between random matrix theory and correlation propagation in quantum dynamics.

Engineering Orbital Angular Momentum in Structured Beams in General Astigmatic Systems via Symplectic Matrix Approach

We studied theoretically and experimentally the propagation of structured Laguerre–Gaussian (sLG) beams through an optical system with general astigmatism based on symplectic ABCD transforms involving geometry of the second-order intensity moments symplectic matrices. The evolution of the coordinate submatrix ellipses accompanying the transformation of intensity patterns at different orientations of the cylindrical lens was studied. It was found that the coordinate submatrix W and the twistedness submatrix M of the symplectic matrix P degenerate in the astigmatic sLG beam with simple astigmatism, which sharply reduces the number of degrees of freedom, while general astigmatism removes the degeneracy. Nevertheless, degeneracy entails a simple relationship between the coordinate element Wxy and the twistedness elements Mxy and Myx of the submatrix M, which greatly simplifies the measurement of the total orbital angular momentum (OAM), reducing the full cycle of measurements of the Hermite–Gaussian (HG) mode spectrum (amplitudes and phases) of the structured beam to the only measurement of the intensity moment. Moreover, we have shown that Fourier transform by a spherical lens enables us to suppress the astigmatic OAM component and restore the original free-astigmatic sLG beam structure. However, with further propagation, the sLG beam restores its astigmatic structure while maintaining the maximum OAM.

Multivariate normal approximation for traces of orthogonal and symplectic matrices

Multivariate normal approximation for traces of orthogonal and symplectic matrices

Factorization of holomorphic matrices and Kazhdan's property (T)

In this article we deduce some algebraic properties for the group Sp2n(O(X)) of holomorphic symplectic matrices on a Stein space X: holomorphic factorization, exponential factorization, and Kazhdan's property (T). In holomorphic factorization we combine a recent result of the third author and K-theory tools to give explicit bounds for the case when X is one-dimensional or two-dimensional. Next we use them to find bounds for exponential factorization. As a further application, we show that the elementary symplectic group Ep2n(O(X)) admits Kazhdan's property (T).

A precise compensation algorithm for visual measurement in non-uniform refractive index environment based on Hamiltonian ray tracing method

In complex large-scale industrial environments, light rays no longer propagate in a straight line due to the influence of non-uniform refractive index, which will lead to large measurement errors in visual measurement. In order to accurately compensate the error, this paper firstly quantitatively analyzes the influence of non-uniform refractive index distribution on visual measurement. Then the symplectic matrix ray-tracing method of Hamiltonian optics is used to reconstruct spatial light in the reverse direction with known light propagation end point and spatial refractive index distribution. Thus the traditional model of light propagation along a straight line in visual measurement can be changed and the measurement error introduced by non-uniform refractive index field can be corrected. Finally, the proposed Hamiltonian ray tracing method was verified through the spatial non-uniform refractive index field reconstructed by Background Oriented Schlieren(BOS) method. The experimental results verified the effectiveness and feasibility of this method.

Symplectic partially hyperbolic automorphisms of 6-torus

We study topological properties of automorphisms of a 6-dimensional torus T6 generated by integer matrices with simple eigenvalues being symplectic with respect to either the standard symplectic structure in R6 or a nonstandard symplectic structure given by an integer skew-symmetric non-degenerate matrix. Such a symplectic matrix generates a partially hyperbolic automorphism of the torus, if its eigenvalues lie both outside and on the unit circle. We study transitive and decomposable cases possible here and present a classification in both cases.