Minimal Geodesics Research Articles

Minimal geodesics of pencils of pairs of projections

Minimal geodesics of pencils of pairs of projections

Circuit complexity for coherent-thermal states in bosonic string theory

In this paper, we first construct thermofield double states for bosonic string theory in the light-cone gauge. We then obtain a coherent-thermal string state and a thermal-coherent string state. We use the covariance matrix approach to calculate the circuit complexity of coherent-thermal string states. In this approach, we generate the optimal geodesics by a horizontal string generator, and then obtain the circuit complexity using the length of the minimal geodesics in the group manifold.

The set of partial isometries as a quotient Finsler space

A known general program, designed to endow the quotient space UA/UB of the unitary groups UA, UB of the C∗ algebras B⊂A with an invariant Finsler metric, is applied to obtain a metric for the space I(H) of partial isometries of a Hilbert space H. I(H) is a quotient of the unitary group of B(H)×B(H), where B(H) is the algebra of bounded linear operators in H. Under this program, the solution of a linear best approximation problem leads to the computation of minimal geodesics in the quotient space. We find solutions of this best approximation problem, and study properties of the minimal geodesics obtained.

Blade Products and Angles Between Subspaces

Principal angles are used to define an angle bivector of subspaces, which fully describes their relative inclination. Its exponential is related to the Clifford geometric product of blades, gives rotors connecting subspaces via minimal geodesics in Grassmannians, and decomposes giving Pl\"ucker coordinates, projection factors and angles with various subspaces. This leads to new geometric interpretations for this product and its properties, and to formulas relating other blade products (scalar, inner, outer, etc., including those of Grassmann algebra) to angles between subspaces. Contractions are linked to an asymmetric angle, while commutators and anticommutators involve hyperbolic functions of the angle bivector, shedding new light on their properties.

Structural Models Based on Minimal Surfaces and Geodesics

This article presents the results of research carried out with respect to the geometric, formal and structural adaptation of minimal surfaces. These surfaces were discretized into strips developable on geodesic curves, and then used for the construction of timber gridshells. For this project, both physical and virtual models derived from the same geometric models were used. The objective was to demonstrate the validity of the use of models and the transformation that modelling is undergoing due to the use of digital tools, both software and hardware. These include, on the one hand, drawing and analysis software and, on the other, digitally controlled fabrication tools. This research focuses specifically on the design and construction of the Scherk Pavilion, a space where the results of various experiments in which the common factor was the use of models was transferred to a real scale.

Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

Let H be a reproducing kernel Hilbert space of functions on a set X. We study the problem of finding a minimal geodesic of the Grassmann manifold of H that joins two subspaces consisting of functions which vanish on given finite subsets of X. We establish a necessary and sufficient condition for existence and uniqueness of geodesics, and we then analyze it in examples. We discuss the relation of the geodesic distance with other known metrics when the mentioned finite subsets are singletons. We find estimates on the upper and lower eigenvalues of the unique self-adjoint operators which define the minimal geodesics, which can be made more precise when the underlying space is the Hardy space. Also for the Hardy space we discuss the existence of geodesics joining subspaces of functions vanishing on infinite subsets of the disk, and we investigate when the product of projections onto this type of subspaces is compact.

The tunneling effect for Schr\xf6dinger operators on a vector bundle

In the semiclassical limit hbar rightarrow 0, we analyze a class of self-adjoint Schrödinger operators H_hbar = hbar ^2 L + hbar W + Vcdot {mathrm {id}}_{mathscr {E}} acting on sections of a vector bundle {mathscr {E}} over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m^1,ldots m^r in M, called potential wells. Using quasimodes of WKB-type near m^j for eigenfunctions associated with the low lying eigenvalues of H_hbar , we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension ell + 1. This dimension ell determines the polynomial prefactor for exponentially small eigenvalue splitting.

Minimal Geodesics of the Isosceles Three Body Problem

The isosceles three-body problem with nonnegative energy is studied from a variational point of view based on the Jacobi–Maupertuis metric. The solutions are represented by geodesics in the two-dimensional configuration space. Since the metric is singular at collisions, an approach based on the theory of length spaces is used. This provides an alternative to the more familiar approach based on the principle of least action. The emphasis is on the existence and properties of minimal geodesics, that is, shortest curves connecting two points in configuration space. For any two points, even singular points, a minimal geodesic exists and is nonsingular away from the endpoints. For the zero energy case, it is possible to use knowledge of the behavior of the flow on the collision manifold to see that certain solutions must be minimal geodesics. In particular, the geodesic corresponding to the collinear homothetic solution turns out to be a minimal for certain mass ratios.

Quantum complexity of time evolution with chaotic Hamiltonians

We study the quantum complexity of time evolution in large-N chaotic systems, with the SYK model as our main example. This complexity is expected to increase linearly for exponential time prior to saturating at its maximum value, and is related to the length of minimal geodesics on the manifold of unitary operators that act on Hilbert space. Using the Euler-Arnold formalism, we demonstrate that there is always a geodesic between the identity and the time evolution operator e−iHt whose length grows linearly with time. This geodesic is minimal until there is an obstruction to its minimality, after which it can fail to be a minimum either locally or globally. We identify a criterion — the Eigenstate Complexity Hypothesis (ECH) — which bounds the overlap between off- diagonal energy eigenstate projectors and the k-local operators of the theory, and use it to argue that the linear geodesic will at least be a local minimum for exponential time. We show numerically that the large-N SYK model (which is chaotic) satisfies ECH and thus has no local obstructions to linear growth of complexity for exponential time, as expected from holographic duality. In contrast, we also study the case with N = 2 fermions (which is integrable) and find short-time linear complexity growth followed by oscillations. Our analysis relates complexity to familiar properties of physical theories like their spectra and the structure of energy eigenstates and has implications for the hypothesized computational complexity class separations PSPACE BQP/poly and PSPACE BQSUBEXP/subexp, and the “fast-forwarding” of quantum Hamiltonians.

Isoperimetric rigidity and distributions of 1-Lipschitz functions

We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our result can be considered as a variant of Cheeger-Gromoll's splitting theorem and also of Cheng's maximal diameter theorem. As an application, we obtain a new isometric splitting theorem for a complete weighted Riemannian manifold with a positive Bakry-Émery Ricci curvature.

Loos symmetric cones

In this paper we consider Loos symmetric spaces on an open cone $$\Omega $$ in the Banach space setting and develop the foundations of a geometric theory based on the (modified) Loos axioms for such cones. In particular we establish exponential and log functions that exhibit many desirable features reminiscent of those of the exponential function from the space of self-adjoint elements to the cone of positive elements in a unital $$C^*$$ -algebra. We also show that the Thompson metric arises as the distance function for a natural Finsler structure on $$\Omega $$ and its minimal geodesics agree with the geodesics of the spray arising from the Loos structure. We close by showing that some familiar operator inequalities can be derived in this very general setting using the differential and metric geometry of the cone.

Holographic reconstruction of bubble spacetimes

We discuss the holographic reconstruction of static thin bubble walls in BTZ black hole geometries. We consider two reconstruction prescriptions suggested in recent years: hole-ography and light-cone cuts, in the context of thin bubble walls, and comment on their applicability in the presence of non-trivial matter in the bulk. We find that while the light-cone cuts prescription goes through within its own limitations, the current hole-ographic approaches are inadequate to describe bubble spacetimes completely. Much like entanglement shadows found around BTZ black holes and conical defects in the bulk, we find that thin bubbles are accompanied by shadows of their own, which are regions of spacetime which are only partially probed by minimal geodesics. We speculate that such shadows might be a generic feature of the presence of matter in the bulk.

Designing metrics; the delta metric for curves

In the first part, we revisit some key notions. Let M be a Riemannian manifold. Let G be a group acting on M. We discuss the relationship between the quotient M∕G, “horizontality” and “normalization”. We discuss the distinction between path-wise invariance and point-wise invariance and how the former positively impacts the design of metrics, in particular for the mathematical and numerical treatment of geodesics. We then discuss a strategy to design metrics with desired properties. In the second part, we prepare methods to normalize some standard group actions on the curve; we design a simple differential operator, called the delta operator, and compare it to the usual differential operators used in defining Riemannian metrics for curves. In the third part we design two examples of Riemannian metrics in the space of planar curves. These metrics are based on the “delta” operator; they are “modular”, they are composed of different terms, each associated to a group action. These are “strong” metrics, that is, smooth metrics on the space of curves, that is defined as a differentiable manifolds, modeled on the standard Sobolev space H2. These metrics enjoy many important properties, including: metric completeness, geodesic completeness, existence of minimal length geodesics. These metrics properly project on the space of curves up to parameterization; the quotient space again enjoys the above properties.

Sphere and projective space of a C *-algebra with a faithful state

Abstract Let 𝒜 be a unital C *-algebra with a faithful state ϕ. We study the geometry of the unit sphere 𝕊 ϕ = {x ∈ 𝒜 : ϕ(x * x) = 1} and the projective space ℙ ϕ = 𝕊 ϕ /𝕋. These spaces are shown to be smooth manifolds and homogeneous spaces of the group 𝒰 ϕ (𝒜) of isomorphisms acting in 𝒜 which preserve the inner product induced by ϕ , which is a smooth Banach-Lie group. An important role is played by the theory of operators in Banach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric in ℙ ϕ , and prove the existence of minimal geodesics, both with given initial data, and given endpoints.

Tunneling for a Class of Difference Operators: Complete Asymptotics

We analyze a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon\mathbf{Z}^d)$, where $V_\varepsilon$ is a multi-well potential and $\varepsilon$ is a small parameter. We derive full asymptotic expansions of the prefactor of the exponentially small eigenvalue splitting due to interactions between two "wells" (minima) of the potential energy, i.e., for the discrete tunneling effect. We treat both the case where there is a single minimal geodesic (with respect to the natural Finsler metric induced by the leading symbol $h_0(x,\xi)$ of $H_\varepsilon$) connecting the two minima and the case where the minimal geodesics form an $\ell+1$ dimensional manifold, $\ell\geq 1$. These results on the tunneling problem are as sharp as the classical results for the Schr\"odinger operator in \cite{hesjo}. Technically, our approach is pseudodifferential and we adapt techniques from \cite{hesjo2} and \cite{hepar} to our discrete setting.

Generating three-qubit quantum circuits with neural networks

A new method for compiling quantum algorithms is proposed and tested for a three qubit system. The proposed method is to decompose a unitary matrix U, into a product of simpler Uj via a neural network. These Uj can then be decomposed into product of known quantum gates. Key to the effectiveness of this approach is the restriction of the set of training data generated to paths which approximate minimal normal subRiemannian geodesics, as this removes unnecessary redundancy and ensures the products are unique. The two neural networks are shown to work effectively, each individually returning low loss values on validation data after relatively short training periods. The two networks are able to return coefficients that are sufficiently close to the true coefficient values to validate this method as an approach for generating quantum circuits. There is scope for more work in scaling this approach for larger quantum systems.

Minimal rays on closed surfaces

Given an arbitrary Riemannian metric on a closed surface, we consider length-minimizing geodesics in the universal cover. Morse and Hedlund proved that such minimal geodesics lie in bounded distance of geodesics of a Riemannian metric of constant curvature. Knieper asked when two minimal geodesics in bounded distance of a single constant-curvature geodesic can intersect. In this paper we prove an asymptotic property of minimal rays, showing in particular that intersecting minimal geodesics as above can only occur as heteroclinic connections between pairs of homotopic closed minimal geodesics. A further application characterizes the boundary at infinity of the universal cover defined by Busemann functions. A third application shows that flat strips in the universal cover of a nonpositively curved surface are foliated by lifts of closed geodesics of a single homotopy class.

Larotonda spaces: Homogeneous spaces and conditional expectations

We define a Larotonda space as a quotient space [Formula: see text] of the unitary groups of [Formula: see text]-algebras [Formula: see text] with a faithful unital conditional expectation [Formula: see text] In particular, [Formula: see text] is complemented in [Formula: see text], a fact which implies that [Formula: see text] has [Formula: see text] differentiable structure, with the topology induced by the norm of [Formula: see text]. The conditional expectation also allows one to define a reductive structure (in particular, a linear connection) and a [Formula: see text]-invariant Finsler metric in [Formula: see text]. Given a point [Formula: see text] and a tangent vector [Formula: see text], we consider the problem of whether the geodesic [Formula: see text] of the linear connection satisfying these initial data is (locally) minimal for the metric. We find a sufficient condition. Several examples are given, of locally minimal geodesics.

Minimal Geodesics Along Volume-Preserving Maps, Through Semidiscrete Optimal Transport

We introduce a numerical method for extracting minimal geodesics along the group of volume-preserving maps, equipped with the $L^2$ metric, which as observed by Arnold [Ann. Inst. Fourier (Grenoble), 16 (1966), pp. 319--361] solve the Euler equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier [Comm. Pure Appl. Math., 44 (1991), pp. 375--417], numerically implemented through semidiscrete optimal transport. It is robust enough to extract nonclassical, multivalued solutions of Euler's equations, for which the flow dimension---defined as the quantization dimension of Brenier's generalized flow---is higher than the domain dimension, a striking and unavoidable consequence of thismodel [A. I. Shnirelman, Geom. Funct. Anal., 4 (1994), pp. 586--620]. Our convergence results encompass this generalizedmodel, and our numerical experiments illustrate it for the first time in two space dimensions.

Global minimizers for Tonelli Lagrangians on the 2-torus

We study action-minimizing orbits in Tonelli Lagrangian systems on the 2-torus on fixed energy levels above Mañé's strict critical value. Our work generalizes the results of Morse, Hedlund and Bangert on minimal geodesics in Riemannian 2-tori. The techniques in the proofs involve classical variational ones, as well as the theories of Mather, Mañé and Fathi, which allow the step from reversible to non-reversible dynamics.