Hilbert Space Theory Research Articles

Quantum scalar field on fuzzy de Sitter space. Part I. Field modes and vacua

We study a scalar field on a noncommutative model of spacetime, the fuzzy de Sitter space, which is based on the algebra of the de Sitter group SO(1, d) and its unitary irreducible representations. We solve the Klein-Gordon equation in d = 2, 4 and show, using a specific choice of coordinates and operator ordering, that all commutative field modes can be promoted to solutions of the fuzzy Klein-Gordon equation. To explore completeness of this set of modes, we specify a Hilbert space representation and study the matrix elements (integral kernels) of a scalar field: in this way the complete set of solutions of the fuzzy Klein-Gordon equation is found. The space of noncommutative solutions has more degrees of freedom than the commutative one, whenever spacetime dimension is d > 2. In four dimensions, the new non-geometric, internal modes are parametrised by S2 × W, where W is a discrete matrix space. Our results pave the way to analysis of quantum field theory on the fuzzy de Sitter space.

Polynomial-based kernel reproduced gradient descent for stochastic optimization

Stochastic gradient descent (SGD) is widely applied to machine learning tasks. However, the existence of gradient variance degrades the convergence performance of SGD, which cannot be tackled well for tasks with one-time data sampling per sample. In this paper, a kernel reproduced gradient descent algorithm is proposed where the gradient of the risk function is estimated by utilizing the Reproducing Kernel Hilbert Space (RKHS) theory. Meanwhile, to ensure a low storage requirement, a kernel function linearization strategy is used to design the polynomial form of the kernel learning algorithm. It is shown that the error resulting from linearization decreases as the model parameter size increases, and the proposed polynomial-like kernel reproduced gradient descent (PKRGD) algorithm can converge faster than SGD. From experiments on models such as the latest image matching model, LightGlue, it is verified that the reproduced gradients have lower variances, and existing optimizers using the reproduced gradients can further accelerate model training.

Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representations and density weights in quantum gravity

Classical General Relativity is a dynamical theory of spacetime metrics of Lorentzian signature. In particular the classical metric field is nowhere degenerate in spacetime. In its initial value formulation with respect to a Cauchy surface the induced metric is of Euclidian signature and nowhere degenerate on it. It is only under this assumption of non-degeneracy of the induced metric that one can derive the hypersurface deformation algebra between the initial value constraints which is absolutely transparent from the fact that the inverse of the induced metric is needed to close the algebra. This statement is independent of the density weight that one may want to equip the spatial metric with. Accordingly, the very definition of a non-anomalous representation of the hypersurface deformation algebra in quantum gravity has to address the issue of non-degeneracy of the induced metric that is needed in the classical theory. In the Hilbert space representation employed in Loop Quantum Gravity (LQG) most emphasis has been laid to define an inverse metric operator on the dense domain of spin network states although they represent induced quantum geometries which are degenerate almost everywhere. It is no surprise that demonstration of closure of the constraint algebra on this domain meets difficulties because it is a sector of the quantum theory which is classically forbidden and which lies outside the domain of definition of the classical hypersurface deformation algebra. Various suggestions for addressing the issue such as non-standard operator topologies, dual spaces (habitats) and density weights have been proposed to address this issue with respect to the quantum dynamics of LQG. In this article we summarise these developments and argue that insisting on a dense domain of non-degenerate states within the LQG representation may provide a natural resolution of the issue thereby possibly avoiding the above mentioned non-standard constructions.

Path signatures for diversity in probabilistic trajectory optimisation

Motion planning can be cast as a trajectory optimisation problem where a cost is minimised as a function of the trajectory being generated. In complex environments with several obstacles and complicated geometry, this optimisation problem is usually difficult to solve and prone to local minima. However, recent advancements in computing hardware allow for parallel trajectory optimisation where multiple solutions are obtained simultaneously, each initialised from a different starting point. Unfortunately, without a strategy preventing two solutions to collapse on each other, naive parallel optimisation can suffer from mode collapse diminishing the efficiency of the approach and the likelihood of finding a global solution. In this paper, we leverage on recent advances in the theory of rough paths to devise an algorithm for parallel trajectory optimisation that promotes diversity over the range of solutions, therefore avoiding mode collapses and achieving better global properties. Our approach builds on path signatures and Hilbert space representations of trajectories and connects parallel variational inference for trajectory estimation with diversity-promoting kernels. We empirically demonstrate that this strategy achieves lower average costs than competing alternatives on a range of problems, from 2D navigation to robotic manipulators operating in cluttered environments.

Nonmesonic Quantum Many-Body Scars in a 1D Lattice Gauge Theory.

We investigate the meson excitations (particle-antiparticle bound states) in quantum many-body scars of a 1D Z_{2} lattice gauge theory coupled to a dynamical spin-1/2 chain as a matter field. By introducing a string representation of the physical Hilbert space, we express a scar state |Ψ_{n,l}⟩ as a superposition of all string bases with an identical string number n and a total length l. For the small-l scar state |Ψ_{n,l}⟩, the gauge-invariant spin exchange correlation function of the matter field hosts an exponential decay as the distance increases, indicating the existence of stable mesons. However, for large l, the correlation function exhibits a power-law decay, signaling the emergence of nonmesonic excitations. Furthermore, we show that this mesonic-nonmesonic crossover can be detected by the quench dynamics, starting from two low-entangled initial states, respectively, which are experimentally feasible in quantum simulators. Our results expand the physics of quantum many-body scars in lattice gauge theories and reveal that the nonmesonic state can also manifest ergodicity breaking.

Learning nonparametric ordinary differential equations from noisy data

Learning nonparametric systems of Ordinary Differential Equations (ODEs) x˙=f(t,x) from noisy data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for f for which the solution of the ODE exists and is unique. Learning f consists of solving a constrained optimization problem in an RKHS. We propose a penalty method that iteratively uses the Representer theorem and Euler approximations to provide a numerical solution. We prove a generalization bound for the L2 distance between x and its estimator. Experiments are provided for the FitzHugh–Nagumo oscillator, the Lorenz system, and for predicting the Amyloid level in the cortex of aging subjects. In all cases, we show competitive results compared with the state-of-the-art.

Time-frequency analysis on flat tori and Gabor frames in finite dimensions

We provide the foundations of a Hilbert space theory for the short-time Fourier transform (STFT) where the flat tori TN2=R2/(Z×NZ)=[0,1]×[0,N] act as phase spaces. We work on an N-dimensional subspace SN of distributions periodic in time and frequency in the dual S0′(R) of the Feichtinger algebra S0(R) and equip it with an inner product. To construct the Hilbert space SN we apply a suitable double periodization operator to S0(R). On SN, the STFT is applied as the usual STFT defined on S0′(R). This STFT is a continuous extension of the finite discrete Gabor transform from the lattice onto the entire flat torus. As such, sampling theorems on flat tori lead to Gabor frames in finite dimensions. For Gaussian windows, one is lead to spaces of analytic functions and the construction allows to prove a necessary and sufficient Nyquist rate type result, which is the analogue, for Gabor frames in finite dimensions, of a well known result of Lyubarskii and Seip-Wallstén for Gabor frames with Gaussian windows and which, for N odd, produces an explicit full spark Gabor frame. The compactness of the phase space, the finite dimension of the signal spaces and our sampling theorem offer practical advantages in some applications. We illustrate this by discussing a problem of current research interest: recovering signals from the zeros of their noisy spectrograms.

Lévy-Leblond equation and Eisenhart–Duval lift in Koopman–von Neumann mechanics

The Koopman–von Neumann (KvN) mechanics is an approach that was formulated long ago to answer the question regarding the existence of a Hilbert space representation of classical mechanics. KvN mechanics is a non-relativistic theory, and it is not clear how spin can be included in it, since spin is widely regarded as a relativistic property. Cabrera et al., in Eur. Phys. J. Spec. Top. 227, 2195 (2019) argued that the Spohn equation [Spohn, Ann. Phys. 282, 420 (2000)] is the correct classical framework for the Koopman–von Neumann theory corresponding to the Dirac equation. However, after Lévy-Leblond’s seminal work on this topic, it became clear that spin naturally arises also from the Galilean invariant wave equations, without any need of relativistic considerations. Inspired by this, we propose another possibility of including spin in the KvN formalism: the Lévy-Leblond equation in the Koopman–von Neumann formalism can be obtained as a null reduction of the massless Dirac equation in the Eisenhart–Duval lift metric. To illustrate the idea, we implement it for a one-dimensional classical system without magnetic interactions.

Experimental and Theoretical Studies of Hyperthermal N + O2 Collisions.

The dynamics of hyperthermal N(4S) + O2 collisions were investigated both experimentally and theoretically. Crossed molecular beams experiments were performed at an average center-of-mass (c.m.) collision energy of ⟨Ecoll⟩ = 77.5 kcal mol-1, with velocity- and angle-resolved product detection by a rotatable mass spectrometer detector. Nonreactive (N + O2) and reactive (NO + O) product channels were identified. In the c.m. reference frame, the nonreactively scattered N atoms and reactively scattered NO molecules were both directed into the forward direction with respect to the initial direction of the reagent N atoms. On average, more than 90% of the available energy (⟨Eavl⟩ = 77.5 kcal mol-1) was retained in translation of the nonreactive products (N + O2), whereas a much smaller fraction of the available energy for the reactive pathway (⟨Eavl⟩ = 109.5 kcal mol-1) went into translation of the NO + O products, and the distribution of translational energies for this channel was broad, indicating extensive internal excitation in the nascent NO molecules. The experimentally derived c.m. translational energy and angular distributions of the reactive products suggested at least two dynamical pathways to the formation of NO + O. Quasiclassical trajectory (QCT) calculations were performed with a collision energy of Ecoll = 77 kcal mol-1 using two sets of potential energy surfaces, denoted as PES-I and PES-II, and these theoretical results were compared to each other and to the experimental results. PES-I is a reproducing kernel Hilbert space (RKHS) representation of multireference configurational interaction (MRCI) energies, while PES-II is a many-body permutation invariant polynomial (MB-PIP) fit of complete active space second order perturbation (CASPT2) points. The theoretical investigations were both consistent with the experimental suggestion of two dynamical pathways to produce NO + O, where reactive collisions may proceed on the doublet (12A') and quartet (14A') surfaces. When analyzed with this theoretical insight, the experimental c.m. translational energy and angular distributions were in reasonably good agreement with those predicted by the QCT calculations, although minor differences were observed which are discussed. Theoretical translational energy and angular distributions for the nonreactive N + O2 products matched the experimental translational energy and angular distributions almost quantitatively. Finally, relative yields for the nonreactive and reactive scattering channels were determined from the experiment and from both theoretical methods, and all results are in reasonable agreement.

A hybrid quantum–classical neural network for learning transferable visual representation

State-of-the-art quantum machine learning (QML) algorithms fail to offer practical advantages over their notoriously powerful classical counterparts, due to the limited learning capabilities of QML algorithms, the constrained computational resources available on today’s noisy intermediate-scale quantum (NISQ) devices, and the empirically designed circuit ansatz for QML models. In this work, we address these challenges by proposing a hybrid quantum–classical neural network (CaNN), which we call QCLIP, for Quantum Contrastive Language-Image Pre-Training. Rather than training a supervised QML model to predict human annotations, QCLIP focuses on more practical transferable visual representation learning, where the developed model can be generalized to work on unseen downstream datasets. QCLIP is implemented by using CaNNs to generate low-dimensional data feature embeddings followed by quantum neural networks to adapt and generalize the learned representation in the quantum Hilbert space. Experimental results show that the hybrid QCLIP model can be efficiently trained for representation learning. We evaluate the representation transfer capability of QCLIP against the classical Contrastive Language-Image Pre-Training model on various datasets. Simulation results and real-device results on NISQ IBM_Auckland quantum computer both show that the proposed QCLIP model outperforms the classical CLIP model in all test cases. As the field of QML on NISQ devices is continually evolving, we anticipate that this work will serve as a valuable foundation for future research and advancements in this promising area.

An evolving space framework for Oseen equations on a moving domain

This article considers non-stationary incompressible linear fluid equations in a moving domain. We demonstrate the existence and uniqueness of an appropriate weak formulation of the problem by making use of the theory of time-dependent Bochner spaces. It is not possible to directly apply established evolving Hilbert space theory due to the incompressibility constraint. After we have established the well-posedness, we derive and analyse a first order time discretisation of the system.

New collocation approaches based on reproducing kernel functions for second order nonlinear boundary value problems

In the previous work, the reproducing kernel Hilbert space (RKHS) theory has been employed mainly to solve linear boundary value problems (BVPs). Reproducing kernel functions (RKFs) based techniques for nonlinear BVPs mainly applied iterative techniques. However, there is no error analysis for these methods. In this letter, we shall propose new collocation algorithms based on RKFs for dealing with second order nonlinear BVPs. And the detailed error analysis is provided.

Orthogonal decomposition of multivariate densities in Bayes spaces and relation with their copula-based representation

Bayes spaces were initially designed to provide a geometric framework for the modeling and analysis of distributional data. It has recently come to light that this methodology can be exploited to construct an orthogonal decomposition of a bivariate probability density into an independence and an interaction part. In this paper, new insights into these results are given by reformulating them using Hilbert space theory, and a multivariate extension is developed using a distributional analog of the Hoeffding–Sobol identity. A connection is also made between the resulting decomposition of a multivariate density and its copula-based representation.

Second Order Two-Species Systems with Nonlocal Interactions: Existence and Large Damping Limits

We study the mathematical theory of second order systems with two species, arising in the dynamics of interacting particles subject to linear damping, to nonlocal forces and to external ones, and resulting into a nonlocal version of the compressible Euler system with linear damping. Our results are limited to the 1 space dimensional case but allow for initial data taken in a Wasserstein space of probability measures. We first consider the case of smooth nonlocal interaction potentials, not subject to any symmetry condition, and prove existence and uniqueness. The concept of solutions relies on a stickiness condition in case of collisions, in the spirit of previous works in the literature. The result uses concepts from classical Hilbert space theory of gradient flows (cf. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, 1973) and a trick used in Brenier et al. (J. Math. Pures Appl. 99(5):577–617, 2013). We then consider a large-time and large-damping scaled version of our system and prove convergence to solutions to the corresponding first order system. Finally, we consider the case of Newtonian potentials - subject to symmetry of the cross-interaction potentials - and external convex potentials. After showing existence in the sticky particles framework in the spirit of Brenier et al. (J. Math. Pures Appl. 99(5):577–617, 2013), we prove convergence for large times towards Dirac delta solutions for the two densities. All the results share a common technical framework in that solutions are considered in a Lagrangian framework, which allows to estimate the behavior of solutions via L^{2} estimates of the pseudo-inverse variables corresponding to the two densities. In particular, due to this technique, the large-damping result holds under a rather weak condition on the initial data, which does not require well-prepared initial velocities. We complement the results with numerical simulations.

On using Reproducing Kernel Hilbert Spaces for the analysis of Replicated Spatial Point Processes

This paper focuses on the use of the theory of Reproducing Kernel Hilbert Spaces in the statistical analysis of replicated point processes. We show that spatial point processes can be observed as random variables in a Reproducing Kernel Hilbert Space and, as a result, methodological and theoretical results for statistical analysis in these spaces can be applied to them. In particular and by way of illustration, we show how we can use the proposed methodology to identify differences between several classes of replicated point patterns using the Box’s M and MANOVA tests, and to classify a new observation, using Discriminant Functions.

Function algebras on the n-dimensional quantum complex space

This paper introduces a (universal) C*-algebra of continuous functions vanishing at infinity on the [Formula: see text]-dimensional quantum complex space. To this end, the well-behaved Hilbert space representations of the defining relations are classified. Then these representations are realized by multiplication operators on an [Formula: see text]-space. The C*-algebra of continuous functions vanishing at infinity is defined by considering an *-algebra such that its classical counterpart separates the points of the [Formula: see text]-dimensional complex space and by taking the operator norm closure of a universal representation of this algebra.

Convergence analysis for kernel-regularized online regression associated with an RRKHS

It is known that the functional reproducing kernel Hilbert space (FRKHS) theory lays the functional analysis foundation for learning non-point-evaluation functional data with kernel-regularized learning. In the present paper, we investigate the convergence of regression learning associated with Radon reproducing kernel Hilbert spaces (RRKHSs) and a parameterized loss. We provide a kind of online learning algorithm and establish an upper bound for the learning rate, and it shows that the learning rate may be improved by adjusting the parameter in the loss function. As incidental conclusions, we provide a method of constructing FRKHSs with Fourier analysis, some RRKHSs are defined for the classes of periodic functions defined on $ [-\pi, \;\pi]^s $; the functions defined on the interval $ [-1, 1] $; the functions defined on $ R_+ = [0, +\infty) $ and the functions defined on the unit sphere.

Stabilization of an interconnected system of Schrödinger and wave equations with boundary coupling

In this paper, we consider the Schrödinger equation coupled by the interface with a wave equation and with a boundary damping. The dissipation is acting on the wave equation through the Neumann boundary condition. We formulate the coupled system as an abstract evolution equation in an appropriate Hilbert space and use linear semigroup theory to show the well-posedness of the system. Then under some assumptions on the geometry of the spatial domain, we prove exponential stability of the solution. The proof of this result is based on a frequency domain approach which consists in verifying that the imaginary axis is included in the resolvent set of the system and analyzing the behavior of the resolvent operator of the system on the imaginary axis. The analysis of the resolvent is carried out by combining contradiction argument with the multipliers technique. This result extends Theorem 3.2 in [13] to multimensional spatial domains.

Eisenhart lift of Koopman-von Neumann mechanics

The Eisenhart lift establishes a fascinating connection between non-relativistic and relativistic physics, providing a space-time geometric understanding of non-relativistic Newtonian mechanics. What is still little known, however, is the fact that there is a Hilbert space representation of classical mechanics (also called Koopman-von Neumann mechanics) that attempts to give classical mechanics the same mathematical structure that quantum mechanics has. In this article, we geometrize the Koopman-von Newmann (KvN) mechanics using the Eisenhart toolkit. We then use a geometric view of KvN mechanics to find transformations that relate the harmonic oscillator, linear potential, and free particle in the context of KvN mechanics.

Pseudo S-spectra of special operators in quaternionic Hilbert spaces

For a bounded quaternionic operator T on a right quaternionic Hilbert space H and ε>0, the pseudo S-spectrum of T is defined asΛεS(T):=σS(T)⋃{q∈H∖σS(T):‖Qq(T)−1‖≥1ε}, where H denotes the division ring of quaternions, σS(T) is the S-spectrum of T and Qq(T)=T2−2Re(q)T+|q|2I. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate several properties of the pseudo S-spectrum and explicitly compute the pseudo S-spectra for some special classes of operators such as upper triangular matrices, self adjoint-operators, normal operators and orthogonal projections. In particular, by an application of S-functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number r if and only if for every ε>0 the pseudo S-spectrum of the operator is the circularization of a closed disk in the complex plane centred at r with the radius ε. Further, we propose a G1-condition for quaternionic operators and prove some results in this setting.