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In this research, we introduce what seems to be a new mathematical object resulting from projecting the $n$-dimensional $Z$-curve onto an $n$-dimensional sphere. The first part presents the algorithm that enables this transition, and the second part focuses on studying the properties of the resulting object.

Trace-free Einstein gravity is a theory of gravity that is an alternative to general relativity, wherein the cosmological constant arises as an integration constant. However, there are no fully diffeomorphism-invariant action principles available that lead to the equations of motion of this theory. Unimodular gravity comes close to this idea, but it relies on action principles that are invariant only under volume-preserving diffeomorphisms. We present a real $BF$-type action principle for trace-free Einstein gravity that is fully diffeomorphism-invariant and does not require any unimodular condition or nondynamical fields. We generalize this action principle by giving another one involving a free parameter., Comment: It matches published version

We compute both analytically and numerically the geometry of the parameter space of the anharmonic oscillator employing the quantum metric tensor and its scalar curvature. A novel semiclassical treatment based on a Fourier decomposition allows to construct classical analogues of the quantum metric tensor and of the expectation values of the transition matrix elements. A detailed comparison is presented between exact quantum numerical results, a perturbative quantum description and the semiclassical analysis. They are shown to coincide for both positive and negative quadratic potentials, where the potential displays a double well. Although the perturbative method is unable to describe the region where the quartic potential vanishes, it is remarkable that both the perturbative and semiclassical formalisms recognize the negative oscillator parameter at which the ground state starts to be delocalized in two wells.

The quantum geometric tensor, composed of the quantum metric tensor and Berry curvature, fully encodes the parameter space geometry of a physical system. We first provide a formulation of the quantum geometrical tensor in the path integral formalism that can handle both the ground and excited states, making it useful to characterize excited state quantum phase transitions (ESQPT). In this setting, we also generalize the quantum geometric tensor to incorporate variations of the system parameters and the phase-space coordinates. This gives rise to an alternative approach to the quantum covariance matrix, from which we can get information about the quantum entanglement of Gaussian states through tools such as purity and von Neumann entropy. Second, we demonstrate the equivalence between the formulation of the quantum geometric tensor in the path integral formalism and other existing methods. Furthermore, we explore the geometric properties of the generalized quantum metric tensor in depth by calculating the Ricci tensor and scalar curvature for several quantum systems, providing insight into this geometric information., Comment: In this version we corrected typos and expanded Section 2. Results unchanged

It has recently been proposed classical analogs of the purity, linear quantum entropy, and von Neumann entropy for classical integrable systems, when the corresponding quantum system is in a Gaussian state. We generalized these results by providing classical analogs of the generalized purities, Bastiaans-Tsallis entropies, R\'enyi entropies, and logarithmic negativity for classical integrable systems. These classical analogs are entirely characterized by the classical covariance matrix. We compute these classical analogs exactly in the cases of linearly coupled harmonic oscillators, a generalized harmonic oscillator chain, and a one-dimensional circular lattice of oscillators. In all of these systems, the classical analogs reproduce the results of their quantum counterparts whenever the system is in a Gaussian state. In this context, our results show that quantum information of Gaussian states can be reproduced by classical information., Comment: 17 pages. 9 figures

We study the formation and evolution of domain walls with initial inflationary fluctuations by numerical lattice calculations that, for the first time, correctly take into account correlations on superhorizon scales. We find that, contrary to the widely-held claim {over the past few tens of years}, the domain wall network exhibits remarkable stability even when the initial distribution is largely biased toward one of the minima. This is due to the fact that the domain wall network retains information about initial conditions on superhorizon scales, and that the scaling solution is not a local attractor in this sense. Our finding immediately implies that such domain walls will have a significant impact on cosmology, including the production of gravitational waves, baryogenesis, and dark matter from domain walls. Applying this result to the axion-like particle domain wall, we show that it not only explains the isotropic cosmic birefringence suggested by the recent analysis, but also predicts anisotropic cosmic birefringence that is nearly scale-invariant on large scales and can be probed by future CMB observations., Comment: v2: 8pages, 13figures, the version submitted to journal

We obtain a classical analog of the quantum covariance matrix by performing its classical approximation for any continuous quantum state, and we illustrate this approach with the anharmonic oscillator. Using this classical covariance matrix, we propose classical analogs of the purity, linear quantum entropy, and von Neumann entropy for classical integrable systems, when the quantum counterpart of the system under consideration is in a Gaussian state. As is well known, this matrix completely characterizes the purity, linear quantum entropy, and von Neumann entropy for Gaussian states. These classical analogs can be interpreted as quantities that reveal how much information from the complete system remains in the considered subsystem. To illustrate our approach, we calculate these classical analogs for three coupled harmonic oscillators and two linearly coupled oscillators. We find that they exactly reproduce the results of their quantum counterparts. In this sense, it is remarkable that we can calculate these quantities from the classical viewpoint., Comment: 16 pages, 3 figures. To be published in Physical Review A

We report new topological invariants in four dimensions that are generalizations of the Nieh-Yan topological invariant. The new topological invariants are obtained through a systematic method along the lines of the one used to get the Nieh-Yan form, but involving an $SO(4,1)$ [$SO(5)$] connection constructed out from an $SO(3,1)$ [$SO(4)$] connection and three $SO(3,1)$ [$SO(4)$] tensor $1$-forms. We give explicit expressions of the new 4-forms that give rise to the new topological invariants for particular choices of these 1-forms and show that the Nieh-Yan form arises as a particular case., Comment: It matches published version

We study the classical analog of the quantum metric tensor and its scalar curvature for two well-known quantum physics models. First, we analyze the geometry of the parameter space for the Dicke model with the aid of the classical and quantum metrics and find that, in the thermodynamic limit, they have the same divergent behavior near the quantum phase transition, as opposed to their corresponding scalar curvatures which are not divergent there. On the contrary, under resonance conditions, both scalar curvatures exhibit a divergence at the critical point. Second, we present the classical and quantum metrics for the Lipkin-Meshkov-Glick model in the thermodynamic limit and find a perfect agreement between them. We also show that the scalar curvature is only defined on one of the system's phases and that it approaches a negative constant value. Finally, we carry out a numerical analysis for the system's finite sizes, which clearly shows the precursors of the quantum phase transition in the metric and its scalar curvature and allows their characterization as functions of the parameters and of the system's size., Comment: 16 pages, 12 figures

We study the quantum metric tensor and its scalar curvature for a particular version of the Lipkin-Meshkov-Glick model. We build the classical Hamiltonian using Bloch coherent states and find its stationary points. They exhibit the presence of a ground state quantum phase transition, where a bifurcation occurs, showing a change of stability associated with an excited state quantum phase transition. Symmetrically, for a sign change in one Hamiltonian parameter, the same phenomenon is observed in the highest energy state. Employing the Holstein-Primakoff approximation, we derive analytic expressions for the quantum metric tensor and compute the scalar and Berry curvatures. We contrast the analytic results with their finite-size counterparts obtained through exact numerical diagonalization and find an excellent agreement between them for large sizes of the system in a wide region of the parameter space, except in points near the phase transition where the Holstein-Primakoff approximation ceases to be valid., Comment: 14 pages

Gioseffo Zarlino reintroduced the Pythagorean paradigm into Renaissance musical theory. In a similar fashion, Nicolaus Copernicus, Galileo Galilei, Johannes Kepler, and Isaac Newton reinvigorated Pythagorean ideas in celestial mechanics; Kepler and Newton explicitly invoked musical principles. Today, the theory of dynamical systems allows us to describe very different applications of physics, from the orbits of asteroids in the Solar System to the pitch of complex sounds. Our aim in this text is to review the overarching aims of our research in this field over the past quarter of a century. We demonstrate with a combination of dynamical systems theory and music theory the thread running from Pythagoras to Zarlino that allowed the latter to construct musical scales using the ideas of proportion known to the former, and we discuss how the modern theory of dynamical systems, with the study of resonances in nonlinear systems, returns to Pythagorean ideas of a Musica Universalis.

The geometry of the parameter space is encoded by the quantum geometric tensor, which captures fundamental information about quantum states and contains both the quantum metric tensor and the curvature of the Berry connection. We present a formulation of the Berry connection and the quantum geometric tensor in the framework of the phase space or Wigner function formalism. This formulation is obtained through the direct application of the Weyl correspondence to the geometric structure under consideration. In particular, we show that the quantum metric tensor can be computed using only the Wigner functions, which opens an alternative way to experimentally measure the components of this tensor. We also address the non-Abelian generalization and obtain the phase space formulation of the Wilczek-Zee connection and the non-Abelian quantum geometric tensor. In this case, the non-Abelian quantum metric tensor involves only the non-diagonal Wigner functions. Then, we verify our approach with examples and apply it to a system of $N$ coupled harmonic oscillators, showing that the associated Berry connection vanishes and obtaining the analytic expression for the quantum metric tensor. Our results indicate that the developed approach is well adapted to study the parameter space associated with quantum many-body systems., Comment: 22 pages

The study of the motion of a rigid body on a plane (RBP motion) is usually one of the most challenging topics that students face in introductory physics courses. In this paper, we discuss a couple of problems which are typically used in basic physics courses, in order to highlight some aspects related to RBP motion which are not usually well understood by physics students. The first problem is a pendulum composed of a rod and disk. The angular frequency of the pendulum is calculated in two situations: disk fixed to the rod and disk free to spin. A detailed explanation of the change in the angular frequency from one case to another is given. The second problem is a ladder which slides touching a frictionless surface. We use this problem to highlight the fact that the contact forces applied by the surface perform translational and rotational work despite that the total mechanical energy of the ladder is conserved., Comment: LaTeX file in RevTeX 4.1 style, 8 pages, 8 figures. Version to be published in Revista Brasileira de Ensino de Fisica

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by $n$-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local $SO(n)$ or $SO(n-1,1)$ transformations, `improved diffeomorphisms', and the `generalization of local translations' of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do {\it not} use Noether's theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the `half off-shell' case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local $SO(3,1)$ or $SO(4)$ off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the `half off-shell' and on-shell cases. We also study Killing vector fields in the `half off-shell' and on-shell cases. The current theoretical framework is illustrated for the `half off-shell' case in static spherically symmetric and Friedmann--Lemaitre--Robertson--Walker spacetimes in four dimensions., Comment: Published version

In the pregnant mother and her fetus, chronic prenatal stress results in entrainment of the fetal heartbeat by the maternal heartbeat, quantified by the fetal stress index (FSI). Deep learning (DL) is capable of pattern detection in complex medical data with high accuracy in noisy real-life environments, but little is known about DL's utility in non-invasive biometric monitoring during pregnancy. A recently established self-supervised learning (SSL) approach to DL provides emotional recognition from electrocardiogram (ECG). We hypothesized that SSL will identify chronically stressed mother-fetus dyads from the raw maternal abdominal electrocardiograms (aECG), containing fetal and maternal ECG. Chronically stressed mothers and controls matched at enrolment at 32 weeks of gestation were studied. We validated the chronic stress exposure by psychological inventory, maternal hair cortisol and FSI. We tested two variants of SSL architecture, one trained on the generic ECG features for emotional recognition obtained from public datasets and another transfer-learned on a subset of our data. Our DL models accurately detect the chronic stress exposure group (AUROC=0.982+/-0.002), the individual psychological stress score (R2=0.943+/-0.009) and FSI at 34 weeks of gestation (R2=0.946+/-0.013), as well as the maternal hair cortisol at birth reflecting chronic stress exposure (0.931+/-0.006). The best performance was achieved with the DL model trained on the public dataset and using maternal ECG alone. The present DL approach provides a novel source of physiological insights into complex multi-modal relationships between different regulatory systems exposed to chronic stress. The final DL model can be deployed in low-cost regular ECG biosensors as a simple, ubiquitous early stress detection and monitoring tool during pregnancy. This discovery should enable early behavioral interventions., Comment: ClinicalTrials.gov registration number: NCT03389178. Code repo: https://code.engineering.queensu.ca/17ps21/ssl-ecg-v2

Bell-type inequalities allow for experimental testing of local hidden variable theories. In the present work we show the violation of Mermin's inequalities in IBM's five-qubit quantum computers, ruling out the local realism hypothesis in quantum mechanics. Furthermore, our numerical results show significant improvement with respect to previous implementations. The circuit implementation of these inequalities is also proposed as a way of assessing the reliability of different quantum computers.

Pre-Columbian Mesoamerica was a fertile crescent for the development of number systems. A form of vigesimal system seems to have been present from the first Olmec civilization onwards, to which succeeding peoples made contributions. We discuss the Maya use of the representational redundancy present in their Long Count calendar, a non-power positional number representation system with multipliers 1, 20, 18$\times$ 20, $\ldots$, 18$\times$ 20$^n$. We demonstrate that the Mesoamericans did not need to invent positional notation and discover zero at the same time because they were not afraid of using a number system in which the same number can be written in different ways. A Long Count number system with digits from 0 to 20 is seen later to pass to one using digits 0 to 19, which leads us to propose that even earlier there may have been an initial zeroless bijective numeration system whose digits ran from 1 to 20. Mesoamerica was able to make this conceptual leap to the concept of a cardinal zero to perform arithmetic owing to a familiarity with multiple and redundant number representation systems.

A permanent challenge in physics and other disciplines is to solve partial differential equations, thereby a beneficial investigation is to continue searching for new procedures to do it. In this Letter, a novel Monte-Carlo Metropolis framework is presented for solving the equations of motion in Lagrangian systems. The implementation lies in sampling the paths space with a probability functional obtained by using the maximum caliber principle. The methodology was applied to the free particle and the harmonic oscillator problems, where the numerically-averaged path obtained from the Monte-Carlo simulation converges to the analytical solution from classical mechanics, in an analogous way with a canonical system where energy is minimized by sampling the state space and computing the average state for each system. Thus, we expect that this procedure can be general enough to solve other differential equations in physics and to be a useful tool to calculate the time-dependent properties of dynamical systems in order to understand the non-equilibrium behavior of statistical mechanical systems.

First-order general relativity in $n$ dimensions ($n \geq 3$) has an internal gauge symmetry that is the higher-dimensional generalization of three-dimensional local translations. We report the extension of this symmetry for $n$-dimensional f(R) gravity with torsion in the Cartan formalism. The new symmetry arises from the direct application of the converse of Noether's second theorem to the action principle of f(R) gravity with torsion. We show that infinitesimal diffeomorphisms can be written as a linear combination of the new internal gauge symmetry, local Lorentz transformations, and terms proportional to the variational derivatives of the f(R) action. It means that the new internal symmetry together with local Lorentz transformations can be used to describe the full gauge symmetry of f(R) gravity with torsion, and thus diffeomorphisms become a derived symmetry in this setting., Comment: It contains a detailed derivation of the generalization of 3-dimensional "local translations" for 4-dimensional first-order general relativity