University of Lorraine

This chapter covers the implementation of field theory in curved manifolds. It begins with General Relativity and its classical tests in the solar system, followed by a brief discussion on gravitational waves. Cosmology is introduced as the study of the Universe’s large-scale structure based on three assumptions: gravitation theory, the nature of matter, and symmetry. The chapter traces modern cosmology’s development from Einstein’s general relativity and the cosmological constant to the Big Bang theory, supported by Hubble’s expanding Universe and cosmic microwave background radiation. Redshift measurements help determine the Universe’s expansion rate and age. The cosmological constant is now crucial for explaining the accelerating expansion. Scalar and spinor fields are adapted to curved spacetime using tetrads, and the coupling between electrodynamics and gravitation is explored. The chapter concludes with the Einstein-Cartan formulation of gravitation and includes exercises for practical applications.

This chapter on conservation laws and group theory in physics focuses primarily on Noether’s theorem, then on the unitary representations of the Galilean group, crucial for describing symmetries in both classical field theories and quantum mechanics. It begins by discussing Noether currents and the role of gauge choices on conserved quantities before addressing the challenge of constructing finite-dimensional unitary representations for non-compact groups like translations in infinite-dimensional spaces, then for compact groups to deal with rotations. The discussion highlights how differential operators represent translations and rotations, illustrating their action on scalar fields and their differential properties in the case of infinite-dimensional unitary representations. Boosts and time translations are similarly treated, emphasizing their differential operator representations. In quantum mechanics, these symmetries require unitary representations that account for phase factors due to the projective nature of quantum states. The chapter concludes with insights into the Bargmann algebra, which extends the Galilean algebra to describe non-relativistic quantum systems, and discusses its implications for physical observables like energy and momentum, linking theoretical group theory with practical applications in quantum mechanics.

This chapter explores alternative relativistic field theories, focusing on scalar field models, higher-dimensional spaces, non-commutative geometry, non-linear theories, and complex scalar fields. It examines electrodynamics within these frameworks, considering extensions like Proca, Born-Infeld, Bopp-Podolsky, Chern-Simons, axions, and Kalb-Ramond field theories, leading to novel physical predictions. The chapter also discusses higher half-integer spinors, the Rarita-Schwinger equation, and alternative gravity theories such as f(R) theories, teleparallel gravity, Einstein-Cartan theory, Weyl’s conformal gravity, and varying G-cosmologies. Additionally, it covers classical relativistic strings and branes, their coupling to Kalb-Ramond fields, and introduces supersymmetric theories, emphasizing their mathematical foundations like Grassmann variables and superspace. The construction of superfields, differences in actions between bosonic and fermionic particles, and supersymmetric transformations are explored, highlighting their role in connecting bosonic and fermionic degrees of freedom. The chapter concludes with discussions on chiral superfields, particularly in models like the Wess-Zumino model, underscoring their significance in modern particle physics and quantum field theory.

The chapter discusses the Path integral formulation of quantum mechanics, pioneered by Feynman, contrasting it with traditional Hamiltonian methods. It explores how quantum systems’ evolution can be understood through a sum over all possible paths, akin to classical trajectories but with quantum amplitudes replacing classical probabilities. Starting from the concept of probability amplitudes and propagators, it derives the Path integral expression using discretization of spacetime, defining the propagator as a sum over infinitesimal paths weighted by the action. This approach encapsulates quantum behavior beyond classical trajectories, highlighting its flexibility and foundational role in quantum field theory. The chapter concludes by connecting back to the Schrödinger equation, demonstrating how the Path integral framework can derive quantum dynamics from classical principles, underscoring its broad applicability in theoretical physics.

This chapter explores fundamental concepts underpinning the geometry of Minkowski spacetime. The discussion begins with a short review of Special Relativity and the notion of tensors and their transformations under the Lorentz group. Representations of the Lorentz group are then built, finite-dimensional representations with their link with SU(2) and infinite-dimensional representations. Spin representations allow for the introduction of Dirac spinors and their interaction with spacetime symmetries in the Minkowski manifold. The discussion is pursued through the construction of Lorentz scalars from Dirac spinors, examining their transformation properties under Lorentz transformations. It establishes the Dirac conjugate and Lorentz vector properties, culminating in a comprehensive exploration of Dirac, Weyl, and Majorana fermions. Key topics include the formulation of the Dirac equation, distinctions between fermion types, and implications for quantum field theory. The chapter concludes by outlining the development of a single-particle wave equation for photons, drawing parallels with Dirac’s formulation for spin-1/2 particles.

The inverted pendulum system, known for its applications across various domains, continues to inspire innovative research owing to its underactuated, nonlinear, inherently un-stable, and multivariable nature. Typically modeled using Euler-Lagrange or Newtonian dynamics, this study focuses on control-ling a Double Inverted Pendulum (DIP) through pole placement, Linear Quadratic Regulator (LQR), and Lyapunov-based control methods. To optimize the LQR control parameters, Particle Swarm Optimization (PSO) is employed. The proposed control approach involves the application of linear control design to a nonlinear system. To illustrate the efficacy of the proposed control strategy, many simulations under diverse condition (comparing the performance of each control technique based on the system rise time, settling time, peak amplitude, and steady-state error) with an 2D animation using Matplotlib© in Python are given.

This chapter explores the Lagrangian formulation of electrodynamics and of relativistic matter fields, specifically focusing on complex scalar Minkowskian fields and their dynamics described through an action principle. The action is formulated as an integral over spacetime, where the Lagrangian density depends on the field and its derivatives, ensuring Lorentz invariance and yielding covariant field equations via the Euler-Lagrange equations. Crucial to maintaining Lorentz covariance is constructing terms, ensuring they form Lorentz scalars consistent with the field’s complex nature. The case of spinor fields is also considered, with special attention paid to the Foldy-Wouthuysen transformation. The chapter also goes into more depth into the use of differential forms, elucidating how these mathematical constructs encapsulate the geometric and algebraic structures essential for understanding the field equations derived from the action principle. Differential forms provide a powerful framework for expressing the underlying geometric properties and symmetries inherent in the field dynamics, enhancing the clarity and depth of the relativistic field theory discussed throughout the chapter.

The chapter delves deeper into advanced gauge field theory using fiber bundles and differential forms, beginning with the notation and symmetry of Christoffel symbols and their mixed indices. It summarizes gauge field theory by focusing on the covariant properties of matter fields under gauge transformations, introducing the covariant exterior derivative and its role in defining gauge covariant matter fields and the curvature 2-form. The exterior covariant derivative’s action on vector-valued and Lie-algebra-valued forms is detailed, followed by exercises to verify these concepts. The text then discusses the Yang-Mills theory in differential forms, presenting the Yang-Mills action, deriving the equations of motion, covering the Bianchi identity and gauge field equations, and briefly touching on the Abelian and non-Abelian cases. Overall, the chapter emphasizes the mathematical framework of fiber bundles and differential forms in gauge field theory, supplemented with exercises for deeper understanding.

This chapter explores the intricate world of classical field theories defined on arbitrary manifolds, introducing the tools of differential geometry. The key concepts of connection and parallel transport of various mathematical objects, scalar, tensor, or spinor fields, are introduced. The discussion begins with an exploration of metric tensors, essential for understanding the geometric properties of spacetime. A detailed methodology for calculating Levi-Civita connection coefficients is presented, emphasizing the transformation of indices and the role of matrix reshaping. This process illuminates the relationship between torsion and curvature in the spacetime manifold. The derivation proceeds with the construction of Christoffel symbols, which are pivotal for understanding the geometric structure and the resulting gravitational effects. Spin connection coefficients are then introduced, showcasing their relevance in describing spacetime symmetries and the behaviour of spinors. The chapter culminates with formulating the Fock-Ivanenko connection, which is crucial for extending the framework to incorporate fermionic fields. Practical computational techniques and symbolic notations, such as Penrose graphical notation, are discussed, providing alternative perspectives on tensor analysis and enhancing comprehension of complex field interactions in gravitational physics.

This chapter explores the insights brought about by the energy minimization approach to formulating physical theories. The discussion emphasizes variational principles and the use of trial functions to approximate energy, illustrating their utility in solving complex physical problems where direct analytical solutions are challenging. Philosophical reflections on scientific theories, including falsifiability and the evolution of scientific truths within the community, are also examined. This comprehensive overview provides insights into foundational concepts in physics, addressing theoretical frameworks and their experimental implications.

This chapter explores classical mechanics of discrete systems, then field theories, reviewing the Lagrangian and Hamiltonian formalisms. The discussion extends to electrodynamics and the coupling of scalar matter fields with electromagnetic fields. It explores the minimal coupling scheme and its application to both Schrödinger and Klein-Gordon equations in the presence of electromagnetic potentials. Insights from prominent physicists are discussed, highlighting the interplay between theoretical elegance and experimental rigor in advancing our understanding of physical laws.

In this chapter, we embark on a comprehensive exploration of gauge field theory, that leads us to the Standard Model of particle physics framework. We begin by investigating Abelian, then non-Abelian symmetries, particularly SU(2) and SU(3), which underpin the electroweak and strong interactions, respectively. Our journey brings us to study historical aspects and deepens with a rigorous examination of the Higgs mechanism, a pivotal concept elucidating how gauge bosons acquire mass through spontaneous symmetry breaking. This mechanism completes the electroweak theory and provides profound insights into the origins of particle masses. We meticulously construct the Standard Model Lagrangian, revealing the intricate interplay between gauge fields, fermionic matter, and the Higgs field. Through this mathematical framework, we uncover the underlying structure of the fundamental forces governing particles and their interactions, showcasing the Standard Model’s remarkable predictive power in describing the dynamics of our universe at the most fundamental levels.

We establish Hölder stability of an inverse hyperbolic obstacle problem. Mainly, we study the problem of reconstructing an unknown function defined on the boundary of the obstacle from two measurements taken on the boundary of a domain surrounding the obstacle.

Background Detection algorithms targeting anatomic landmarks in three‐dimensional (3D) ultrasound (US) volume (three‐dimensional US) appear to be a relevant and easy‐to‐implement option to address junior and occasional operators' difficulties in probe positioning for two‐dimensional (2D) fetal biometry. Objectives This study assesses the feasibility of complete automation for fetal biometry and the resulting agreement with standard 2D (US) measurements. The secondary objectives were to assess the impact of software‐driven measurement on image quality scoring, reproducibility, and agreement with human‐driven measurements issued from the same volumes. Methods Datasets were collected from a consecutive sample of women attending standard US follow‐up (singleton, 16–30 weeks of gestation). Each dataset contained 2D measurements for reference (head and abdomen circumference and femoral length) and 3D US volume acquisitions of the fetal head, abdomen, and thigh. Both algorithm‐based and operator‐based detection of the targeted plans and calipers positioning were applied to the 3D volumes to produce software‐driven and human‐driven measurements. The resulting 3D measurements were assessed for completion rates, image quality, and reproducibility. Results On 175 datasets collected, completion rates in achieving software‐driven 3D measurements ranged between 94% (abdomen) and 100% (head). A modest weakening in quality (of uncertain clinical significance) was notable for the head and abdomen measurements. Compared to the 2D measurements, the software‐driven tended to slightly overestimate the estimated fetal weight (EFW; e.g., 95% confidence interval ranging from 445 to 635 g for a 525 g‐sized fetus at 22 weeks of gestation). The random error tended to be inflated for fetuses >700 g. Intra‐ and inter‐operator reproducibility were appropriate (intraclass correlation coefficient intervals ranged from 0.8 to 0.99). Conclusion Complete automation of US biometry appears feasible and presents appropriate reproducibility and image quality scoring, but third‐trimester biometry needs improvement. Before clinical implementation, it is time to assess the impact of point‐of‐care use on large populations.

Background Though promising, the implementation of crisis resolution teams has been unequal across Europe. In France, their deployment is currently receiving interest but there is to date no national policy and the research is scarce. Methods In the present study a psychiatric service converted one of its two inpatient wards into a crisis resolution team (EPSIAD) enabling a quasi-experimental naturalistic design. Variables on admissions, length of hospital stay and patient satisfaction were collected and analysed in the year preceding and the year following the conversion. Results In the year following the EPSIAD implementation, there were more admissions of female patients (41.0% vs 49.5%, p = 0.0262), a five-day decrease in the length of hospital stay (p < 0.0001) and increased patient satisfaction, with a particular increase in clarity of information, quality of the relationship with the care staff and service and feeling involved in medical decisions (p < 0.0001). Conclusions The results of the present study indicate that the combination of a hospital ward with a crisis resolution team has the potential to increase global quality of care by providing a complementary mental health service. Crisis resolution teams may provide a viable alternative to hospitalisation that increases patient satisfaction and allows new patients to receive intensive care, with women especially benefiting from care at home. There is a need to cater for patients refusing psychiatric care altogether and hospital inpatient wards might specialise in involuntarily-admitted patient care.

Apples and their derivatives are among the most widely consumed fruit products in the world and iconic examples of food‐safety issues. By using a systematic search in the PubMed, Web of Science, and Embase databases, we extracted 1374 publications on pesticides, mycotoxins, and heavy metal contents in apple products, which represented 44%, 48%, and 26% of publications on fruit, respectively. We selected 90 articles in which we were able to assess compliance with the European Food Safety Authority's (EFSA) regulations and found a 42.8% overall rate of checks exceeding the Maximum Residue Limit (MRL), a 51.6% rate for pesticides, a 42.55% rate for heavy metals, and a 40.2% rate for mycotoxins. Over 60% of the 92 pesticides considered were banned by the European Union. The rate of noncompliance was much higher in the Middle East (65.2%), Africa (50%), Asia (43.9%), Europe (37.5%), and South America (33.3%) than in North America (12.5%). We observed an influence of the climate Köppen classification and the 2024 Human Development Index (HDI) on the rate of exceeding MRLs. Our data raise questions about the compliance with production regulation requirements and the efficacy of controls. According to the criteria that define MRLs, we also question non‐negligible public health issues generated by the high rate of noncompliance.

To demonstrate for the first time the potential of stereolithography‐printed, architected, bio‐based carbon‐supported Ni catalysts for CO2 methanation, three honeycomb carbon structures with different textural properties were prepared, impregnated with 15 wt.% of nickel metal particles, and studied to correlate their catalytic performances with their textural properties and surface chemistry. Compared with the non‐activated monolith and the steam‐activated monolith, the CO2‐activated monolith achieved a higher CO2 conversion of 62 % with a CH4 selectivity of 73 % at 460 °C. Our comparative study demonstrated that the CO2‐activated carbon support, although having fewer basic sites on the surface, exhibits greater dispersion of the Ni phase, enabling an increase in H2 chemisorption. These results are of interest for further studies related to the optimization of catalytic performance through the design of the architectural and textural properties of such macrostructures.