**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Lagrangian (field theory)

Summary

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
One motivation for the development of the Lagrangian formalism on fields, and more generally, for classical field theory, is to provide a clean mathematical foundation for quantum field theory, which is infamously beset by formal difficulties that make it unacceptable as a mathematical theory. The Lagrangians presented here are identical to their quantum equivalents, but, in treating the fields as classical fields, instead of being quantized, one can provide definitions and obtain solutions with properties compatible with the conventional formal approach to the mathematics of partial differential equations. This enables the formulation of solutions on spaces with well-characterized properties, such as Sobolev spaces. It enables various theorems to be provided, ranging from proofs of existence to the uniform convergence of formal series to the general settings of potential theory. In addition, insight and clarity is obtained by generalizations to Riemannian manifolds and fiber bundles, allowing the geometric structure to be clearly discerned and disentangled from the corresponding equations of motion. A clearer view of the geometric structure has in turn allowed highly abstract theorems from geometry to be used to gain insight, ranging from the Chern–Gauss–Bonnet theorem and the Riemann–Roch theorem to the Atiyah–Singer index theorem and Chern–Simons theory.
In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a Riemannian manifold.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (2)

Related concepts (108)

Related courses (15)

PHYS-324: Classical electrodynamics

The goal of this course is the study of the physical and conceptual consequences of Maxwell equations.

EE-201: Electromagnetics II : field computation

Ce cours traite de l'électromagnétisme dans le vide et dans les milieux continus. A partir des principes fondamentaux de l'électromagnétisme, on établit les méthodes de résolution des équations de Max

PHYS-202: Analytical mechanics (for SPH)

Présentation des méthodes de la mécanique analytique (équations de Lagrange et de Hamilton) et introduction aux notions de modes normaux et de stabilité.

Static forces and virtual-particle exchange

Static force fields are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces arising from the interactions between two bodies mediated by virtual particles, particles that exist for only a short time determined by the uncertainty principle. The virtual particles, also known as force carriers, are bosons, with different bosons associated with each force.

Frame fields in general relativity

A frame field in general relativity (also called a tetrad or vierbein) is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by and the three spacelike unit vector fields by . All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field.

Field (physics)

In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each point on the map; the temperature can be considered at a certain point in time or over some interval of time, to study the dynamics of temperature change. A surface wind map, assigning an arrow to each point on a map that describes the wind speed and direction at that point, is an example of a vector field, i.

Related lectures (117)

Path Integral: FundamentalsPHYS-314: Quantum physics II

Covers the fundamentals of path integral and its applications in physical scenarios.

Classical Field Theory: Lagrangian and Hamiltonian FormulationPHYS-431: Quantum field theory I

Explores classical field theory, focusing on Lagrangian formulation and the Euler-Lagrange equations, emphasizing the property of locality in spacetime.

Quantum Field Theory: Conservation LawsPHYS-431: Quantum field theory I

Explores deriving conserved currents in classical and quantum field theory, emphasizing symmetries and equations of motion.

Quantum Field Theory(QFT) as one of the most promising frameworks to study high energy and condensed matter physics, has been mostly developed by perturbative methods. However, perturbative methods can only capture a small island of the space of QFTs.QFT in hyperbolic space can be used to link the conformal bootstrap and massive QFT. Conformal boundary correlators also can be studied by their general properties such as unitarity, crossing symmetry and analicity. On the other hand, by sending the curvature radius to infinity we reach to the flat-space limit in hyperbolic space. This allows us to use conformal bootstrap methods to study massive QFT in one higher dimension and calculate observables like scattering amplitudes or finding bounds on the couplings of theory. The main goal of my research during my Ph.D. would be to study QFTs in hyperbolic space to better understand strongly coupled QFTs.Hamiltonian truncation is a numerical method to study strongly coupled QFTs by imposing a UV cutoff. We use this method to study strongly coupled QFT in hyperbolic space background. For simplicity, we start with scalar field theory in 2-dimensional AdS. We expect to extract the spectrum of our theory as a function of AdS curvature and find the boundary correlation functions.

Thermal and quantum phase transitions of some rare earth compounds (LiErF4, LiYbF4, LiGdF4 and LiTmF4) are established using the mean field theory. These preliminary calculations allowed evidencing the existence of a novel high-field antiferromagnetic phase in LiErF4, and a still unexplained symmetry breaking in LiGdF4. But the discrepancies with experimental results impel a more sophisticated method. We then present analytical and numerical evidence for the validity of an effective approach to the description of the dipolar coupled antiferromagnet LiErF4. We show that the approach, when implemented in mean field calculations, is able to capture both the qualitative and quantitative aspects of the physics of LiErF4 at small external field and low temperature, yielding results that agree with those obtained in the full Hilbert space using mean field theory. This model nevertheless still fails to describe the LiHoF4 system and needs to be improved. We finally use this toy model as a basis for classical Monte Carlo simulations of LiErF4, which allows the calculation of thermodynamical quantities of the system, as well as the evolution of the order parameters as a function of field H and temperature T. These calculations yield results that are much closer to the experiments than those based on the mean field approximation. Although the theoretical critical temperature is still overestimated by 34%, the critical exponents computed from this effective model correspond to those found experimentally.

2012