nLab Euler-Lagrange equation

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Contents

Context

Variational calculus

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Equality and Equivalence

Contents

Idea

In variational calculus the Euler-Lagrange equations of a nonlinear functional arising from transgression of a local Lagrangian density characterize the extrema of that functional, hence its critical locus (the “shell”).

This originates from and is mainly used in physics, specifically in Lagrangian field theory, where the functional in question is the action functional of a physical system, and where its critical points encode the physically realized field histories by the principle of extremal action.

Details

See at A first idea of quantum field theory this def.

Examples

References

Named after Leonhard Euler and Joseph-Louis de Lagrange.

Textbook account in the context of gauge theory:

See also:

Discussion in the convenient context of smooth sets:

Last revised on April 14, 2024 at 15:42:25. See the history of this page for a list of all contributions to it.