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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 .
A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories.
In Lagrangian mechanics, the generalized coordinates form a discrete set of variables that define the configuration of a system. In classical field theory, the physical system is not a set of discrete particles, but rather a continuous field ϕ(r, t) defined over a region of 3D space.
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations . Formally, the Lagrangian is invariant under these transformations.
The star is surrounded by a very strong magnetic field (10 13 G), and birefringence is expected from the vacuum polarization described by the Euler–Heisenberg Lagrangian. A degree of polarization of about 16% was measured and was claimed to be "large enough to support the presence of vacuum birefringence, as predicted by QED".
For a scalar field theory with D spacetime dimensions, the only dimensionless parameter g n satisfies n = 2D ⁄ (D − 2). For example, in D = 4, only g 4 is classically dimensionless, and so the only classically scale-invariant scalar field theory in D = 4 is the massless φ 4 theory.
Lagrangian (field theory), a formalism in classical field theory; Lagrangian point, a position in an orbital configuration of two large bodies; Lagrangian coordinates, a way of describing the motions of particles of a solid or fluid in continuum mechanics; Lagrangian coherent structure, distinguished surfaces of trajectories in a dynamical system
For example, renormalization in QED modifies the mass of the free field electron to match that of a physical electron (with an electromagnetic field), and will in doing so add a term to the free field Lagrangian which must be cancelled by a counterterm in the interaction Lagrangian, that then shows up as a two-line vertex in the Feynman diagrams.