The Lagrangian density

In field theory the Lagrangian density is referred to simply as the Lagrangian, using the same symbol L. The variation of L is written explicitly as... 

A covariant Lagrangian density (i.e. an equation that looks the same in all Lorentz frames) will be obtained if the Lagrangian density is defined in terms of a relativistic scalar density, as in... 

The electromagnetic field may now formally be interpreted as the gauge field which must be introduced to ensure invariance under local U( 1) gauge transformation. In the most general case the field variables are introduced in terms of the Lagrangian density of the field, which itself is gauge invariant. In the case of the electromagnetic field, as before,... 

In the most general case the Lagrangian density of a field suffers a reduction of symmetry at some critical value of an interaction parameter. Suppose that... 

The two field equations may be generated from the Lagrangian density... 

We now apply the generalized Matsubara formalism, discussed earlier, to a fermionic theory aiming to discuss effects of simultaneous spatial confinement and finite temperature. We consider the Wick-ordered massive Gross-Neveu model in a D-dimensional Euclidean space, described by the Lagrangian density (D.J. Gross et.al., 1974)... 

The two SU(2) theories can be represented as the block diagonals of the SU(4) gauge theory. The Lagrangian density for the system is then... 

Equation (256) serves to define the Lagrangian density, L, corresponding to Euler density p. 

A major complication exists for constructing the Lagrangian density of a pair of particles diffusing relative to each other. The diffusion (Euler) equation is dissipative and the density of the diffusing species is not conserved. The Euler density, p, would lead to a space—time invariant, Sfr, which would not be constant. This difficulty requires the same approach as that used to handle the Schrodinger equation. Morse and Feshbach [499] define a reverse or backward diffusion equation where time goes backwards compared with that in eqn. (254)... 

It is of special interest to make the connection of this with the Lagrangian density. Consider this expression multiplied by — 1/2, and set... 

This approach to defining the Lagrangian density with the aid of both forward and backward Euler densities ip and ip uses the neat construct that ip ip is time-invariant. This is as true in the quantum mechanical analogy. 

So far, the Lagrangian density for a homogenous problem (no sink or source term in the diffusion equation) has been considered, subject to the requirement that the approximate trial function, ip, can be forced to satisfy the boundary conditions. In this sub-section, these limitations are removed and the Lagrangian density for the Green s function developed. The Green s functions for the forward and backward time process satisfy the equations... 

The Lagrangian density produced by these scalar fields is, as we have seen... 

The origin of this problem traces back to the fact that, because the Lagrangian density is degree zero in the temporal ordering parameter, it is then invariant with respect to any transformation of this parameter that preserves the ordering. 

The geodesic equations in the space with the metric tensor (6) can be obtained, in the usual way, by defining the Lagrangian density... 

A scalar field is defined from the Lagrangian density L and the action S =... 

One can do the same thing for a scalar field, except that since the field is defined everywhere in space, one uses the Lagrangian density L,... 

Note that since we start from the Fourier components of the field, a few differences arise as compared to the study of a field in real space. The Lagrangian density is given by... 

QED theory is based on two distinct postulates. The first is the dynamical postulate that the integral of the Lagrangian density over a specified space-time region is stationary with respect to variations of the independent fields Atl and ijr, subject to fixed boundary values. The second postulate attributes algebraic commutation or anticommutation properties, respectively, to these two elementary fields. In the classical model considered here, the dynamical postulate is retained, but the algebraic postulate and its implications will not be developed in detail. 

Noether s theorem will be proved here for a classical relativistic theory defined by a generic field , which may have spinor or tensor indices. The Lagrangian density (, 9/x) is assumed to be Lorentz invariant and to depend only on scalar forms defined by spinor or tensor fields. It is assumed that coordinate displacements are described by Jacobi s theorem S(d4x) = d4x 9/xgeneral variation of the action integral, evaluated over a closed space-time region 2, is... 

These formulas are valid for the nonrelativistic one-electron Schrodinger equation. The Lagrangian density is... 

For the Dirac field in an externally determined Maxwell field, the Lagrangian density including a renormalized mass term is... 

As in the case of the electromagnetic self-mass, the implied dynamical mass increment is infinite unless perturbation-theory sums are truncated by a renormalization cutoff procedure. In analogy to electrodynamics, each fermion field acquires an incremental dynamical mass through interaction with the gauge field. This implies in electroweak theory that neutrinos must acquire such a dynamical mass from their interaction with the SUIT) gauge field. For a renormalized Dirac fermion in an externally determined SUIT) gauge field, the Lagrangian density is... 

In the presence of an externally determined fermion gauge current, the Lagrangian density for the 5(7(2) gauge field is [435]... 

The Lagrangian density for a massless fermion field interacting with the S(J(2) gauge field is... 

Because field quantization falls outside the scope of the present text, the discussion here has been limited to properties of classical fields that follow from Lorentz and general nonabelian gauge invariance of the Lagrangian densities. Treating the interacting fermion field as a classical field allows derivation of symmetry properties and of conservation laws, but is necessarily restricted to a theory of an isolated single particle. When this is extended by field quantization, so that the field amplitude rjr becomes a sum of fermion annihilation operators, the theory becomes applicable to the real world of many fermions and of physical antiparticles, while many qualitative implications of classical gauge field theory remain valid. 

Special techniques are required to describe the symmetry of fields. Since fields are defined in terms of continuous variables it is desirable to formulate suitable transformations of dynamic variables pertaining to fields, in terms of continuous parameters. This is done by using Hamilton s principle and defining quantities such as momentum densities for any field. The most useful parameter to quantify the symmetry of a field is the Lagrangian density (T 3.3.1). 

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