Lagrangian density

Another possibility to represent the quantum mechanical Lagrangian density is using the logarithm of the amplitude X = Ina, a = e. In that particular representation, the Lagrangean density takes the following symmetrical fomi... 

We thus obtain a Lagrangean density, whieh is equivalent to Eq. (149) for all solutions of the Dirac equation, and has the structure of the nonrelativistic Lagrangian density, Eq. (140). Its variational derivations with respect to v / and v / lead to the solutions shown in Eq. (152), as well as to other solutions. 

These field equations are derivable from the following 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 formalism can be extended for a quantum Jield with the TFD Lagrangian density given by t = — , where is a replica of for the tilde fields so leading to similar equations of motion. For the purpose of our applications, we shall restrict our analysis to free massless fields. Thus, considering the free-massless boson (Klein-Gordon) field, the two-point Green function in the doubled space is given by... 

The generalized Lagrangian density of the non-linear er-w-model in the RMF approximation used for modeling the phase of uniform nuclear matter containing interacting neutrons, protons, muons and electrons can be written as... 

Variational calculus with this Lagrangian density leads [17] to the field equation ... 

In quantum field theory, the gauge field is determined by its Lagrangian density, and the fermion field, by the Dirac Lagrangian density ... 

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... 

Equation (261) for the general Lagrangian density is rather unwieldy and unlikely to be of much direct use. Instead, the steady state version (G = G ) can be written forg = /f Gdf0... 

In the steady-state Lagrangian density of eqn. (261), 17 = 0 and the delta function term and sink terms have been dropped. The quantities 0 and A have to be evaluated for each boundary and are noted above. The invariant is... 

Lagrangian density of eqn. (262)]. As only diffusion of the fluorophors is of interest, then... 

The non-simply connected U(l) vacuum is considered first to illustrate the method as simply as possible. This is defined as earlier in this review by the globally invariant Lagrangian density... 

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

As discussed in Section I.C we will say that two scalars are dual or that they form a dual pair if they verify the duality constraint (15) or, equivalently, (119) for any given time.] According to the method of the Lagrange multipliers, let us vary, as independent fields, the two scalars < ) and 0 in the modified Lagrangian density... 

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. 

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