Coherence condition

Coherence conditions in the sense of physics occur in field theory.

If a Lagrangian of a field is known, then in the general case there exists a system of differential equations for the field potential. In each class of solutions of those equations, an integral coherence condition is valid.

This is more simply understood in examples. An introductory case is the Lagrangian for a scalar field with potential

s(x)

in one-dimensional space: take the following

There is an equation system and coherence condition for this field.

then

The coherence condition in this case is the following:

<math>0=\int_{0}^{\infty}[\dot s^2\delta s+2s\dot s\delta \dot s]\,dx.<math>

In the class of linear functions, the variation

<math>\delta s=\delta b+x\delta a<math>

where

<math>\delta a<math>, and <math>\delta b<math>, are non-fixed numbers.

<math>0=\int_{0}^{\infty}[a^2(\delta b+x\delta a)+2a(b+ax)\delta a]\,dx.<math>

Then

a = 0, b =constant.

When the field equation

is valid for interacted field then by physical meaning this state is the vacuum state of the field.

then the coherence condition in this state is

<math>0=\int_{0}^{\infty} s \dot s \delta \dot s\,dx<math><math>0=\int_{0}^{\infty} [b-\ln(a+x) {1\over (a+x)}\,dx,<math>

where the integration constant a remains unknown.

then the coherence condition is

<math>o=\int_{0}^{\infty}\dot s^2\delta s\,dx=a^2\int_{0}^{\infty}{\delta b+x\delta a\over (1+2s)^2}\,dx.<math>

In this state a = 0 and the field is in its vacuum state.

When the variation of the Lagrangian is made in the usual way,

and the coherence condition is absent.

This field has two vacuum states, one coherent state and one standard state.

Differential equations with coherence conditions set up the full equation system for field potentials.

Categories: Theoretical physics