Derrick’s theorem(2)

Related post : Derrick’s theorem

Consider a theory in D spatial dimensions involving one or more scalar fields $\phi_a$, with a Lagrangian density of the form $$L= \frac{1}{2} G_{ab}(\phi) \partial_\mu \phi_a \partial^\mu \phi_b- V(\phi)$$ where the eigenvalues of G are all positive definite for any value of $\phi$, and V = 0 at its minima. Any finite energy static solution of the field equations is a stationary point of the potential energy $$E = I_K + I_V ,$$ where $$I_K[\phi]= \frac{1}{2} \int d^Dx G_{ab}(\phi) \partial_j \phi_a \partial_j\phi_b$$ and $$I_V = \int d^Dx V(\phi)$$ are both positive. Since the solution is a stationary point among all configura- tions, it must, a fortiori, also be a stationary point among any subset of these ̄ configurations to which it belongs. Therefore, given a solution $\phi(x)$, consider the one-parameter family of configurations, $$f_\lambda(x)= \bar{\phi}(\lambda x)$$ that are obtained from the solution by rescaling lengths. The potential energy of these configurations is given by

\begin{align} E_\lambda &= I_K(f_\lambda) + I_V(f_\lambda)\\ > &=\lambda^{2-D} I_K[\bar\phi]+\lambda^{-D} I_V[\bar\phi]\tag{1} > \end{align}

My query from the above statement:

1. Can you explain The $G_{ab}$

2. how equation (1) came from the previous equation.

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1) Explain what about the $G_{ab}$? 2) What have you tried so far? –  user1504 May 1 '13 at 15:22
I mean why It arises in the equation?> I mean the significance? –  Unlimited Dreamer May 1 '13 at 15:32
$G_{ab}$ is just part of the definition of whatever model you want to consider. For (2) hint: you had the integration variables $x$, now you have the integration variables $\lambda x$.... –  twistor59 May 1 '13 at 16:06
$G_{ab}$ has the interpetation of a metric on the space of fields. You're probably used to fields which are part of a linear space. In that case $G_{ab}$ is a constant matrix which can be removed by field redefinitions. But you can also allow the fields to live in an abstract curved manifold. Think of $\phi_a$ as coordinates in field space like $x_\mu$ are spacetime coordinates in GR. So this is just a generalisation of what you are used to. –  Michael Brown May 1 '13 at 16:12
\begin{align} E_\lambda &= I_K(f_\lambda) + I_V(f_\lambda)\\ &=\lambda^{2-D} I_K[\bar\phi]+\lambda^{-D} I_V[\bar\phi]\tag{1} \end{align} How they considered this dimension? I mean how 2nd line came from the first line? –  Unlimited Dreamer May 2 '13 at 8:50