## Ex nihilo nihil fit ~ Parmenides

In this post, in order to solve the Wheeler-deWitt problem-of-time I discussed in my last post, and since string/M-theoretic braneworld cosmology offers the only solution in a unified C*-Heisenberg-algebraic sense, I must first analyse a field theory on a D-brane sigma world-volume and study the Dirac-Born-Infeld (DBI) action. Let me start by giving a proof that the DBI action has a symmetry under the transformation

of full

on a generalized Riemannian structure ${C_ + } \subset \Im M$

First, I must prove the following relation

with $g$ the Riemannian metric on $\Im M$

and $\not S\widehat {\not F}$ the metric on ${\not L_{\widehat {\not F}}}$

Now, since $\varphi _\Phi ^ *$ is the pullback of the embedding map defined by the field $\Phi$, we have

Thus, the determinants on the l. h. s. are those of the $D \times D$ matrices, while the determinant on the r.h.s. is that of the $\left( {p + 1} \right) \times \left( {p + 1} \right)$ matrix distinguished by the index $ab$. Now, the r. h. s. of

is the Lagrangian ${L_{DBI}}$ of the Dirac-Born-Infeld  action and can be proven by combining the following relations of various determinants

1) Let $s$ be the symmetric part of $t$ given by

and

and ${t^{ij}} = {E^{ij}}$ the inverse matrix of ${E_{ij}}$, then we have

2) From

and the definition of $\not S\widehat {\not F}$, we have

3) Now, by using the explicit expression for ${t_{\widehat {\not F}}}$, it follows that

Using these relations in 1), 2), 3), one can prove the representation of the DBI action given in

which is

The integral of the scalar density agrees with the DBI action

when evaluated on the leaf $\varphi \,\Phi \left( \Sigma \right)$ of ${L_{\widehat {\not F}}}$ at ${x^i} = {\Phi ^i}(x)$,  hence, the DBI action is invariant under the world-volume diffeomorphism on the D-brane.

Now I am in a position to prove that the DBI action is not only invariant under the world-volume diffeomorphism but also under the full target space diffeomorphism and the B-field gauge transformation. Towards this end,

can be re-written as an integral over the target space $M$ as

where

is a Dirac delta function interpreted as a distribution along ${x^i}$ directions. The infinitesimal transformation of the full diffeomorphism and the B-field gauge transformation are parametrized by $\varepsilon + \Lambda$.  So the transformation of the integrand ${\widehat {\not L}_{DBI}}$ is gotten from that of $\det g$

and that of $\not S\widehat {\not F}$

And the glorious result is

with

being central to full diffeomorphism. This is the expected result since ${\widehat {\not L}_{DBI}}d{x^0} \wedge ... \wedge d{x^p}$ is a section of ${\rm{det}}\left( {{\Delta ^ * }} \right)$

On the other hand, the delta function transforms as

Now, combining, we get

as desired, hence, the transformation of the integrand in the DBI action

is a total derivative and the DBI action is invariant under full target space diffeomorphisms and B-field gauge transformations and the invariance within the static gauge is what will allow us to further analyse the Wheeler-deWitt problem-of-time.