Homeomorphism between the space of all Ashtekar connections and spacetime?

Question

Excerpt from an essay of mine:

Let $\Psi(\varsigma)$ be the wavefunction in the loop representation, where $\varsigma:[0,1]\to\mathcal{M}$, where $\mathcal{M}$ is spacetime. Then, let $\mathcal{A}$ be the Ashtekar connection and $\mathcal{W}_\varsigma[\mathcal{A}]$ be the Wilson loop of the connection $\mathcal{A}$. Give the connection the action $S[\mathcal{A}]=\oint_\varsigma -i\mathcal{A}$. One then has \begin{multline} \Psi(\varsigma)\Psi(\varsigma_{1})=\langle \mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_1}[\mathcal{A}_1]\rangle=\langle0| \mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_1}[\mathcal{A}_1]|0\rangle=\\ \int D\mathcal{A}\int D\mathcal{A}\left(\operatorname{Tr}\left(\mathcal{P}\exp\left(\oint_\varsigma -i\mathcal{A}\right)\right)\right)^2\mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_{1}}[\mathcal{A}_1] \end{multline} This, as is written in the equation, has the form of a 2 Wilson loop correlation function. For simplicity, we assume an interaction of the form $(\mathcal{W}_\varsigma[\mathcal{A}])^2$. Let $|\Theta\rangle$ be the ground state of loop quantum gravity. Then, one has the S-matrix elements of loop quantum gravity in the space of all Ashtekar connections as \begin{equation} \langle\Theta|\mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_1}[\mathcal{A}_1]|\Theta\rangle=\sum^\infty_{n=0}(-i\lambda)^n\int\mathrm{d}^4x_n\langle0|\mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_1}[\mathcal{A}_1]\prod^n_{j=1}(\mathcal{W}_{\varsigma}[\mathcal{A}^j])^2|0\rangle=\sum^\infty_{n=0}\Theta^{(n)} \end{equation} Here, $\lambda$ is the coupling constant of LQG. $\mathcal{A}^j$ stands for different connections, i.e., $\mathcal{A}^1=\mathcal{B},\mathcal{A}^2=\mathcal{C}$, e.t.c. The Feynman diagrams come from $\langle\Theta|\mathcal{W}_\varsigma[\mathcal{A}]\mathcal{W}_{\varsigma_1}[\mathcal{A}_1]|\Theta\rangle=D_F(\mathcal{A},\mathcal{A}_1)+\mbox{\emph{all possible contractions}}$. Here, ``all possible contractions'' means all possible pairings between the connections of which the Wilson loop is a functional. We choose to study only a limited number of contractions.

We write two contractions for $\Theta^{(1)}$: 1)$D_F(\mathcal{A},\mathcal{A}_1)\left(D_F(\mathcal{B,B})\right)^2$

2) $D_F(\mathcal{A,B})D_F(\mathcal{A}_1,\mathcal{B})D_F(\mathcal{B,B})$

We write four contractions for $\Theta^{(2)}$: 1) $D_F(\mathcal{A,A}_1)\left(D_F(\mathcal{B,B})\right)^2D_F(\mathcal{B,C})\left(D_F(\mathcal{C,C})\right)^2$

2) $D_F(\mathcal{A},\mathcal{B})D_F(\mathcal{A}_1,\mathcal{B})D_F(\mathcal{B,B})D_F(\mathcal{B,C})\left(D_F(\mathcal{C,C})\right)^2$

3)$D_F(\mathcal{A},\mathcal{B})D_F(\mathcal{A}_1,\mathcal{C})\left(D_F(\mathcal{B,B})\right)^2\left(D_F(\mathcal{C,C})\right)^2$

4) $D_F(\mathcal{A,C})D_F(\mathcal{A}_1,\mathcal{C})\left(D_F(\mathcal{B,B})\right)^2D_F(\mathcal{B,C})D_F(\mathcal{C,C})$

Note that this list does not contain every single propagator.

In the space of all Ashtekar connections (which is a subset of the space of all principal connections of spacetime), the interactions of LQG are given by finding out all possible contractions of the Feynman propagators and drawing diagrams in the space of all Ashtekar connections.

My question thus is: Is it possible to somehow find a homeomorphism between the space of all Ashtekar connections and spacetime (so that the interactions of LQG can be formulated on spacetime itself)?

This post imported from StackExchange Physics at 2015-10-20 21:59 (UTC), posted by SE-user Sanath Devalapurkar

Comments

The short answer is: no. Why? We have two different topological invariants -- the dimension of the "space of all Ashtekar connections" is infinite-dimensional, whereas spacetime is 4-dimensional. And $4\neq\infty$.

This post imported from StackExchange Physics at 2015-10-20 21:59 (UTC), posted by SE-user Alex Nelson