- #1
jfy4
- 649
- 3
Hi,
There is clearly an analogous relationship between Yang-Mills theories and LQG, for example the covariant derivative, the field strength tensor, the field and its conjugate momentum etc...
I am wondering how far this analogy extends. Specifically, is the Lagrangian for LQG the following?
[tex]
\mathcal{L}=-\frac{1}{4}F^{\,i}_{ab}F^{\,ab}_{i}
[/tex]
where
[tex]
F^{\,i}_{ab}=\partial_a A_{b}^{i}-\partial_b A_{a}^{i}+\epsilon^{i}{}_{jk}A_{a}^{j}A_{b}^{k}
[/tex]
There is clearly an analogous relationship between Yang-Mills theories and LQG, for example the covariant derivative, the field strength tensor, the field and its conjugate momentum etc...
I am wondering how far this analogy extends. Specifically, is the Lagrangian for LQG the following?
[tex]
\mathcal{L}=-\frac{1}{4}F^{\,i}_{ab}F^{\,ab}_{i}
[/tex]
where
[tex]
F^{\,i}_{ab}=\partial_a A_{b}^{i}-\partial_b A_{a}^{i}+\epsilon^{i}{}_{jk}A_{a}^{j}A_{b}^{k}
[/tex]