Ashtekar variables

In theoretical physics, Ashtekar (new) variables (named after Abhay Ashtekar who invented them) represent an unusual way to rewrite the metric on the three-dimensional spatial slices in terms of a SU(2) gauge field and its complementary variable. Ashtekar variables are the key building block of loop quantum gravity.

The basic relation is

$(\mathrm{det}_{3\times 3} g) g^{ij} = 8\pi G_{\mathrm{Newton}}\gamma\sum_{a=1}^{3} E_a^i E_a^j,$

where the densitized drei-bein $E_a^i$ is the dual variable of a three-dimensional SU(2) gauge field

$[E_a^i, A_j^b] \sim i\hbar \delta_a^b\delta_j^i.$

In the first equation, γ is the Immirzi parameter, a factor that renormalizes Newton's constant G_Newton. While the field redefinition above is faithful locally on the configuration space of the three-dimensional metric tensor, it introduces new periodicities (the Wilson loops of all gauge fields take values in the space of complex units) and quantization laws that cannot be derived from the metric itself.

In loop quantum gravity, these are manifested as the area quantization rules. These rules do not follow from the metric tensor and its quantization, but rather from the special global properties of Ashtekar's field redefinition. A different field redefinition could "predict" the quantization of other quantities than the area.

Further reading

• Ashtekar, Abhay (1986). "New Variables for Classical and Quantum Gravity". Physical Review Letters 57 (18): 2244–2247. Bibcode 1986PhRvL..57.2244A. doi:10.1103/PhysRevLett.57.2244. PMID 10033673.