Ashtekar variables
Encyclopedia
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
where the densitized drei-bein
is the dual variable of a three-dimensional SU(2) gauge field
In the first equation,
is the Immirzi parameter
, a factor that renormalizes Newton's constant
. While the field redefinition above is faithful locally on the configuration space of the three-dimensional metric tensor
, it introduces new periodicities (the Wilson loop
s 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.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, Ashtekar (new) variables (named after Abhay Ashtekar
Abhay Ashtekar
Abhay Vasant Ashtekar is an Indian theoretical physicist. He is the Eberly Professor of Physics and the Director of the Institute for Gravitational Physics and Geometry at Pennsylvania State University. As the creator of Ashtekar variables, he is one of the founders of loop quantum gravity and its...
who invented them) represent an unusual way to rewrite the metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a 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
Loop quantum gravity
Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
.
The basic relation is
where the densitized drei-bein
is the dual variable of a three-dimensional SU(2) gauge fieldIn the first equation,
is the Immirzi parameterImmirzi parameter
The Immirzi parameter is a numerical coefficient appearing in loop quantum gravity, a nonperturbative theory of quantum gravity. The Immirzi parameter measures the size of the quantum of area in Planck units...
, a factor that renormalizes Newton's constant
. While the field redefinition above is faithful locally on the configuration space of the three-dimensional metric tensorMetric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
, it introduces new periodicities (the Wilson loop
Wilson loop
In gauge theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop...
s 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.