Why is the gauge group for Ashtekar variable theory for General

In summary, the Ashtekar variable theory for General Relativity is based on the idea of using a new set of variables, known as Ashtekar variables, to describe the geometry of spacetime. These variables are derived from the traditional metric variables used in General Relativity, but they have the advantage of being more closely related to the underlying geometric structure of spacetime. This allows for a more elegant and powerful formulation of the theory, particularly when it comes to the quantization of gravity. Additionally, the Ashtekar variables are associated with a gauge group, which allows for a deeper understanding of the symmetries and constraints of the theory. This makes the gauge group an essential component of the Ashtekar variable theory
  • #1
yyoon@fas.harvard.edu
As far as I understand, the gauge group for Ashtekar variable theory
for General Relativitiy is SU(2), because its group is three-
dimensional. But, why is this SU(2) instead of SO(3)? Are there any
plausible reasonings behind this? And, are there any phenomenological
differences if the gauge group were SO(3) instead of SU(2)?

Thanks in advance

Youngsub
 
Physics news on Phys.org
  • #2
The Lie group SU(2) is isomorphic to SO(3), that is, they're structurally equivalent.
 

1. Why is the gauge group important in Ashtekar variable theory for General Relativity?

The gauge group in Ashtekar variable theory is crucial because it allows for the description of the gravitational field in terms of connections rather than metric tensors. This leads to a more elegant and mathematically consistent formulation of General Relativity.

2. How does the gauge group affect the dynamics of General Relativity in Ashtekar variable theory?

The gauge group plays a central role in the dynamics of General Relativity in Ashtekar variable theory. It provides the framework for the equations of motion and determines the physical observables that can be measured in the theory.

3. What is the significance of the SU(2) gauge group in Ashtekar variable theory?

The SU(2) gauge group is important in Ashtekar variable theory because it corresponds to the local rotations of the spatial frame. This allows for the description of spatial diffeomorphisms, which are crucial for the covariance of General Relativity.

4. Why is the gauge group in Ashtekar variable theory non-Abelian?

The non-Abelian nature of the gauge group in Ashtekar variable theory is necessary for the consistency of the theory. It allows for the formulation of the Gauss constraint, which is one of the fundamental equations of motion in the theory.

5. How does the gauge group in Ashtekar variable theory relate to other theories of gravity?

The gauge group in Ashtekar variable theory is closely related to other theories of gravity, such as loop quantum gravity and spin foam models. It provides a connection between these theories and offers a different perspective on the geometric and quantum properties of space-time.

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