How to use Ashtekar's variables in classical gravitational physics?
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First, some general info on Ashtekar variables.
Ashtekar-Barbero connection is an su2su2 valued spatial connection
A=Aiaτidxa,A=Aaiτidxa,
which is by definition related to the so3,1∼sl2,Cso3,1∼sl2,C valued spin connection ωω by
Aija=εijkω0ka+γ−1ωija,Aaij=εijkωa0k+γ−1ωaij,
where γγ is a dimensionless constant called the Immirzi parameter.
The other part of Ashtekar's variables is given by the densitized triad field
Eai=12εabcεijkejbekc=|e|⋅eai.Eia=12εabcεijkebjeck=|e|⋅eia.
There's two exceptional properties which make this objects so useful in canonical General Relativity:
It transforms as a su2su2 valued spatial connection, e.g. under coordinate-dependent spatial rotations Ω(x)∈SU(2)Ω(x)∈SU(2),
A→ΩAΩ−1+ΩdΩ−1.A→ΩAΩ−1+ΩdΩ−1.
Note that this property is non-trivial. The second summand of AA indeed transforms like a connection, but the first summand transforms like a 3-vector. But the transformation law of the gauge connection allows us to add an arbitrary vector to it. So AA ends up being a su2su2 connection on the spatial slice.
AA and EE come naturally as a canonical conjugate pair for the Holst action
SHolst[ω,e]=∫MeI∧eJ(⋆+γ−1)FIJ,SHolst[ω,e]=∫MeI∧eJ(⋆+γ−1)FIJ,
where ⋆⋆ is an internal automorphism of so3,1so3,1 (aka the electro-magnetic duality operator) defined as
⋆FIJ=12εIJKLFKL,⋆FIJ=12εIJKLFKL,
and F=dω+ω∧ωF=dω+ω∧ω is the curvature of the spin connection. It is straightforward to check that the canonical conjugate of AA is EE. Since Holst action is classically equivalent to General Relativity (because the equations of motion coming from it are equivalent to Einstein's equations), this allows for re-interpretation of the phase space variables of General Relativity in terms of a su2su2connection AA and its canonical conjugate densitized triad field EE.
You might be wondering why this is important for Quantum Gravity. Well, in the quantum theory states are naturally functions over 1/21/2 of the phase space variables (this is sometimes called polarization in geometric quantization). This opens an interesting possibility of using certain functionals called cylindrical functions over the space of connections as describing states of the Quantum Gravity theory. These functionals can only be built over the space of connections – which is true for the Ashtekar connection AA. After defining the constraints as quantum operators on this space, the formal definition of a Quantum Gravity theory is complete. This is in fact exactly the outline of the Canonical Loop Quantum Gravity programme.
Ashtekar-Barbero connection is an su2su2 valued spatial connection
A=Aiaτidxa,A=Aaiτidxa,
which is by definition related to the so3,1∼sl2,Cso3,1∼sl2,C valued spin connection ωω by
Aija=εijkω0ka+γ−1ωija,Aaij=εijkωa0k+γ−1ωaij,
where γγ is a dimensionless constant called the Immirzi parameter.
The other part of Ashtekar's variables is given by the densitized triad field
Eai=12εabcεijkejbekc=|e|⋅eai.Eia=12εabcεijkebjeck=|e|⋅eia.
There's two exceptional properties which make this objects so useful in canonical General Relativity:
It transforms as a su2su2 valued spatial connection, e.g. under coordinate-dependent spatial rotations Ω(x)∈SU(2)Ω(x)∈SU(2),
A→ΩAΩ−1+ΩdΩ−1.A→ΩAΩ−1+ΩdΩ−1.
Note that this property is non-trivial. The second summand of AA indeed transforms like a connection, but the first summand transforms like a 3-vector. But the transformation law of the gauge connection allows us to add an arbitrary vector to it. So AA ends up being a su2su2 connection on the spatial slice.
AA and EE come naturally as a canonical conjugate pair for the Holst action
SHolst[ω,e]=∫MeI∧eJ(⋆+γ−1)FIJ,SHolst[ω,e]=∫MeI∧eJ(⋆+γ−1)FIJ,
where ⋆⋆ is an internal automorphism of so3,1so3,1 (aka the electro-magnetic duality operator) defined as
⋆FIJ=12εIJKLFKL,⋆FIJ=12εIJKLFKL,
and F=dω+ω∧ωF=dω+ω∧ω is the curvature of the spin connection. It is straightforward to check that the canonical conjugate of AA is EE. Since Holst action is classically equivalent to General Relativity (because the equations of motion coming from it are equivalent to Einstein's equations), this allows for re-interpretation of the phase space variables of General Relativity in terms of a su2su2connection AA and its canonical conjugate densitized triad field EE.
You might be wondering why this is important for Quantum Gravity. Well, in the quantum theory states are naturally functions over 1/21/2 of the phase space variables (this is sometimes called polarization in geometric quantization). This opens an interesting possibility of using certain functionals called cylindrical functions over the space of connections as describing states of the Quantum Gravity theory. These functionals can only be built over the space of connections – which is true for the Ashtekar connection AA. After defining the constraints as quantum operators on this space, the formal definition of a Quantum Gravity theory is complete. This is in fact exactly the outline of the Canonical Loop Quantum Gravity programme.
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