Ashtekar variables

Short description: Variables used in general relativity

Main articles: Physics:Frame fields in general relativity, spin connection, and Self-dual Palatini action

In the ADM formulation of general relativity, spacetime is split into spatial slices and a time axis. The basic variables are taken to be the induced metric qab(x) on the spatial slice and the metric's conjugate momentum Kab(x), which is related to the extrinsic curvature and is a measure of how the induced metric evolves in time.[1] These are the metric canonical coordinates.

In 1986 Abhay Ashtekar introduced a new set of canonical variables, Ashtekar (new) variables to represent an unusual way of rewriting the metric canonical variables on the three-dimensional spatial slices in terms of an SU(2) gauge field and its complementary variable.[2]

## Overview

Ashtekar variables provide what is called the connection representation of canonical general relativity, which led to the loop representation of quantum general relativity[3] and in turn loop quantum gravity and quantum holonomy theory.[4]

Let us introduce a set of three vector fields  Eja , j=1,2,3 that are orthogonal, that is,

δjk=qab Eja Ekb.

The  Eia are called a triad or drei-bein (German literal translation, "three-leg"). There are now two different types of indices, "space" indices  a,b,c that behave like regular indices in a curved space, and "internal" indices  j,k,ℓ which behave like indices of flat-space (the corresponding "metric" which raises and lowers internal indices is simply  δjk ). Define the dual drei-bein Eaj as

Eaj=qab Ejb.

We then have the two orthogonality relationships

δjk=qab Eaj Ebk ,

where qab is the inverse matrix of the metric  qab (this comes from substituting the formula for the dual drei-bein in terms of the drei-bein into  qab Eaj Ebk and using the orthogonality of the drei-beins).

and

Eja Ebj =δba

(this comes about from contracting  δjk=qab Ekb Eja with  Ecj and using the linear independence of the  Eak ). It is then easy to verify from the first orthogonality relation, employing  Eja Ebj=δba , that

qab=∑j, k=13δjk Eja Ekb=∑j=13Eja Ejb ,

we have obtained a formula for the inverse metric in terms of the drei-beins. The drei-beins can be thought of as the 'square-root' of the metric (the physical meaning to this is that the metric  qab , when written in terms of a basis  Eja , is locally flat). Actually what is really considered is

(det(q)) qab=∑j=13E~ja E~jb ,

which involves the "densitized" drei-bein E~ia instead (densitized as  E~ja=det⁡(q) Eja ). One recovers from  E~ja the metric times a factor given by its determinant. It is clear that  E~ja and  Eja contain the same information, just rearranged. Now the choice for  E~ja is not unique, and in fact one can perform a local-in-space rotation with respect to the internal indices  j without changing the (inverse) metric. This is the origin of the  SU(2) gauge invariance. Now if one is going to operate on objects that have internal indices, one needs to introduce an appropriate derivative (covariant derivative), for example the covariant derivative for the object  Vib will be

Da Vjb=∂aVjb−Γajk Vkb+Γacb Vjc

where  Γacb is the usual Levi-Civita connection and  Γajk is the so-called spin connection. Let us take the configuration variable to be

Aaj=Γaj+β Kaj

where Γaj=Γakℓ ϵkℓj and Kaj=Kab E~bj/det⁡(q) . The densitized drei-bein is the conjugate momentum variable of this three-dimensional SU(2) gauge field (or connection)  Abk , in that it satisfies the Poisson bracket relation

{ E~ja(x), Abk(y) }=8π GNewton β δba δjk δ3(x−y).

The constant β is the Immirzi parameter, a factor that renormalizes Newton's constant  GNewton. The densitized drei-bein can be used to reconstruct the metric as discussed above, and the connection can be used to reconstruct the extrinsic curvature. Ashtekar variables correspond to the choice  β=−i (the negative of the imaginary number,  i );  Aaj is then called the chiral spin connection.

The reason for this choice of spin connection was that Ashtekar could much simplify the most troublesome equation of canonical general relativity – namely the Hamiltonian constraint of LQG. This choice made its formidable second term vanish, and the remaining term became polynomial in his new variables. This simplification raised new hopes for the canonical quantum gravity programme.[5] However, it did present certain difficulties: Although Ashtekar variables had the virtue of simplifying the Hamiltonian, it has the problem that the variables become complex.[6] When one quantizes the theory it is a difficult task to ensure that one recovers real general relativity, as opposed to complex general relativity. Also the Hamiltonian constraint Ashtekar worked with was the densitized version, instead of the original Hamiltonian; that is, he worked with H~=det⁡(q)H.

There were serious difficulties in promoting this quantity to a quantum operator. In 1996 Thomas Thiemann was able to use a generalization of Ashtekar's formalism to real connections (β takes real values) and in particular devised a way of simplifying the original Hamiltonian, together with the second term. He was also able to promote this Hamiltonian constraint to a well defined quantum operator within the loop representation.[7][8]

Lee Smolin & Ted Jacobson, and Joseph Samuel, independently discovered that there exists in fact a Lagrangian formulation of the theory by considering the self-dual formulation of the tetradic Palatini action principle of general relativity.[9][10][11] These proofs were given in terms of spinors. A purely tensorial proof of the new variables in terms of triads was given by Goldberg[12] and in terms of tetrads by Henneaux, Nelson, & Schomblond (1989).[13]

## References

1. ↑ Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald. Gravitation. New York: W. H. Freeman and Company.
2. ↑ Ashtekar, A (1986). "New variables for classical and quantum gravity". Physical Review Letters 57 (18): 2244–2247. doi:10.1103/physrevlett.57.2244. PMID 10033673. Bibcode: 1986PhRvL..57.2244A.
3. ↑ Rovelli, C.; Smolin, L. (1988). "Knot Theory and Quantum Gravity". Physical Review Letters 61 (10): 1155–1158. doi:10.1103/physrevlett.61.1155. PMID 10038716. Bibcode: 1988PhRvL..61.1155R.
4. ↑ J. Aastrup; J. M. Grimstrup (2015). "Quantum Holonomy Theory". Fortschritte der Physik 64 (10): 783. doi:10.1002/prop.201600073. Bibcode: 2016ForPh..64..783A.
5. ↑ For more details on this and the subsequent development, see  Lectures on Non-Perturbative Canonical Gravity (1st ed.). World Scientific Publishing. 1991.
6. ↑ See Baez, John; Muniain, Javier P. (1994). Gauge Fields, Knots and Gravity (1st ed.). World scientific Publishing. part III, chapter 5.
7. ↑ Thiemann, T. (1996). "Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity". Physics Letters B (Elsevier BV) 380 (3-4): 257–264. doi:10.1016/0370-2693(96)00532-1. ISSN 0370-2693.
8. ↑ For an account of these developments see Baez, John. "The Hamiltonian constraint in the loop representation of quantum gravity". University of California, Riverside. https://math.ucr.edu/home/baez/hamiltonian/hamiltonian.html.
9. ↑ Samuel, J. (April 1987). "A Lagrangian basis for Ashtekar's formulation of canonical gravity". Pramana – Journal of Physics (Indian National Science Academy) 28 (4): L429-L432. https://www.ias.ac.in/describe/article/pram/028/04/0000-0000.
10. ↑ Jacobson, Ted; Smolin, Lee (1987). "The left-handed spin connection as a variable for canonical gravity". Physics Letters B (Elsevier) 196 (1): 39–42. doi:10.1016/0370-2693(87)91672-8. ISSN 0370-2693.
11. ↑ Jacobson, T; Smolin, L. (1988-04-01). "Covariant action for Ashtekar's form of canonical gravity". Classical and Quantum Gravity 5 (4): 583–594. doi:10.1088/0264-9381/5/4/006. ISSN 0264-9381.
12. ↑ Goldberg, J.N. (1988-04-15). "Triad approach to the Hamiltonian of general relativity". Physical Review D (American Physical Society (APS)) 37 (8): 2116–2120. doi:10.1103/physrevd.37.2116. ISSN 0556-2821.
13. ↑ Henneaux, M.; Nelson, J.E.; Schomblond, C. (1989-01-15). "Derivation of Ashtekar variables from tetrad gravity". Physical Review D (American Physical Society (APS)) 39 (2): 434–437. doi:10.1103/physrevd.39.434. ISSN 0556-2821.

## Further reading

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

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