Editing Ashtekar variables

{{Short description|Variables used in general relativity}}
{{main|Frame fields in general relativity|spin connection|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]] <math>q_{ab} (x)</math> on the spatial slice and the metric's conjugate momentum <math>K^{ab} (x)</math>, which is related to the [[extrinsic curvature]] and is a measure of how the induced metric evolves in time.<ref>{{cite book |title=[[Gravitation (book)|Gravitation]] |first=Charles W. |last=Misner |first2=Kip S. |last2=Thorne |first3=John Archibald |last3=Wheeler |publisher=W. H. Freeman and Company |location=New York |isbn= }}</ref> 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.<ref>{{cite journal | last1 = Ashtekar | first1 = A | year = 1986 | title =  New variables for classical and quantum gravity| journal = Physical Review Letters | volume = 57 | issue = 18| pages = 2244–2247 | doi=10.1103/physrevlett.57.2244 | pmid=10033673|bibcode = 1986PhRvL..57.2244A }}</ref>

==Overview==
Ashtekar variables provide what is called the connection representation of canonical general relativity, which led to the loop representation of quantum general relativity<ref>{{cite journal | last1 = Rovelli | first1 = C. | last2 = Smolin | first2 = L. | year = 1988| title =  Knot Theory and Quantum Gravity| journal = Physical Review Letters | volume = 61 | issue = 10| pages = 1155–1158 | doi=10.1103/physrevlett.61.1155| pmid = 10038716 |bibcode = 1988PhRvL..61.1155R }}</ref> and in turn [[loop quantum gravity]] and [[quantum holonomy]] theory.<ref>{{cite journal |author1= J. Aastrup |author2=J. M. Grimstrup |year= 2015 |title=Quantum Holonomy Theory  |journal=Fortschritte der Physik |volume=64 |issue=10 |pages=783 |arxiv = 1504.07100|bibcode=2016ForPh..64..783A |doi=10.1002/prop.201600073 }}</ref>

Let us introduce a set of three vector fields <math>\ E^a_j\ ,</math> <math>\ j = 1,2,3\ </math> that are orthogonal, that is,

:<math>\delta_{jk} = q_{ab}\ E_j^a\ E_k^b ~.</math>

The <math>\ E_i^a\ </math> are called a triad or ''drei-bein'' (German literal translation, "three-leg"). There are now two different types of indices, "space" indices <math>\ a,b,c\ </math> that behave like regular indices in a curved space, and "internal" indices <math>\ j,k,\ell\ </math> which behave like indices of flat-space (the corresponding "metric" which raises and lowers internal indices is simply <math>\ \delta_{jk}\ </math>). Define the dual ''drei-bein'' <math>\ E^j_a\ </math> as

:<math>\ E^j_a = q_{ab}\ E^b_j ~.</math>

We then have the two orthogonality relationships

:<math>\ \delta^{jk} = q^{ab}\ E^j_a\ E^k_b\ ,</math>

where <math>q^{ab}</math> is the inverse matrix of the metric <math>\ q_{ab}\ </math> (this comes from substituting the formula for the dual ''drei-bein'' in terms of the ''drei-bein'' into <math>\ q^{ab}\ E^j_a\ E^k_b\ </math> and using the orthogonality of the ''drei-beins'').

and

:<math>\ E_j^a\ E^j_b\ = \delta_b^a\ </math>

(this comes about from contracting <math>\ \delta_{jk} = q_{ab}\ E_k^b\ E_j^a\ </math> with <math>\ E^j_c\ </math> and using the [[linear independence]] of the <math>\ E_a^k\ </math>). It is then easy to verify from the first orthogonality relation, employing <math>\ E_j^a\ E^j_b = \delta_b^a\ ,</math> that

:<math>\ q^{ab} ~=~ \sum_{j,\ k=1}^{3}\; \delta_{jk}\ E_j^a\ E_k^b ~=~ \sum_{j=1}^{3}\; E_j^a\ E_j^b\ ,</math>

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 <math>\ q^{ab}\ ,</math> when written in terms of a basis <math>\ E_j^a\ ,</math> is locally flat). Actually what is really considered is

:<math>\ \left( \mathrm{det} (q) \right)\ q^{ab} ~=~ \sum_{j=1}^{3}\; \tilde{E}_j^a\ \tilde{E}_j^b\ ,</math>

which involves the ''"densitized"'' ''drei-bein'' <math>\tilde{E}_i^a</math> instead {{big|(}}''densitized'' as <math display="inline">\ \tilde{E}_j^a = \sqrt{ \det (q)\ }\ E_j^a\ </math>{{big|)}}. One recovers from <math>\ \tilde{E}_j^a\ </math> the metric times a factor given by its determinant. It is clear that <math>\ \tilde{E}_j^a\ </math> and <math>\ E_j^a\ </math> contain the same information, just rearranged. Now the choice for <math>\ \tilde{E}_j^a\ </math> is not unique, and in fact one can perform a local in space [[rotation]] with respect to the internal indices <math>\ j\ </math> without changing the (inverse) metric. This is the origin of the <math>\ \mathrm{ SU(2) }\ </math> 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 <math>\ V_i^b\ </math> will be

:<math>\ D_a\ V_j^b = \partial_a V_j^b - \Gamma_{a \;\; j}^{\;\; k}\ V_k^b + \Gamma^b_{ac}\ V_j^c\ </math>

where <math>\ \Gamma^b_{ac}\ </math> is the usual [[Levi-Civita connection]] and <math>\ \Gamma_{a \;\; j}^{\;\; k}\ </math> is the so-called [[spin connection]]. Let us take the configuration variable to be

:<math>\ A_a^j = \Gamma_a^j + \beta\ K_a^j\ </math>

where <math>\Gamma_a^j = \Gamma_{ak\ell}\ \epsilon^{k \ell j}</math> and <math display="inline">K_a^j = K_{ab}\ \tilde{E}^{bj} / \sqrt{\det (q)\ } ~.</math> The densitized ''drei-bein'' is the conjugate momentum variable of this three-dimensional SU(2) gauge field (or connection) <math>\ A^k_b\ ,</math> in that it satisfies the Poisson bracket relation

:<math>\ \{\ \tilde{E}_j^a (x) ,\ A^k_b (y)\ \} = 8\pi\ G_\mathsf{Newton}\ \beta\ \delta^a_b\ \delta^k_j\ \delta^3 (x - y) ~.</math>

The constant <math>\beta</math> is the [[Immirzi parameter]], a factor that renormalizes [[Newton's constant]] <math>\ G_\mathsf{Newton} ~.</math> The densitized ''drei-bein'' can be used to re construct the metric as discussed above and the connection can be used to reconstruct the extrinsic curvature. Ashtekar variables correspond to the choice <math>\ \beta = -i\ </math> (the negative of the [[imaginary number]], <math>\ i\ </math>), <math>\ A_a^j\ </math> 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.<ref>For more details on this and the subsequent development, see {{cite book |title=Lectures on Non-Perturbative Canonical Gravity |edition=1st |year=1991 |publisher=World Scientific Publishing}}</ref> 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 number|complex]].<ref>See {{cite book |at=part&nbsp;III, chapter&nbsp;5 |title=Gauge Fields, Knots and Gravity |first1=John |last1=Baez |first2=Javier P. |last2=Muniain |edition=1st |year=1994 |publisher=World scientific Publishing }}</ref> When one quantizes the theory it is a difficult task to ensure that one recovers [[real number|real]] general relativity, as opposed to [[complex number|complex]] general relativity. Also the Hamiltonian constraint Ashtekar worked with was the densitized version, instead of the original Hamiltonian; that is, he worked with <math display="inline">\tilde{H} = \sqrt{\det (q)} H ~.</math>

There were serious difficulties in promoting this quantity to a [[quantum operator]]. In 1996 [[Thomas Thiemann]] who was able to use a generalization of Ashtekar's formalism to real connections (<math>\beta</math> 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.<ref>{{cite journal | last=Thiemann | first=T. |author-link=Thomas Thiemann | year=1996 | title=Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity | journal=[[Physics Letters B]] | publisher=Elsevier BV | volume=380 | issue=3-4 | pages=257–264 | issn=0370-2693 | doi=10.1016/0370-2693(96)00532-1 | arxiv=gr-qc/9606088 }}</ref><ref>For an account of these developments see {{cite web |first=John |last=Baez |author-link=John Baez |type=academic personal webpage |url=https://math.ucr.edu/home/baez/hamiltonian/hamiltonian.html |title=The Hamiltonian constraint in the loop representation of quantum gravity |website=ucr.edu |publisher=[[University of California, Riverside]] }}</ref>

Lee Smolin & Ted Jacobson, and Joseph Samuel independently discovered that there exists in fact a [[Lagrangian (field theory)|Lagrangian]] formulation of the theory by considering the self-dual formulation of the [[tetradic Palatini action]] principle of general relativity.<ref>{{cite journal |first=J. |last=Samuel |date= April 1987 |title=A Lagrangian basis for Ashtekar's formulation of canonical gravity |journal=Pramana – Journal of Physics |volume=28 |issue=4 |page=L429-L432 |publisher=[[Indian National Science Academy]] |url=https://www.ias.ac.in/describe/article/pram/028/04/0000-0000 |via=ias.ac.in}}</ref><ref>{{cite journal | last=Jacobson | first=Ted | last2=Smolin | first2=Lee | year=1987 | title=The left-handed spin connection as a variable for canonical gravity | journal=[[Physics Letters B]] | publisher=Elsevier | volume=196 | issue=1 | pages=39–42 | issn=0370-2693 | doi=10.1016/0370-2693(87)91672-8 }}</ref><ref>{{cite journal | last=Jacobson | first=T | last2=Smolin | first2=L. | date=1988-04-01 | df=dmy-all | title=Covariant action for Ashtekar's form of canonical gravity | journal=[[Classical and Quantum Gravity]] | volume=5 | issue=4 | pages=583–594 | issn=0264-9381 | doi=10.1088/0264-9381/5/4/006}}</ref> These proofs were given in terms of spinors. A purely tensorial proof of the new variables in terms of triads was given by Goldberg<ref>{{cite journal | last=Goldberg | first=J.N. | date=1988-04-15 | df=dmy-all | title=Triad approach to the Hamiltonian of general relativity | journal=[[Physical Review D]] | publisher=[[American Physical Society]] (APS) | volume=37 | issue=8 | pages=2116–2120 | issn=0556-2821 | doi=10.1103/physrevd.37.2116 }}</ref> and in terms of tetrads by Henneaux, Nelson, & Schomblond (1989).<ref>{{cite journal | last=Henneaux | first=M. | last2=Nelson | first2=J.E. | last3=Schomblond | first3=C. | date=1989-01-15 |df=dmy-all | title=Derivation of Ashtekar variables from tetrad gravity | journal=Physical Review D | publisher=[[American Physical Society]] (APS) | volume=39 | issue=2 | pages=434–437 | issn=0556-2821 | doi=10.1103/physrevd.39.434 }}</ref>

==References==
{{Reflist}}

==Further reading==
*{{cite journal |last=Ashtekar |first=Abhay |year=1986 |title=New Variables for Classical and Quantum Gravity |journal=Physical Review Letters |volume=57 |issue=18 |pages=2244&ndash;2247 |doi=10.1103/PhysRevLett.57.2244 |pmid=10033673 |bibcode=1986PhRvL..57.2244A}}

[[Category:Loop quantum gravity]]