Loop Variables for General Relativity

In General > s.a. kaluza-klein theories.
* Idea: A formulation based on a set of variables inspired by the loop states for canonical quantum gravity.
* Small T-algebra: The Wilson loop configuration variables and the momentum variables for a loop C are, respectively,

T_C(A):= \(1\over2\)tr P exp{κ ∫_C A} ,   T ^a[α](s):= \(1\over2\)tr[P exp{κ ∫_C A} E^a(α(s)) ] .

* Smeared momentum variables:

T_S(A):= ∫_S dS^ab η_abc T ^c_{C{·, τ)}(σ,τ),   where   T ^a_{C{·, τ)} := tr h_{C{·, τ)}(σ,τ)[A] E^a(α(σ,τ)) .

* Full T-algebra: Defined by

T ^ab[C](s, s'):= \(1\over2\)tr [P exp{κ ∫_s'^s A} E^a(C(s)) P exp{κ ∫_s^s' A} E^b(C(s'))] ,

and similarly for a higher number of indices.
> In quantum gravity: see loop representation; 3D quantum gravity.

References > s.a. supergravity.
@ General: Mensky in(84)?; Rovelli & Smolin PRL(88), NPB(90); Manojlović CQG(90); Goldberg et al CMP(92), Boström CQG(92) [degeneracy in small algebra]; Loll CQG(93)gq [inequalities on traces of holonomies]; Venkatesh a1305; Gambini et al a2003 [and the algebra of non-local observables].
@ With holonomy of metric connection: Refs in Bezerra & Letelier CQG(91).
@ Similar variables: Barrett GRG(89) [Lorentz group holonomies]; Newman & Rovelli PRL(92) [lines of force]; Venkatesh a1211 [from loop algebras].
> Online resources: see Wikipedia page [loop representation in gauge theories and quantum gravity].

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