Approaches to Quantum Gravity

Spacetime symmetries in histories canonical gravity 81The first step in the application of the histories formalism to loop quantum gravityis to develop the histories analogue of the connection formalism of GeneralRelativity. The original formulation of the programme involved the considerationof self-dual SL(2, C) connections on spacetime, together with a field of tetradsfor a Lorentzian metric [26; 4]. However, the mainline approach in loop quantumgravity finds it more convenient to employ in quantisation a real SU(2) connection,proposed by Barbero [5]. The Barbero connection may be obtained as a variable ina state space, extending that of canonical General Relativity; or it may be obtainedfrom a Lagrangian action (the Holst Lagrangian by [11]). However, the latter procedureinvolves gauge fixing, and it is not clear whether the connection may bedefined in its absence – see [19; 20] for related discussions.The histories description for classical gravity in term of the Holst Lagrangian hasbeen developed in [25]. The basic variables at the covariant level is an SL(2, C)connection and a field of tetrads on spacetime M, together with their conjugatevariables. The corresponding history space carries a symplectic action of the groupDi f f (M) of spacetime diffeomorphisms. The introduction of an equivariant foliationfunctional allows the translation of the spacetime description into that ofan one-parameter family of canonical structures. The results of the metric-basedtheory can be fully reproduced in this construction: the set of constraints correspondingto the Holst Lagrangian is invariant under the action of the spacetimeDi f f (M) group. Hence the generators of the spacetime diffeomorphisms groupcan also be projected onto the reduced state space.The next step would involve choosing the basic variables for quantisation. Followingthe spirit of loop quantum gravity, we may try to identify a loop algebra,and then construct a histories Hilbert space by studying its representation theory.The obvious place to start would be the loop algebra corresponding to thespacetime SL(2, C) connection of the covariant description. This, however, wouldinvolve a representation theory for loop variables with a non-compact gauge group,which to the best of our knowledge has not yet been fully developed. Moreover,we would have to identify a new role for the tetrad fields, because at this level theycommute with the connection variables.It may be more profitable to work with ‘internal’ fields, namely the ones thatcorrespond to one-parameter families of the standard canonical variables. Thiswould allow the consideration of connections with compact gauge group. However,a complication arises, because of the gauge-dependence of the definition ofthe Barbero connection. A gauge-fixing condition, at this level, breaks the backgroundindependence of the theory. In [25] we show that a connection sharing allproperties of the Barbero connection can be defined in a gauge invariant way, albeitin a larger space than the one usually employed.A history quantisation may be therefore envisioned that will employ variablesdefined with support on a two-dimensional cylinder – giving a history analogue of