Approaches to Quantum Gravity

82 N. Savvidouthe T 0 variables – and a three-dimensional space S × R (where S is a spatial twosurface),as a history analogue of the T 1 variables. The price of gauge-invariance isthat additional canonical variables have to be quantised: they correspond to a unittimelike vector field that determines the possible ways that the group SU(2) canbe embedded into SL(2, C) –see[14] for the quantisation of a related structure inthe context of quantum field theories.The above is a potential point of divergence between histories quantisation andthe canonical loop approach, which is necessary in light of the strong restrictionsplaced by our requirement of full spacetime general covariance. At present theresearch is focussed on finding a proper algebra for quantisation. An interestingpossibility is that the histories formalism may provide spacetime geometric operators:for example, an operator for spacetime volume; or ‘length’ operators thatdistinguish between spacelike and timelike curves.AcknowledgementThe preparation of this publication was supported by the EP/C517687 EPSRCgrant.References[1] J. Ambjorn, J. Jurkiewicz, & R. Loll, A non-perturbative Lorentzian path integral forgravity. phys. Rev. Lett. 85 (2000) 924.[2] C. Anastopoulos, Continuous-time histories: observables, probabilities, phase spacestructure and the classical limit. J. Math. Phys. 42 (2001) 3225.[3] C. Anastopoulos & K. N. Savvidou, Minisuperspace models in history theory. Class.Quant. Grav. 22 (2005) 1841.[4] A. Ashtekar, New variables for classical and quantum gravity. Phys. Rev. Lett.,57(18) (1986) 2244–2247.[5] J. F. Barbero, Reality conditions and Ashtekar variables: a different perspective.Phys.Rev. D51 (1995) 5498.[6] L. Bombelli, J. H. Lee, D. Meyer, & R. Sorkin, Spacetime as a causal set. Phys. Rev.Lett. 59 (1989) 521.[7] A. Burch, Histories electromagnetism. J. Math. Phys. 45(6) (2004) 2153.[8] M. Gell-Mann & J. B. Hartle, Quantum mechanics in the light of quantumcosmology. In Complexity, Entropy and the Physics of Information, ed. W. Zurek.(Addison Wesley, Reading, 1990).[9] R. B. Griffiths, Consistent histories and the interpretation of quantum mechanics.J. Stat. Phys. 36 (1984) 219.[10] J. Hartle, Spacetime quantum mechanics and the quantum mechanics of spacetime.In Proceedings on the 1992 Les Houches School,Gravitation and Quantisation(1993).[11] S. Holst, Barbero’s Hamiltonian derived from a generalized Hilbert–Palatini action.Phys. Rev. D53 (1996) 5966.[12] C. Isham, Structural issues in quantum gravity. Plenary Talk at GR14 conference,(1995) gr-qc/9510063.