100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Loop Quantum Cosmology 389sically diverging) which requires a non-perturbative treatment. Moreover,classically we expect space to degenerate at the singularity, for instance asingle point in a closed isotropic model. This means that we cannot take thepresence of a classical geometry to measure distances for granted, which istechnically expressed as background independence. A non-perturbative andbackground independent quantization of gravity is available in the form ofloop quantum gravity, 14,15,16 which by now is understood well enough inorder to be applicable in physically interesting situations.Here, we only mention salient features of the theory which will turn outto be important for cosmology; for further details see [17]. The first oneis the kind of basic variables used, which are the Ashtekar connection 18,19describing the curvature of space and a triad (with density weight) describingthe metric by a collection of three orthonormal vectors in eachpoint. These variables are important since they allow a background independentrepresentation of the theory, where the connection A i a is integratedto holonomies∫h e (A) = P exp A i aτ i ė a dt (5)ealong curves e in space and the triad Eia to fluxes∫F S (E) = Ei a τ i n a d 2 y (6)Salong surfaces S. (In these expressions, ė a denotes the tangent vector to acurve and n a the co-normal to a surface, both of which are defined withoutreference to a background metric. Moreover, τ j = − 1 2 iσ j in terms of Paulimatrices). While usual quantum field theory techniques rest on the presenceof a background metric, for instance in order to decompose a field in itsFourier modes and define a vacuum state and particles, this is no longeravailable in quantum gravity where the metric itself must be turned into anoperator. On the other hand, some integration is necessary since the fieldsthemselves are distributional in quantum field theory and do not allow awell-defined representation. This “smearing” with respect to a backgroundmetric has to be replaced by some other integration sufficient for resultingin honest operators. 20,21 This is achieved by the integrations in (5) and (6),which similarly lead to a well-defined quantum representation. Usual Fockspaces in perturbative quantum field theory are thereby replaced by theloop representation, where an orthonormal basis is given by spin networkstates. 22