Approaches to Quantum Gravity

Loop quantum gravity 239In what follows we will denote the Hilbert space H ω by H Kin in order to indicatethat it is a Hilbert space of kinematical states, i.e. the constraints have not yet beenimplemented and the states are therefore not gauge invariant.IV. Implementation of the constraints.The crucial question is whether the constraints can be realised in this representationas densely defined and closable (the adjoint is also densely defined) operators. This isnon-trivial, especially in field theories such as General Relativity due to the followingreasons.1. The constraints are usually defined as functions of certain limits of elements of P.For instance, if M is a cotangent bundle then P consists of smeared configurationand momentum variables, say S(q) := ∫ d3 xS ab h ab , P(s) := ∫ d3 xs ab π abfor GR where s, S are smooth, symmetric tensor (densities) of compact support.However, the (smeared) constraints of GR are not polynomials of P(s), S(q),rather they are non-polynomial expressions of the local functions h ab (x), π ab (x)and their first and second derivatives. Obviously, one can get those functions bytaking a limit in which S ab , s ab become Dirac distributions, however, since only thesmeared fields are defined as operators on H Kin , it is a highly non-trivial questionwhether the constraints are densely defined at all. Technically, the un-smeared fieldsbecome operator valued distributions and it is difficult to make sense out of productsof those located at the same points. Thus, one may be facing ultraviolet problems.2. Notice that all but the at most linear functions face the so called operator orderingproblem: It makes a difference whether we identify the function f 1 f 2 ∈ C ∞ (M)(which does not belong to P) with ( f 1 , f 2 ) or ( f 2 , f 1 ) in A. If f 1 , f 2 are real valued,then one may choose a symmetric ordering [( f 1 , f 2 ) + ( f 2 , f 1 )]/2, however,it is not possible to rescue all the classical relations to the quantum level, at leastin irreducible representations, which is the content of the famous Groenewald–vanHowe theorem [11]. This may be an obstacle especially for constraint quantisation,because we may pick up what are called anomalies: While the classical constraintsform a closed subalgebra (possibly with structure functions), the quantum constraintsmay not. This could imply that the physical Hilbert space, discussed below,is too small.V. Solving the constraints and physical Hilbert space.Let us assume that we are given some set of real valued constraints C I where I takesa range in some index set and suppose that they form a first class system, that is,{C I , C J } = f K IJ C K where f K IJ may be non-trivial, real valued functions onphase space. This is precisely the situation in GR where the index set stands for somecountable system of smearing functions I = (N, ⃗N) called lapse and shift functions.Suppose that we have successfully quantised the constraints and structure functionsas operators Ĉ I , fˆK IJ on H Kin as specified in step IV. The first possible problem isthat the point zero is not contained in the spectrum of some of the Ĉ I in which casethe physical Hilbert space is empty. In that case the quantisation of those operatorsor the kinematical Hilbert space is invalid and must be changed. Let us assume thatthis problem has been circumvented. If the point zero is not contained in the point