Approaches to Quantum Gravity

Loop quantum gravity 243∂ F τ f,T /∂τ I . They define physical Hamiltonians, i.e. Dirac observables. For the relativisticparticle it is easy to see that this generator coincides with √ δ ab p a p b + m 2 asexpected.Unfortunately, for sufficiently complicated dynamical systems such as GR, the expressions(13.10) are rather complex and formal in the sense that little is known aboutsufficient criteria for the convergence of the series involved in (13.10) and whetherthey can be quantised on H Kin crucially depends on a judicious choice of the T I .However, at least in principle there is a guideline to address the Problem of Time.This concludes the outline of the canonical quantisation programme for arbitraryconstrained systems. We will apply it in the next section to General Relativity.13.3 Loop quantum gravityThe classical canonical framework was developed by ADM in the 1960s and inthe previous subsection we have outlined the canonical quantisation algorithm.Hence we should now start to systematically apply it to GR. Unfortunately this isnot directly possible because for the ADM formulation it has not been possible tofind background independent representations of the algebra P generated from thefunctions S(q), P(s) discussed in the previous subsection which support the constraints.Therefore, the canonical programme was stuck for decades until the mid1980s and all the results obtained before that date are at best formal. Without a representationone cannot tell whether the algebraic objects that one is dealing with aredensely defined at all, what the spectra of operators are, whether formal solutionsto the constraint equations are indeed generalised eigenvectors, etc. For instance,the function x ↦→ exp(kx), k ∈ R −{0} certainly is formally an eigenfunction ofthe operator id/dx on L 2 (R, dx), however, it is neither a proper eigenvector (sinceit is not normalisable) nor a generalised eigenvector because it cannot appear inthe spectral resolution of the self-adjoint operator id/dx (because exp(kx) has aformally imaginary eigenvalue). Hence a representation is indispensable in orderto construct a viable theory.13.3.1 New variables and the algebra PProgress was made due to a switch to new canonical variables [17; 18] which wenow describe. We will be brief, the interested reader can find the details 6 in [1; 2].6 As a historical aside, it was believed that the theory for ι = ±i is distinguished because the Hamiltonianconstraint (13.19) then simplifies and even becomes polynomial after multiplying by √ |det(E)|. Unfortunatelythe representation theory for this theory could never be made sense of because the connection then is complexvalued and one obtains the non-polynomial reality conditions A+ A = 2Ɣ. It was then believed that one should