Quantum Spin Dynamics ( QSD ) IIT

We continue here the analysis of the previous paper of the Wheeler-DeWitt constraint operator for four-dimensional, Lorentzian, non-perturbative, canon-ical vacuum quantum gravity in the continuum. In this paper we derive the complete kernel, as well as a physical inner product on it, for a non-symmetric version of the Wheeler-DeWitt operator. We then deene a symmetric version of the Wheeler-DeWitt operator. For the Euclidean Wheeler-DeWitt operator as well as for the generator of the Wick transform from the Euclidean to the Lorentzian regime we prove existence of self-adjoint extensions and based on these we present a method of proof of self-adjoint extensions for the Lorentzian operator. Finally we comment on the status of the Wick rotation transform in the light of the present results. 1 Complete physical Hilbert space and observ-ables In this section we will compute the complete kernel of both the Diieomorphism and the non-symmetric Euclidean and Lorentzian Hamiltonian constraint (for the symmetric Hamiltonian operator, see the next section). The kernel turns out to be spanned by distributions which do not only involve cylindrical functions which live on at most two-valent graphs or on graphs containing vertices with arbitrary valence but such that at each vertex the tangents of incident edges are co-planar. These solutions involve vertices of arbitrary valence and intersection characteristics, do take the curvature term F ab of the classical constraint fully into account and are not necessarily annihilated by the volume operator. Also they are sensitive to whether they belong to the kernel of the Euclidean or Lorentzian Hamiltonian constraint. This space of distributional solutions inherits a natural inner product coming from H via the group averaging method and it turns out that it coincides with the one given in 9]. Furthermore, we will deene the notion of an observable and give explicit, non-trivial examples of those.