We construct a large number of coherent states satisfying the resolution of unity with a positive weight function, obtained through analytic solutions of the Stieltjes moment problem (coherent states on a plane) and the Hausdorff moment problem (coherent states on a disk). These solutions are obtained through the method of inverse Mellin transform. In addition, these coherent states induce a deformation of the metric that has been calculated analytically.

Covariant scalar field quantization, nicknamed $({\phi }^r{)}_n$, where $r$ denotes the power of the interaction term and $n=s+1$ where $s$ is the spatial dimension and 1 adds time. Models such that $r<2n/(n-2)$ can be treated by canonical quantization, while models such that $r>2n/(n-2)$ are nonrenormalizable, leading to perturbative infinities, or, if treated as a unit, emerge as ‘free theories’. Models such as $r=2n/(n-2)$, e.g., $r=n=4$, again using canonical quantization also become ‘free theories’, which must be considered quantum failures. However, there exists a different approach called affine quantization that promotes a different set of classical variables to become the basic quantum operators and it offers different results, such as models for which $r>2n/(n-2)$, which has recently correctly quantized $({\phi }^{12}{)}_3$. In the present paper we show, with the aid of a Monte Carlo analysis, that one of the special cases where $r=2n/(n-2)$, specifically the case $r=n=4$, can be acceptably quantized using affine quantization.

Quantum constraints of the type Q|ψ_phys = 0 can be straightforwardly implemented in cases where Q is a self-adjoint operator for which zero is an eigenvalue. In that case, the physical Hilbert space is obtained by projecting onto the kernel of Q, i.e. H_phys = ker Q = ker Q*. It is, however, non-trivial to identify and project onto H_phys when zero is not in the point spectrum but instead is in the continuous spectrum of Q, because then ker Q = ∅. Here, we observe that the topology of the underlying Hilbert space can be harmlessly modified, namely, loosely speaking, in the direction perpendicular to the constraint surface. Consequently, Q becomes non-self-adjoint, which then allows us to conveniently obtain H_phys as the proper Hilbert subspace H_phys = ker Q* on which one can project as usual. In the simplest case, the necessary change of topology amounts to passing from an L² Hilbert space to a Sobolev space.

Recent progress in the quantization of nonrenormalizable scalar fields has found that a suitable non-classical modification of the ground state wave function leads to a result that eliminates term-by-term divergences that arise in a conventional perturbation analysis. After a brief review of both the scalar field story and the affine quantum gravity program, examination of the procedures used in the latter surprisingly shows an analogous formulation which already implies that affine quantum gravity is not plagued by divergences that arise in a standard perturbation study. Additionally, guided by the projection operator method to deal with quantum constraints, trial reproducing kernels are introduced that satisfy the diffeomorphism constraints. Furthermore, it is argued that the trial reproducing kernels for the diffeomorphism constraints may also satisfy the Hamiltonian constraint as well.

Affine quantization is a relatively new procedure, and it can solve many new problems. This essay reviews this new, and novel, procedure for particle problems, as well as those of fields and gravity. New quantization tools, which are extremely close to, and even constructed from, the tools of canonical quantization, are able to fully solve selected problems that using the standard canonical quantization would fail. In particular, improvements can even be found with an affine quantization of fields, as well as gravity.

The central principle of affine quantum gravity is securing and maintaining the strict positivity of the matrix {ĝ_ab(x)} composed of the spatial components of the local metric operator. On spectral grounds, canonical commutation relations are incompatible with this principle, and they must be replaced by noncanonical, affine commutation relations. Due to the partial second-class nature of the quantum gravitational constraints, it is advantageous to use the recently developed projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational operator constraints is formulated quite naturally by means of a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. It is anticipated that skills and insight to study this formulation can be developed by studying special, reduced-variable models that still retain some basic characteristics of gravity, specifically a partial second-class constraint operator structure. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models. Finally, developing a procedure to pass to the genuine physical Hilbert space involves several interconnected steps that require careful coordination.