# Other Papers

- 001 Regularization independence of finite states in four dimensional quantized gravity
This is one of a series of works designed to address a major criticism concerning the mathematical rigor of the generalized Kodama states. The present paper analyzes the criterion for finiteness due to cancelation of the ultraviolet divergences stemming from the quantum Hamiltonian constraint, in the full theory. We argue that any reliable state must be independent of the regulating functions and parameters utilized to extract finite results. Using point-splitting regularization, we show that the results, typically regarded either as being purely formal or meaningless, are indeed mathematically rigorous and consistent with the axioms of field theory and regulator independence. Our analysis is carried out at the level of the quantum constraint solutions, and does not consider the algebra of constraints.

- 002 Chang--Soo variables coupled to the inflaton field in anisotropic minisuperspace
In this paper we illustrate some of the dynamics of inflation by coupling a Klein–Gordon scalar field to a new set of variables named after Chang–Soo/CDJ in anisotropic minisuperspace reduced to the isotropic sector. We provide further arguments for the semiclassical quantum correspondence in conjunction with a prescription for computing the dynamics of inflation for a general self-interaction potential. We compute these dynamics to third order and provide physically motivated arguments for various parameters governing the initial conditions of the universe based upon our model. Comparisons are made to the FRW model, and its congruity with the SQC noted.

- 003 Proposed general solution for initial value problem for vacuum GR in Chang--Soo/CDJ variables in anisotropic minisuperspace: preview into inflation
In this paper we illustrate the dynamics of the Chang–Soo/CDJ variables in the description of vacuum GR in anisotropic minisuperspace, uncovering a new class of general solutions in both the degenerate and the nondegenerate sectors. One result of these new variables is the existence of an unbridgeable gap between these sectors. The gravitational dynamics self-consistently hold down to arbitrarily small times, implying a semiclassical-quantum correspondence as well as an elimination of the big bang singularity in the nondegenerate sector. We also present an algorithm for constructing general solutions for the spacetimes labelling a general quantum state in these variables.

- 004 Vacuum GR in Chang--Soo variables: Hilbert space structure in anisotropic minisuperspace
In this paper we address the major criticism of the pure Kodama state, namely its normalizability and its existence within a genuine Hilbert space of states, by recasting Ashtekar’s general relativity into set of new variables attributed to Chang–Soo/CDJ. While our results have been shown for anisotropic minisuperspace, we reserve a treatment of the full theory for a following paper which it is hoped should finally bring this issue to a close. We have performed a canonical treatment of these new variables from the level of the classical/quantum algebra of constraints, all the way to the construction of the Hilbert space of states, and have demonstrated their relevance to the principle of the semiclassical-quantum correspondence. It is hoped that these new variables and their physical interpretation should provide a new starting point for investigations in classical and quantum GR and in the construction of a consistent quantum theory.

- 005 Proposal for the Hilbert space structure for unconstrained vacuum general relativity
There is a new set of variables, which from our analysis appear naturally adapted to a Hilbert space description on the reduced phase space for vacuum GR. In this paper we present the Hilbert space, which features a basis of coherent-like states labeled by the algebraic classification of the corresponding spacetime. These wavefunctions, which satisfy the semiclassical-quantum correspondence, correspond to the solution of the quantum Hamiltonian constraint on the space of gauge-invariant, diffeomorphism invariant states in these variables, and are free of field-theoretical singularities.

- 006 Vacuum general relativity in the Instanton representation: the quantum theory
This paper quantizes the physical space of vacuum GR in the full theory using the instanton representation, a new method. We have constructed a Hilbert space of normalizable wavefunctions labeled by the eigenvalues of the Weyl curvature, which encodes the Petrov classification of spacetime. For zero cosmological constant the states are labeled by two arbitrary functions of position in 3-space, comprising the physical degrees of freedom. For nonzero cosmological constant one of the continuous labels degenerates into a discrete label in steps of the Planck length squared in comparison to the characteristic linear dimension of the universe. Additionally, we have provided a possible resolution to the normalizability of the Kodama state within this representation.

- 007 Solution to the quantum constraints of four dimensional quantized gravity by dimensional reduction. Part II
In this paper we present a general solution to the quantum constraints of general relativity from the perspective of dimensional reduction, in continuation from Part I. First we present the constraints in terms of the polar decomposition of the CDJ matrix, noting that some kinematic degrees of freedom consequently become ignorable. In the second half of the paper we perform a dimensional reduction to the level the the dynamical subspace using a Cartesian decomposition and write down a general solution featuring the dynamics. A main result of Part II is to further the interpretational third-quantized analogy to second-quantized spin network states.

- 008 Solution to the constraints of four dimensional quantized gravity by dimensional extension. Part I
The purpose of Part I of this paper is to compute the ingredients necessary to invert the kinetic operator for the system of equations determining the generalized Kodama states. We perform this computation from the dimensionally extended viewpoint which treats the kinematic and the dynamic components as individual subspaces of a larger total space. The kinematic subspace provides a new structure having the interpretation as the analogue in infinite dimensional functional spaces as the edge of a generalized spin network state carrying a representation of the kinematic gauge group of gauge transformations and diffeomorphisms. We illustrate a technique to compute the ‘dressed’ third-quantized propagator on the dynamics subspace, relegating its interpretation in analogy to spin network states as well as a more thorough analysis of the dimensionally reduced case to Part II.

- 009 Instanton representation of Plebanski gravity: XIX. Reality conditions
In this paper we implement reality conditions on the instanton representation of Plebanski gravity using adjointness relations in the quantum theory. The result is an explicit parametrization for the Ashtekar connection by three degrees of freedom which guarantee the reality of the Ashtekar densitized triad. The results of this paper are limited to the diagonal sector of the full theory.

- 010 Instanton representation of Plebanski gravity: XVIII. Quantization and proposed resolution of the Kodama state
In this paper we implement reality conditions on the instanton representation of Plebanski gravity using adjointness relations in the quantum theory. The result is an explicit parametrization for the Ashtekar connection by three degrees of freedom which guarantee the reality of the Ashtekar densitized triad. The results of this paper are limited to the diagonal sector of the full theory.

- 011 Generalized Kodama partition functions: A preview into normalizability for the generalized Kodama states
In this paper we outline the computation of the partition function for the generalized Kodama states (GKod) of quantum gravity using the background field method. We show that the coupling constant for GKod is the same dimensionless coupling constant that appears in the partition function of the pure Kodama state (Chern–Simons functional) and argue that the GKod partition function is renormalizable as a loop expansion in direct analogy to Chern–Simons perturbation theory. The GKod partition function contains an infinite set of 1PI vertices uniquely fixed, as a result of the semiclassical-quantum correspondence, by the first-order vertex. This implies the existence of a well-defined effective action for the partition function since the ‘phase’ of the GKod, provided a finite state exists, is equivalent to this effective action. Additionally, the separation of the matter from the gravitational contributions bears a resemblance to the infinite dimensional analogue to Kaluza–Klein theory. Future directions of research include extension of the computations of this paper to the norm of the GKod as well as to examine the analogue of the Chern–Simons Jone’s polynomials and link invariants using the GKod as a measure.

- 014 Instanton representation of Plebanski gravity XVI: Hamiltonian and Hamilton–Jacobi dynamics on superspace (April 28, 2010)
In this paper we focus on the Hamiltonian dynamics on the kinematic phase space of the instanton representation, in the full theory. The general solutions for pure gravity are reduced to quadratures and fixed point iteration both for vanishing and nonvanishing cosmological constant, with convenient physical interpretations. A Hamilton–Jacobi analysis is performed of the semiclassical orbits and their relation to the quantum theory motivated. We have constructed Hamilton–Jacobi functionals mimicking the classical dynamics. A main result is emergence of a natural time variable on the configuration space with respect to which the remaining variables evolve..

- 015 A brief survey of the renormalizability of four dimensional gravity for generalized Kodama states. Hamiltonian and Hamilton–Jacobi dynamics on superspace (May 15, 2008)
We continue the line of research from previous works in assessing the suitability of the pure Kodama state both as a ground state for the generalized Kodama states, as well as characteristic of a good semiclassical limit of general relativity. We briefly introduce the quantum theory of fluctuations about DeSitter spacetime, which enables one to examine some perturbative aspects of the state. Additionally, we also motivate the concept of the cubic tree network, which enables one to view the generalized Kodama states in compact form as a nonlinear transformation of the pure Kodama states parametrized by the matter content of the proper classical limit. It is hoped that this work constitutes a first step in addressing the nonperturbative renormalizability of general relativity in Ashtekar variables. Remaining issues to address, including the analysis of specific matter models, include finiteness and normalizability of the generalized Kodama state as well as reality conditions on the Ashtekar variables, which we relegate to separate works.

- 016 Instanton representation of Plebanski gravity XIII Canonical structure of the Petrov classification of nondegenerate spacetimes (Apr 27, 2010)
The instanton representation of Plebanski gravity admits a natural canonical structure where the (densitized) eigenvalues of the CDJ matrix are the basic momentum space variables. Canonically conjugate configuration variables exist for six distinct configurations in the full theory, referred to as quantizable configurations. The CDJ matrix relates to the Petrov classification and principal null directions of spacetime, which we directly correlate to these quantizable degrees of freedom. The implication of this result is the ability to perform a quantization procedure for spacetimes of Petrov Type I, D, and O, using the instanton representation.

- 017 Wavefunction of the universe, observables and the issue of time (Part I) (Apr 23, 2010)
In this paper we show how it is possible for states to undergo a nontrivial time evolution under the totally constrained Hamiltonian of general relativity. This evolution occurs via nonvanishing Poisson brackets with the state functionals, restricted to a polarization on configuration space.

- 018 A systematic approach to the solution of the constraints of quantum gravity: The full theory (Feb 12, 2008)
This is the third paper in a series outlining an algorithm to construct finite states of quantum gravity in Ashtekar variables. In this paper we treat the case of the Klein–Gordon field quantized with gravity on the same footing. We address the full theory, outlining the solution to the constraints and the construction of the corresponding wavefunction of the universe. The basic method for the full theory is to expand the constraints relative to the solution for the pure Kodama state and rewrite them in the form of a generalized nonlinear group transformation of the CDJ matrix, viewed as a nine-dimensional vector. We then outline a prescription for finding the fixed point of the flow, and the corresponding generalized Kodama state for the full theory is constructed. The final solution is expressed in an asymptotic series in powers of model-specific matter inputs, suppressed by a small dimensionless constant, relative to the pure Kodama state. We discuss this expansion from different perspectives. Lastly, we explicitly show how the solution to the quantized constraints establishes a wavefunction of the universe with a predetermined semiclassical limit built in as a boundary condition on quantized gravity in the full theory.

- 019 Nonconventional functional calculus techniques in quantum field theories and in quantum gravity (Apr 7, 2008)
In this excerpt, we introduce the concept of the ‘nonconventional’ calculus, outlining a prescription for making sense of the ultraviolet singularities which occur in quantum field theories with a view to addressing quantum gravity. We first show how these singularities arise when interpreting the canonical commutation relations for a general quantum field theory and then discuss a new interpretation which deals with them without using regularization procedures. We then discuss the relative commutativity of functional with spacetime variation and its implications for addressing the issue of time in quantum gravity in contrast to nongravitational field theories. We end with a brief introduction into how these concepts may be applied to the quantization of gravity in Ashtekar variables coupled to matter fields.

- 020 Instanton representation of Plebanski gravity: IX Hamiltonian minisuperspace dynamics in undensitized momentum space variables (Apr 23, 2010)
In this paper we illustrate the dynamics of the instanton representation in the description of vacuum GR in minisuperspace for undensitized variables. We uncover a new class of general solutions in both the degenerate and the nondegenerate sectors of the theory. Additionally, the individual sectors are preserved under Hamiltonian evolution. Finally, we present an algorithm for constructing general solutions by expansion about the isotropic sector of the instanton representation.

- 021 Instanton representation of Plebanski gravity: VIII Initial value and Gauss’ law constraints in polar form. (Apr 22, 2010)
In this paper we provide a prescription for solving the initial value constraints of GR for algebraically general spacetimes in the polar representation. The prescription uses the Ashtekar magnetic field and two eigenvalues of the CDJ matrix as inputs, and provides as an output the SO(3,C) frame solving the Gauss’ law constraint, in the form of a triplet of SO(3,C) angles. The two eigenvalues are the physical degrees of freedom, which fix the third eigenvalue through the diffeomorphism and Hamiltonian constraints. We provide two fixed point iteration procedures for determining the angles, as well as an analysis of the algebraically special cases.

- 022 Instanton representation of Plebanski gravity: VII value and Gauss' law constraints in rectangular form (Apr 22, 2010)
We provide an algorithm for solving the initial value constraints of GR using the rectangular parametrization of the CDJ matrix. This paper focuses on the Gauss’ law constraint and the criteria for its integrability. The Gauss’ law constraint takes as its input the chosen configuration, and for an output reduces the CDJ matrix to its physical degrees of freedom. This in turn provides an input into the Hamiltonian constraint for which we provide a formal solution by ex- pansion about the Kodama state in the general case. We establish criteria for integrability of the Gauss’ law constraint for given starting configurations and we provide several examples.

- 023 Instanton representation of Plebanski gravity: VI Induced geometric structures (Apr 22, 2010)
IIn this paper we present various geometric structures induced by the instanton representation of Plebanski gravity, to solidify the conceptual foundations which will be needed for future papers in this series. The main theme is the relation of these structures to integrability as regards the Gauss’ law constraint and the existence of holonomic coordinates on configuration space.

- 024 Instanton representation of Plebanski gravity: V Riemannian structure and relation to metric GR (Apr 22, 2010)
In this paper we solidify the relation of the instanton representa- tion of Plebanski gravity to Einstein’s metric theory, using the relation between nonabelian gauge theory and the intrinsic geometry of gauge invariant variables. It is found that this representation corresponds to a 3-dimensional space with nonmetricity and torsion. By imposing metricity by way of the Gauss’ law constraint, we correlate this 3-dimensional space to a 4-dimensional spacetime geometry which implies the Einstein–Hilbert action. We have generalized the Einstein spaces derived by previous authors, within the purely Yang–Mills context, to incorporate gravitational degrees of freedom.

- 025 The canonical versus path integral quantization approach to generalized Kodama states (Part I) (Feb 1, 2008)
This is the fifth paper in the series outlining an algorithm to consistently quantize four-dimensional gravity. We derive the pure Kodama state by path integration, in analogy to the no-boundary proposal for constructing quantum gravitational wavefunctions, checking at each stage of the process the equivalence of the canonical and path integral approaches. A family of additional pure Kodama states is identified via the canonical approach and a criterion for their suitability as a basis of states is examined. We provide an interpretation for the problem of time within the context of generalized Kodama states and propose a possible method of resolution. We also develop different techniques for solving the Gauss’ law constraints at the kinematical level, in preparation for future work in this series.