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Generalized Kodama states. V. Evidence of the

normalizability and renormalizability of 4D QGRA

(Part II).

Eyo Eyo Ita III

February 24, 2019

Department of Applied Mathematics and Theoretical Physics

Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road

Cambridge CB3 0WA, United Kingdom

eei20@cam.ac.uk

Abstract

This is the sixth paper in the series outlining an algorithm to con-sistently quantize four-dimensional gravity. In this work we transformthe pure Kodama state into the metric representation for the BianchiIX minisuperspace model. Previous such calculations have been carriedout to semiclassical order for a particular choice of gauge, revealing theexistence of five topologically distinct states in the metric representa-tion. We have performed our calculation to all orders in perturbationtheory by maintaining the gauge degrees of freedom explicit, revealingthe existence of a sixth state. We propose a resolution to the issue ofreality conditions for the Ashtekar variables, and also for the recast-ing of 4-dimensional general relativity as a renormalizable field theory,stemming from the nonperturbative result obtained from this work.We also address the issue of normalizability of the Kodama state.

1

1 Introduction

It has been demonstrated that the Kodama state ΨKod[A] is a particluarsoltution of the quantum constraints in the Ashtekar connection representa-tion of general relativity. All that remains is to determine the reality condi-tions on the connection, which is an outstanding problem [3]. Notwisthand-ing the classical equivalence of Ashtekar to metric relativity, it may be inter-esting to ascertain whether the quantum limit of the latter as a real theorycan be obtained from the former. It appears that the best way to address theissue of reality consitions, which are difficult to implement in the Ashtekarvariables, problem is to transform ΨKod[A] into the metric representationwhere contact can be made more closely to experiment and to establishedresults in relativity.

It was demsonstrated in [18] and [19] that under a certain gauge inthe Bianchi IX universe, the Kodama state transforms into 5 independentstates in the metric representation. The states wer determined in the semi-classical approximation, and each state corresponded to a critical point ofthe Chern-Simons action, which corresponds to a different contour of in-tegration in connection space. The five topologically inequivalent contourscorrespond to the five states, which where compared to the Hartle-Hawkingand the Vilenkin states [1],[2]. Through motivation by the spirit of thiswork, it seems logical of all constraints upon classical and quantum relativ-ity available in the metric representation of a wavefunction, some (if not all)of these should translate into reality conditions upon the Ashtekar variablesthemselves. What is currently lacking is the existence of nonperturbativelyexact solutions in the latter representation. However, the existence of statessatisfying the SQC may be a way to bridge this gap.

We shall demonstrate in this paper how to determine all states of theBianchi IX model with cosmological constant to all orders beyond the semi-classical approximation, first in minisuperspace in this work and in the fulltheory in subsequent works. Furthermore, we will show a systematic methodof enumerating the states. Hartle and Hawking [1] made a proposal for thewavefunction of the universe in metric variables in terms of a path integral,due in part to the formal solution of the path integral to the constraints ofrelativity. In [11] the analogue of the Hartle-Hawking state was defined, sim-ilarly by path integral, in the Ashtekar variables, resulting in the pure andthe generalized Kodama states. It will be interesting to examine the mani-festation of the pure and generalized Kodama states in the metric and other(including the loop and spin network) representations. This work broachesthe concept with regard to minisuperspace models, but the results shouldbe generalizable to the full theory, at least for states satisfying the SQC.

1

2 Transformation of the Bianchi IX minisuperspace

model

For a homogeneous 3-manifold Σ, where exists an invariant basis of one-forms ωa = ωa

i dxi satisfying the Mauran-Cartan structure equation

dωa +1

2fabcω

b ∧ ωc = 0. (1)

The structure constants for a general group of isometries of Σ were classifiedby Bianchi into one of several types, based on the trace of the structureconstants.

fabc = maeǫebc + δa[bwc], (2)

where mae is symmetric. Within the sub-class wa = 0, the Bianchi IX modelis particularly useful in that due to the fact that it allows a Gaussian integralto be performed for the semiclassical part of the Kodama state, as opposedto a group for which the structure constants vanish.

The Kodama pure state is given by ΨKod[A] = exp[−κ−1ICS [A]], whereICS is the Chern-Simons functional

ICS[A] =

∫

Σtr(A ∧ dA+

2

3A ∧A ∧A). (3)

In (3) the trace is taken over the indices corresponding to the appropriateBianchi group structure constants. Using (3) in conjunction with (1) in thecase of a homogeneous connection (homogeneous with respect to the left-invariant basis of one-forms, and not to the Cartesian system of Σ), thedependence upon the connection can be factored out and the correspondingvolume integrals performed

∫

Σω ∧ ω ∧ ω = V olΣ. (4)

Absorbing this V ol factor into the constant κ, (3) can be written, in com-ponents, as

ICS [A] ∝ (−AiamijAja +

1

3ǫijkǫabcA

aiA

bjA

ck). (5)

To transform the pure Kodama state for this model into the metric repre-sentation one must append a term

2

(~G)−1

∫

ΣσiaA

ai . (6)

which amounts to performing a canonical transformation on the minisuper-space of the connection representation. The factor of (~G)−1 is necessaryto make the argument of the exponential of (6) dimensionless. In [6] Pater-noga and Graham find a metric representation of the pure Kodama state byperforming an integral

ΨCS[σ] =

∫

Σ9

d9Apaexp[ 2κ

(κpaApa −

1

2Apam

pqAqa + det(Apa))]

(7)

over a 9-dimensional complex contour in A space. There were six identifiedgauge and three physical degrees of freedom in the variable of integrationApq. The integration was performed over the gauge degrees, all Gaussianintegrals, by expanding about the critical points of the exponent. The finalintegration over the three physical variables was highly non-Gaussian andcould only approximately be performed in the steepest-descents approxima-tion, corresponding to five integration contours for the five critical points,each leading to a different state.

In [17] the metric representation of the state Ψ[σ] was determined byChopin Soo in the full theory by Feynman diagramatic expansion about theGaussian portion, treating the metric as a source current for Aa

i (x). Thecalculation was performed to one-loop order by Fadeev-Poppov gauge-fixingand Chern-Simons perturbation theory.

A drawback to gauge-fixing in the path integral of quantum gravity isthat it can obscure the interpretation of the state in its corresponding source-current representation, particularly since the computation can usually beperformed only to finite order in perturbation theory. Gauge fixing hasdemonstrably been appropriate for a renormalizable theory such as Yang-Mills theory, for which the equivalence among different gauges can directlybe related to experimental results and established physical principles. How-ever, quantum general relativity is not as straightforward to deal with.

In this paper we will redo the calculation of [6], but from a differentperspective. We will express the integrand explicitly in terms of the gaugedegrees of freedom, and path integrate just the physical degrees into themetric representation. One may balk at the potential appearance of gaugedegrees of freedom in the state, but this is significantly easier to carry outthe nonperturbative determinination of ΨKod[σ]. Furthermore, we alreadyknow a-priori that ΨKod[A] is invariant under variations of these gauge de-grees of freedom, as a result of solving the Gauss’ law and diffeomorphismconstraints (by variations we mean variations that correspond to small gaugetransformations, connected to the identity within a gauge equivalence class)-hence so should it be in the metric representation.

3

Let us parametrize the connection explicitly in terms of the gauge chosen.

Aia = PilDldQda, (8)

where Pil represents a diffeomorphism (three gauge DOF), and Qda repre-sents a SU(2) rotation (three gauge DOF), each of which can be parametrizedby three Euler angles for a total of six angles. The physical degrees of free-dom can be represented by a diagonal matrix Dld = alδld, given by.

Dld =

a1 0 00 a2 00 0 a3

.

Let us now compute the quantities needed to perform the integral

AiamijAja =

∑

l

∑

m

(PilalQla)mij(PjmamQma) =

∑

l

∑

m

(P T )limijPjmQma(Q

T )alalam

=∑

l

∑

m

(P TmP )lm(QQT )mlalam =∑

l

∑

m

Mlmalam = Mlmalam ≡ Mlmalam(~θP , ~θQ). (9)

Note that the Einstein summation convention is applicable only for the righthand side of the last line. Moving on to the cubic term,

ǫijkǫabcAaiA

bjA

ck

∑

l,m,n

ǫijkǫabc(PilalQla)(PjmamQmb)(PknanQnc)

=∑

l,m,n

(detP )(detQ)(ǫlmn)2alaman = 6(detP )(detQ)a1a2a3

≡ g(~θP , ~θQ)a1a2a3, (10)

where M and g will serve as a ’manifestly’ gauge-dependent propagator andvertex for the corresponding Feynman diagram expansion. Putting it to-gether, the Chern-Simons functional in terms of the physical and unphysicaldegrees of freedom reads

SCS [a, J ] = κ−1(−∑

l,m

Mlmalam +∑

j

ajJj + ga1a2a3

). (11)

Usually when one performs the path integral of an expression like (11), oneintegrates the Gaussian part exactly relative to the origin ~a = a1 = a2 =a3) = (0, 0, 0) and then treats the cubic term as a perturbation in order toobtain a Feynman-diagrammatic expansion in powers of g. By the steepestdescents method, one expands about the critical points of the action in orderto obtain an asymptotic approximation.

We will adopt a ’best of both worlds’ approach. First we will contour-integrate the Gaussian, taking into account the source current, not relative

4

only to the origin, but also relative to the critical points of the full, unap-proximated action. Then we will obtain the full nonperturbative result byFeynman diagrams relative to each critical point. Despite the fact that ~a iscomplex, we can choose an integration contour along the appropriate axisfrom −∞ to ∞ as long as we deform the contour through the critical pointsin question.

It was found in [4],[18],[19] five critical points in a simple gauge choicein the one-loop expansion. We will attempt to do two at least things: (i)Find a nonperturbative expression for the metric representation of the state,valid to all orders, and (ii) Find out if there were any hidden critical points,and correspondingly hidden quantum states due to the chosen gauge.

First, we find the critical points of the action for J = 0. Note that thecritical points correspond to flat connections, which in the full theory canbe expressed in terms of moduli spaces. In minisuperspace the condition offlatness is an algebraic condition on the connection that holds for all Aa

i .However, since we are saving the degrees of freedom that we have chosen toidentify as gauge, we need only evaluate the critical points corresponding tothe physical degrees of freedom in connection space. Hence,

δS

δAai

= 0 ∀a, i −→ δS

δai= 0 for i = 1, 2, 3. (12)

This leads to three equations

−2M11a1 − 2M12a2 − 2M13a3 + ga2a3 = 0

−2M22a2 − 2M12a1 − 2M23a3 + ga1a3 = 0

−2M33a3 − 2M13a1 − 2M23a2 + ga1a2 = 0

(13)

We can eliminate a3 from this system via

1

2a3 =

M11a1 +M12a2ga2 − 2M13

=M22a2 +M12a1ga1 − 2M23

=g

4M33a1a2−

M13

2M33a1−

M23

2M33a2

(14)to produce a system of two algebraic equations in a1 and a2, namely

M11g(a1)2+2(M12M13−M11M23)a1 = M22g(a2)

2+2(M12M23−M22M13)a2(15)

which is the equation for a rotated hyperbola, translated from the origin of(a1, a2) space and

a2

( g2

2M33(a1)

2−2gM23

M33a1−2

(M232

M33−M22

))= g

M13

M33(a1)

2−2(M23M13

M33−M22

)a1

(16)

5

is a curve which for a given a2 intersects the a1 axis twice, correspondingto the roots of the numerator, has in the worst case two asymptotes, cor-responding to the roots of the denominator, and then levels off at ±∞. Inthe worst case, naively, the two branches of the hyperbola (15) could con-ceivably intersect (16) in 8 points, resulting in 8 topologically inequivalentcontours and 8, rather than 5 states. [4] left open the possibility that theremay exist additional states, which we will attempt to find here. Our resulthas to do with the parametrization of the state in terms of all of its degreesof freedom rather than only the physical ones.

It is clear that (16), being linear in a2, can be subtituted into (15) toyield a 6th degree polynomial equation of the form

α1u2(a1)

6 + (2α1uv + β1u2)(a1)

5 +(α1(v

2 + 2uw) + 2β1uv − β2xu)(a1)

4

+(2α1vw + β1(v

2 + 2uw)− β2(xv + yu))(a1)

3

+(α1w

2 + 2β1vw − α2x− (xw + yv))(a1)

2 + (β1 − α2y − β2yw)a1 = 0, (17)

where we have made the following identifications

α1 = M11g; β1 = M12M13 −M11M23;

α2 = M22g; β2 = M12M23 −M22M13;

x = gM13

M33; y =

M23M13

M33−M12;

u =g2

2M33; v = 2g

M23M33

M33; w =

M232

M33−M22. (18)

a1 = 0 is an obvious solution, which collapses the algebraic equation toa quintic polynomial equation in a1. By the theorem of Galois it is notpossible to solve a this quintic explicitly in terms of radicals. However, weknow that in the worst case of gauge there are five distinct solutions, notincluding a1 = 0, which can be found by various methods. So we expect atotal of 6 critical points for a1, which uniquely determine the critical pointsfor (a2, a3). Thus it can be concluded that there exist 6 topologically distinctsectors of the the Ashtekar connection, which correspond to 6 states in themetric representation.

The point of this derivation is to show the pitfall of attempting to quan-tize general realtivity in a particular gauge. By maintaining the gauge de-pendence explicit we have determined that there must exist an additionalquantum state which was missed due to a convenient gauge choice. Eachgauge choice corresponds to a different pair of curves (15) and (16). By con-tinuously varying the gauge parameters (~θP , ~θQ) one expects the curves to aswell vary continously, as well as their intersection points. Likewise, the stateas seen in the metric representation concievably may vary contiously as well.

6

However, since the state is gauge invariant we should expect the state as wellto remain invariant to all orders in perturbation theory irrespective of therepresentation. Any percieved variation is an artifact of the semiclassical ap-proximation stemming from the steepest descents method, particularly if theSQC does not hold for the state. The SQC was demonstrated ([8],[8][9],[10])for the pure Kodama state in the connection representation. It seems, inprinciple, that the SQC should not be broken simply due to transformationinto its metric counterpart or into any other representation.

It is possible as well, by a suitable choice of parameters, that one maycollapse the sixth-degree polynomial to one of a lesser degree, resulting infewer solutions. From this one could arrive at an erroneous conclusion con-cerning the number of states. Even worse, perhaps even the parameterscan be chosen such that there appear to be no states at all! So a strategy,when dealing with quantum general relativity, is to maintain the gauge de-pendence explicit- hopefully in a convenient parametrization. One needn’tworry since the state is already known to be gauge/diffeomorphism invariant(under transformations connected to the identity). Thus one has a criterionof the state valid nonperturbatively to all orders, and one can regard suchunphysical variables as ignorable: ”The gauge degrees of freedom are notgoing anywhere”.

Let us assume that there are 6 critical points of the action α(n), forn = 1, ...6. We shall now compute the corresponding states in the metricrepresentation exactly. Note, as observed in [6], that attempting to evaluate(7) via the steepest descents method allows one to perform only the first6 integrations exactly via the Gaussian method. A Gaussian integral per-formed over all degrees of freedom about the trivial critical point of (7) canbe done by inspection, yielding

ΨCS[σ] =

∫

Σ9

d9Apaexp[ 2κ

(κpaApa −

1

2Apam

pqAqa))]

= κ9/2(Detmij)−1/2exp

[κ−1κpa(m

−1)pqκqa

]. (19)

The exact state then is given by a Feynman perturbation series, with thereplacement Aa

i → δ/δκia. The coupling constant κ is rescaled and the resultis

ΨKod[κia] = κ9/2(Detmij)

−1/2exp[2√κdet

∂

∂κia

]exp[κ−1κpa(m

−1)pqκqa

].

(20)(20) is formally an exact result to all orders based upon the gauge choice(a1, a2, a3)crit = (0, 0, 0), which enables one to probe beyond the steepestdescents semiclassical approximation to any order desired, in order to obtainperturbative information about that particular. However, we would like to

7

be able to repeat the calculation for all six states. Furthermore, in the form(20) the gauge degrees of freedom are manifestly entangled with the physicalones, possibly, making the interpretation of the state not as clear.

Let us transform the metric explicitly into its physical degrees of freedom.All that is required is to transform the Fourier coupling term (6) into thebasis parametrized by Aa

i .

∫

ΣσiaA

ai =

∫

ΣσiaQ

abPliAbl =

∑

b

∫

ΣσiaQ

abPbiab, (21)

where in (21) the Einstein summation convention applies to all indices exceptthe b index. From (21) it is clear that the physical degrees of freedom ofthe metric representation σi

a must be those canonically conjugate to thecorresponding ones chosen in the connection representation Aa

i . So oneshould define the physical metric components correpsonding to the chosenconnection components ab as

σb =∑

i,a

QabPbiσia (22)

then we have σb = σb(~θP , ~θQ). Another aspect of (21) pertains to the res-olution of the reality conditions in Ashtekar variables. Since we requirethe state in the metrix representation to correspond to a real metric, therequirement is that only the real part of the connection Aa

i couple to thedensitized triad σi

a. Recall that in the steepest descents method, the inte-grand is transformed into Gaussian form and integrated with respect to areal variable of integration in order that the integral converge. In the caseof Ashtekar varibales the variable of integration is given by

Aai = Γa

i − iKai , (23)

where Γai is the spin connection compatible with the triad and Ka

i relatesto the extrinsic curvature. Consider the analogous variables in our newparametrization, three physical degrees of freedom

ai = γi − iki for i = 1, 2, 3. (24)

In the steepest descents approximation the actual variable of Gaussian in-tegration is related to the modulus

qi =∣∣ai−αi

∣∣ ∼((γi−Re[α

(n)i ])2+(ki−Im[α

(α)i ])2)

)for each i = 1, 2, 3 (25)

where α(n) label the distinct critical points. The the Gaussian part of thesteepest descents method is carried out over the variable qi for each i. Inorder to ensure that the metric is real in the metric representation, all that is

8

required in the Fourier tansform component term (21) is for the densitizedtriad σa be coupled to qi or to some other aspect that encapsulates thereality of the variables. We will investigate this in more depth in futurework. The claim is that the reality conditions on the Ashtekar connectionare precisely that the conditions for the (STM) steepest descent method(constant imaginary part of the Chern-Simons functional corresponding tothe maximum of the real part) be satisfied, and that the source currentbeing coupled to the connection as well be real as described above. Therequirements for applicability the STM are that the Chern-Simons functionalbe analytic, which is not too stringent a requirement.

We are now ready to transform the pure Kodama state into the con-nection representation for all 6 topological sectors. We wish to perform theintegral

ΨP,Q[~σ] =

∫da1da2da3exp

[V olΣµ

~GΛ

(−M ijaiaj + g′a1a2a3

)+

V olΣ~G

σiai

].

(26)Here we have defined g′ = µ−1g, where µ is a mass scale defined such as tomake the matrix Mij dimensionless. As the connection ai is of unit massdimension, then g′ is of mass dimension −1, assuming g to be dimensionless.By defining the dimensionful coupling constant λ′ by

λ′ =(µV olΣ

~GΛ

)(27)

and rescaling the connection to dimensionless variables by ai → λ′−1/2ai,we are reduced to the integral (over connection 3-space)

ΨP,Q[~σ] = λ′−3/2∫

da1da2da3exp[−M ijaiaj+µ−1Λλ′−1/2

σiai+g′λ′−1/2a1a2a3

].

(28)Note that as a result of [λ′] = −2, the coupling constant multiplying thecubic term in (30) is dimensionless. This is the equivalent of recastingquantum general relativity into a renormalizable theory! Only a few mod-ifications need be made to convert this minisuperpsace model into the fulltheory. One merely replaces the matrix M−1 with a propagator, namelythe operator d−1 in the position representation. One needn’t balk at gaugeinvariance, since as indicated previously, one should never integrate alonggauge directions when quantizing general realitivity, without fair compen-sation for the insight lost. Thus the operator d−1 is by definition invertibleon the physical subpace of the phase space. This allows us to reproduce theresults of this paper for the full theory, something we will also carry out inthis series.

Now define the new dimensionless coupling constants

9

k = g′λ′−1/2; λ = µ−1λ′−1/2

Λ =(ΛV olΣ

~Gµ

)(29)

We are then reduced to evaluating the integral

ΨP,Q[~σ] = λ′−3/2∫

da1da2da3 exp[−M ijaiaj + λσiai

]exp(ka1a2a3

)(30)

Note that all coupling constants are parametrized by the gauge degrees offreedom. So by choosing a particular gauge in which to make observationsof the theory, one is by definition setting a scale for perturbation theory.(30) is the starting point for our analysis.

Let us evaluate the state first for the (a1, a2, a3)crit = (0, 0, 0) sector.Assuming the matrix M ij is positive definite, the Gaussian part can bedone by inspection to yield

ΨGauss[~σ, ~θP , ~θQ] = λ′−3/2∫

da1da2da3 exp[−M ijaiaj + λσiai

]

= λ′−3/2(Det−1/2Mij)exp

[λ2

4σi(M

−1)ij σi

]. (31)

Note that the dependence of the Gaussian part of the state may possibly beexplicitly gauge dependent, since it is only a semiclassical approximation.However, for the exact state, even though one can write and expressionof the form Ψexact[~σ, ~θP , ~θQ], we surmise it to be explicitly independent of

(~θP , ~θQ) (within a given topological sector) to all orders. The correspondingnonperturbatively exact state for this sector is then given by

Ψexact[~σ, ~θP , ~θQ] = Ψ~0[~σ]

= λ′−3/2(Det−1/2Mij)exp

[k

∂

∂σ1

∂

∂σ2

∂

∂σ3

]exp[λ2

4σi(M

−1)ij σi

](32)

The notation Ψ~0 signifies that we are evaluating the trivial topological sectorof the theory. Note that we have carried around all constants that onewould normally discard in field theory. This is because they now carry alabel, based upon the gauge parametrized. One must carry these numerical’constants’ to any order in the Feynman diagram expansion calculated. Inthis way one can perform everything one does in quantum mechanics. Inour case, calculating norms, observables, finding probabilities for varioustopological sectors, etc.

To ensure that the state is not a-priori unnormalizable, the argument ofthe exponential of the Gaussian portion must be negative, or λ2 < 0. Inorder for this to be the case, there are two main possibilities. Recall theMaurer-Cartan expression for the invariant one-forms.

10

dωa +1

2

(maeǫebc + δa[bwc]

)ωb ∧ ωc. (33)

If in (33) the one-forms ω have unit mass dimension, then we must have[mab] = [wa] = 0. However, assuming [ω] = 0, then mab carries the mass-dimension of the detivative, which is unity. This is the reason why weassigned it a mass scale µ above. Now g is dimensionless, as is the quantity~GΛ. A cursory galnce at the coupling constants for the state reveals thesituation. Define the length scale of the universe by L ∝ (V olΣ)

1/3. Thenthe quantity lµ is dimensionless, just like Planck’s constant. We have

k = ±g√~GΛ(µL)−3/2; λ = ±

√~GΛ(µL)3/2(~G)−1µ−2. (34)

Since λ is squared in the argument of the Gaussian, the normalizability ofthe state would naively imply that the cosmological constant Λ must benegative! The ramification of this would mean that the ground state of theuniverse corresponds to a anti deSitter (as opposed to deSitter [16],[7][20])spacetime, which would imply that the Kodama state in the connectionrepresentation is given by

ΨKod[A] ∝ e6(~GΛ)−1ICS [A]. (35)

Since the observational data on cosmology suggest that Λ is slightly positive,another possibility consistent with this is to require µ to be negative. µ isjust a quantity of unit mass dimension, so it sets a mass scale. But thereis no requirement that it correspond to a positive number. The alternativeis to keep both Λ and µ positive, and to live with an unnormalizable state-this would defeat the purpose of our investigation, therefore we shall adoptthe Λ > 0, µ < 0 case.

Another item of note is that the coupling constants (34) are very small,in addition to being dimensionless. And in particular, g and mmn are gauge-dependent. Recall their expressions

g = 6(detP )(detQ); Mmn = (P TmP )lm(QQT )lm (36)

with no summation over l,m. This means that one conceivably can picka gauge in which g is extremely small. This is not too foreign a concept,actually. The closest analogy in the usual quantum field theories, is when onevaries the energy scale via the renormalization group equation, as necessaryto create an asymptotically free theory as in quantum chromodynamics. Therenormalization group can be though of as a gauge group in its own right. Bychoosing a different energy scale one is choosing a different gauge in which toexamine the theory. However, due to the lack of ability to parametrize thegauge explicitly for the whole range of energies (for example, the low-energysector of QCD), one accepts an effective desription of the scale limited to.

11

So in a certain sense, the SU(2)∗Diff group of relativity corresponds to therenormalization group of field theory! From this perspective, there alreadyexists a renormalizable quantum theory of gravity.

On a separate note, what we have constructed for the Ψ~0 sector hasapplications in the loop representation and also in spin networks. If onemakes the identification

σ −→∫

γdxi ≡ lγ , (37)

where lγ is the length along a curve in the loop representation. Then one hasa representation of the expectation value of the Wilson loop. One, of course,can perform this calculation in the nonminisuperspace theory in order tomake contact with the current results on knot theory and spin-networks.We will save this for a separate work, but a lot can also be deduced inthe minisuperspace analogue of knot theory. The implication is that theKodama state can be transformed into the loop representation

Ψ[γ] =

∫DA

[P e

R

γAΨKod[A]

](38)

for one of the 6 topological sectors, and all the results of loop quantumgravity can be carried over into the ground state for quantum gravity. Like-wise, the results of the work of Witten on the relation of the 3-dimensionalChern-Simons partition function to 2-dimensional conformal theory can becarried over. If there exists a closed form expression for the Chern-Simonspartition function in terms of conformal blocks, then this can be used toderive results (and vice-versa) regarding Ψ[σ] both in minisuperspace andnon minisuperspace.

Given that we can explicitly write an expression for the zero sector, it isstraightforward to extend it to any of the other topological sectors. Let uslabel the critical connections of this sectors by ~α(n). In the steepest descentsapproach one parametrizes the deviation from the critical connections interms of a magnitude (which we think of as the Fourier conjugate to thevariable σi in the conjugate (metric) representation, and a phase for eachcritical point.

Z = ai − α(n)i = X + iY = |Z|eiθ(n) (39)

where in (39) θ(n) is the phase of the path of steepest descent approaching

the critical point α(n). The Chern-Simons action can be written in thefollowing form

ICS [A] =

∫

Σtr(A ∧ F − 1

3A ∧A ∧A

). (40)

12

Suspend the minisuperspace model for a moment and imagine that we arein the full theory. Then for a flat connection we would have F = 0, whereF is the curvature, which means that the connection must be pure gauge,of the form, A = θ = g−1dg for a group element g of the group G. Thus,the action, evaluated on the critical point, becomes

ICS [A] = −1

3

∫

Mθ ∧ θ ∧ θ =

8π2

3DegG, (41)

where DegG is the degree of the map from the gauge group to the manifoldΣ, which is labeled by the integers n. So one can identify the constantpart of the steepest descent approximation, the zeroth order term in thesteepest descents Taylor expansion, with this topological quantity. But theimplications are much deeper. If the quantum gravitational wavefunctionsof the metric representation of fall into one of six topological sectors, thenit implies a limit on n of 6, at least for minisuperspace. For the full theorythere should be no limit on n.

Anyway, define (40) for the minisuperspace Bianchi IX model on thecritical connections by

ICS[~α(n)] = π(n)(Σ). (42)

Looking back at (39) we need the magnitude of Z to range from 0 to ∞.This corresponds to taking

∞ ≤ Re[ai − α(n)i ] ≤ −∞. (43)

I believe that this is the reality condition long searched for for the Ashtekarvariables. It amounts to the requirement that the Gaussian integral convergein order to produce real metric relativity in the metric representation. Thereason is the following. If one were to evaluate the Gausian part of thewavefunction by the steepest descent method, then one would write downthe expression making the shift ~a′ = ~a− ~α(n)

ΨGauss[~σ, ~θP , ~θQ] =

∫da′1da

′

2da′

3 exp[−M ijaiaj

]

=∣∣DetMij

∣∣−1/2exp[−M ijα

(n)i α

(n)j

]eiθ(n)

∫∞

−∞

dt e−12t2

=√2π∣∣DetMij

∣∣−1/2exp[−M ijα

(n)i α

(n)j

]eiθ(n) =

√2π∣∣DetMij

∣∣−1/2exp[−π(n)(Σ)](44)

However, we are including a source current term

exp[∑

i

∫

Σaiσi

](45)

13

In order to translate this into the language of complex variables theory,in short, to be able to perform the analogous integral along with a sourcecurrent σi, in the t = |z| direction from −∞ to ∞, only the real parti of aican contribute. Hence,

∑

i

∫

Σaiσi =

∑

i

∫

Σ|Zi|σi =

∑

i

∫

Σtiσi (46)

So the Gaussian portion of the steepest descent path integral becomes,schematically,

ΨGauss(n)[~σ, ~θP , ~θQ] =

∫∞

−∞

dt e−1/2~aM~a+~σ·~a

=√2π∣∣DetMij

∣∣−1/2eiθ(n)exp

[−π(n)(Σ)]exp

[λ2

4σi(M

−1)ij σi

](47)

Overall, we can write down the nonperturbative expression for the state forany of the topological sectors as

Ψ(n)[~σ, ~θP , ~θQ]

=√2π∣∣DetM

∣∣−1/2eiθ(n)exp

(−π(n)(Σ)

)exp[k

∂

∂σ1

∂

∂σ2

∂

∂σ3

]exp[λ2

4~σM−1~σi

].(48)

The interpretation is that this is the wavefunctional for the nth topologicalsector. Even though the arguments are indicated in terms of gauge degreesof freedom, one expects that the wavefunction is explicitly independent ofthese degrees of freedom. Thus

∂

∂ ~θPΨ(n)[~σ, ~θP , ~θQ] =

∂

∂ ~θQΨ(n)[~σ, ~θP , ~θQ] = 0, (49)

although this may not necessarily be the case for the Gaussian part.

3 Perturbative expansion of the nonperturbative

Kodama wavefunction

Now that we have derived a nonperturbative expression for the full quantumgravitational state in the metric representation for the Bianchi IX model,we wish to assess whether or not it is normalizable, in order to establishwhether or not there exists a consistent quantum theory of gravity for ouruniverse. But before one can assess normalizability, it would be nice if onecould write a nonperturbative expression for the state. This is the only way

14

one can be 100 percent sure: by carrying out the expansion all orders inperturbation theory.

Omitting the numerical coefficients for simplicity, We must find

exp[k

∂

∂σ1

∂

∂σ2

∂

∂σ3

]exp[λ2

4~σM−1~σi

]. (50)

in order to determine the state. We will try to be as explicit as possiblein all steps and we will establish for convenience a few conventions. First,since M ij and λ are dimensionless quantities, we will absorb the latter intothe definition of the former whenver convenenient by the replacement

λ2

4(M−1)ij −→ (M−1)ij . (51)

Note that M ij can either be positive or negative, depending upon the choiceof the relative sign between the cosmological constant and the mass scale ofthe Biachi IX structure constants (we require it to the negative to ensurethe wavefunction is not a-priori unnormalizable). Another item of note isthat by ’pushing forward’ the gauge degrees of freedom from the connectionrepresentation into the metric representation, we have achieved considerablesimplification of the partition function. The tradeoff is that the ’propagator’M−1 and the vertex k have become explicitly gauge dependent. But we shallnot let this deter us from our goal. It amounts to the simplification

ǫijkǫabc δ

δσia

δ

δσjb

δ

δσkc

−→ ∂

∂σ1

∂

∂σ2

∂

∂σ3≡ ∂1∂2∂3. (52)

Before we proceed, let us fix a few more conventions. Make the definition

Ψ = exp∑

ij

σi(M−1)ij σj = e

~eσM−1~

eσ ≡ eΦ (53)

Here we have defined the argument of the Gaussian as a ’phase’ Φ. Nowlet us examine the action of some derivatives on the Gaussian part of thestate.

δ

δσiexp[σM−1~σi

]= 2σij(M

−1)jexp[σM−1~σi

]= 2(~σM−1)ie

~eσM−1~

eσ ≡ 2φiφeΦ

(54)Note that only the symmetric part of M−1 contributes. Another rule is thefollowing

∂i∂jΦ = 2(M−1)ij (55)

Now that we have defined our quantum gravitational Feynman rules let usacquire some practice with a few terms

15

∂iΨ = 2φiΨ (56)

Acting again.

∂j∂iΨ =(2(M−1)ij + 4φiφj

)Ψ (57)

And a third time,

∂k∂j∂iΨ =(4(M−1)ikφj + 4(M−1)jkφi + 4(M−1)ijφk + 8φiφjφk

)Ψ (58)

We are now ready to begin. We wish to compute

Ψ[~σ] = exp[k∂1∂2∂3

]eΦ (59)

Some possible hurdles to the progress in a quantum field theory are (i)Nonrenormalizability. In this case, one lives with an effective theory in agiven energy regime, and parametrizes their ignorance above that regime.In our case, the coupling constant k is dimensionless, therefore the theory (ifwe were performing this calculation for in the full theory) would be renor-malizable. (ii) Non finiteness. Even if a quantum theory is perturbativelyrenormalizable, there is no guarantee that the full perturbative series con-verges to a finite answer. In general, one would need to verify requirementsof Borel summability. We will attempt to deal with these issues in thispaper.

Performing a Taylor expansion, we have

Ψ =∑

m,n

km

m!(∂1∂2∂3)

m 1

n!Φn =

∑

m,n

km

m!n!(∂1)

m(∂2)m(∂3)

mΦn (60)

Note that partial derivatives communte, ∂i∂j = ∂j∂i as did the originalfunctional derivatives, so it should not matter in what sequence we performthe operations. Two more identities before we get started

∂m

∂xmxn = θ(n−m)

n!

(n −m)!, (61)

where θ(t) is the Heaviside step function, given by

θ(t) = 0 for t < 0; θ(t) = 1 for t > 0. (62)

And one more identity,

∂m

∂xmF (x)G(x) =

∑

k

(n

k

)∂k

∂xkF

∂m−k

∂xm−kG (63)

Proceeding from (60), we have

16

Ψ =∑

m,n

km

m!n!(∂1)

m(∂2)m[(∂3)

mΦn]=∑

m,n

km

m!n!(∂1)

m(∂2)m( n!

(n−m)!Φn−m(2φ3)

)

(64)Making use of (63), we have

Ψ =∑

m,n

km

m!n!(∂1)

m[ n!

(n−m)!

[(∂2)

mΦn−m](2φ3)

+n!

(n−m)!

(m

1

)(∂2)

m−1Φn−m∂2(2φ3)]

(65)

Note that since φ2 is only first-order in ~σ, only the first two terms of (63)contribute. Plodding along,

Ψ =∑

m,n

km

m!n!(∂1)

m[ n!

(n−m)!

( (n −m)!

(n− 2m)!Φn−2m

)(4φ2φ3)

+n!

(n−m)!

(m

1

)(n−m)!

(n− 2m+ 1)!Φn−2m+1(4φ2(M

−1)23)]

(66)

It will be convenient to separate the expression into two terms, and to dealwith each term individually. Note that we have manifestly maintained var-ious the combinatorial factors, just to keep organized. We will cancel theseout later. We have two terms.(i) First we have A (not to be confused with the Ashtekar SU(2) potential)

A =∑

m,n

km

m!n!

[n!

(n−m)!

(n−m)!

(n− 2m)!

((∂1)

mΦn−2m(4φ2φ3

+

(m

1

)(∂1)

m−1Φn−2m∂1(4φ2φ3)

)+

(m

2

)(∂1)

m−2Φn−2m(∂1)2(4φ2φ3

))]

(67)

and B (not to be confused with the Ashtekar magnetic field)

B =∑

m,n

km

m!n!

(m

1

)n!

(n−m)!

(n−m)!

(n− 2m+ 1)!

[(∂1)

mΦn−2m+1(4φ2(M

−1)23

+

(m

1

)(∂1)

m−1Φn−2m+1∂1(4φ2(M−1)23

](68)

Continuing with A, we have

17

A =∑

m,n

km

m!n!

n!

(n −m)!

(n−m)!

(n− 2m)!

[(n− 2m)!

(n− 3m)!Φn−3m8φ1φ2φ3

+

(m

1

)(n− 2m)!

(n− 3m+ 1)!Φn−3m+18φ1

((M−1)12φ3 + (M−1)13φ2

)

+

(m

2

)(n− 2m)!

(n − 3m+ 2)!Φn−3m+216φ1

[((M−1

12 )(M−1)13)]

(69)

were in the last line we have used

∂1((M−1)12φ3 +(M−1)13φ2

)= (M−1)13(M

−1)13 + (M−1)13(M−1)12. (70)

Continuing with B, we have

B =∑

m,n

km

m!n!

n!

(n−m)!

(n−m)!

(n− 2m+ 1)!

[(n− 2m+ 1)!

(n− 3m+ 1)!Φn−3m+18(M−1)23φiφ2

+

(m

1

)(n− 2m+ 1)!

(n− 3m+ 2)!Φn−3m+2

(8(M−1)12(M

−1)23φ1

)](71)

It is now that we start cancelling terms

A =∑

m,n

km

m!

8

(n− 3m)!Φn−3m

[φ1φ2φ3

+

(m

1

)(n− 3m+ 1)−1Φ

((M−1)12φ1φ3 + (M−1)13φ1φ2

+

(m

2

)(n− 3m+ 1)−1(n− 3m+ 2)−1Φ2(M−1)12(M

−1)13φ1

](72)

and

B =∑

m,n

km

m!

8

(n− 3m+ 1)!Φn−3m+1

[(M−1)23φ1φ2

+2m

n− 3m+ 2Φ(M−1)12(M

−1)23φ1

](73)

Consolidating terms and splitting the sums, we have for A

18

A = 8φ1φ2φ3

∑

m

km

m!

∑

n

Φn−3m

(n− 3m)!

+8((M−1)12φ1φ3 + (M−1)13φ1φ2

)∑

m

km

(m− 1)!

∑

n

Φn−3m+1

(n− 3m+ 1)!

+16(M−1)12(M−1)13φ1

∑

m

km

(m− 2)!

∑

n

Φn−3m+2

(n − 3m+ 2)!, (74)

and for B,

B = 8(M−1)23φ1φ2

∑

m

km

m!

∑

n

Φn−3m+1

(n− 3m+ 1)!

+8M−1)12(M−1)23φ1

∑

m

km

m!

∑

n

Φn−3m+2

(n− 3m+ 2)!(75)

whereupon we observe a remakable property. Each term in the sums can bewritten as the exponential function by either relabeling indices or absorbingfactors of k as necessary. The final result is

A+B =

(8φ1φ2φ3 ++8(M−1)23φ1φ2 + 8(M−1)12(M

−1)23φ1

8((M−1)12φ1φ3 + (M−1)13φ1φ2

)k + 16

((M−1)12(M

−1)13φ1

)k2

)ek+Φ (76)

Let us recapitulate. We have evaluated the entire perturbation series forthe Bianchi IX minisuperspace model and have obtained the non perturba-tive result of summing the series to all orders in perturbation theory, as adirect result of maintaing the gauge parametrization of the state explicit.Furthermore, it is remarkable that the result is merely a cubic polynomialtimes the original state, a Gaussian, with the ’phase’ shifted by the dimen-sionless coupling constant k! In full-blown equation form, maintaining the’propagator’ in its rescaled form,

Ψexact[~σ, ~θP , ~θQ] ∝ exp[k

∂

∂σ1

∂

∂σ2

∂

∂σ3

]exp[σi(M

−1)ij σi

]

=(w0 + w1k + w2k

2)ekexp[σi(M

−1)ij σi

](77)

where w0, w1 and w2 are polynomial functions of ~σ of at most cubic degree.

19

4 Discussion: Norm of the nonperturbative Ko-dama wavefunction

We can now write down explicitly the full wavefunction for any of the topo-logically distinct sectors, rescaling Mij as

Ψ(n)[~σ, ~θP , ~θQ] =√2πλ′−3/2∣∣DetM

∣∣−1/2eiθ(n)exp

(−π(n)(Σ)

)W exp

[λ2

4~σM−1~σi + k

].(78)

To calculate the norm of the wavefunction, we could use the usual quantummechanical definition

Norm =⟨Ψ(n)

∣∣Ψ(n)

⟩=

∫d3σ∣∣Ψ∣∣2 (79)

We shall lay down the foundations to compute the essential parts, savingthe numerical coefficients until afterwards. There are a few things to note:(i) As in the transformation from the connection into the metric represen-tation, we will not integrate over any of the gauge degrees of freedom inthe determination of the norm, only the physical ones. (ii) An interestingquestion is what measure one would need for integration such that the statesfrom two topologically distinct sectors are orthogonal. The only parametersavailable that distinguish one ’state’ from another are the phase of steepestapproach θ(n) and the ’minisuperspace winding number’ π(n)(Σ). For exam-ple, one could always find a prescription to gaurantee orthonormalizability.One possibility is to rescale the phase θ(n) of the path of steepest descent,using the rescaling factor as a variable of integration, whereupon using thedelta-function orthonormalizability of a plane wave, one could would findthis to be the case. For example, one could take θ(n) → θ(n)ǫ, and definethe part of the state corresponding to the phase factor by

Ψ(n)[ǫ, ~σ] ≡ Ψ(n)[0, ~σ]exp(−2π(n)(Σ)

)exp(iǫθ(n)), (80)

where Ψ(n)[0, ~σ] is the part of the wavefunction not involving any topologicalquantities, and find a measure such that

⟨Ψ(m)

∣∣Ψ(m)

⟩≡∫

dµ(ǫ)ei(ǫθ(m)−ǫθ(n))exp(−2(π(n)(Σ) + π(m)(Σ))

)= δ(m)(n).

(81)Since the value of the integral is to a large extent independent of the

path (when the Cauchy-Riemann conditions for analyticity are met withina topological sector), there is considerable leeway in fixing a well-definedcriterion. We will take this up in future work. For now, let us explicitlycompute the ingredients needed for the norm, for which the phase factorsθ(n) cancel out in the modulus squared.

20

Since we have expressed the prefactor of the wavefunction in terms ofthe quantities Φi, it will be convenient to use this as the integration variable

σi(M−1)ij = φj −→ σi = MijΣj (82)

Hence we have the following replacements for the ’phase’ (rescaling it again)and the measure

Φ = φiMijφj ; d3σ = (detM)d3φ (83)

In computation of the norm we will make use of a well known property ofthe Gaussian integral, namely that its integral over an odd (linear) func-tion vanishes, but the the integral over an even function gives a nontrivialresult. This can be summarized in Wick’s theorem. For variables xi in ann-dimensional Euclidean space,

∫∞

−∞...∫∞

−∞dx1...dxN e−1/2~xM~xxixj ...xkxl∫

∞

−∞...∫∞

−∞dx1...dxNe−1/2~xM~x

=⟨xixj...xkxl

⟩=∑

Wick

(M−1)ab...(M−1)cd (84)

The most inconvenient factor to deal with is the W factor. Also, we willassume that the conditions upon the cosmological constant Λ and the scaleµ are such that the wavefunction is a Gaussian. We need to find, in the ~φvariables,

∫d3φ(detM)

∣∣W 2∣∣exp

[−φiMijφj

](85)

Here, we have rescaled M by a factor of 2, which modifies the couplingconstant accordingly. We can do this since M is dimensionless. A cursoryglance at the expression for W

W = 8φ1φ2φ3 ++8(M−1)23φ1φ2 + 8(M−1)12(M−1)23φ1

8((M−1)12φ1φ3 + (M−1)13φ1φ2

)k + 16

((M−1)12(M

−1)13φ1

)(86)

indicates that all cross terms in its square are linear in at least one variable.Therefore the cross terms will not contribute to the norm. We need onlyconsider the diagonal squared terms

|W |2 = 64[φ21φ

22φ

23(M

−1)223φ21φ

22 +

((M−1

12 )(M−1)23)2

+ k2((M−1

12)2φ2

1φ23

+((M−1)13

)2φ21φ

22

)+ 4k4

((M−1)12(M

−1)13)2φ21

].(87)

21

All that is necessary is to apply Wick’s theorem to each term, and we willhave the norm. Starting with the simplest term,

⟨φ21

⟩∝ (M−1)11 (88)

Moving to the next,

⟨Φ21φ

22

⟩∝ (M−1)11(M

−1)22 + 2(M−1)12(M−1)12⟨

Φ21φ

23

⟩∝ (M−1)11(M

−1)33 + 2(M−1)13(M−1)13. (89)

Moving on to the last and most difficult contribution. We should get 15terms

⟨φ21φ

22φ

23

⟩∝ (M−1)11(M

−1)22(M−1)33 + ... (90)

We will save the full-blown calculation for the appendix of one of the futureworks in this series. It will be of the form

⟨∣∣W∣∣2⟩ = W0 +W2k

2 +W4k4 (91)

where the w’s are functions of the matrix elements of M−1

A fundamental question remains, that given that there is explicit gaugedependence in the matrix M , assuming that no errors were made in thederivation: are the expectation values of observables O independent of thegauge parametrization? One possible criterion that the derivative with re-spect to the gauge degrees of freedom vanish, is what establishes conven-tional definitions of an observable. Thus the criterion is for

−→ ∂

∂Qab

⟨O⟩=−→ ∂

∂Pab

⟨O⟩= 0. (92)

(92) can then be used to provide constraints on the set of observables inquantum gravity, making them easier to find.

The full-blown norm, in terms of all paramters, is given by

∣∣Ψ∣∣2 = 2πλ′−3∣∣DetM

∣∣−1exp(−2π(n)(Σ)

)∫

d3φ∣∣W∣∣2(detM)exp

[−8

λ2~φM−1~φi + 2Re[k]

]. (93)

Expanding out the relevant terms,

∣∣Ψ∣∣2 ∝ exp

(−2π(n)(Σ)

)e2Re[k]λ′−3

Det1/2(λ2

M

)[W0 +W2k

2 +W4k4]= N(n)

(94)

22

The only adjustible parameters in the computation of the norm are thegauge parameters in the matrix M , the scale µ and the sice of the universelΣ. (94)can be set equal to the normalization factor of the wavefunction, andthen the normalized states are determined by dividing by N(n). So we haveseen explicitly that the normalization factor of the metric representation ofthe Kodama state contains topological information. This can in turn beused to deduce more about the state in the connection representation.

Various objections have been raised throught the literature as to the nor-malizability of the Kodama state. As we have seen, there exists a criteriafor explicitly determining its nomralizability. Furthermore, the normaliz-abilty depends upon the contour of integration required, which in terun setsthe reality conditions on the Ashtekar connection. Given a Hilbert space ofstates and a norm, one can now calculate observables for the theory.

5 Discussion/ Conclusion

The action of the exponentiated cubic operator arising from the pure Ko-dama state in the connection representation is no more severe than actingthrice with a creation operator on the ground state of a simple harmonic os-cillator (creating the analogous third-order Hermite polynomial as an eigen-value). In a nutshell, the gravitational wavefunction in the metric represen-tation, when evaluated to all orders in perturbation theory, is a harmonicoscillator. Since the third- excited state of the SHO is normalizable, then somust be the quantum gravitational wavefunction- to all orders in perturba-tion theory. It is concluded that former computations were obscured by theprocess of gauge-fixing. The claim is that all five states found in [18] and in[19] are manifestations of the Gaussian state with the effects of the gaugechoice arried through to all finite orders of perturbation theory.

This result as well appears to be in support of the SCC [8]. ΨKod satisfiesthe SQC exactly in the connection representation. Therefore it is logical tosurmise that it does so as well in the metric representation. The harmonicoscillator states are essentially as one can get to the SQC in the metricrepresentation. If ΨKod solved all constraints for a given operator ordering,then it stands to reason that there exists an operator ordering of the ADMWheeler deWitt equation in the Bianchi IX universe for which our derivedstate is an exact solution to all orders.

We have found normalizable states of quantum gravity in the metric rep-resentation. It is hoped that this addresses the issue of the normalizabilityof the Kodama state [12],[13], and the reality conditions on the Ashtekarvariables [3]. In what sense is ΨKod normalizable? We have seen that nor-malizability is based upon the Ashtekar connection’s satisfying the reality

23

conditions necessary for the path integral into the metric representation toconverge. We have demonstrated convergence by evaluating the full pertur-bative series to all orders.

Future work will include extending the calculations in [17], performingthe loop or metric transformation while maintaining the gauge degrees offreedom explicit. It is expected that these gauge degrees of freedom can beutilitzed to narrow the field of search for observables. Prior to this treatmentparts I,III and IV regarding the path integral versrus canonical approach togeneralized Kodama states will be released.

References

[1] Hartle and Hawking ‘Wave function of the universe’ Phys. Rev. D28(1983)2960

[2] Alexander Vilenkin ‘Boundary conditions in quantum cosmology’ Phys.Rev. D12 (1986)3560

[3] Ahbay Ashtekar ‘New perspectives in canonical quantum gravity’ (Bib-liopolis, Napoli, 1988)

[4] Graham and Csordas ‘Quantum states on supersymmetric minisuper-space with a cosmological constant’ gr-qc/9506002

[5] Graham and Csordas ‘Hartle-Hawking state in supersymmetric minisu-perspace’ gr-qc/9506074

[6] Graham and Csordas ‘The Chern-Simons state for the non-diagonalBianchi IX model’ gr-qc/9802009v1

[7] E. E. Ita ‘Existence of generalized semiclassical Kodama states. I. TheAshtelkar–Klein–Gordon model’ gr-qc/0703052v1

[8] E. E. Ita ‘Existence of generalized semiclassical Kodama states. II. Theminisuperspace Ashtelkar–Klein–Gordon model’ gr-qc/0703056v1

[9] E. E. Ita ‘Existence of generalized semiclassical Kodama states. III. Anew approach to finite, full, quantum gravity’ gr-qc/0703057

[10] E. E. Ita ‘Existence of generalized semiclassical Kodama states. IV. Thesearch for a quantization of 4-dimensional gravity’ gr-qc/0704.0367v1

[11] Eyo Ita ‘The canonical versrus path integral quantization approach togeneralized Kodama states. V(Part I)’ to appear

24

[12] Gullermo A.Mena Marugan ‘Is the exponential of the Chern-Simonsaction a normalizable state?’ gr-qc/9402034

[13] Edward Witten ‘A note on the Chern-Simons and Kodama wavefunc-tions’ gr-qc/0306083

[14] Bryce S. DeWitt ‘Quantum theory of gravity I. The canonical theory’Phys. Rev. Volume 160, number 5 (1967)1113

[15] Paul Dirac ‘Lectures on quantum mechanics’

[16] Lee Smolin ‘Quantum gravity with a positive cosmological constant’hep-th/0207079

[17] Chopin Soo, ‘Wavefunction of the universe and Chern-Simons pertur-bation theory’ gr-qc/0109046

[18] Robert Paternoga and Robert Graham, ‘Triad representation of theChern-Simons state in quantum gravity’ gr-qc/0003111

[19] Robert Paternoga and Robert Graham, ‘Physical states of Bianchi typeIX quantum cosmologies described by the Chern-Simons functional’ gr-qc/9603027

[20] Hideo Kodama ‘Holomorphic wavefunction of the universe’ Phys.Rev.D42 (1990)2548

[21] Friedel and Smolin ‘The linearization of the Kodama state’ hep-th/0310224v3

[22] Andrew Randono ‘Generalizing the Kodama State I: Construction’ gr-qc/0611073

[23] Andrew Randono ‘Generalizing the Kodama State II: Properties andPhysical Interpretation’ gr-qc/0611074

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