Dr Alan Carson
email: a.carson@ed.ac.uk.

Jon Stone

Carina Hibberd

Gordon Murray

Rod Duncan

Richard Coleman

Charles Warlow

Richard Roberts

Anthony Pelosi

Jonathon Cavanagh

Hibberd

Warlow

Hansen

Disability, distress and unemployment in neurology outpatients with symptoms 'unexplained by organic disease'

Introduction

We know that one third of neurology outpatients have symptoms, such as pain, weakness and sensory disturbance, that are not explained by recognised 'organic' disease [START_REF] Carson | Do medically unexplained symptoms matter? A prospective cohort study of 300 new referrals to neurology outpatient clinics[END_REF][START_REF] Reid | Medically unexplained symptoms in frequent attenders of secondary care: a retrospective study[END_REF][START_REF] Fink | Somatoform disorders among first time referrals to a neurology service[END_REF][START_REF] Lempert | Psychogenic disorders in neurology: frequency and clinical spectrum[END_REF]. But are these patients really ill? In other words, do they just have symptoms, or are such symptoms actually associated with disability and distress, and is this reflected in their employment status and receipt of disability-related state financial benefits?

A previous study we conducted of symptoms unexplained by organic disease suggested that neurology patients with this problem were at least as disabled and distressed as patients with neurological disease but it was of only a small sample (90 patients with unexplained symptoms) recruited from only one service [START_REF] Carson | Do medically unexplained symptoms matter? A prospective cohort study of 300 new referrals to neurology outpatient clinics[END_REF]. In this study we aimed to determine the disability, distress, and employment status of more than 1,000 consecutive cases with symptoms unexplained by disease, who were prospectively identified as part of a large multicentre study of new neurology outpatients, and to compare these with those of controls from the same cohort whose symptoms were explained by neurological disease.

Methods

The Scottish Neurological Symptoms Study was a prospective, multi-centre, cohort study of NHS neurology outpatients in Scotland. Ethical approval for the study was granted by a Multi-centre Research Ethics Committee. This paper reports a case control analysis of study baseline data.

Participating clinics

Thirty six out of thirty eight consultant neurologists, working across all four Scottish NHS neurology centres, participated. Patients were recruited from their general neurology clinics (including their supervised trainee clinics) in the main Scottish

Patients

All newly referred patients at the participating clinics were potentially eligible for inclusion. The exclusion criteria were: age less than 16, cognitive or physical impairment of a degree that precluded informed consent, inability to read English, or if the neurologist identified the patient as unsuitable for the study (e.g. too distressed, terminally ill). New patients included patients with existing neurological diagnoses who had been re-referred from primary care. Patients were sent information about the study prior to their appointment with the neurologist. After the consultation they were invited by their neurologist to speak to a research assistant. Consent was obtained from patients willing to participate Immediately following the initial consultation neurologists were asked to complete a four point Likert scale:

'To what extent do you think this patient's clinical symptoms are explained by organic disease? -"not at all", "somewhat", "largely" or "completely" [START_REF] Carson | Do medically unexplained symptoms matter? A prospective cohort study of 300 new referrals to neurology outpatient clinics[END_REF]. Operational criteria were provided to guide ratings (web figure 1).

Measures

Following the initial neurological consultation, all participating patients completed a self-report questionnaire which included the following measures:

Demographics including age and sex.

Reported disability (Health status): using the Medical Outcomes Study Short Form 12-item Scale (SF-12) [START_REF] Ware | A 12-Item Short-Form Health Survey: construction of scales and preliminary tests of reliability and validity[END_REF]) which examines health status across eight domains and in two composite scores of physical and mental health status. We calculated the 'raw' score in each of the domains as a percentage and also the two composite scores as recommended by the authors.

Emotional distress using the Hospital Anxiety and Depression scale (HAD)[8] [9]

Employment Status and receipt of state financial benefits by patient self-report. We asked specifically whether patients were (a) in paid employment (and, if not, whether this was because of health problems); (b) in receipt of income support, unemployment benefits, disability living allowance or incapacity benefit.

Analysis

First, we calculated the number of patients in each of the four 'organicity' groups. We then amalgamated those patients whose symptoms were rated as 'not at all explained' and 'somewhat' explained by organic disease and classified them as having symptoms 'unexplained' by disease (cases) and patients who were rated as having symptoms 'largely' and 'completely' explained by disease as having 'explained' symptoms (controls) . We then compared the cases and controls using difference in means and odds ratios.

Results

Patients were recruited between 16/12/2002 and 26/02/2004. During this period a total of 4,299 new patients attended the designated clinics. Recruitment to the study is summarised in figure 1. The final sample was of 3,781 patients (88% of all attendees and 91% of all eligible outpatients) of whom 1,144 patients (30% of the total) were rated by neurologists as having symptoms 'not at all' (n=446; 12%) or only 'somewhat explained' (n=698; 18%) by disease; they were the 'unexplained' cases.

The medical diagnoses given to the cases have been described in detail elsewhere [START_REF] Stone | Who is referred to neurology clinics?-The diagnoses made in 3781 new patients[END_REF] [START_REF] Stone | Symptoms 'unexplained by organic disease' in 1144 new neurology outpatients: how often does the diagnosis turn out to be wrong?[END_REF]. The remaining 2,637 patients had symptoms which were 'largely' (n=940; 25%) or 'completely explained' (n=1697; 45%) by organic disease; they were the 'explained' controls. (We have also analysed the data across all four groups and include this as supplementary web table 1).

[Figure 1 about here] Cases were, on average, four years younger and more likely to be female than controls (Table 1). They had lower scores in all SF-12 individual domains, including all the disability domains including physical and social function and physical and emotional role function. Cases also had a higher level of emotional distress on the SF-12 mental health scale and on the HADS.

[ Table 1 about here] A similarly large proportion of both cases and controls reported not being in paid employment (50% vs 50%). However of those unemployed cases were more likely to report that they were not working because of ill health (26% vs 18%). This difference was reflected in the finding that cases were also more likely to be in receipt of disability related state financial benefits (27% v 22%: OR 1.3 , 95% confidence interval 1.1 to 1.5).

Discussion

We found substantial self-reported disability, distress, and unemployment in patients with symptoms 'unexplained' by 'organic' disease. All these variables were more severe in cases than controls. Furthermore the cases were more likely to report unemployment for reasons of ill health and to be in receipt of disability related state financial benefits.

These findings clearly indicate that patients with symptoms 'unexplained' by organic disease are not merely the 'worried well'. Rather they have substantive self-reported disability, distress and associated unemployment. Whilst their degree of disability and distress was actually statistically greater than that of controls with neurological disease, this difference is around the minimum considered clinically significant [START_REF] Samsa | Determining clinically important differences in health status measures: a general approach with illustration to the Health Utilities Index Mark II[END_REF], [START_REF] Puhan | The minimal important difference of the hospital anxiety and depression scale in patients with chronic obstructive pulmonary disease[END_REF]. A more conservative interpretation of our data would therefore be that they are similar to patients with neurological disease on these characteristics.

Although cases had greater rates of health related unemployment and disability related state financial benefits than controls the majority of patients with symptoms unexplained by organic disease were actually in work. This observation does not support anecdotal suggestions that all such patients are motivated by work avoidance and receipt of benefits.

This study has the strength of being of a large and representative sample of neurology patients. However it also has limitations: Although almost all Scottish neurologists participated in the study not all their clinics were sampled and specialised clinics such as neurovascular and memory clinics were not included; consequently patients with these disorders may be under-represented. Similarly we cannot be certain that Scottish neurological practice is similar to neurological practice round the world, although the prevalence rates of the common neurological disorders in patients attending the clinics sampled would suggest it was.

Although disability related state financial benefits are subject to independent assessment and in a small number of cases anti-fraud investigations, much of the information used to assess such claims is still based upon self-report and cannot therefore be regarded as completely objective. Furthermore, absolute rates of health related unemployment and the uptake of financial benefits may be influenced by economic conditions as well as by illness.

Among other limitations was our approach of 'lumping' patients with symptoms unexplained by organic disease into one group. Whilst we believe that there are sound theoretical reasons for doing this [START_REF] Wessley | Functional somatic syndromes: one or many?[END_REF] others may prefer to 'split' this group into many separate conditions. Furthermore we did not we seek to determine the factors causing the observed disability; these are likely to include a variety of factors including but not exclusively psychiatric illness. Finally our 'unexplained' cases were slightly younger and more likely to be female, this may influence the reported rates of disability and distress.

Previous studies of specific diagnostic groups including psychogenic seizures [START_REF] Reuber | Outcome in psychogenic nonepileptic seizures: a 1 to 10 year follow up of 164 patients[END_REF]15,[START_REF] Lempert | Natural history and outcome of psychogenic seizures:a clinical study in 50 patients[END_REF], psychogenic sensory loss [START_REF] Stone | the 12 year prognosis of unilateral functional weakness and sensory disturbance[END_REF] and psychogenic movement disorders [START_REF] Anderson | Impact of psychogenic movement disorders versus Parkinson's on disability, quality of life, and psychopathology[END_REF][START_REF] Feinstein | Psychiatric outcome in patients with a psychogenic movement disorder Neuropsych[END_REF] have also found that these patients have a degree of physical disability similar to that of patients with neurological disease. A retrospective Dutch study of a consultation liaison psychiatry service [START_REF] Thomassen | Somatoform disorders in consultation-liaison psychiatry: a comparison with other mental disorders[END_REF] included 544 patients who had been diagnosed with a somatoform disorder, of whom 215 had conversion disorder, and found a high rate of unemployment amongst the older patients with somatoform symptoms but not amongst younger ones. Crimlisk et al in a study of 64 patients with motor conversion disorder at a tertiary centre found that 77% of their cohort had held jobs prior to symptom onset but only 11% were still working at six years follow up [START_REF] Crimlisk | Slater revisited: a 6 year follow up study of patients with medically unexplained motor symptoms[END_REF]. A recent cohort study of psychogenic seizures in the West of Scotland described very high rates of unemployment (90%) and uptake of benefits (62%) [START_REF] Mckenzie | Early outcomes and predictors in 260 patients with psychogenic nonepileptic attacks[END_REF]. We found a lower rate of unemployment in our cohort. We suspect this was due to two factors: First we distinguished between those whose unemployment was due to their health complaint and those who were unemployed for other reasons and second we studied a sample more representative of general neurology clinic attenders.

In summary, our date indicates that neurology patients with symptoms unexplained by disease are not merely the 'worried well' but are demonstrably ill by the usually applied criteria of disability and distress. Our data make a strong case the

  neurological centres -Aberdeen, Dundee, Edinburgh, and Glasgow and associated peripheral clinics in Airdrie, East Kilbride, Falkirk, Inverness, Perth, Stirling, Vale of Leven, and Wishaw -between December 2002 and February 2004. All the clinics sampled took mainly general practice referrals with patients allocated by medical records staff according to availability of appointment. Tertiary clinics, where patients required a verified diagnosis to attend (such as multiple sclerosis clinics), and emergency clinics were excluded.

  development and implementation of targeted interventions for this group of patients.It is however imperative that such, potentially complex, interventions are properly tested in well designed randomised controlled trials before they are implemented.Researchers should consider including economic measures into trials as successful treatment could not only improve the quality of patients lives, but also help them to return to work.

Figure 1 -

 1 Figure 1-Flow Chart of patient recruitment into study.

Table 1 .

 1 Disability, distress and employment status of cases with symptoms 'unexplained by organic disease' and controls with symptoms 'explained by organic disease'. Sample size varies between 1134 and 1143 in the not at all/somewhat explained group and between 2612 and 2633 in the largely/completely explained group. b Note-denominator is those who reported not being in paid employment n= 1806

		Did not attend, n= 926 Clinic cancelled, n=137 'Misreferral' n=4 Not a new patient n= 3
		Patients excluded, n= 138 Cognitively impaired, (80), Language difficulties (17), Considered by doctor as unsuitable (15), Too physically disabled or ill (12), No reason found (10), Behavioural problems (3), Too young (1)
		Refused to participate, n= 269
	3892 patients gave consent
		Patient did not complete assessment, n=101 Neurologist diagnoses not traceable, n=10
	3781 patients in study sample
	1144 patients with symptoms unexplained by 'organic' disease	2637 patients with symptoms explained by 'organic' disease

a

Heading-Supplementary web figure 1. Operationalised criteria for 'organicity' rating WHAT DO WE MEAN BY 'ORGANIC DISEASE' ?

The following table is meant as a guide for this study and we are aware that any divisions like this are imperfect.

Many patients have a mixture of symptoms, syndromes or disease and the final coding is your decision based on these guidelines NOT 'Organic Disease' for the purposes of this study 'Organic Disease' for the purposes of this study


M Q Rahman

S S Abeysinghe

S Kelly

N S Roskell

P R Shannon

D M I Montgomery
email: donald.montgomery@ggc.scot.nhs.uk

Persistence of glaucoma medical therapy in the Glasgow Glaucoma Database

Keywords: Glaucoma, Database, Persistence Word count: 1702 Glaucoma, Database, Persistence

Aims: To report the persistence of glaucoma medical therapy in a database of 1006 patients with ocular hypertension (OHT), normal tension glaucoma (NTG) and primary open angle glaucoma (POAG) attending the Glaucoma Clinic at Glasgow Royal Infirmary, Glasgow, U.K. Method: Analyses have been carried out using specially written queries to generate reports relating to initial treatment choice and persistence for individual drugs. Queries were investigated in the database time period from 16 th February 1982 to 11 th February 2009. When investigating drug persistence, the results from the database were split into two distinct time periods from 1997 to 2001 and from 2002 to 2009 to reflect the available treatment options used. Results: The number of patients with each diagnosis was as follows: POAG 608; OHT 246; NTG 152. The Kaplan-Meier estimate for mean persistence from 1997 to 2001 (time to treatment discontinuation) of latanoprost was 58.8 + 1.95 months, timolol was 41.8 + 3.94 months, brimonidine was 24.1 + 3.05 months, and betaxolol was 22.9 + 2.04 months. The Kaplan-Meier estimate for mean persistence from 2002 to 2009 of latanoprost (time to treatment discontinuation) was 52.0 + 2.26 months, bimatoprost was 25.8 + 2.89 months, and travoprost was 23.0 + 1.27 months. The Kaplan-Meier estimate for mean persistence of latanoprost (time to treatment change) was 37.5 + 2.47 months, travoprost was 30.2 + 2.70 months, and bimatoprost was 17.5 + 2.88 months.

Conclusion:

The introduction of the first prostaglandin analogue, latanoprost dramatically improved treatment persistence for glaucoma patients. In the current prostaglandin-rich treatment environment, our data does not show any significant difference between prostaglandins with respect to treatment persistence.

Introduction

Glaucoma is estimated to affect about 67 million people throughout the world,(1) with a prevalence of 2% of the adult population in developed countries [START_REF] Coffey | Prevalence of glaucoma in the west of Ireland[END_REF][START_REF] Klein | Prevalence of glaucoma. The Beaver Dam Eye Study[END_REF][START_REF] Reidy | Prevalence of serious eye disease and visual impairment in a north London population: population based, cross sectional study[END_REF]. Medical therapy has been the mainstay of treatment for several years and previous studies have demonstrated that the lowering of intraocular pressure (IOP) decreases the risk of visual field loss [START_REF] Heijl | Reduction of intraocular pressure and glaucoma progression: results from the Early Manifest Glaucoma Trial[END_REF][START_REF] Anderson | Collaborative normal tension glaucoma study[END_REF].

There are several terms used in the literature to describe the interaction between the physician and patient with regard to medical therapy for glaucoma. In order to maintain the maximum benefit from treatment, patients must take medications as directed and this is known as compliance. Perhaps more appropriately, an agreement between both patients and physician regarding treatment decisions is important and this is known as concordance. Once initiated on treatment, patients must also continue on medications over the long term and this is known as persistence. This is of particular importance in patients with glaucoma in whom disease may develop insidiously, and previous studies of compliance and persistence among glaucoma patients have shown these to be poor. [START_REF] Rotchford | Compliance with timolol treatment in glaucoma[END_REF][START_REF] Mackean | Compliance with treatment of patients with chronic open-angle glaucoma[END_REF][START_REF] Konstas | Compliance and viewpoint of glaucoma patients in Greece[END_REF][START_REF] Patel | Compliance in patients prescribed eyedrops for glaucoma[END_REF] We have previously described, in this journal, a large and unique computerised database established at the glaucoma clinic at Glasgow Royal Infirmary, with data now spanning 27 years [START_REF] Rahman | Surveillance of glaucoma medical therapy in a Glasgow teaching hospital: 26 years' experience[END_REF]. We now present our findings on treatment persistence in patients on glaucoma medical therapy.

Subjects and methods

The Glasgow Glaucoma Database

The database was designed using Microsoft Access with the intention of collecting information on all patients who had undergone, or were undergoing, medical treatment for ocular hypertension (OHT), normal tension glaucoma (NTG), or primary open angle glaucoma (POAG). In 1999 the database was populated with retrospective data abstracted from medical records dating from as early as 1981, according to the algorithm in Figure 1a Only monotherapies that were first prescribed before or at each period onset and administered to more than 5% of patients during that period were included in the analyses. Our definition of initial treatment was the first visit at which the patient was administered glaucoma medication. Furthermore, the analysis population was restricted to patients that received identical treatment to both eyes at their first visit.

We assumed that new patients in the clinic were treatment-naive at the first visit recorded in the database. First visit records that were blank were ignored and subsequent medications were not analysed.

Prescribing policies

We had an unrestricted hospital formulary. The treatment hierarchies historically used in our practise are shown in figure 1b. In the mid 1990s several novel topical intraocular pressure (IOP) lowering agents were introduced. The initial treatment protocol that was implemented in our clinic in 1997 attempted to arrange the newer agents in a hierarchy, with brimonidine selected as the favoured second line agent following beta blockers. Initial audit using the database, however, quickly revealed an unacceptable increase in the discontinuation rate due to adverse effects in the years that followed. It was clear that brimonidine and dorzolamide were particularly implicated and these findings informed the introduction of a new protocol in 2000.

The average interval between treatment change periods was 3 months.

Defining time to failure

End of initial treatment persistence was defined in two ways:

• Time to initial treatment discontinuation.

• Time to initial treatment change.

Initial treatment discontinuation occurred once initial treatment in either eye was discontinued or surgery in either eye was performed. Surgery, discontinuation or the first record of a new treatment addition in either eye indicated a treatment change event. Patients that did not change/discontinue initial treatment on/before 11/02/09 or were recorded as dead, lost to follow up, or moved away were considered censored observations.

Statistical analysis

Relative persistence was assessed using Cox proportional hazards models [START_REF] Cox | Regression models and life tables[END_REF]. This approach allows the adjustment of relative treatment estimates by the potentially important co-variates: treatment, sex, age, diagnosis, IOP, and comorbidities (including diabetes and cardiovascular disease). Covariates were included based on the likelihood of their association with persistence. By accounting for the potential influence of these covariates, we could make more reliable and accurate estimates of relative treatment persistence. The Cox proportional hazards model assumes that the hazards of discontinuation and change were proportional between treatments over time. Kaplan-Meier survival curves and Kaplan-Meier mean estimates were calculated for each medication analysed. The survival curves revealed no significant violation of the assumption of proportionality between treatments.

Results

The mean age of the database population was 66.5 years, ranging from 18 to 93 years. Figure 2a shows the initial treatments by the year that they were started, and figure 2b shows all treatment prescriptions by year. The most frequently started drug administered to at least 1% of the database population was latanoprost in 302 patients, followed by betaxolol in 211 patients, timolol in 151 patients, travoprost in 77 patients, brimonidine in 56 patients, bimatoprost in 33 patients, and lastly dorzolamide which was initiated in 22 patients.

Persistence for medications from 1997-2001

Cox proportional hazard ratios for time to treatment discontinuation for medications most frequently used in the period [1997][1998][1999][2000][2001] There did not appear to be an unequivocal difference between the three prostaglandin analogues when comparing either time to discontinuation or time to initial treatment change.

Discussion

Assessment of patients in a 'real world' setting, as with our database, allows more accurate conclusions to be drawn compared with the strictly controlled regimes in a clinical trial. We believe that our database following up patients over a 27 year period is unique in the United Kingdom.

The analysis from 1997 -2001, demonstrates clearly that patients on latanoprost had superior persistency compared to the beta blockers and the alpha agonist brimonidine.

Patients on latanoprost were less likely to either discontinue use or switch to a subsequent therapy than other classes of glaucoma medications. Although definitions of persistency may differ slightly between studies, these findings are similar to those found in large, retrospective studies in the literature [START_REF] Dasgupta | Population-based persistency rates for topical glaucoma medications measured with pharmacy claims data[END_REF][START_REF] Nordstrom | Persistence and adherence with topical glaucoma therapy[END_REF][START_REF] Reardon | Patient persistency with topical ocular hypotensive therapy in a managed care population[END_REF][START_REF] Shaya | Discontinuation rates of topical glaucoma medications in a managed care population[END_REF]. This maybe due to much lower rates of adverse effects with latanoprost than other medications as we have reported in a previous study [START_REF] Rahman | Surveillance of glaucoma medical therapy in a Glasgow teaching hospital: 26 years' experience[END_REF]. Latanoprost continues to be the most prescribed drug in our clinic as can be seen in figure 2b.

Our analysis from 2002 to 2009 comparing persistency between prostaglandin analogues failed to show a clear advantage of one medication over another. These findings therefore agree with the results of Wilensky et al who found similar persistency rates between latanoprost, travoprost and bimatoprost, in their retrospective study [START_REF] Wilensky | Measurement of persistence and adherence to regimens of IOP-lowering glaucoma medications using pharmacy claims data[END_REF]. However, two other studies in America have shown different results, with both finding that latanoprost had a superior persistency than either travoprost or bimatoprost [START_REF] Reardon | Patient persistency with ocular prostaglandin therapy: a population-based, retrospective study[END_REF][START_REF] Reardon | Persistence on prostaglandin ocular hypotensive therapy: an assessment using medication possession and days covered on therapy[END_REF].

There are several possible reasons to explain this difference in findings. Both American papers were retrospective studies of pharmacy and medical claims data that used proxy measures for persistency such as identifying index drug refills. Such methods may be subject to potential confounding errors and bias. Conversely all of our patients in the database were assessed in the clinic by surgeons, and the persistency of each drug was recorded directly.

It is also important to consider that in the USA, medication costs both to patients and health insurance companies, are potential sources of bias in the reporting of drug persistency rates in the literature. This is less of a problem in the U.K. National Health Service where medications remain free at the point of use for patients.

Our patients on prostaglandin analogues have been followed up continuously for up to 7 years, whereas both of the American studies only examined pharmacy and medical claims data for patients for approximately one year [START_REF] Reardon | Patient persistency with ocular prostaglandin therapy: a population-based, retrospective study[END_REF][START_REF] Reardon | Persistence on prostaglandin ocular hypotensive therapy: an assessment using medication possession and days covered on therapy[END_REF]. Caution must always be used when extrapolating the results from short term studies to long term clinical outcomes. Hypersensitivity reactions to brimonidine, for example, were underestimated by early short term studies [START_REF] Schuman | A 1-year study of brimonidine twice daily in glaucoma and ocular hypertension. A controlled, randomized, multicenter clinical trial. Chronic Brimonidine Study Group[END_REF][START_REF] Leblanc | Twelve-month results of an ongoing randomized trial comparing brimonidine tartrate 0.2% and timolol 0.5% given twice daily in patients with glaucoma or ocular hypertension. Brimonidine Study Group 2[END_REF]. A previous long term study from our database highlighted the true, higher incidence of this problem with brimonidine [START_REF] Osborne | Alphagan allergy may increase the propensity for multiple eye-drop allergy[END_REF]. It is therefore possible that the shorter study period for these papers overestimated the persistency rates of latanoprost.

A potential criticism of this study is the smaller number of patients compared with the much larger populations from the American studies, but we feel that the long length of prospective follow up in our population collectively representing over 7000 treatment years, balances this and gives our study sufficient statistical power.

In conclusion, our study has found that the introduction of the first prostaglandin analogue, latanoprost, as initial monotherapy, dramatically improved treatment persistence for glaucoma patients in the time period from 1997-2001. In the current prostaglandin-rich treatment environment from 2002 -2009, our data do not show any significant difference between prostaglandins with respect to treatment persistence, when used as initial monotherapy. Greater numbers of patients, and longer periods of follow up are required to give meaningful information on the relative treatment persistence for the newer prostaglandin analogues.

  . Information collected for each patient at start of treatment included the following: patient identification number; date of birth; sex; diagnosis; starting treatment and date commenced. If a subsequent visit resulted in no alteration of treatment, the date of the visit was simply logged. If an alteration did occur, this was documented in the database along with detailed reasons for the change. Since 1999, the database has been updated prospectively for all new and existing patients at every clinic visit. It currently contains complete treatment histories on 1006 patients, representing over 7000 patient treatment years. Data are saved securely on a central server and may be accessed simultaneously from multiple clinic based computers. Analyses have been carried out using specially written queries to generate reports relating to initial treatment choice and persistence for individual drugs. Patient Selection Queries were investigated in the database time period from 16 th February 1982 to 11 th February 2009. When investigating drug persistence, the results from the database were split into two distinct time periods from 1997 to 2001 and from 2002 to 2009.

Female

  patients made up the majority of patients comprising 58% of the total. The number of patients with each diagnosis was as follows: POAG 608; OHT 246; NTG 152. The intraocular pressure (IOP) range was 11 to 59 with a mean of 23.4.

Figure 1a :

 1a Figure 1a: Glasgow Royal Infirmary glaucoma database patient information

Figure 1b :

 1b Figure 1b: The treatment hierarchies used in 1997 and 2000.

Figure 2a :

 2a Figure 2a: Initial treatments by year.

Figure 2b :

 2b Figure 2b: All treatments administered once by year.

Figure 3a :

 3a Figure 3a: Kaplan-Meier survival function for time to interval treatment discontinuation for patients receiving initial treatment on or after 1 st January 1997.

Figure 3b :

 3b Figure 3b: Kaplan-Meier survival function for time to interval treatment change for patients receiving initial treatment on or after 1 st January 1997.

Figure 3d :

 3d Figure 3d: Kaplan-Meier survival function for time to interval treatment change for patients receiving initial treatment on or after 1 st January 2002.

Table 1a :

 1a are shown in table 1a: Cox proportional hazard ratios for time to treatment discontinuation for

	Medications	Hazard ratio (95% confidence interval)
	Betaxolol vs. latanoprost	2.59 (2.02, 3.32)
	Brimonidine vs. latanoprost	2.55 (1.76, 3.69)
	Betaxolol vs. timolol	1.74 (1.21, 2.50)
	Medications Brimonidine vs. timolol	Hazard ratio (95% confidence interval) 1.71 (1.10, 2.66)
	Timolol vs. latanoprost	1.49 (1.05, 2.11)
	Betaxolol vs. latanoprost	4.35 (3.26, 5.81)
	Brimonidine vs. latanoprost	3.74 (2.43, 5.75)
	Timolol vs. latanoprost	2.31 (1.57, 3.42)
	Betaxolol vs. timolol	1.88 (1.28, 2.76)
	medications most frequently used in the period 1997-2001
	The Kaplan-Meier survival curve for time to initial treatment discontinuation is
	shown in figure 3a. The Kaplan-Meier estimate for mean persistence of latanoprost
	was 58.8 + 1.95 months, timolol was 41.8 + 3.94 months, brimonidine was 24.1 +
	3.05 months, and betaxolol was 22.9 + 2.04 months.
	Cox proportional hazard ratios for time to initial treatment change are shown in table
	1b.

Table 1b :

 1b Cox proportional hazard ratios for time to initial treatment change in the

	Persistence for medications from 2002-2009
	Cox proportional hazard ratios for time to treatment discontinuation and time to initial
	treatment change for prostaglandin analogues used in the period 2002-2009 are shown
	in table 2.
	Medication	Hazard ratio for time to	Hazard ratio for time to
		treatment discontinuation	initial treatment change
		(95% confidence	(95% confidence interval)
		interval)
	period 1997-2001. Bimatoprost vs. latanoprost 1.06 (0.52, 2.16)	1.18 (0.71, 1.97)
	Travoprost vs. latanoprost	0.94 (0.57, 1.57)	0.75 (0.51, 1.12)
	The Kaplan Meier-survival curve for time to initial treatment change is shown in
	figure 3b. The Kaplan-Meier estimate for mean persistence of latanoprost was 43.2 + Travoprost vs. bimatoprost 0.89 (0.41, 1.97) 0.64 (0.36, 1.15)
	2.28 months, timolol was 32.0 + 3.60 months, brimonidine was 16.8 + 2.64 months,
	and betaxolol was 15.8 + 1.68 months.
	It appeared that patients receiving latanoprost as an initial treatment were the least
	likely to discontinue their treatment or to have their treatment regimen altered
	followed by timolol.

Table 2 :

 2 Cox proportional hazard ratios for time to treatment discontinuation and time to initial treatment change for prostaglandin analogues used in the period 2002-2009.

	The Kaplan-Meier survival curve for time to initial treatment discontinuation is
	shown in figure 3c. The Kaplan-Meier estimate for mean persistence of latanoprost
	was 52.0 + 2.26 months, bimatoprost was 25.8 + 2.89 months, and travoprost was
	23.0 + 1.27 months.
	5 +

The Kaplan Meier-survival curve for time to initial treatment change is shown in figure

3d

. The Kaplan-Meier estimate for mean persistence of latanoprost was 37.

Licence for Publication

The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence (or non exclusive for government employees) on a worldwide basis to the BMJ Publishing Group Ltd to permit this article (if accepted) to be published in BJO and any other BMJPGL products and sublicences such use and exploit all subsidiary rights, as set out in our licence (http://bjo.bmj.com/ifora/licence.pdf).

Funding: The study was funded by Pfizer Ltd.

Competing Interest: None declared.

Provenence and peer review: Not commissioned; externally peer reviewed.


T Christodoulakis

G Doulis
email: gdoulis@phys.uoa.gr

Petros A Terzis
email: pterzis@phys.uoa.gr

E Melas

Th Grammenos

G O Papadopoulos

A Spanou
email: aspanou@central.ntua.gr

Towards canonical quantum gravity for 3+1

Keywords: 04, 60, Ds, 04, 60, Kz

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Introduction

Dirac's seminal work on his formalism for a self-contained treatment of systems with constraints [1], [2], [3], [START_REF] Dirac | Lectures on Quantum Mechanics[END_REF] has paved the way for a systematic treatment of constrained systems. Some of the landmarks in the study of constrained systems have been the connection between constraints and invariances [START_REF] Bergmann | [END_REF], the extension of the formalism to describe fields with half-integer spin through the algebra of Grassmann variables [6] and the introduction of the BRST formalism [7]. All the classical results obtained so far have made up an armoury prerequisite for the quantization of gauge theories and there are several excellent reviews studying constraint systems with a finite number of degrees of freedom [START_REF] Sudarshan | Classical Dynamics: A Modern Perspective[END_REF] or constraint field theories [START_REF] Hanson | Constrained Hamiltonian Systems, Accademia Nazionale dei Lincei[END_REF], as well as more general presentations [START_REF] Sundermeyer | Constrained Dynamics[END_REF], [START_REF] Gitman | Quantization of Fields with Constraints[END_REF], [START_REF] Govaers | Hamiltonian Quantization and Constrained Dynamics[END_REF], [START_REF] Henneaux | Quantization of Gauge Systems[END_REF], [START_REF] Wipf | Hamilton's Formalism for Systems with Constraints[END_REF]. In particular, the conventional canonical analysis approach of quantum gravity has been initiated by P.A.M. Dirac [START_REF] Dirac | [END_REF], P.G. Bergmann [16] and B.S. deWitt [17]. For a modern account see [START_REF] Th | Modern Canonical Quantum General Relativity[END_REF]. In the absence of a full theory of quantum gravity, it is reasonably important to address the quantization of (classes of) simplified geometries. The most elegant way to achieve a degree of simplification is to impose some symmetry. For example, the assumption of a G 3 symmetry group acting simply transitively on the surfaces of simultaneity, i.e. the existence of three independent space-like Killing Vector Fields (KVF), leads to classical and subsequently quantum homogeneous cosmology (see e.g. [START_REF] Ryan | Homogeneous Relativistic Cosmologies[END_REF], [START_REF] Christodoulakis | Quantum Cosmology[END_REF]). The imposition of lesser symmetry, e.g. fewer KVF's, results in the various inhomogeneous cosmologies [START_REF] Krasinski | Inhomogeneous Cosmological Models[END_REF]. The canonical analysis under the assumption of spherical symmetry, which is a G 3 acting multiply transitively on two-dimensional space-like subsurfaces of the three-slices, has been first considered in [START_REF] Thomi | [END_REF], [23]. Quantum black holes have been treated, for instance, in [24] while in [25], [START_REF] Kiefer | Quantum Gravity[END_REF] a lattice regularization has been employed to deal with the infinities arising due to the ill-defined nature of the quantum operator constraints. In this work we consider the quantization of all geometries admitting two-dimensional surfaces of maximal symmetry, i.e. spheres (constant positive curvature), planes (zero curvature) and Gauss-Bolyai-Lobachevsky (henceforth GBL) spaces (constant negative curvature). In the second section we give the reduced metrics, the space of classical solutions and the Hamiltonian formulation of the reduced Einstein-Hilbert action principle, resulting in one (quadratic) Hamiltonian and one (linear) momentum first class constraint. In the third section we consider the quantization of this constraint system following Dirac's proposal of implementing the quantum operator constraints as conditions annihilating the wave-function [START_REF] Dirac | Lectures on Quantum Mechanics[END_REF]. Our guide-line is a conceptual generalization of the quantization scheme developed in [START_REF] Hajicek | [END_REF], [28] for the case of constraint systems with finite degrees of freedom, to the present case, which is enabled through the use of a specific re-normalization Assumption and an associated Requirement. After the symmetry reduction, the system still represents an one-dimensional field theory since all remaining metric components depend on time and the radial coordinate. Nevertheless, we manage to extract a Wheeler-deWitt equation in terms of three smooth scalar functionals of the reduced geometries. The exploitation of a residual freedom left by the imposition of the Requirement enables us to acquire the entire solution space to this equation. Finally, some concluding remarks are included in the discussion, while the proof of the existence for the third functional is given in the Appendix.

Possible Metrics, Classical Solutions and Hamiltonian Formulation

Our starting point is the two-dimensional spaces of positive, zero and negative constant curvature. Their line elements are respectively: 2 , ds 2 = dθ 2 + θ 2 dφ 2 , ds 2 = dθ 2 + sinh 2 θ dφ 2 (2.1)

ds 2 = dθ 2 + sin 2 θ dφ
with an obvious range of the coordinates for each case. The corresponding (maximal) symmetry groups are generated by the following KVF's:

ξ 1 = ∂ ∂φ , ξ 2 = -cos φ ∂ ∂θ + cot θ sin φ ∂ ∂φ , ξ 3 = sin φ ∂ ∂θ + cot θ cos φ ∂ ∂φ (2.
2)

ξ 1 = ∂ ∂φ , ξ 2 = -cos φ ∂ ∂θ + sin φ θ ∂ ∂φ , ξ 3 = sin φ ∂ ∂θ + cos φ θ ∂ ∂φ (2.3) ξ 1 = ∂ ∂φ , ξ 2 = -cos φ ∂ ∂θ + coth θ sin φ ∂ ∂φ , ξ 3 = sin φ ∂ ∂θ + coth θ cos φ ∂ ∂φ (2.4)
We next promote these KVF's to four-dimensional fields by adding to each and every of them the zero-sum 0 ∂ ∂ t + 0 ∂ ∂ r . We then enforce these vector fields as symmetries of a generic space-time metric g IJ (t, r, θ, φ), i.e. we demand that its Lie derivative with respect to all three fields for each family vanishes. We thus arrive at three classes of metrics, collectively described by the following line element:

ds 2 = -α(t, r) 2 + β(t, r) 2 γ(t, r) 2 dt 2 + 2 β(t, r) d t d r + γ(t, r) 2 d r 2 + ψ(t, r) 2 d θ 2 + ψ(t, r) 2 f (θ) 2 d φ 2 (2.5)
where f (θ) = sin θ in the case of spherical symmetry, f (θ) = θ in the case of plane symmetry, and f (θ) = sinh θ in the case of the GBL symmetry.

In order to attain the classical space of solution for these geometries one can, exploiting the freedom to change coordinates in the (t, r) subspace, bring the upper left block of the metric in conformally flat form and readily solve the vacuum Einstein's field equations. The result can be compactly summarized in the light-cone coordinates u, v by the following line elements:

ds 2 = 2 A + 2 ψ(u, v) 4 ψ(u, v) du dv + ψ(u, v) 2 dθ 2 + ψ(u, v) 2 f (θ) 2 dφ 2 , ( 2.6)
where

ψ(u, v) = - A 2 1 + ProductLog(- exp -λ(A+u+ v) A A ) , = ±1 and λ = ±1 for f (θ) = sin θ, λ = ±i for f (θ) = sinh θ; ProductLog(z)
is the principal solution for w to the equation z = w exp w . The corresponding to the plane symmetric case f (θ) = θ line-element is given by:

ds 2 = 2 1 ψ(u, v) du dv + ψ(u, v) 2 dθ 2 + ψ(u, v) 2 θ 2 dφ 2 , ( 2.7)
where

ψ(u, v) = ± √ 2u + 2 v.
A somewhat more detailed derivation of the entire classical solution space, for Λ = 0 as well, is presented in Appendix A: As we see, for each type of symmetry, there exists one solution in which the metric components depend only on the radial coordinate ("pointlike") and another with only time dependence ("cosmological"). For example, in the case of spherical symmetry and Λ = 0, we have the well known Schwarzschild and Kantowski-Sachs metrics respectively.

Notice that in the cases of spherical and GBL symmetry the only degree of freedom left, apart from the cosmological constant Λ, is the constant A. This constant ( which is to be identified with e.g. the Schwarzschild Mass in the corresponding case) is essential, as far as geometry is concerned: it can not be eliminated or even altered by a coordinate transformation, which is the "gauge" of any geometrical theory. In this sense this constant can, and must, be regarded as a dynamical degree of freedom, despite that there is no time evolution of it; the dynamical character of A is emanating from the fact that different values of it correspond to distinct geometries. The remaining case of plane symmetry is not so rich, since it does not contain any such constant, and thus constitutes an isolated point in the classical vacuum solution space.

One might wonder how this interpretation of "dynamical character" can be represented at the quantum level. The answer can be found in the local characterization of A in terms of invariant relations among geometrical quantities constructed out of the Riemmann tensor and its covariant derivatives, which by the way ensure the essential character of A: Expressing e.g. (A.6) in phase-space and constructing its quantum operator analogue, we have a quantum dynamical evolution of A when we apply this operator on the states we find. This gives another type of evolution to empty spherical symmetric configurations, which does not make use of external matter source operators.

The Hamiltonian formulation of the system (2.5) proceeds, briefly, as follows (see, e.g., chapter 9 of [START_REF] Sundermeyer | Constrained Dynamics[END_REF]): We first define the vectors

η I = 1 α(t, r) (1, - β(t, r) γ(t, r) 2 , 0, 0), F I = η J ;J η I -η I ;J η J ,
where I, J = 0, 1, 2, 3 are space-time indices and " ; " stands for covariant differentiation with respect to (2.5). Then, utilizing the Gauss-Codazzi equation, we eliminate all second time-derivatives from the Einstein-Hilbert action and arrive at an action quadratic in the velocities,

I = d 4 x √ -g(R -2Λ -2 F I ;I
). The application of the Dirac algorithm results firstly in the primary constraints

P α ≡ δL δ α ≈ 0, P β ≡ δL δ β ≈ 0 and the Hamiltonian H = N o H o + N i H i dr, (2.8)
where 2 , N 2 = 0, N 3 = 0 and H o , H i are given by

N o = α(t, r), N 1 = β(t, r) γ(t, r)
H o = 1 2 G αβ π α π β + V, ( 2.9a
)

H 1 = -γ π γ + ψ π ψ , H 2 = 0, H 3 = 0, ( 2
G αβ =      γ 4 ψ 2 - 1 4 ψ - 1 4 ψ 0      , (2.10) while the potential V is V = -2 γ + 2Λγ ψ 2 -2 ψ 2 γ + 4 ψ ψ γ (2.11)
with = {1, 0, -1} for the families (2.5) of two-dimensional subspaces with positive, zero or negative constant curvature, respectively. The requirement for preservation, in time, of the primary constraints leads to the secondary constraints

H o ≈ 0, H 1 ≈ 0 (2.12)
At this stage, a tedious but straightforward calculation produces the following open Poisson bracket algebra of these constraints:

{H o (r), H o (r)} = 1 γ 2 (r) H 1 (r) + 1 γ 2 (r) H 1 (r) δ (r, r), {H 1 (r), H o (r)} = H o (r)δ (r, r), (2.13
)

{H 1 (r), H 1 (r)} = H 1 (r)δ (r, r) -H 1 (r)δ(r, r) ,
indicating that they are first class and also signaling the termination of the algorithm. Thus, our system is described by (2.12) and the dynamical Hamilton-Jacobi equations

d π γ d t = {π γ ,
E IJ ≡ R IJ - 1 2 g IJ R + Λg IJ = 0 I, J = 0, 1, 2, 3. (2.14)
We end up this section by noting a few facts concerning the transformation properties of γ(t, r), ψ(t, r) and their derivatives under changes of the radial variable r of the form r → r = h(r). As it can easily be inferred from (2.5):

γ(r) = γ(r) d r d r , ψ(r) = ψ(r), d ψ(r) d r = d ψ(r) d r d r d r , ( 2.15)
where the t-dependence has been omitted for the sake of brevity. Thus, under the above coordinate transformations, ψ is a scalar, while γ, ψ are covariant rank 1 tensors (one-forms), or, equivalently in one dimension, scalar densities of weight -1. Therefore, the scalar derivative is not

d d r but rather d γ d r
. Finally, if we consider an infinitesimal transformation r → r = r -η(r), it is easily seen that the corresponding changes induced on the basic fields are:

δ γ(r) = (γ(r) η(r)) , δψ (r) = ψ (r) η(r) (2.16)
i.e., nothing but the one-dimensional analogue of the appropriate Lie derivatives. With the use of (2.16), we can reveal the nature of the action of H 1 on the basic configuration space variables as that of the generator of spatial diffeomorphisms:

γ(r) , dr η(r) H 1 (r) = (γ(r) η(r)) , ψ(r) , dr η(r) H 1 (r) = ψ (r) η(r).
(2.17)

Thus, we are justified to consider H 1 as the representative, in phase-space, of an arbitrary infinitesimal re-parametrization of the radial coordinate. In the same manner we can also see that the action of the quadratic constraint H o on the basic configuration space variables is identical to an infinitesimal change of the time coordinate, see e.g. p. 21 in [START_REF] Carlip | Quantum Gravity in 2+1 dimensions[END_REF]. The qualitative difference in the corresponding proof is that the dynamical equations giving the time derivatives of the momenta are explicitly used; hence the terms generator of "time deformations" or "dynamical evolution" used for H o . Thus, the linear and quadratic constraints are seen to be the generators of space-time diffeomorphisms, i.e. they represent in phase space the local "gauge" coordinate transformations; therefore the imposition of their proper quantum analogues will guarantee the invariance of the ensuing quantum theory under the aforementioned "gauge". Our study is limited, at the present state, to this goal and is thus not concerned with global issues like boundary terms.

Quantization

We are now interested in attempting to quantize this Hamiltonian system following Dirac's general spirit of realizing the classical first class constraints (2.12) as quantum operator constraint conditions annihilating the wave function. The main motivation behind such an approach is the justified desire to construct a quantum theory manifestly invariant under the "gauge" generated by the constraints. To begin with, let us first note that, despite the simplification brought by the imposition of the symmetry (2.2), (2.3), (2.4), the system is still a field theory in the sense that all configuration variables and canonical conjugate momenta depend not only on time ( as is the case in homogeneous cosmology), but also on the radial coordinate r. Thus, to canonically quantize the system in the Schrödinger representation, we first realize the classical momenta as functional derivatives with respect to their corresponding conjugate fields

πγ (r) = -i δ δ γ(r) , πψ (r) = -i δ δ ψ(r) .
We next have to decide on the initial space of state vectors. To elucidate our choice, let us consider the action of a momentum operator on some function of the configuration field variables, say πγ (r)γ(r) 2 = -2iγ(r)δ(r, r).

The Dirac delta-function renders the outcome of this action a distribution rather than a function. Also, if the momentum operator were to act at the same point as the function, i.e. if r = r, then its action would produce a δ(0) and would therefore be ill-defined. Both of these unwanted features are rectified, as far as expressions linear in momentum operators are concerned, if we choose as our initial collection of states all smooth functionals (i.e., integrals over r) of the configuration variables γ(r), ψ(r) and their derivatives of any order. Indeed, as we infer from the previous example,

πγ (r) drγ(r) 2 = -2i drγ(r)δ(r, r) = -2iγ(r);
thus the action of the momentum operators on all such states will be well-defined (no δ(0)'s) and will also produce only local functions and not distributions. However, even so, δ(0)'s will appear as soon as local expressions quadratic in momenta are considered, e.g., πγ (r) πγ (r) drγ(r

) 2 = πγ (r)(-2i drγ(r)δ(r, r)) = πγ (r)(-2iγ(r)) = -2δ(0).
An other problem of equal, if not greater, importance has to do with the number of derivatives (with respect to r) considered: A momentum operator acting on a smooth functional of degree n in derivatives of γ(r), ψ(r) will, in general, produce a function of degree 2n, e.g., πγ (r) drγ (r) 2 = -2i drγ (r)δ (r, r) = -2iγ (4) (r).

Thus, clearly, more and more derivatives must be included if we desire the action of momentum operators to keep us inside the space of integrands corresponding to the initial collection of smooth functionals; eventually, we have to consider n → ∞. This, in a sense, can be considered as the reflection to the canonical approach, of the non-renormalizability results existing in the so-called covariant approach. Loosely speaking, the way to deal with these problems is to regularize (i.e., render finite) the infinite distribution limits, and re-normalize the theory by, somehow, enforcing n to terminate at some finite value.

In the following, we are going to present a quantization scheme of our system which:

(a) avoids the occurrence of δ(0)'s (b) reveals the value n = 1 as the only natural (i.e. without ad-hoc cut-offs) possibility to obtain a closed space of state vectors (c) extracts a finite-dimensional Wheeler-deWitt equation governing the quantum dynamics.

The scheme closely parallels, conceptually, the quantization developed in [START_REF] Hajicek | [END_REF], [28] for finite systems with one quadratic and a number of linear first class constraints. Therefore, we deem it appropriate, and instructive, to present a brief account of the essentials of this construction.

To this end, let us consider a system described by a Hamiltonian of the form

H ≡ µX + µ i χ i = µ 1 2 G AB (Q Γ )P A P B + U A (Q Γ )P A + V (Q Γ ) + µ i φ A i (Q Γ )P A , ( 3.1)
where A, B, Γ . . . = 1, 2 . . . , M count the configuration space variables and i = 1, 2, . . . , N < (M -1) numbers the super-momenta constraints χ i ≈ 0, which along with the super-Hamiltonian constraint X ≈ 0 are assumed to be first class:

{X, X} = 0, {X, χ i } = XC i + C j i χ j , {χ i , χ j } = C k ij χ k , ( 3.2)
where the first (trivial) Poisson bracket has been included only to emphasize the difference from (2.13).

The physical state of the system is unaffected by the "gauge" transformations generated by (X, χ i ), but also under the following three changes:

(I) Mixing of the super-momenta with a non-singular matrix

χi = λ j i (Q Γ )χ j
(II) Gauging of the super-Hamiltonian with the super-momenta

X = X + κ (Ai (Q Γ )φ B) i (Q Γ )P A P B + σ i (Q Γ )φ A i (Q Γ )P A (III) Scaling of the super-Hamiltonian X = τ 2 (Q Γ )X.
Therefore, the geometrical structures on the configuration space that can be inferred from the super-Hamiltonian are really equivalence classes under actions (I), (II) and (III); for example (II), (III) imply that the super-metric G AB is known only up to conformal scalings and additions of the super-momenta coefficients ḠAB = τ 2 (G AB + κ (Ai φ B) i ). It is thus mandatory that, when we Dirac-quantize the system, we realize the quantum operator constraint conditions on the wave-function in such a way as to secure that the whole scheme is independent of actions (I), (II), (III). This is achieved by the following steps:

(1) Realize the linear operator constraint conditions with the momentum operators to the right

χi Ψ = 0 ↔ φ A i (Q Γ ) ∂ Ψ(Q Γ ) ∂ Q A =
0, which maintains the geometrical meaning of the linear constraints and produces the M -N independent solutions to the above equation q α (Q Γ ), α = 1, 2, . . . , M -N called physical variables, since they are invariant under the transformations generated by the χi 's.

(2) In order to make the final states physical with respect to the "gauge" generated by the quadratic constraint X as well:

Define the induced structure

g αβ ≡ G AB ∂ q α ∂ Q A ∂ q β ∂ Q B and
realize the quadratic in momenta part of X as the conformal Laplace-Beltrami operator based on g αβ . Note that in order for this construction to be self consistent, all components of g αβ must be functions of the physical coordinates q γ . This can be proven to be so by virtue of the classical algebra the constraints satisfy (for specific quantum cosmology examples see [START_REF] Christodoulakis | Quantum Cosmology[END_REF]).

We are now ready to proceed with the quantization of our system, in close analogy to the scheme above outlined. In order to realize the equivalent to step 1, we first define the quantum analogue of H 1 (r) ≈ 0 as

Ĥ1 (r)Φ = 0 ↔ -γ(r) ( δ Φ δ γ(r) ) + ψ (r) δ Φ δ ψ(r) = 0. (3.3)
As explained in the beginning of the section, the action of Ĥ1 (r) on all smooth functionals is well defined, i.e., produces no δ(0)'s. It can be proven that, in order for such a functional to be annihilated by this linear quantum operator, it must be scalar, i.e. have the form

Φ = γ(r) L Ψ (0) , Ψ (1) , . . . , Ψ (n) dr (3.4a) Ψ (0) ≡ ψ(r), Ψ (1) ≡ ψ (r) γ(r) , . . . , Ψ (n) ≡ 1 γ(r) d dr . . . n-1 ψ(r) (3.4b)
where L is any function of its arguments. We note that, as it is discussed at the end of the previous section, ψ γ is the only scalar first derivative of ψ, and likewise for the higher derivatives. The proof of this statement is analogous to the corresponding result concerning full gravity [START_REF] Thomas | The Differential Invariants of Generalized Spaces[END_REF]: consider an infinitesimal re-parametrization of r r = r-η(r). Under such a change, the left-hand side of (3.4a), being a number, must remain unaltered. If we calculate the change induced on the right-hand side we arrive at

0 = Lδγ + γ δL δγ δγ + γ δL δψ δψ dr = [γ Ĥ1 (L)]η(r)dr, (3.5)
where use of (2.16) and a partial integration has been made. Since this must hold for any η(r), the result sought for is obtained. We now try to realize step 2 of the programm previously outlined. We have to define the equivalent of Kuchař's induced metric on the so far space of "physical" states Φ described by (3.4a) which are the analogues, in our case, of Kuchař's physical variables q α . Let us start our investigation by considering one initial candidate of the above form.

Then, generalizing the partial to functional derivatives, the induced metric will be given by

g ΦΦ = G αβ δΦ δ x α δΦ δ x β , where x α = {γ, ψ} (3.6)
and G αβ is given by (2.10). Note that this metric is well defined since it contains only first functional derivatives of the state vectors, as opposed to any second order functional derivative operator that might be considered as a quantum analogue of the kinetic part of H o . Nevertheless, g ΦΦ is a local function and not a smooth functional. It is thus clear that, if we want the induced metric g ΦΦ to be composed out of the "physical" states annihilated by Ĥ1 , we must establish a correspondence between local functions and smooth functionals. A way to achieve this is to adopt the following ansatz:

Assumption: We assume that, as part of the re-normalization procedure, we are permitted to map local functions to their corresponding smeared expressions e.g., ψ(r) ↔ drψ(r).

Let us be more specific, concerning the meaning of the above Assumption. Let F be the space which contains all local functions, and define the equivalence relations

∼: {f 1 (r) ∼ f 2 (r), r = g(r)}, ≈: {h 1 (r) ≈ h 2 (r) d r d r , r = g(r)} (3.7)
for scalars and densities respectively. Now let F o = {f ∈ F, mod (∼, ≈)} and F I the space of the smeared functionals. We define the one to one maps G, G -1

G : F o → F I : ψ(r) → ψ(r) dr, G -1 : F I → F o : ψ(r) dr → ψ(r) (3.8)
The necessity to define the maps G, G -1 on the equivalence classes and not on the individual functions, stems out of the fact that we are trying to develop a quantum theory of the geometries (2.5) and not of their coordinate representations. If we had tried to define the map G from the original space F to F I we would end up with states which would not be invariant under spatial coordinate transformations (r-reparameterizations). Indeed, one can make a correspondence between local functions and smeared expressions, but smeared expressions must contain another arbitrary smearing function, say s(r).

Then the map between functions and smeared expressions is one to one (as is also the above map) and is given by multiplying by s(r) and integrating over r; while the inverse map is given by varying w.r.t. s(r). However, this would be in the opposite direction from that which led us to the states (3.4a) by imposition of the linear operator constraint.

As an example consider the action of this operator on one particular case of the states (3.4a), containing the structure s(r) : Ĥ1 (r) s(r) γ(r) ψ(r) dr = -s (r) γ(r) ψ(r) = 0 for arbitrary s(r) (3.9)

Thus, every foreign to the geometry structure s(r) is not allowed to enter the physical states. Now, after the correspondence has been established, we can come to the basic property the induced metric must have. In the case of finite degrees of freedom the induced metric depends, up to a conformal scaling, on the physical coordinates q α by virtue of (3.2). In our case, due to the dependence of the configuration variables on the radial coordinate r, the above property is not automatically satisfied; e.g. the functional derivative δ δψ(r) acting on a functional whose integrand contains Ψ (n) will produce, upon partial integration of the n th derivative of the Dirac delta function, a term proportional to Ψ (2n) . Therefore, since L in (3.4a) contains derivatives of ψ(r) up to Ψ (n) , the above mentioned property must be enforced. The need for this can also be traced to the substantially different first Poisson bracket in (2.13), which signals a non trivial mixing between the dynamical evolution generator H o and the linear generator H 1 . Thus, according to the above reasoning, in order to proceed with the generalization of Kuchař's method, we have to demand that: Requirement: L Ψ (0) , . . . , Ψ (n) must be such that g ΦΦ becomes a general function, say F γ(r) L(Ψ (0) , . . . , Ψ (n) ) of the integrand of Φ, so that it can be considered a function of this state:

g ΦΦ Assumption ≡ F γ(r) L(Ψ (0) , . . . , Ψ (n) )dr = F (Φ).
At this point, we must emphasize that the application of the Requirement in the subsequent development of our quantum theory will result in very severe restrictions on the form of (3.4a). Essentially, all higher derivatives of ψ(r) (i.e Ψ (2) . . . Ψ (n) )) are eliminated from Φ (see (3.10), (3.20) bellow). This might, at first sight, strike as odd; indeed, the common belief is that all the derivatives of the configuration variables should enter the physical states. However, before the imposition of both the linear and the quadratic constrains there are no truly physical states. Thus, no physical states are lost by the imposition of the Requirement; ultimately the only true physical states are the solutions to (3.24). Of course, if one insists, one can keep higher derivatives, say k, in the functional. But then, in order to enforce the Requirement one would have to eliminate by hand all derivatives higher than k from the components of the induced metric. However, to our view, such an action would seem very un-natural. In addition, although we do not have a concrete rigorous proof, we believe that this ad-hoc elimination would-in the present approach-break the space-time covariance of the quantum theory we are trying to built. Having clarified the way in which we view the Assumption and Requirement above, we now proceed to the restrictions implied by their use.

Let us now turn to the degree of derivatives (n) of ψ(r). As we argued before, the functional derivatives δ δψ(r) and δ δγ(r) acting on a functional containing in its integrand Ψ (n) will, upon partial integration of the n th derivative of the Dirac delta function, produce a term proportional to Ψ (2n) and Ψ (2n-1) respectively. More precisely g ΦΦ = . . . + 2 G 12 δΦ δγ(r) δΦ δψ (r) .

Where the functional derivatives are:

δΦ δψ = . . . + γ ∂L ∂Ψ (n) δΨ (n) δψ dr = . . . + γ ∂L ∂Ψ (n) 1 γ d dr . . . n-1 δ(r, r) dr = = . . . - d dr ∂L ∂Ψ (n) 1 γ d dr . . . n-2 δ(r, r) dr = = . . . -γ ∂ 2 L ∂ (Ψ (n) ) 2 Ψ (n+1) 1 γ d dr . . . n-2 δ(r, r) dr = . . . = . . . + (-1) n γ(r) ∂ 2 L ∂ (Ψ (n) ) 2 Ψ (2n) δ(r, r) dr = = . . . + (-1) n γ ∂ 2 L ∂ (Ψ (n) ) 2 Ψ (2n)
and

δΦ δγ = . . . + γ ∂L ∂Ψ (n) δΨ (n) δγ dr = . . . + γ ∂L ∂Ψ (n) 1 γ d dr . . . n-2 - δ(r, r) γ(r) 2 ψ (r) dr = = . . . + γ ∂L ∂Ψ (n) 1 γ d dr . . . n-2 - δ(r, r) γ(r) Ψ (1) dr = . . . - d dr ∂L ∂Ψ (n) 1 γ d dr . . . n-3 - δ(r, r) γ(r) Ψ (1) dr = = . . . -γ ∂ 2 L ∂ (Ψ (n) ) 2 Ψ (n+1) 1 γ d dr . . . n-3 - δ(r, r) γ(r) Ψ (1) dr = . . . = . . . + (-1) n-1 ∂ 2 L ∂ (Ψ (n) ) 2 Ψ (2n-1) Ψ (1) δ(r, r) dr = = . . . + (-1) n-1 ∂ 2 L ∂ (Ψ (n) ) 2 Ψ (2n-1)
Ψ (1) .

Therefore

g ΦΦ = . . . - γ 2ψ (-1) 2n-1 ∂ 2 L ∂ (Ψ (n) ) 2 2 Ψ (1) Ψ (2n-1) Ψ (2n) ,
where the . . . stand for all other terms, not involving Ψ (2n) . Now, according to the aforementioned Requirement we need this to be a general function, say F (γL), and for this to happen the coefficient of Ψ (2n) must vanish, i.e. 1) . Now, the term in Φ corresponding to L 1 is, up to a surface term, equivalent to a general term depending on Ψ (0) , . . . , Ψ (n-1) only: indeed,

∂ 2 L ∂ (Ψ (n) ) 2 = 0 ⇔ L = L 1 Ψ (0) , . . . , Ψ (n-1) Ψ (n) + L 2 Ψ (0) , . . . , Ψ (n-
Φ 1 = γ(r)L 1 1 γ(r) d dr Ψ (n-1) dr,
which upon subtraction of the surface term

A = dr d dr dΨ (n-1) L 1
produces a smooth functional with arguments up to Ψ (n-1) only. Since a surface term in Φ does not affect the outcome of the variational derivatives δ Φ δ ψ(r) and δ Φ δ γ(r) , we conclude that only L 2 is important for the local part of Φ. The entire argument can be repeated successively for n-1, n-2, . . . , 2; therefore all Ψ (n) 's are suppressed from L except when n = 1. The case n = 1 needs separate consideration since, upon elimination of the linear in Ψ (2) term we are left with a local function of Ψ (1) , and thus the possibility arises to meet the Requirement by solving a differential equation for L. In more detail, if Φ ≡ γ(r)L ψ, Ψ (1) dr, (3.10)

g ΦΦ reads g ΦΦ = γ 4ψ 2 L -Ψ (1) ∂L ∂Ψ (1) L -Ψ (1) ∂L ∂Ψ (1) -2 ψ ∂L ∂ψ -Ψ (1) ∂ 2 L ∂ψ ∂Ψ (1) + + γ 2ψ L -Ψ (1) ∂L ∂Ψ (1)
∂ 2 L ∂(Ψ (1) ) 2 Ψ (2) .

(3.11)

Through the definition

H ≡ L -Ψ (1)
∂L ∂Ψ (1) (3.12)

we obtain

∂H ∂ψ = ∂L ∂ψ -Ψ (1) ∂ 2 L ∂ψ ∂Ψ (1) , ∂H ∂Ψ (1) = -Ψ (1) ∂ 2 L ∂ (Ψ (1) ) 2 .
Thus (3.11) assumes the form (2) , which upon addition, by virtue of the Assumption, of the surface term (1) H ∂H ∂Ψ (1) dΨ (1) gives

g ΦΦ = γ 4ψ 2 H 2 -2 ψ H ∂H ∂ψ - 2 ψ Ψ (1) H ∂H ∂Ψ (1) Ψ
A = d dr 1 2ψΨ
g ΦΦ = γ 4ψ 2 H 2 -2 ψ H ∂H ∂ψ + 4ψ 2 Ψ (1)
∂ ∂ψ 1 2ψΨ (1) H ∂H ∂Ψ (1) dΨ (1) .

(3.13)

Since in the last expression we have only a multiplicative γ(r), it is obvious that the Requirement

g ΦΦ = F (γ L)
can be satisfied only by

g ΦΦ = κ γ L, ( 3.14)
with g ΦΦ given by (3.13) and κ any constant. Upon differentiation of this equation with respect to Ψ (1) we get ∂ ∂ψ 1 2ψΨ (1) H ∂H ∂Ψ (1) dΨ (1) = κ ∂L ∂Ψ (1) .

Multiplying the last expression by Ψ (1) and subtracting it from (3.14) (with g ΦΦ given by (3.13)) we end up with the autonomous necessary condition for H(ψ, Ψ (1) ):

H 1 4ψ 2 H - 1 2ψ ∂H ∂ψ -κ = 0,
where (3.12) was also used. The above equation can be readily integrated giving

H = 0, H = - 4κψ 2 3 + ψ a(Ψ (1) ),
where a(Ψ (1) ) is an arbitrary function of its argument. The first possibility gives according to (3.12) L = λ Ψ (1) which, however, contributes to Φ a surface term, and can thus be ignored. Inserting the second solution into (3.12) we construct a partial differential equation for L, namely L -Ψ (1) ∂L ∂Ψ (1) = -4κψ 2 3 + ψ a(Ψ (1) ), which upon integration gives L = -4κψ 2 3 -ψ Ψ (1) a(Ψ (1) ) Ψ (1) 2 dΨ (1) + c 1 (ψ) Ψ (1) .

Since this form of L emerged as a necessary condition, it must be inserted (along with H) in (3.14). The result is that c 1 (ψ) = 0. Thus L reads 1) a(Ψ (1) ) Ψ (1) 2 dΨ (1) .

L = - 4κψ 2 3 -ψ Ψ (
(3.15)

By assuming that the Ψ (1) -dependent part of L equals b(Ψ (1) ), i.e.

-Ψ (1) a(Ψ (1) ) Ψ (1) 2 dΨ (1) = b(Ψ (1) ),

we get, upon a double differentiation with respect to Ψ (1) , the ordinary differential equation a (Ψ (1) ) Ψ (1) = b (Ψ (1) ) with solution a(Ψ (1) ) = b(Ψ (1) ) + κ 1 -Ψ (1) b (Ψ (1) ), where κ 1 is a constant. Substituting this equation into (3.15) and performing a partial integration we end up with ) ) .

L = - 4κψ 2 3 + κ 1 ψ + ψ b(Ψ ( 1
(3.16)

κ, κ 1 and b(Ψ (1) ) being completely arbitrary and to our disposal; the two simplest choices κ = 0, b(Ψ (1) ) = 0 and κ 1 = 0, b(Ψ (1) ) = 0 lead respectively to the following two basic ultra-local smooth functionals:

q 1 = drγ(r) ψ(r), q 2 = drγ(r)ψ(r) 2 .
The next simpler choice κ = 0, κ 1 = 0 and b(Ψ (1) ) arbitrary leads to a generic q 3 = drγ(r) √ ψ b(Ψ (1) ). However, it can be proven that, for any choice of b(Ψ (1) ), the corresponding renormalized induced metric

g AB = G αβ δq A δx α δq B δx β where A, B = 1, 2, 3
is singular. The calculation of g AB gives:

g 11 = G αβ δq 1 δx α δq 1 δx β = 0 Assumption ⇐⇒ g 11 ren = 0, g 12 = G αβ δq 1 δx α δq 2 δx β = - 3 8 γ ψ Assumption ⇐⇒ g 12 ren = - 3 q 1 8 , g 22 = G αβ δq 2 δx α δq 2 δx β = - 3 4 γ ψ 2 Assumption ⇐⇒ g 22 ren = - 3 4 q 2 , g 13 = G αβ δq 1 δx α δq 3 δx β = 1 4 γ Ψ (2) b = d dr 1 4 b Assumption ⇐⇒ g 13 ren = 0, g 23 = G αβ δq 2 δx α δq 3 δx β = 1 8 γ ψ -3 b + 3Ψ (1) b + 2 ψ Ψ (2) b Assumption ⇐⇒ g 23 ren = 1 8 drγ ψ -3 b + 3Ψ (1) b + 2 ψ Ψ (2) b -dr d dr 1 4 dΨ (1) ψ 3/2 b = = - 3 8 drγ ψ b = - 3 q 3 8 , g 33 = G αβ δq 3 δx α δq 3 δx β = 1 2 γ b -Ψ (1) b Ψ (2) b Assumption ⇐⇒ g 33 ren = 1 2 drγ b -Ψ (1) b Ψ (2) b -dr d dr 1 2 dΨ (1) b -Ψ (1) b b = 0,
where by we denote differentiation with respect to Ψ (1) . Thus the re-normalized induced metric reads g AB ren (q 1 , q 2 , q 3 ) = -

3 8       0 q 1 0 q 1 2 q 2 q 3 0 q 3 0       .
Effecting the transformation (q 1 , q2 , q3 ) = q 1 , q 2 , f q 3 q 1

we bring g AB ren into a manifestly degenerate form:

g AB ren (q 1 , q 2 ) = - 3 8       0 q 1 0 q 1 2 q 2 0 0 0 0       .
So, it seems that, as far as the ultra local part of the functionals is concerned, the renormalized metric is given by the upper left 2 × 2 block of the above expression. It is interesting to observe that the integrants of q 1 , q 2 form a base in the space spanned by γ, ψ. It is convenient to change these two integrants (namely γ √ ψ and γ ψ 2 ) to γ and γψ 2 respectively, since the latter choice complies with the ultra local parts of the potential (2.11). One might wonder if this action is permitted, namely if the new re-normalized metric, resulting from the choice of the new functionals,

y 1 = γ(r)dr, y 2 = γ(r)ψ(r) 2 dr
is equivalent to the previous. Interestingly enough, the answer is yes. Indeed, following the line of thoughts leading to g AB ren (q 1 , q 2 ) one arrives at

g AB ren (y 1 , y 2 ) = - 1 4     - (y 1 ) 2 y 2 y 1 y 1 3 y 2     ,
which is related to the upper left 2 × 2 block of g AB ren (q 1 , q 2 ) through the transformation (y 1 , y 2 ) = (q 1 ) 4/3 (q 2 ) 1/3 , q 2 . Quite unexpectedly, this transformation is identical to the transformation connecting the integrands of the two pairs of functionals γ √ ψ, γψ 2 , (γ, γψ 2 ). This is a strong indication that the use of the Assumption preserves the geometry of the re-normalized manifold. We thus adopt, without any loss of generality the ultra local functionals:

y 1 = γ(r)dr, y 2 = γ(r)ψ(r) 2 dr (3.17)
One might think that this preservation of the geometry of the re-normalized manifold is due to the ultra local nature of the integrands but, as we shall subsequently see, this state of affairs continuous to hold even when functionals with integrands which contain derivatives of the configuration variables are considered. Indeed it is quite essential to have a functional that contains first derivative of ψ(r), since a term of this kind (namely

ψ 2 γ
) does appear in the potential (2.11). Thus, it is clear that this is not the end of our investigation for a suitable space of state vectors: the caveat is that the argument leading to y 1 , y 2 crucially depends upon the original choice of one initial candidate smooth scalar functional (3.10). Therefore, to complete the search we must close the circle by starting with the two already secured smooth functionals (y 1 , y 2 ), and a third of the general form y 3 = dr γ(r) L(Ψ (1) )

(since the ψ dependence has already been fixed to either 1 or ψ 2 ). The calculation of the, related to y 3 , components of the induced metric g AB gives:

g 13 = γ 4ψ 2 L -Ψ (1) L + ψ Ψ (2) L Assumption ⇐⇒ g 13 ren = dr γ 4ψ 2 L -Ψ (1) L + ψ Ψ (2) L -dr d dr 1 4 dΨ (1) L ψ = = 1 4 dr γ ψ 2 L Assumption = 1 4 γ γ ψ 2 γL Assumption = 1 4 drγ drγ ψ 2 drγL = y 1 y 3 4 y 2 , g 23 = γ 4 -L + Ψ (1) L + ψ Ψ (2) L Assumption ⇐⇒ g 23 ren = dr γ 4 -L + Ψ (1) L + ψ Ψ (2) L -dr d dr 1 4 dΨ (1) ψ L = = - 1 4 drγ L = - y 3 4 , g 33 = γ 4ψ 2 L -Ψ (1) L 2 + γ 2ψ L -Ψ (1) L Ψ (2) L Assumption ⇐⇒ g 33 ren = dr γ 4ψ 2 L -Ψ (1) L 2 + γ 2ψ L -Ψ (1) L Ψ (2) L - - dr d dr dΨ (1) L -Ψ (1) L L 2ψ g 33 ren = γ 4ψ 2 L -Ψ (1) L 2 -Ψ (1) dΨ (1)
Ψ (1) ∂ ∂Ψ (1) L -Ψ (1) L 2 dr.

(3.18)

The expression inside the square brackets of g 33 ren above, being a generic function of Ψ (1) , can also be considered as a function of L, say W (L(Ψ (1) )). It is thus clear that the Requirement is satisfied for any L(Ψ (1) ):

Assumption ⇐⇒ g 33 ren = ( γdr) 2 4 γψ 2 dr W ( γLdr γdr ) = (y 1 ) 2 4y 2 W ( y 3 y 1 ).
Let this expression W (L(Ψ (1) )) be parameterized as

L Ψ (1) 2 - 4 F [L Ψ (1) ] 2 3 F [F [L (Ψ (1) )]] 2 . (3.19)
This "peculiar" parametrization of the arbitrariness in L Ψ (1) has been chosen in order to facilitate the subsequent proof that the freedom in the choice of L (left by the imposition of the Requirement) is a pure general coordinate transformation (gct) of the induced re-normalized metric.

The reduced re-normalized manifold is thus parameterized by the following three smooth scalar functionals: 2 dr, y 3 = γ(r)L(Ψ (1) )dr. (3.20) Any other functional, say y 4 = dr γ(r) K ψ(r), Ψ (1) (r) , can be considered as a function of y1 , y 2 , y 3 ; indeed, since the scalar functions appearing in the integrands of y 2 , y 3 form a base in the space spanned by ψ, Ψ (1) , we can express the generic K in y 4 as

y 1 = γ(r)dr, y 2 = γ(r)ψ(r)
K γψ 2
γ , Ψ (1) , which (through the Assumption) gives y 4 = y 1 K

y 2 y 1 , L -1 y 3 y 1 .
The geometry of this space is described by the induced re-normalized metric

g AB ren (y 1 , y 2 , y 3 ) = -       3 y 2 (y 1 ) 4   (y 1 ) 2 - (y 3 ) 2 F » F " y 3 y 1 «-2 F " y 3 y 1 " 2   -1 y 1 3 y 2 y 3 F » F " y 3 y 1 «-2 (y 1 ) 3 F " y 3 y 1 " 2 -1 y 1 -1 y 2 0 3 y 2 y 3 F » F " y 3 y 1 «-2 (y 1 ) 3 F " y 3 y 1 " 2 0 - 3 y 2 F » F " y 3 y 1 «-2 (y 1 ) 2 F " y 3 y 1 " 2              . (3.21)
Any function Ψ(y 1 , y 2 , y 3 ) on this manifold is of course annihilated by the quantum linear constraint, i.e.

Ĥ1 Ψ(y 1 , y 2 , y 3 ) = ∂Ψ(y 1 , y 2 , y 3 ) ∂y 1 Ĥ1 y 1 + ∂Ψ(y 1 , y 2 , y 3 ) ∂y 2 Ĥ1 y 2 + ∂Ψ(y 1 , y 2 , y 3 ) ∂y 3 Ĥ1 y 3 = 0 since the derivatives with respect to r are transparent to the partial derivatives of Ψ (which are, just like the y A 's, r-numbers). The covariant metric (3.21) describes a three dimensional conformally flat geometry, since the corresponding Cotton-York tensor vanishes. The Ricci scalar is R = 3 8 y 2 , indicating that the arbitrariness in F (and thus also in L) is a pure gauge. The change of coordinates

(y 1 , y 2 , y 3 ) = (e -1 8 (5 Y 1 +3 Y 3 ) , e Y 1 +Y 2 +Y 3 , e -1 8 (5 Y 1 +3 Y 3 ) F -1 (e
the manifestly conformally flat form:

g AB ren (Y 1 , Y 2 , Y 3 ) =         e Y 1 +Y 2 +Y 3 0 0 0 - 4 3 e Y 1 +Y 2 +Y 3 0 0 0 -e Y 1 +Y 2 +Y 3         , (3.23)
in which all the F dependence has indeed disappeared. The final restriction on the form of Ψ will be obtained by the imposition of the quantum analog of the quadratic constraint H o . According to the above exposition we postulate that the quantum gravity of the geometries given by (2.5) will be described by the following partial differential equation (in terms of the Y A 's)

Ĥo Ψ ≡ [- 1 2 2 c + V ren ] Ψ(Y 1 , Y 2 , Y 3 ) = 0 (3.24) with 2 c = 2 + d -2 4 (d -1) R (3.25)
being the conformal Laplacian based on g AB ren (Y 1 , Y 2 , Y 3 ), R the Ricci scalar, and d the dimensions of g AB ren . The metric (3.23) is conformally flat with Ricci scalar

R = 3 8 e -Y 1 -Y 2 -Y 3
, and its dimension is d = 3. The re-normalized form of the potential (2.11) offers us the possibility to introduce, in a dynamical way, topological effects into our wave functional: Indeed, under our Assumption, the first two terms become -2 y 1 and 2 Λ y 2 , respectively, while the last, being a total derivative, becomes

A T ≡ 4 ψ ψ γ | β α (if α < r < β).
In the spirit previously explained we should drop this term, however one could also keep it. The re-normalized form of the remaining, third, term of the potential can be obtained as follows

y 3 = γ L(Ψ (1) ) ⇔ L(Ψ (1) ) = y 3 γ Assumption ⇐⇒ L(Ψ (1) ) = y 3 y 1 ⇔ Ψ (1) = L -1 y 3 y 1 , thus finally ψ γ = L -1 y 3 y 1
and the third term becomes -2 y 1 L -1 y 3 y 1

2

. Finally, effecting the transformation (3.22) the form of the re-normalized potential is

V ren = -2 e -1 8 (5 Y 1 +3 Y 3 ) -2 e -1 8 (5 Y 1 +3 Y 3 ) L -1 F -1 (e 1 24 (-9 Y 1 +8 Y 2 -15 Y 3 ) ) 2 + 2 Λ e Y 1 +Y 2 +Y 3 + A T (3.26)
and the Wheeler-deWitt equation is given as

-2 e 1 8 (3 Y 1 +5 Y 3 )+Y 2 Ψ(Y 1 , Y 2 , Y 3 ) + 2 Λ e 2(Y 1 +Y 2 +Y 3 ) Ψ(Y 1 , Y 2 , Y 3 ) - 2 e 1 8 (3 Y 1 +5 Y 3 )+Y 2 L -1 F -1 (e 1 24 (-9 Y 1 +8 Y 2 -15 Y 3 ) ) 2 Ψ(Y 1 , Y 2 , Y 3 ) + A T e Y 1 +Y 2 +Y 3 Ψ(Y 1 , Y 2 , Y 3 ) - 3 128 Ψ(Y 1 , Y 2 , Y 3 ) - 1 4
∂Ψ(Y 1 , Y 2 , Y 3 ) ∂Y 1 + 3 16
∂Ψ(Y 1 , Y 2 , Y 3 ) ∂Y 2 + 1 4
∂Ψ(Y 1 , Y 2 , Y 3 ) ∂Y 3 - 1 2
∂ 2 Ψ(Y 1 , Y 2 , Y 3 ) ∂ (Y 1 ) 2 + 3 8
∂ 2 Ψ(Y 1 , Y 2 , Y 3 ) ∂ (Y 2 ) 2 + 1 2 ∂ 2 Ψ(Y 1 , Y 2 , Y 3 ) ∂ (Y 3 ) 2 = 0.
Since F -1 is an arbitrary function of its arguments, we may contemplate the choice:

F -1 e 1 24 (-9 Y 1 +8 Y 2 -15 Y 3 ) = L e 1 24 (-9 Y 1 +8 Y 2 -15 Y 3 ) - . ( 3.27)
Of course there is a question of existence for such a choice: since F which appears in (3. [START_REF] Ryan | Homogeneous Relativistic Cosmologies[END_REF]) is nothing but a convenient parametrization of the expression inside the square brackets of (3.18), any demand that F has a specified form (much more in terms of L) constitutes an implicit restriction on the form of L itself. Subsequently, at least the existence of such an L must be secured. Indeed, in the Appendix B it is shown that an appropriate L exists; its form is given by (B.4):

L(Ψ (1) ) = m + (Ψ (1) ) 3/2 ((Ψ (1) ) 2 -) 13/16 e k- 3 16
((Ψ (1) ) 2 -) dΨ (1) where c 1 m + c 2 + c 3 e k = 0.

This choice for F reduces the Wheeler-deWitt equation to the final almost-separable form

2 Λ e 2(Y 1 +Y 2 +Y 3 ) Ψ(Y 1 , Y 2 , Y 3 ) -2 e 4 3 Y 2 Ψ(Y 1 , Y 2 , Y 3 ) + A T e Y 1 +Y 2 +Y 3 Ψ(Y 1 , Y 2 , Y 3 ) - 3 128 Ψ(Y 1 , Y 2 , Y 3 ) - 1 4 ∂Ψ(Y 1 , Y 2 , Y 3 ) ∂Y 1 + 3 16
∂Ψ(Y 1 , Y 2 , Y 3 ) ∂Y 2 + 1 4
∂Ψ(Y 1 , Y 2 , Y 3 ) ∂Y 3 - 1 2 ∂ 2 Ψ(Y 1 , Y 2 , Y 3 ) ∂ (Y 1 ) 2 + 3 8
∂ 2 Ψ(Y 1 , Y 2 , Y 3 ) ∂ (Y 2 ) 2 + 1 2 ∂ 2 Ψ(Y 1 , Y 2 , Y 3 ) ∂ (Y 3 ) 2 = 0. (3.28)
This equation is separable for Λ = 0 and A T = 0. In this case it can readily be solved:

assuming Ψ(Y 1 , Y 2 , Y 3 ) = Ψ 1 (Y 1 ) Ψ 2 (Y 2 ) Ψ 3 (Y 3
) and dividing (3.28) by Ψ we get the three ordinary differential equations:

1 4 Ψ 1 (Y 1 ) dΨ 1 (Y 1 ) dY 1 + 1 2 Ψ 1 (Y 1 ) d 2 Ψ 1 (Y 1 ) d (Y 1 ) 2 = m, 3 16 Ψ 2 (Y 2 ) dΨ 2 (Y 2 ) dY 2 + 3 8 Ψ 2 (Y 2 ) d 2 Ψ 2 (Y 2 ) d (Y 2 ) 2 -2 e 4 3 Y 2 = n, 1 4 Ψ 3 (Y 3 ) dΨ 3 (Y 3 ) dY 3 + 1 2 Ψ 3 (Y 3 ) d 2 Ψ 3 (Y 3 ) d (Y 3 ) 2 - 3 128 = m -n,
where m and n are separation constants. Their solutions are:

Ψ 1 (Y 1 ) = c 1 e 1 4 ( -1- √ 1+32 m)Y 1 + c 2 e 1 4 ( -1+ √ 1+32 m)Y 1 , Ψ 2 (Y 2 ) = c 3 e -Y 2 /4 I - √ 3 8 √ 3+128 n 2 √ 3 e 2 Y 2 /3 + c 4 e -Y 2 /4 I √ 3 8 √ 3+128 n 2 √ 3 e 2 Y 2 /3 Ψ 3 (Y 3 ) = c 5 e 1 8 (-2- √ 7+128 m-128 n)Y 3 + c 6 e 1 8 (-2+ √ 7+128 m-128 n)Y 3 ,
where

I ± √ 3 8 √ 3+128 n 2
√ 3 e 2 Y 2 /3 are modified Bessel functions of the first kind and noninteger order.

Discussion

We have considered the canonical analysis and subsequent quantization of the (3+1)dimensional action of pure gravity plus a cosmological constant term, under the assumption of the existence of two-dimensional (spacelike) surfaces of maximal symmetry. At the classical level, the application of the Dirac algorithm results in one linear and one quadratic first class constraints. The linear constraint is shown to correspond to arbitrary changes of the radial coordinate. The quadratic constraint is the generator of the time evolution. Adopting the Schrödinger picture for the quantum momentum operators, we are led to choose as our initial collection of state vectors all smooth (integrals over the radial coordinate r) functionals, in order to avoid an ill-defined action of these operators. The quantum linear constraint entails a reduction of this collection to all smooth scalar functionals. At this stage the need emerges to somehow obtain an induced metric on the so far "physical" states, which is composed out of these states. This leads us to firstly adopt a particular (formal) re-normalization prescription and secondly impose the Requirement. As a result, the final collection of state vectors is reduced to the three (essentially unique) smooth scalar functionals (y 1 , y 2 , y 3 ). The quantum analogue of the kinetic part of the quadratic constraint is then realized as the conformal Laplace-Beltrami operator based on the induced re-normalized metric. After the interpretation (through the Assumption) of the potential part of H o a Wheeler-deWitt equation emerges. In order to analytically solve this equation we exploit the freedom in the choice of L appearing in y 3 , which is left by the imposition of Requirement and which is shown to be a pure general coordinate transformation on the re-normalized manifold. Effecting an appropriate change of variables the metric is put in conformally flat form. Then, the aforementioned freedom is used to make the equation separable.

Generally (and somewhat loosely) speaking, our goal is, at a first stage, to assign a unique number between 0 and 1 to each and every geometry of the families considered, in a way that is independent of the coordinate system used to represent the metric. Of course, at the present status of things we cannot do this, since the following two problems remain to be solved: i) render finite the three smooth functionals and ii) select an appropriate inner product.

The first will need a final regularization of y 1 , y 2 , y 3 , but most probably, the detailed way to do this will depend upon the particular geometry under consideration.

For the second, a natural choice would be the determinant of the induced re-normalized metric, although the problem with the positive definiteness may dictate another choice.

At this point it is instructive to reflect upon the fate of the classical "open" Poisson bracket algebra (2.13) of the constraints H o , H 1 within the context of the present quantization scheme. It is true that it is very difficult to investigate the quantum realization of this Poisson algebra in the space spanned by the configuration functions and their derivatives of arbitrary order. The way we have chosen to overcome this difficulty is to generalize Kuchar's construction; an essential element of this construction is the reduction of the initial configuration space. This reduction consists in restricting to the space spanned by the "physical" degrees of freedom, annihilated by the linear constraints. Thus, even in the case of finite degrees of freedom, the classical algebra is not, at the quantum level, realized in the original configuration space, and therefore its realization can not be faithful in this space. As a second step in this construction, the reduced space of "physical" states is endowed with a metric (induced by the super-metric) and the kinetic part of the quadratic constraint is realized as the Laplace-Beltrami operator based on this metric. The consistency of the scheme is guaranteed by the existence of the reduce space of "physical" states, the fact that, by virtue of the first class algebra (3.2), the induced metric is composed out of the "physical" degrees of freedom q α , and standard existence theorems concerning second order partial differential equations. Of course, the ultimate finding of solutions to this equation can also serve as proof of consistency.

In the case we address, the first step of the generalization of Kuchar's construction is smoothly carried out, resulting in the "physical" states (3.4a), (3.4b). It is the generalization of the second step that can not be implemented without the use of the central idea of our work, i.e. the Assumption and the Requirement. It is only after their acceptance that it is possible to construct the finally reduced physical space spanned by the three functionals (3.20) and endowed with the physical metric (3.21). All the requirements for consistency of our quantum theory are thus met: The reduced space of "physical" states exists, and the postulated operator equation (3.24) admits solutions. Furthermore, the entire solution space (for Λ = 0, A T = 0) is given.

There is also a similarity with string theory which we think is worth pointing out: Indeed, in string theory the Virasoro algebra is made consistent by adjusting the dimensions of the target space. In our quantum theory, the initial degrees of freedom are infinite and any possible realization of the open Poisson algebra (2.13) is bound to be plagued by infinities (anomalies). In this respect, the function of the Assumption and the Requirement leads to a unique choice of the dimensions of the physical space i.e. three. Thus the well known anomaly cancelation in string theory is appearing in a natural way in our scheme, hence the parallelism.

Finally, we would like to comment upon the relation of the results here obtained to the results presented in our previous work [START_REF] Christodoulakis | [END_REF]. There, the 2+1 action of pure gravity plus a cosmological constant term was quantized in a similar manner, under the assumption of existence of one space-like Killing vector field. We would like to point out the quite interesting fact that, although the systems considered are different, the resulting re-normalized manifolds, their geometry and the corresponding Wheeler-deWitt equations are strikingly similar. To our view, this constitutes a very strong indication that the imposition of the Assumption and the Requirement is not simply an elegant way to reduce the number of spatial derivatives of the configuration fields involved in the scalar functionals, but is rather a tool for unraveling the underlying geometrical structure of Quantum Gravity (in the approximation considered, of course).

For Λ = 0 (A.3) is the well-known Schwarszchild metric [32]. The constants A, Λ appearing in (A.3),(A.4),(A.5) are essential: They can not be eliminated or even altered by a coordinate transformation, and thus different values of them define distinct geometries. Their character as essential can be verified by the use of the Cartan-Karlhede equivalence method [33], or the use of a criterion developed in [34], or by enclosing them in invariant relations satisfied among geometrical objects constructed by the Riemmann tensor and its covariant derivatives of an appropriate order. Such relations typically arise when we eliminate the coordinates between the expressions giving a sufficient number of curvature scalars. For metrics (A.3), (A.4) a pair of such relation is given by:

R -4Λ = 0, (3S1 -8Λ 2 ) 6 (27 A 2 S2 + 64Λ 2 + 144 A 2 Λ 3 ) -3(3S1 -8Λ 2 ) 5 (27 A 2 S1S2 + 32ΛS2 + 27 A 2 S3) + 3 (3S1 -8Λ 2 ) 4 ((36A 2 Λ + 12 )S2 2 + 81A 2 S1S3 + 80 ΛS3) - 18 (3S1 -8Λ 2 ) 3 (A 2 S2 3 + (36A 2 Λ + 10 )S2S3) + 9 (3S1 -8Λ 2 ) 2 (18A 2 S2 2 S3 + (108A 2 Λ + 25 )S3 2 ) -X486 A 2 S2S3 2 (3S1 -8Λ 2 ) + 486 A 2 S3 3 = 0, (A.6)
where = -1, 1 for spherical and GBL symmetry respectively, S1 is the Kretschmann scalar curvature, S2 its Laplacian, and S3 the square of the measure of its gradient:

S1 ≡ R IJKL R IJKL , S2 ≡ g IJ S1 ;I;J , S3 ≡ g IJ S1 ;I S1 ;J .
For metric (A.5) the constant κ is not essential as it can be seen by applying the criterion of [34]. It is included only because the value κ = 0 gives a space different from the family with κ = 0. Thus, in this case, one invariant relation is needed (in order to prove the essentiality of Λ). This could be the common to all metrics R = 4 Λ or, the alternative relation:

(3S1 -8Λ 2 )(S2 -6S1Λ + 16Λ 3 ) -3S3 = 0. (A.7)
The case K > 0 is completely analogous : F is now time-like and can thus be reduced to the form F = (1, 0), indicating that ψ(t, r) is to be identified to the new time coordinate T ≡ ψ(t, r). Again the freedom in choosing the new radial coordinate can be used to eliminate the off-diagonal element of g 2 µν . Therefore, the metric corresponding to (A.1) is

ds 2 = -A(T, R) dT 2 + B(T, R) d r 2 + T 2 d θ 2 + T 2 f (θ) 2 d φ 2 . (A.8) When this line-element is inserted into (2.14) the constraint equation E 0 1 ≡ -∂ R A(T,R) T A(T,R) 2 = 0 is obtained. Insertion of the solution to this equation A(T, R) = α(T ) into (2.14) makes the combination T α(T )(E 1 1 -E 0 0 ) ≡ ∂ T α(T ) α(T ) + ∂ T B(T,R) B(T,R) = 0 easy to integrate, resulting in B(T, R) = β(R)
α(T ) . At this stage, the line-element (A.8) reads:

ds 2 = -α(T ) dT 2 + β(R) α(T ) d r 2 + T 2 d θ 2 + T 2 f (θ) 2 d φ 2 (A.9)
revealing that it is spatially homogeneous, since β(R) can be absorbed by a redefinition dρ ≡ β(R) 1/2 dR. This result can be considered as the counterpart to Birkhoff's theorem. In these standard coordinates the only remaining functionally independent equation E 0 0 = 0 is readily solved, resulting in the following three cosmological solutions

ds 2 = -(1 - A T - Λ T 2 3 ) -1 d T 2 + (1 - A T - Λ T 2 3 ) d ρ 2 + T 2 d θ 2 + T 2 (sin θ) 2 d φ 2 (A.10) ds 2 = -(-1+ A T - Λ T 2 3 ) -1 d T 2 +(-1+ A T - Λ T 2 3 ) d ρ 2 +T 2 d θ 2 +T 2 (sinh θ) 2 d φ 2 (A.11) ds 2 = -( κ T - Λ T 2 3 ) -1 d T 2 + ( κ T - Λ T 2 3 ) d R 2 + T 2 d θ 2 + T 2 θ 2 d φ 2 (A.12)
For Λ = 0 (A.10) is the well-known Kantowski-Sachs metric [35]. The character of the constants A, Λ, κ is the same as in the previous case (K > 0) and their (locally) defining invariant relations are identical to (A.6), (A.7). Finally, the case K = 0 does not produce any new, non-trivial solution: The investigation is most easily carried out in the conformal gauge and light-cone coordinates for metric (2.5) ds 2 = 2 A(u, v) d u d v + ψ(u, v) 2 d θ 2 + ψ(u, v) 2 f (θ) 2 d φ 2 . Since K ≡ 2 ∂uψ(u,v) ∂vψ (u,v) A(u,v) = 0 we have that ψ(u, v) = h(u) or k(v). In either case, with an appropriate redefinition of u or v the previous line-element can be cast into the form ds 2 = 2 A(U, v) d U d v+U 2 d θ 2 +u 2 f (θ) 2 d φ 2 . The case with v 2 in the spatial part is completely equivalent. The constraint equations now become E 0 0 ≡ Λ + U 2 = 0, E 0 1 ≡ 2∂ U A(U,v) UA(U,v) 2 = 0 where = -1, 1, 0 for the spherical, GBL or plane symmetry respectively. The system is compatible only for = 0 (plane symmetry), its solution being Λ = 0, A(U, v) = A1(v) which, however, is nothing but the Minkowski space-time.

Notice that in the families of metrics (A.3), (A.4), (A.10), (A.11) Minkowski spacetime is attained with A = 0, Λ = 0, while we can not set κ = 0, Λ = 0 in (A.5), (A.12). Also, setting A = 0, κ = 0 in all metrics results in maximally symmetric space-times. Lastly, for Λ = 0 all solutions can be concisely summarized, in the conformal light-cone gauge, by (2.6), (2.7).

B Appendix: Existence of L

In this Appendix we show that an L(Ψ (1) ) exists for which (3.18) is equal to (3.19) for the particular choice of F [L(Ψ (1) )] given by (3.27). To begin with let us change coordinates from Ψ (1) to ω ≡ Ψ (1) 2 -. The term inside the square brackets in the r.h.s of (3.18) becomes

L(ω) - + ω 2 ω L (ω) 2 -2 √ + ω 2 L(ω) -+ω 2 ω L (ω) L(ω) -+ω 2 ω L (ω) √ + ω 2 dω,
where the prime now denotes differentiation with respect to the variable ω. On the other hand (3.19), through the choice F = L -1 (ω) becomes

L(ω) 2 - 4 ω 2 3 L (ω) 2 .
We thus have to prove the existence of an L(ω) which secures the equality between the above two expressions, namely that We have now to prove that there is a choice of the constants k, m for which F = 0, so that A = 0 ⇒ I = 0. Our strategy is to confine, through integrability conditions for (B.5), as much as possible the form of F (k, m). As a first step, we must dispose off the double integral (in the variable ω) appearing in A. By differentiating this new form of B with respect to k, and repeating the procedure described just above, we can eliminate the remaining integral that appears in B (and in fact all the ω-dependence). Thus, we end up with

I ≡ L(ω) - + ω 2 ω L (ω) 2 -2 √ + ω 2 L(ω) -+ω 2 ω L (ω) L(ω) -
B = 2 F -3 ∂F ∂k + ∂ 2 F ∂k 2 =

  .9b) while the indices {α, β} take the values {γ, ψ} and = d d r . The reduced Wheeler-deWitt super-metric G αβ reads

1 )- 3 ( 4 )( 3 2 -ω 2

 134322 Let us assume that (B.1) holds. Then, the expression -3ω 2 I -+ω 2 + ω 2 ) 2 L (ω) 2 + 4 ω 2 (2 + ω 2 )L (ω) 2 + 8 ω 3 ( + ω 2 )L (ω)L (ω) = 0. (B.2)The case L(ω) = const. does not concern us since it corresponds to the functional y 1 . Therefore, dividing (B.2) by L (ω) 2 and defining a(ω) ≡ L (ω)/L(ω) we obtain8 ω 3 ( + ω 2 ) a (ω) + ω 4 + 2 ω 2 -3 2This expression for L(ω) emerged as an integrability condition for the integro-differential equation (B.1). It is therefore necessary to insert (B.4) into this equation. The result is the following expression for A ≡ -4( + ω 2 ) -1/2 I: 10 ω 2 -ω 4 ) ω 37/8 ( + ω 2 ) 1/4 e k-dω dω.Surprisingly enough the above expression is ω-independent, i.e ∂A/∂ω = 0. Therefore B ≡ A -F (k, m) = 0. (B.5)

2 -( 3 2 - 2 -( 3 2 - 3 .

 2322323 0, which has the following solutions: F (k, m) = λ 1 (m) e k + λ 2 (m) e 2 k . (B.6) Inserting (B.6) into (B.5) and differentiating with respect to m we get 8 ( + ω 2 ) 3/4 ω 13/8 e k-3 16 ω 10 ω 2 -ω 4 )ω 37/8 ( + ω 2 ) 1/4 e k-3 16 ω 2 dω -λ 1 (m) e k -λ 2 (m) e 2 k = 0,which by differentiation with respect to k gives8 ( + ω 2 ) 3/4 ω 13/8 e k-3 16 ω 10 ω 2 -ω 4 ) ω 37/8 ( + ω 2 ) 1/4 e k-3 16 ω 2 dω -λ 1 (m) e k -2 λ 2 (m) e 2 k = 0.Subtracting these last two equations we haveλ 2 (m) = 0 ⇒ λ 2 (m) = cIf we insert this result together with (B.6) in (B.5) and we double differentiate with respect to m we will getλ 1 (m) = 0 ⇒ λ 1 (m) = c 1 m + c 2 .Therefore, (B.6) becomesF (k, m) = c 1 m e k + c 2 e k + c 3 e 2 kand we are finally able to conclude that the choice m = -c 2 +c 3 e k c 1 satisfies (B.1).

  To do this we differentiate B with respect to k and solve the resulting expression for the aforementioned double integral. By inserting the outcome of this operation into (B.5) we get -10 ω 2 -ω 4 ) ω 37/8 ( + ω 2 ) 1/4 e k-3 16 ω 2 dω = 0.

	B =	∂F ∂k	-2 F +	8 m ( + ω 2 ) 3/4 ω 13/8	e k-3 16 ω 2 -m	(3 2

(-9 Y 1 +8 Y 2 -15 Y 3 ) )) (3.22)(where F -1 denotes the function inverse to F , i.e F -1 (F (x)) = x) brings the metric to

Acknowledgments

One of the authors (G. O. Papadopoulos) is a Killam Postdoctoral Fellow and acknowledges the relevant support from the Killam Foundation.

A Appendix: The Classical Solution Space

In this Appendix we outline the investigation of the solution space to (2.14) for the family of geometries described by (2.5). We first observe that the block-diagonal form of this metric is preserved under the (restricted) coordinate transformations (t, r) → (T (t, r), R(t, r)). Under any such change the 2 × 2 (t, r) part of (2.5) transforms as a second rank covariant tensor and can thus serve as a two-dimensional metric, say g (2) µ ν (of signature (-, +)), while ψ(t, r) transforms as a scalar. Therefore, the doublet F ≡ ( ∂ψ (t,r) ∂t , ∂ψ(t,r) ∂r ) is an one-form under the above mentioned restricted transformations and its measure K = g (2)µ ν F µ F ν is invariant.We can thus distinguish three cases corresponding to K being positive, negative or zero respectively. When these cases are combined with the three types of symmetry, they produce nine possibilities which have to be considered.

Let us take the case K > 0. The one-form F is space-like and therefore can, by virtue of a standard theorem of differential geometry, be reduced to the form F = (0, 1) by a change of coordinates in the (t, r) subspace. Because of the scalar character of ψ(t, r) we conclude that it has to be identified to the new radial coordinate R ≡ ψ(t, r). At this stage, a further transformation of the time coordinate, keeping R unchanged, can be employed to eliminate the off-diagonal component of g (2) µ ν . The line-element (2.5) is thus, without loss of generality, reduced to the form

When this line-element is inserted in (2.14) the constraint equation

β(R) . At this stage, the line-element (A.1) reads:

The form of this metric reveals that it is static, since α(T ) can be absorbed by a time redefinition dτ ≡ α(T ) 1/2 dT . For Λ = 0 and spherical symmetry this result is known as Birkhoff's theorem. Using these standard coordinates we can easily integrate the only functionally independent equation E 0 0 obtaining the following three "point-like" solutions for the cases of spherical, GBL and plane symmetry respectively:

3)

)