Professor Geoffrey Grimmett

Jean-François Laslier

Friedrich Pukelsheim

Victoriano Ramírez González

Richard Rose

Wojciech Słomczyński

Martin Zachariasen

RESPONSIBLE Karol Życzkowski

Ms Cristina Castagnoli

CITIZENS' RIGHTS AND CONSTITUTIONAL AFFAIRS CONSTITUTIONAL AFFAIRS THE ALLOCATION BETWEEN THE EU MEMBER STATES OF THE SEATS IN THE EUROPEAN PARLIAMENT Cambridge Compromise

EXECUTIVE SUMMARY

The Committee on Constitutional Affairs has commissioned a Symposium of Mathematicians to recommend a mathematical "formula" for the apportionment of the seats of the European Parliament between the Member States. The purpose of the reform is "to eliminate the political bartering which has characterised the distribution of seats so far", and the required formula should be "durable, transparent and impartial to politics".

The apportionment formula is required to allocate a minimum of 6 seats to each Member State, and a maximum of 96 seats. The Parliament-size is to be no greater than 751. The recommended formula should satisfy a condition of degressive proportionality.

The Cambridge Apportionment Meeting took place in January 2011. Its principal recommendation is that seats in the European Parliament be apportioned in such a way that each Member State receives a base of 5 seats, and the remaining seats are divided between the Member States in proportion to their populations (subject to the constraints). The recommended "base+prop" formula rounds fractional allocations of seats upwards, thereby guaranteeing a minimum of 6 seats to every Member State. Tables displaying the application of this formula to the current European Union, as well as after accessions by Croatia and Iceland, may be found in Section 9.

Associated with the principal recommendation is a recommendation to amend the definition of "degressive proportionality" in order to accommodate mathematical methods for the apportionment of Parliament. The reason is that mathematical methods generally result in allocations failing to satisfy the definition presented by the former Rapporteurs Alain Lamassoure and Adrian Severin in their 2007 Report.

It is further recommended that consideration be given to the value of the minimum number of seats, currently 6, in the light of future accessions to the European Union, and that the functioning of the maximum number, currently 96, be reviewed prior to each quinquennial apportionment of Parliament.

The final recommendation of the Cambridge Apportionment Meeting is that Eurostat be asked to review the methods used by Member States in calculating their populations.

INTRODUCTION

The author was invited by the Committee on Constitutional Affairs (AFCO) to write a Briefing Note and to organize a Symposium of Mathematicians in order to recommend a mathematical "formula" for the apportionment of the seats of the European Parliament. The background to this Symposium is explained in the following extracts from a note [START_REF] Duff | Mathematical Symposium[END_REF] prepared for the Committee on Constitutional Affairs by the Rapporteur, Mr Andrew Duff:

The aim of the symposium is to discuss and, if possible, to propose to the Committee on Constitutional Affairs a mathematical formula for the redistribution of the 751 seats in the European Parliament. The formula should be as transparent as possible and capable of being sustained from one Parliamentary mandate to the next.

The purpose of the reform is to eliminate the political bartering which has characterised the distribution of seats so far by enabling a smooth reallocation of seats once every five years which takes account of migration, demographic shifts and the accession of new Member States.

The Cambridge Apportionment Meeting convened on 28-29 January 2011 at the Centre for Mathematical Sciences, University of Cambridge. This is its Report.

CAMBRIDGE APPORTIONMENT MEETING

Terms of reference

These are the Terms of Reference for the Briefing Note, received from the Committee on Constitutional Affairs.

Background

The European Parliament is discussing a proposal for a modification of the Act concerning the election of the Members of the European Parliament by direct universal suffrage of 20 September 1976.

On 19 May 2010 the Rapporteur on the Constitutional Affairs Committee of the European Parliament, Mr Andrew DUFF, presented his working document on the proposal and on 5 July 2010 the draft report on the same issue.

One of the core issues of the report concerns the principle of degressive proportionality for the distribution of seats.

The problem of fitting everyone in fairly beneath the ceiling of 751 according to the principle of degressive proportionality lies directly at the door of the European Parliament which, under the Treaty of Lisbon, acquires the right (and the duty) to initiate changes in the composition of the House during each mandate. Parliament will also be responsible for proposing the temporary addition of MEPs from accession states during a five-year mandate.

Aim and content of the briefing note

Against this background, the requested briefing note shall identify a mathematical formula for the distribution of seats which will be durable, transparent and impartial to politics.2

Methodology and language

The analysis to be undertaken shall be objective and all sources of information must be indicated.

The note shall be drawn up in English.

Delivery conditions

The final version of the briefing note must be handed over to the European Parliament at the latest on 31 March 2011. In Paragraph 10 of the "Explanatory Statement" of the above Resolution, it is written: 10. Within this context, how to go forward? The ideal alternative would be to agree on an undisputed mathematical formula of "degressive proportionality" that would ensure a solution not only for the present revision but for future enlargements or modifications due to demographic changes.

CONSTRAINTS ON THE ALLOCATION METHOD

Thus, "degressive proportionality" is to include two requirements:

DP1. no smaller State shall receive more seats than a larger State, DP2. the ratio population/seats shall increase as population increases.

Condition DP1 is easy to accept on the basis of rationality, but Condition DP2 poses a serious practical difficulty, and has in addition been violated in recent Parliamentary apportionments. It has been noted [START_REF]Mathematics and Democracy, Recent Advances in Voting Systems and Collective Choice[END_REF] that there are hypothetical instances of apportionment for which there exists no solution satisfying both DP1 and DP2.

It is in the nature of mathematicians to distil the essence of an assertion, and thus the following example is included here. Imagine a Parliament comprising 105 seats that is to contain representatives of just five States as follows.

Member

Our recommendation

Two approaches to degressive proportionality were discussed. A. Adopt a method whose outcomes satisfy Condition DP2 but with a possibly reduced Parliament-size. B. Propose a change to the Lamassoure-Severin definition of degressive proportionality that lies within existing law and allows greater flexibility and transparency. A method ("iterative interpolation") achieving A was summarized at the Symposium, and is included in Section 6.2. We noted that this method, while mathematically sound, does not generally achieve the given Parliament-size, and secondly that its formulation is markedly less transparent than the method ultimately recommended and described in Section 4.

It was agreed that Alternative B is the better way forward, and can be achieved easily via a minor amendment of the wording of the European Parliament's Resolution on "Proposal to amend the Treaty provisions concerning the composition of the European Parliament" [START_REF] Balinski | Fair Representation[END_REF] . We recommend that degressive proportionality be interpreted by amending Paragraph 6 of the Resolution through the addition of the bold text as follows:

[The European Parliament] Considers that the principle of degressive proportionality means that the ratio between the population and the number of seats of each

CAMBRIDGE COMPROMISE

This Section contains our recommendation for the apportionment of the seats in the European Parliament. 6

Base+prop method

The so-called base+prop method proceeds in two stages. At the first stage, a fixed base number of seats is allocated to each Member State. At the second stage, the remaining seats are allocated to States in proportion to their population-sizes (subject to rounding, and capping at the maximum). In order to achieve the given Parliament-size, one introduces a further ingredient called the divisor.

The base+prop method is formulated as follows [START_REF] Martínez-Aroza | Several methods for degressively proportional allocations. A case study[END_REF]

Cambridge Compromise apportionments in practice

In Section 9 are presented Tables indicating apportionments of the European Parliament following the Cambridge Compromise, for the current 27 Member States, and with the additions of Croatia and Iceland in that order. Population figures are taken from the Eurostat website 9 . [START_REF] Zachariasen | On the relationship between base and rounding-method in divisorbased apportionment[END_REF] Our recommendation is named in harmony with the Jagiellonian Compromise proposal for voting within the Council of the European Union. See references [8,10] at the end of the Report. [START_REF] Martínez-Aroza | Several methods for degressively proportional allocations. A case study[END_REF] For a discussion of this method see reference [9], where it is referred to as the "fix+prop" method. [START_REF] Życzkowski | Voting in the European Union: The square-root system of Penrose and a critical point[END_REF] There will generally be a range of possible values for the final divisor; see Question 8.7 in Section 8. 9 http://epp.eurostat.ec.europa.eu/

APPORTIONMENT METHODS

Methods of apportionment

We concentrated on apportionment methods comprising the following two choices: of an apportionment function A, and of a rounding method.

1. Choose an apportionment function A sufficiently general to be capable of accommodating the given constraints on minimum, maximum, and Parliament-size, 2. allocate to a Member State with population p the fractional number A(p) seats, 3. round A( p) to obtain a whole number [ A(p)] , 4. adjust the function in such a way as to satisfy the given constraints. General accounts of such functions may be found in articles contained in references [3,4] at the end of the Report.

The apportionment function is to be chosen in such a way that allocations are nondecreasing in population, and satisfy degressive proportionality (and possibly other criteria allied to principled approaches to apportionment). The specific functions considered at the Cambridge Apportionment Meeting satisfy the amended version of degressive proportionality proposed in Section 3.2, but each can result in violations of the Lamassoure-Severin (2007) condition. For this reason, we resolved to recommend the amended version of degressive proportionality.

We considered two categories of apportionment function, basically those comprising linear elements, and those with non-linear elements.

• Non-linear functions, including parabolic and power functions [START_REF] Słomczyński | Jagiellonian Compromise: an alternative voting system for the Council of the European Union[END_REF] .

Advantages: smoothness (in the mathematical sense).

Disadvantages: non-linear (thus, with a diminished potential for proportionality), unprincipled use of a power, non-robustness with respect to certain population distributions, non-transparent, relative difficulty of computation. • Linear functions, including base+prop and spline methods [START_REF] Słomczyński | On bounds for the allocation of seats in the European Parliament[END_REF] .

Advantages: transparent, greater potential for proportionality, ease of computation.

Disadvantages: non-smoothness of allocation function at the maximum.

We noted that, as the European Union is progressively enlarged (the constraints being unchanged), the non-linear allocation functions become closer to linear.

Having taken this analysis into consideration, we resolved to restrict attention to allocation functions of linear type. The simplest of these applies a single linear function, subject to adjustments that guarantee the minimum and maximum constraints.

Following a discussion of the merits of base+prop and other linear-type methods such as spline methods, we resolved that base+prop is amongst the most transparent methods, and is preferable to others on the grounds of degressive proportionality.

We noted that the base+prop method can be interpreted as one in which the base is an allocation to Member States, and the remaining seats (prop) are proportional to population (subject to capping at the maximum). We felt that this interpretation might find favour with the Committee on Constitutional Affairs, in view of the founding principles of the EU, enshrined in the Treaty, that the Union is made up both of Member States (enjoying equality in international law) and of citizens (enjoying democratic equality). Notwithstanding, we recommend the base+prop method on the basis of the remit of Section 1.2, in particular for its durability, transparency, impartiality, and its adherence to degressive proportionality.

Rounding methods

In many situations in life, one may be confronted with a fraction x when a whole number is required. There are a variety of methods for 'rounding' x to a nearby whole number. This situation is commonplace in apportionment problems given the integral nature of the unit being distributed.

We concentrate here on the three principal methods of rounding:  rounding downwards: for example, 5.1  5, 5.9  5,

 "standard rounding" to the nearest whole number: 5.1  5, 5.9  6,  rounding upwards: 5.1  6, 5.9  6.

There is an extensive literature on rounding methods for apportionment, especially in the context of the US House of Representatives [START_REF] Ramírez González | Degressive proportionality. Composition of the European Parliament. The parabolic method[END_REF] . Previous analysis indicates that, other things being equal, downwards (respectively, upwards) rounding tends to favour larger (respectively, smaller) States, and standard rounding is generally regarded as being fairly neutral to State size.

Choosing the base and rounding method

Implementation of the base+prop method requires a choice of base. This choice is informed by the demand that the smallest States have a minimum of 6 seats, and by the principle of degressive proportionality. Since the base allocation is made without regard to populationsize,

• a larger base tends to favour smaller States,

• a smaller base tends to favour larger States.

The relationship between the smaller and larger States is relevant to choice of both base and rounding method, and these two choices are therefore inter-related. The relevant factors are (a) the minimum, and (b) degressive proportionality. In the interests of transparency, we favoured a whole number base rather than a fraction. Two candidate methods propose themselves given the minimum number 6 of seats.

• 6 + standard. Base = 6, with standard rounding to the nearest whole number.

• 5 + upwards. Base = 5, with rounding upwards to a whole number. Each of these methods guarantees a minimum of 6 seats to all States, and is degressively proportional in the amended sense proposed in Section 3.2.

We argued as follows in reaching Recommendation 3 (namely, to adopt the method 5 + upwards). Since each of the above methods satisfies the amended form of degressive proportionality, we concentrated on the durability of the minimum. We were advised by the AFCO representatives in attendance of an expectation that the smallest States in the current EU would indeed be allocated exactly 6 seats. With the current population figures, the smallest four Member States are as follows: By reason of the increase in its population, Malta gains a seventh seat under the method 6 + standard. We viewed this as a sign of non-robustness in the minimum, and for this reason we have recommended the use of 5 + upwards. Does there invariably exist a value of the divisor d that results in the exact value of the given Parliament-size? This general question has received attention in the mathematical literature of apportionment (see reference [5] at the end of the Report, for example). An issue can arise as a result of unlikely but conceivable coincidences in the factors of population-sizes, but these are very rare in an apportionment problem of the scale of the European Parliament. We feel that this possibility can be disregarded 15 .

The iterative-interpolation method of apportionment

Details are presented of a further apportionment method of linear type. This is included in this Report as an example of a method that results invariably in an allocation satisfying the Lamassoure-Severin degressive proportionality conditions DP1 and DP2 (see Section 3.1). We do not recommend the adoption of this method, for reasons including its relative lack of transparency. We do not believe that adherence to the Lamassoure-Severin (2007) definition of degressive proportionality should emerge as a greater priority than transparency.

First, we review the spline method 16 . It has allocation function [START_REF] Schuster | Seat biases of apportionment methods for proportional representation[END_REF] Base+prop is an example of a so-called "divisor method". For the sake of clarity for the lay reader, we might take the initial divisor as the total population of the European Union divided by the maximum number of seats minus the base allocation nb. This choice does not take into account the capping of allocations at the maximum, and is for illustration only. 15 See reference [5] for an analysis of "ties". 16 Readers are referred to reference [7] for a fuller discussion.

where is the population of the smallest Member State. It differs from the base+prop method in that it is designed to allocate the given minimum to the smallest State before rounding.

The iterative-interpolation method is summarized as follows.

1. The spline method is applied to the entire set of Member States, and the largest value k is found such that the smallest k States have allocations satisfying the Lamassoure-Severin (2007) conditions. The allocations to the first k States are fixed. 2. The allocation to the (k+1)th State is reduced by one and then fixed. 3. The spline method is now applied to the (k+1)th and larger States (with adjusted values of the minimum and of ), and the above steps are iterated until all States have received their fixed allocations.

It may be checked that apportionment by the method of iterated interpolation:

 inevitably satisfies the Lamassoure-Severin (2007) definition of degressive proportionality,  does not necessarily achieve the exact Parliament-size, but may have smaller total.

Choosing the minimum and the base

The minimum number of seats per Member State is currently set to 6; how should this minimum vary as the number of Member States changes? We were not asked to report on this question, but nevertheless we embarked on a discussion as an aid to understanding the role of the minimum.

We considered ways in which the minimum might be required to vary. As the minimum varies, so must the base. Two possible approaches are:

A. determine a minimum that varies with the number of States, and then choose an appropriate (possibly fractional) base, B. choose a base that decreases formulaically as the number of States increases, and observe how the minimum varies, in the presence of a given rounding method. We felt that, regardless of which approach is adopted, the issue of transparency dictates that the minimum be clearly defined.

Scheme A

Our approach to Scheme A is as follows. In the current setting, there are 27 Member States with a minimum of 6 seats, so that the total number of seats involved in the minimum is 6  27 = 162, a proportion of 162/751 ≈ 22%. Our favoured approach, having heard the advice of the Rapporteur, is to cap this proportion at a suitable figure, and to choose the minimum as large as possible subject to this cap. The effect of a cap of 25% would be as follows: a minimum of 6 seats with 27-31 Member States, a minimum of 5 with 32-37 Member States, and so on. The adoption of a proportion lower than 25% would result in a swifter reduction of the minimum.

At the transition points, a number of seats are transferred out of the base allocation into the proportional allocation. This has the effect of adjusting the relationship between the smaller and the larger Member States, since more seats are distributed according to population.

As the minimum varies, so must the base. We considered two ways in which the base could follow the minimum.

A1. The base is set to one less than the minimum. Combined with upwards rounding, this ensures that the minimum constraint is achieved in any situation (as currently pertains amply) in which some State has population-size smaller than the divisor. A2. The base is set to the smallest value (which will generally be fractional) such that the smallest Member State receives the minimum number of seats exactly. (For illustration, subject to the current population figures and rounding upwards, this results in a base of approximately 4.5.) We discussed without clear conclusion the relative merits of the simplicity, and hence transparency, of Method A1, and the mathematical appeal of an implicit definition such as that of Method A2. If transparency is paramount then Method A1 should be adopted.

Scheme B

In Scheme B, the base b is determined as a function of the number n of Member States. We discussed formulae of the type b = 135/n , which is in agreement with the Cambridge Compromise recommendation to take b = 5 when n = 27. Taken together with upwards rounding and with current population figures, this would imply a minimum of 6 seats in the current European Union and for the two enlargements shown in Section 9.

This approach has the attractive property that the base, being generally fractional, decreases in a more gradual fashion. This contrasts with the downward unit-steps of Scheme A1. On the other hand, the minimum allocation descends in a manner that may be considered over-sensitive to the smallest population-size. This approach may, therefore, lie beyond the spirit of the current terms of reference.

RECOMMENDATION A IS THAT DUE CONSIDERATION BE GIVEN TO THE MANNER IN WHICH THE MINIMUM, CURRENTLY 6, AND BASE SHOULD VARY IN THE LIGHT OF FUTURE CHANGES IN THE NUMBER OF MEMBER STATES OF THE EUROPEAN UNION.

Choosing the maximum

The base+prop method takes account of both Member States and Citizenry: each Member State receives 5 seats of right, plus one or more further seats in proportion to its population. This proportionality is valid over the entire range of populations except where the maximum of 96 is invoked. The maximum of 96 is dictated within our terms of reference. For any Member State sufficiently large that its allocation is capped at 96, there is a loss of proportionality. The effect of the maximum constraint is highlighted by the analysis of this Report. As the architecture of the European Union develops through accessions, the significance of the maximum will diminish until it ceases to be operative.

RECOMMENDATION B IS THAT THE EUROPEAN PARLIAMENT SHOULD REVIEW THE MANNER OF FUNCTIONING OF THE MAXIMUM

CONSTRAINT ON NUMBER OF SEATS, CURRENTLY 96, PRIOR TO FUTURE APPORTIONMENTS.

Official population statistics

Census data on population is usually collected only once a decade; hence, official data about the current population is an estimate based on updates of this data. Both the year of the census and the methods of updating can vary between countries, thus introducing inconsistencies between countries. There can also be different national methods of determining who is counted as a resident. Census data is key to the determination of the apportionment of the European Parliament.

We were told by the AFCO representatives that the Commission is considering proposing a statistics regulation which would allow Eurostat to verify more accurately than at present the consistency and comparability of national data.

RECOMMENDATION C IS

A non-linear allocation function might be a smoother approach to the capping at the maximum. Why have you not recommended such a function?

Answer: We considered the use of linear functions to be more transparent than the use of non-linear functions, and to be closer to the concept of proportionality. We noted that the capping at 96 disturbs the linearity at the upper end, and we regretted this from a mathematical point of view since we would have preferred to use a smooth allocation function. However, the maximum of 96 was dictated by our terms of reference, and we considered it preferable to retain linearity for Member States not at the maximum, rather than disturb it throughout the entire list.

13

 13

6. FURTHER DISCUSSION 6 . 1 .

 61 Base+prop for the mathematician A mathematical formulation of the base+prop method of apportionment is presented next. Let m be the minimum and M the maximum number of seats per Member State, and H the target Parliament-size. Write b for the base, and d for the free parameter called the divisor. The allocation function A is given effectively by where p denotes the population of a State. The total number of seats allocated to n Member States with populations is where [x] is a rounding of the fraction x, and the divisor d is chosen 14 in such a way that The European Parliament has currently m = 6, M = 96, H = 751. We recommend taking the base b = 5 and rounding upwards.

3 .

 3 If Croatia were to accede during the legislative period, Parliament could be temporarily enlarged by 5 + 6 = 11 seats (since 4 425 747/819 000 ≈ 5.4  6).

Figure 1 .

 1 Figure 1. The Cambridge Compromise allocations with 27 Member States. States marked in red have allocations in violation of Condition DP2 of Section 3.1 (considered in order of decreasing population-size). The black line indicates the allocation function before rounding (with capping at the maximum).

3. DEGRESSIVE PROPORTIONALITY 3.1. Background

	This Report takes note of the following constraints:
	 the Parliament size shall not exceed 751, We discussed the issue of degressive proportionality in depth, with specific reference to the  there shall be a minimum threshold of 6 seats per Member State, European Parliament Resolution on "Proposal to amend the Treaty provisions concerning  no Member State shall receive more than 96 seats, the composition of the European Parliament" adopted on 11 October 2007 (Report of the  no smaller State shall receive more seats than a larger State, Rapporteurs Alain Lamassoure and Adrian Severin to the Committee on Constitutional  the allocation shall respect the principle of "degressive proportionality". Affairs) 3 . In Paragraphs 4 and 6 of the Motion, it is written:
	We were advised by the AFCO representatives in attendance that the first three constraints 4. [The European Parliament] Notes that Article [9a] of the Treaty on European
	are not absolutely rigid, but nevertheless there exists a general expectation in Parliament Union as incorporated in the draft amending treaty provides for a framework
	that its total size should not be less than 751, and that the smallest States should receive comprising an overall ceiling of 750, a maximum of 96 for the most populous
	an allocation not greater than 6 seats. Member State and a minimum of 6 for the least populous Member State, and
	that it lays down the principle of representation of European citizens in
	The issue of "degressive proportionality" is addressed in detail in Section 3 of this Report. accordance with degressive proportionality, without defining this term in any
	more precise way;
	6. [The European Parliament] Considers that the principle of degressive In reaching our conclusions, we have taken into account the following additional proportionality means that the ratio between the population and the number of observations concerning the general structure of the European Parliament: seats of each Member State must vary in relation to their respective populations  the EU has currently 27 Member States,  the smallest population (as published officially by Eurostat) is currently 412 970, in such a way that each Member from a more populous Member State represents
	and the largest 81 802 257, more citizens than each Member from a less populous Member State and
	 future accessions may include a number of States with a spread of sizes, conversely, but also that no less populous Member State has more seats than a
	 there will be migration and demographic changes, more populous Member State.
	 Member States' population figures (as published by Eurostat) will be used as input
	to the formula.

  Each State must receive exactly 21 seats if the two conditions are to be satisfied. Imagine now that the given Parliament-size is 106 rather than 105. If Condition DP1 is to be satisfied, the extra seat must be allocated to Greece. However, this violates Condition DP2,

	State	Population	Apportionment
	Greece	11 305 118	21
	Belgium	10 839 906	21
	Portugal	10 637 713	21
	Czech Republic	10 506 813	21
	Hungary	10 014 324	21
	Totals	53 303 874	105

since That is, a MEP from the largest Member State, Greece, represents fewer citizens than a MEP from a smaller Member State, Belgium. This difficulty arises in situations where there are a number of Member States having similar populations. While the above example is hypothetical, it illustrates the reason why, in typical apportionments by mathematical means, there are violations of Condition DP2.

Conditions DP1 and DP2, acting together, can force States with similar populations to have equal numbers of seats. As further States become similar in size through, for example, accession, migration, or demographic change, they may be coerced into near equality. One may note that this occurs because of pair-wise comparisons between pairs of States when ordered by population-size.

  Member State before rounding to whole numbers must vary in relation to their respective populations in such a way that each Member from a more populous Member State represents more citizens than each Member from a less populous Member State and conversely, but also that no less populous Member State has more seats than a more populous Member State.Recommendation 1 would not have been timely at the adoption in 2007 of the European Parliament Resolution on "Proposal to amend the Treaty provisions concerning the composition of the European Parliament", since seats were then allocated by negotiation rather than by formula. Recommendation 1 may be viewed as a minor adaptation of the 2007 resolution, motivated by the search for robustness.

	RECOMMENDATION 1 IS THAT THE EUROPEAN PARLIAMENT ADOPT
	THIS REVISED DEFINITION OF DEGRESSIVE PROPORTIONALITY.

Choice of base and rounding method RECOMMENDATION 3 IS TO SET THE BASE b = 5 AND TO USE ROUNDING UPWARDS.

  : 1. assign to each Member State a fixed number of seats, called the base and denoted as b, 2. for a given divisor d, assign to a Member State with population p a further quotient p/d, resulting in the seat share b + p/d, 3. perform a rounding of the seat share b + p/d into a whole seat number [b + p/ d ] , 4. if the seat number [ b + p/d] exceeds the maximum allocation, replace it by this maximum, 5. adjust the divisor d in such a way that the sum of the seat numbers of all Member States equals the given Parliament-size.

8

RECOMMENDATION 2 IS TO ADOPT A BASE+PROP METHOD OF APPORTIONMENT.

4.2.

  Suppose that the population of Malta increases by 8000, and the other populations remain unchanged. The amended Table is as follows:

	Member State Population 5 + upwards 6 + standard Now
	24 Estonia	1 340 127	7	8	6
	25 Cyprus	803 147	6	7	6
	26 Luxembourg	502 066	6	7	6
	27 Malta	412 970	6	6	6
	Against each State is reported its current allocation ('Now'), and its hypothetical allocations
	under the two methods under consideration.
	Member State Population 5 + upwards 6 + standard Now
	24 Estonia	1 340 127	7	8	6
	25 Cyprus	803 147	6	7	6
	26 Luxembourg	502 066	6	7	6
	27 Malta	420 970	6	7	6

THAT THE EUROPEAN COMMISSION BE ENCOURAGED TO ENSURE THAT EUROSTAT REVIEWS THE METHODS USED BY MEMBER STATES IN CALCULATING THEIR CURRENT POPULATIONS, IN ORDER TO ENSURE ACCURACY AND CONSISTENCY. 8. QUESTIONS ARISING 1. Why have you recommended base 5 with rounding up, rather than an alternative such as base 6 and standard rounding to the nearest whole

	number?
	Answer: See Section 5.3. The recommended method is more robust with respect to
	the minimum.
	2.

Why have you not recommended a method with a greater degree of proportionality, with States allocated fewer than 6 seats being raised to 6?

  Answer: This refers perhaps to the divisor method that allocates seats proportionally to population (subject to rounding and capping) while raising the allocation where necessary to 6. States with population-sizes that are insufficient to merit 6 seats are raised to exactly 6. The effect is that a substantial number of Member States receive exactly the minimum number 6 of seats. Strict proportionality is reinforced for the larger States where uncapped, and the prominence of the maximum constraint tends to increase. We viewed the decrease in allocations to smaller States as being in tension with the spirit of degressive proportionality.

3. Why do you cap the allocation to the largest Member State?

Answer: Because a maximum of 96 was included in our terms of reference.

4.

Why have you not allocated the non-base seats proportionally to population-size, over the entire range of Member States including the

	largest?
	Answer: Given the upper cap of 96, this approach would result in a substantially
	reduced Parliament-size, and incidentally, a large number of Member States would
	receive smaller allocations than under the Cambridge Compromise. There is some
	freedom in the choice of such a method. If one plots a straight line of seats against
	population, with the smallest and largest States receiving 6 and 96 seats
	respectively, the ensuing Parliament-size is 703 using standard rounding.
	Furthermore, 18 Member States receive smaller allocations than under the
	Cambridge Compromise.

6. How robust is the proposed method with respect to the possible accession of a large State?

  Answer: The proposed method is very robust in this regard. An acceding large Member State would attract a substantial allocation of seats. Fewer seats would be available to the other Member States, and this would tend to render inoperative the capping of the allocation to the largest Member States. This conclusion is supported by preliminary calculations based on current population figures.

7.

What happens when a new State accedes to the Union during a legislative period?

  Answer: The Parliament could be enlarged temporarily to accommodate the number of further seats allocated to the new State. If the divisor was d, and the new population p, this would amount to [b + p/d] seats (capped, if appropriate, at the maximum). For this purpose, a divisor should be determined and published at the time of every apportionment, and the number of seats for an acceding State settled within the Treaty of Accession. 17 Notes 1. The divisor 819 000 is such that the seat total is 751. (See Section 4.1). 2. The calculation is illustrated by the case of Sweden. With the divisor 819 000, Sweden gains 5 base seats plus 9 340 682/819 000 ≈ 11.4 further seats. The total is 16.4, which is rounded upwards to 17.

Duff, A. Mathematical Symposium, Cambridge,

28-29 January 2011. European Parliament, 2011.

Italics by the current author.

European Parliament Resolution on "Proposal to amend the Treaty provisions concerning the composition of the European Parliament" adopted on 11 October 2007 (INI/2007/2169).

Noted in reference[START_REF] Martínez-Aroza | Several methods for degressively proportional allocations. A case study[END_REF] at the end of the Report. See also references[START_REF] Pukelsheim | Putting citizens first: Representation and power in the European Union[END_REF][START_REF] Ramírez González | Degressive proportionality. Composition of the European Parliament. The parabolic method[END_REF][START_REF] Ramírez González | Degressively proportional methods for the allotment of the European Parliament seats amongst the EU Member States[END_REF].

European Parliament Resolution on "Proposal to amend the Treaty provisions concerning the composition of the European Parliament" adopted on 11 October 2007 (INI/2007/2169).The allocation between the EU Member States of the seats in the European Parliament

See references[START_REF] Słomczyński | On bounds for the allocation of seats in the European Parliament[END_REF][START_REF] Ramírez González | Degressive proportionality. Composition of the European Parliament. The parabolic method[END_REF][START_REF] Ramírez González | Degressively proportional methods for the allotment of the European Parliament seats amongst the EU Member States[END_REF] at the end of the Report.

See references[START_REF] Balinski | Fair Representation[END_REF][START_REF] Martínez-Aroza | Several methods for degressively proportional allocations. A case study[END_REF][START_REF] Pukelsheim | Putting citizens first: Representation and power in the European Union[END_REF].

See reference[START_REF] Balinski | Fair Representation[END_REF] for a survey and critique, and also reference[START_REF] Schuster | Seat biases of apportionment methods for proportional representation[END_REF].

After the Symposium was completed, one of our tentative conclusions was confirmed, namely that 5 + upwards results inevitably in the same allocation as 6 + downwards, and indeed as 5.5 + standard. See reference[START_REF] Zachariasen | On the relationship between base and rounding-method in divisorbased apportionment[END_REF].

As noted in Footnote 8, there is generally a range of values for the divisor that result in the same seat allocations. The divisor should be published at the time of an apportionment.

Population figures are taken from the Eurostat website (OJ 22.12.2010 L 338/47).

The divisor 835 000 is such that the seat total is 751.

If Iceland were to accede during the legislative period, Parliament could be temporarily enlarged by 5 + 1 = 6 seats (since 317 630/835 000 ≈ 0.4  1).20 Eurostat 1.1.2011.

This document was requested by the European Parliament

The divisor 844 000 is such that the seat total is 751.


Alejandro Perez
email: perez@cpt.univ-mrs.fr

Daniele Pranzetti
email: daniele.pranzetti@cpt.univ-mrs.fr

On the regularization of the constraints algebra of Quantum Gravity in 2 + 1 dimensions with non-vanishing cosmological constant

We use the mathematical framework of loop quantum gravity (LQG) to study the quantization of three dimensional (Riemannian) gravity with positive cosmological constant (Λ > 0). We show that the usual regularization techniques (successful in the Λ = 0 case and widely applied in four dimensional LQG) lead to a deformation of the classical constraint algebra (or anomaly) proportional to the local strength of the curvature squared. We argue that this is an unavoidable consequence of the non-local nature of generalized connections.

I. INTRODUCTION

Three dimensional quantum gravity can be defined from a number of different points of view. The first of these was the Ponzano-Regge [START_REF] Ponzano | Semi-classical limit of Racah coefficients[END_REF] model of quantum gravity on a triangulated 3-manifold which provides a quantization of Regge calculus. The Ponzano-Regge model is a state sum model for 3-dimensional euclidean quantum gravity without cosmological constant using the Lie group SU [START_REF] Turaev | sum invariants of 3-manifolds and quantum 6j-symbols[END_REF]. Let M be a triangulated compact 3-manifold, a state of the model is an assignment of an irreducible representation of SU (2) to each edge of the triangulation. For each state there is a certain weight, a real number. The weight is given by the local formula: W = interior edges (-1) 2j (2j + 1) interior triangles (-1) j1+j2+j3 tetrahedra j 1 j 2 j 3 j 4 j 5 j 6 ,

where the weight for a tetrahedron is a 6j-symbol and for each triangle the admissibility conditions are given by the requirement that j 1 + j 2 + j 3 ∈ Z, while j 1 + j 2 -j 3 ≥ 0, j 1 + j 3 -j 2 ≥ 0, and j 2 + j 3 -j 1 ≥ 0. The partition function, or state sum, is obtained by summing over all possible values of the spin on every edge in the interior of the manifold, subject to fixed values on the boundary:

Z = j1,j2,..jn W . (2)
Since the set of irreducible representations of SU( 2) is infinite, for some triangulations this gives a finite sum, whereas for some other triangulations this is a divergent infinite sum 1 .

In the late eighties, early nineties mathematicians were looking for manifold invariants with the hope that this could help to classify them and this led Turaev and Viro [START_REF] Turaev | sum invariants of 3-manifolds and quantum 6j-symbols[END_REF] to the definition of a state sum which, under many aspects, was very similar to that of Ponzano and Regge. The two main differences were first that Turaev and Viro explicitly showed their model to be triangulation independent and secondly they replaced the Lie group SU (2) with its quantum deformation U q SL(2). When the deformation parameter q is a root of unity, then there are only a finite number of irreducible representations, which means that the edge lengths are not summed up to infinite values, and the partition function is always well-defined. A very important consequence of this is that the answer obtained is finite, and so the model appears to be a regularized version of the Ponzano-Regge model.

In particular, for q = e iπ r , with integer r ≥ 3, the weight W q for a given assignment of spins is given by a formula analogous to that for Ponzano-Regge in which every factor depends on q. Crucially, the spins are limited to the range 0 ≤ j ≤ (r -2)/2, and there is an extra admissibility condition j 1 + j 2 + j 3 ≤ r -2 .

(

) 3
The 6j-symbol is replaced by a quantum 6j-symbol, and the factor for each edge by a quantum dimension. The Turaev-Viro model is defined by the partition function:

Z = N -v q (r-2)/2
j1,j2,..jn=0

W q , (4)
where N q is a constant depending on q, and v is the number of internal vertices. Since the sum is finite, this is always well-defined. Moreover, the partition function Z q depends only on the boundary triangulation, the boundary data, and the topology of the manifold. The natural question is then in what way the Turaev-Viro state sum is connected to Quantum Gravity? The answer, first exhibited by Witten [START_REF] Witten | 2+1)-Dimensional Gravity as an Exactly Soluble System[END_REF], and then rigorously proven by other authors ([4]- [START_REF] Turaev | Topology of shadows[END_REF]), was that the Turaev-Viro state sum is equivalent to a Feynman path integral with the Chern-Simons action for SU (2) k ⊗ SU (2) -k , where k is the level, and then the connection with gravity follows from the fact that the Chern-Simons action for this group product is related to the Einstein-Hilbert action for gravity with cosmological constant Λ ( [START_REF] Ooguri | Discrete and Continuum Approaches to Three-Dimensional Quantum Gravity[END_REF], [START_REF] Williams | Discrete quantum gravity: The Regge calculus approach[END_REF]) if k 2 = 4π 2 /Λ. Quantum groups also enter the quantization of Chern-Simons theory in the so-called combinatorial quantization approach ( [START_REF] Alekseev | Combinatorial quantization of the Hamiltonian Chern-Simons theory[END_REF]), where a quantum deformation of the structure group is introduced as an intermediate regularization.

From the perspective of loop quantum qravity (LQG) only the case of vanishing cosmological constant is clearly understood. The quantization is in this case a direct implementation of Dirac's quantization program for gauge systems. The basic unconstrained phase space variables are represented as operators in an auxiliary Hilbert space (or kinematical Hilbert space H kin spanned by spin network states) where the constraints are represented by regularized quantum operators. A nice feature of the regularization (which is both natural but also unavoidable in the context of LQG) is that it leads to regulated quantum constraints satisfying the appropriate quantum constraint algebra. There is no anomaly. This feature together with the background independent nature of the whole treatment allows for the removal of regulators and the definition of the physical Hilbert space and complete set of gauge invariant observables.

We give some additional details on the construction of the physical Hilbert space in the Λ = 0 case, as they will allow us to relate the results of LQG in this simple case with what we have mentioned above concerning other approaches. The physical inner product and the physical Hilbert space H phys of 2 + 1 gravity with Λ = 0 can be defined by introducing a regularization of the formal expression for the generalized projection operator into the kernel of curvature constraint ( [START_REF] Rovelli | The projector on physical states in loop quantum gravity[END_REF], [START_REF] Reisenberger | A left-handed simplicial action for euclidean general relativity[END_REF]):

P = " x∈Σ δ( F (A(x))" = D[N ]exp i Σ T r[N F (A)] . (5)
In [START_REF] Noui | Three dimensional loop quantum gravity: Physial scalar produt and spin foam models[END_REF] it has been shown how, introducing a regularization as an intermediate step for the quantization, this projector can be given a precise definition leading to a rigorous expression for the physical inner product of the theory which can be represented as a sum over spin foams whose amplitudes coincide with those of the Ponzano-Regge model. Moreover, the divergences mentioned above do not appear in the LQG treatment as the problematic triangulations (leading to infinities in the Ponzano-Regge state sum) simply do not contribute to physical transition amplitudes.

The previous discussion provides motivation to investigate the possible quantization, in the context of LQG, of the non vanishing cosmological constant case. If such program could be achieved one would expect to be able to understand the Turaev-Viro amplitudes as the physical transition amplitudes or physical inner product between kinematical spin network states. Unfortunately, at the moment there is little evidence supporting this expectation. Consequently, our work concentrates on the very basic starting point of the Dirac program: the study of the quantization of the constraints and their associated constraint algebra. Our results reveals a puzzling feature that, we argue, is proper to the nature of the kind of regularizations admitted by the LQG mathematical framework, namely the appearance of quantization anomalies. As we will show, the nice interplay between symmetry and the regularizing nature of the LQG representation of basic kinematical observables, present in the Λ = 0 case, is lost in the case of non vanishing cosmological constant. Thus, the generic appearance of anomalies in the quantum constraint algebra appears as a serious difficulty for the completion of the LQG program for this model.

The paper is organized as follows: In Section II we start with a brief discussion on the UV problem in LQG. The analysis of the constraint quantization and the associated algebra is presented in sections III-IV A.

II. UV DIVERGENCES, THEIR REGULARIZATION, AND AMBIGUITIES IN LQG

In standard background dependent QFT in order to make sense of products of operator valued distributions one has to provide a regularization prescription. Removing the regulator is a subtle task involving the tuning of certain terms in the Lagrangian (counter terms) that ensure finite results when the regulator is removed. However, any of these regularization procedures is intrinsically ambiguous. Ambiguities associated to the UV regularization allows to classify a theory as a renormalizable QFT if there are finitely many ambiguities which can be fixed by a finite number of renormalization conditions, i.e., one selects the suitable theory by appropriate tuning of the finite number of ambiguity parameters in order to match observations.

Removing UV divergences by a regularization procedure is intimately related to the appearance of ambiguities in the quantum theory. This problem is intrinsic to the formalism of QFT and is also present in the context of LQG although in a disguised way [START_REF] Perez | On the regularization ambiguities in loop quantum gravity[END_REF] 2 . In LQG one aims at the canonical quantization of gravity in the connection formulation. General relativity (Riemannian in 3d and both Riemannian or Lorentzian in 4d) admits an unconstrained phase space formulation in terms of an SU (2) connection A i a and its non Abelian conjugate electric field E a i . At the unconstrained level, the phase space is isomorphic to that of an SU (2) Yang-Mills theory. However, there is fundamental difference with Yang-Mills theory in that general relativity contains diffeomorphisms as part of the gauge group. This additional structure motivates the introduction of a representation of the phase space basic variables as operators in an auxiliary Hilbert space where diffeomorphisms are unitarily represented. This important feature is crucial for the implementation of diffeomorphism invariance (which in addition fixes uniquely-up to unitarily equivalence-the representation of the basic unconstrained (or kinematic) phase space variables as operators in a Hilbert space [17]). However, a prize to be paid with this choice of quantization is the impossibility of representing the connection A i a as a quantum operator but only its holonomy (or parallel transport) along one dimensional paths (the so called generalized connection). In this way the basic quantum object is not the configuration variable but a non local object constructed from it. As a consequence of this, local quantities entering the definition of the constraints (such as the curvature tensor F (A)) do not admit a direct quantum analog and must be replaced by expressions written in terms of holonomies or Wilson loops along extended paths. This process eliminates the potential UV divergences associated to non linearities in the constraints-as it provides a natural point-split-like regularization of local quantities-but generically introduces infinite dimensional ambiguities. As explained in more detail below, the reason for this is roughly the same that allows for infinitely many possible ways of defining a lattice action for lattice Yang-Mills theory when replacing the Yang-Mills connection by its parallel transport in the lattice action.

The fact that these non linear quantities require regularization is not surprising as it is a standard feature of quantum field theories and the difficulty associated to the definition of products of operator valued distributions at the same point. However, what is special of LQG is that its kinematical structure obliges one to use a regularization in terms of holonomies as these are the fundamental quantum observables in the auxiliary Hilbert space (the generalized connections). Even the connection itself (the classical configuration field) must be 'approximated' by a regulated version in terms of holonomies along infinitesimal paths. The fact that this process is ambiguous can be illustrated by the following simple example. Consider a one parameter family of contractible loops α around the point p ∈ Σ for > 0 such that the limit → 0 one has α → p. If we denote by W the holonomy of the connection A around the loop α , then the following is an infinite dimensional family of regularizations of the local curvature ab F i ab (A)| p :

f [τ i W ] = n/2 j=1/2 c j χj (τ i W ) j(j+1)(2j+1) 2 n/2 k=1/2 c k , (6)
for arbitrary coefficients c j and some positive integer n, and where τ i are the standard generators of the su(2) Lie algebra, and χ j (g) is the character of g ∈ SU (2) in the unitary irreducible representation of spin j. The fact that for arbitrary c j 's and n these are good regularizations of the curvature component ab F i ab (A)| p at the point p comes from the fact that for smooth configurations of the connection one has

f [τ i W ] = ab F i ab (A)| p + O( 2 )
, independently of the freedom in choosing the parameters of the family. Moreover, among the parameters of this family one should also consider the functional arbitrariness entering in defining the above homotopy of loops around p. This is a particular example that illustrates the nature of the ambiguities involved when replacing the local connection by its parallel transport.

This (infinite dimensional) regularization ambiguity associated to the need to replace connections by generalized connections are present in the LQG quantization of three dimensional gravity for Λ = 0. In fact there is an infinite dimensional set of quantizations of the constraints all of which are anomaly free at the regulated level. The requirement of the quantum constraints to reproduce the appropriate gauge symmetry algebra does not reduce the ambiguities in this case. However, this by itself does not necessarily mean that the quantum theory is ill defined: a detailed analysis [START_REF] Perez | On the regularization ambiguities in loop quantum gravity[END_REF] shows that when the regulator is removed in the construction of the generalized projection operator P (of equation ( 5)) the physical Hilbert space is uniquely determined: all physical quantities are independent of the regularization ambiguities.

A second source of ambiguities, intimately related to the replacement of connections with (group valued) generalized connections, is that the conjugate momenta E a i become non commutative [16]. This introduces additional ordering ambiguities in cases where one is confronted with the quantization of non linear functionals of the electric field. These ambiguities are not present in the quantization of three dimensional gravity with vanishing cosmological constant as, in that case, all the constraints are at most linear in E a i . However, the ordering ambiguity will arise in the Λ = 0 case where constraints are quadratic in E a i . We shall see that using the classical constraints algebra as a guiding principle the ordering ambiguities can indeed be reduced (see Section IV A).

Finally, there is the ambiguity associated to the details of the extended structures used for regulating the basic local quantities defining the constraints. For example, the shape of loops used in the definition of Wilson loops used to regulate the curvature strength, or the choice of co-dimension one surfaces used when the local E a i field are replaced by flux operators admitting quantization in the LQG framework. Although these ambiguities are not really relevant in the Λ = 0 case, they have a strong effect on the constraint algebra in the Λ = 0 case as well as in more general settings.

III. CANONICAL THREE DIMENSIONAL GRAVITY WITH Λ = 0

We are interested in (Riemannian) three dimensional gravity with cosmological constant in the first order formalism. The space-time M is a three dimensional oriented smooth manifold and the action is given by

S[e, ω] = M tr[e ∧ F (ω) + Λ 3 e ∧ e ∧ e] ( 7 )
where e is a su(2) Lie algebra valued 1-form, F (ω) is the curvature of the three dimensional connection ω and T r denotes a Killing form on su [START_REF] Turaev | sum invariants of 3-manifolds and quantum 6j-symbols[END_REF]. Assuming the space-time topology to be M = ΣR where Σ is a Riemann surface of arbitrary genus, the phase space is parametrized by the pull back to Σ of ω and e. In local coordinates we can express them in terms of the 2-dimensional connection A i a and the triad field E b j = bc e k c η jk where a = 1, 2 are space coordinate indices and i, j = 1, 2, 3 are su [START_REF] Turaev | sum invariants of 3-manifolds and quantum 6j-symbols[END_REF] indices and ab = -ba with 12 = 1 (similarly ab =ba with 12 = 1). The Poisson bracket among these variables is given by

{A i a (x), E b j (y)} = δ b a δ i j δ (2) (x, y) . (8)
Due to the underlying SU (2) and diffeomorphism gauge invariance the phase space variables are not independent and satisfy the following set of first class constraints. The first one is the analog of the familiar Gauss law of Yang-Mills theory, namely

G i ≡ D a E a i = 0 , (9)
where D a is the covariant derivative with respect to the connection A. The constraint ( 9) is called the Gauss constraint.

It encodes the condition that the connection be torsion-less and it generates infinitesimal SU (2) gauge transformation. The second constraint reads

C i = ab (F i ab (A) + Λ i jk e j a e k b ) = 0 = ab F i ab (A) + Λ cd ijk E c j E d k = 0 , (10)
where in the first line we have written the constraint in terms of the triad field, while in the second line we have used the electric field. This second set of first class constraints is associated to the diffeomorphism invariance of three dimensional gravity. In order to exhibit the underlying (infinite dimensional) gauge symmetry Lie algebra it is convenient to smear the constraints ( 10) and ( 9) with arbitrary test fields α and N , which we assume not depending on the phase space variables3 , they read:

G(α) = Σ α i G i = Σ α i D a E a i = 0 (11)
and

C(N ) = Σ N i C i = Σ N i (F i (A) + Λ ijk E j E k ) = 0 . ( 12
)
The constraints algebra is then

{C(N ), C(M )} = Λ G([N, M ]) {G(α), G(β)} = G([α, β]) {C(N ), G(α)} = C([N, α]), (13)
where [a, b] i = i jk a j b k is the commutator of su [START_REF] Turaev | sum invariants of 3-manifolds and quantum 6j-symbols[END_REF]. For future use it will be convenient to split the constraint C(N ) as

C(N ) = F (N ) + E(ΛN ), (14)
where

F (N ) = Σ N i (F i (A)), E (ΛN ) = Σ Λ ijk N i E j E k . (15)
A. Canonical Quantization

The canonical quantization of the kinematics (i.e. the definition of the auxiliary Hilbert space where the constraints are to be quantized) is well understood. Details can be found in [19]. Following Dirac's quantization procedure one first finds a representation of the basic variables in an auxiliary Hilbert space H aux . The key ingredient is the background independent construction of this auxiliary Hilbert space. The main input is to replace functionals of the connection (taken as configuration variables) by functionals of holonomies along paths (the so-called generalized connections) γ ⊂ Σ: these are the basic excitations in terms of which the Hilbert space is constructed. Given a connection A and a path γ, one defines the holonomy h γ [A] by

h γ [A] = P exp γ A . ( 16
)
The conjugate momentum (densitized triad) E a i field is associated to its flux across co-dimension one surfaces. One promotes these basic variables to operators acting on an auxiliary Hilbert space where constraints are to be represented by quantum operators satisfying the operator quantum analog of [START_REF] Noui | Three dimensional loop quantum gravity: Physial scalar produt and spin foam models[END_REF].

The auxiliary Hilbert space is defined by the Cauchy completion of the space of cylindrical functionals Cyl, on the space of (generalized) connections Ā4 . The space Cyl is defined as follows: any element of Cyl, Ψ Γ,f [A], is a functional of A labeled by a finite graph Γ ⊂ Σ and a continuous function f : SU (2) N (Γ) → C where N (Γ) is the number of links of the graph Γ. Such a functional is defined as follows

Ψ Γ,f [A] = f (h γ1 [A], • • • , h γ N (Γ) [A]) (17)
where h γi [A] is the holonomy along the link γ i of the graph Γ. If one considers a new graph Γ such that Γ ⊂ Γ , then any cylindrical function Ψ Γ,f [A] defined on Γ can be promoted to a cylindrical function Ψ Γ ,f [A] defined on Γ in a natural way [20]. Given any two cylindrical functions Ψ Γ1,f [A] and Ψ Γ2,g [A], their inner product is defined by the Ashtekar-Lewandowski measure

< Ψ Γ1,f , Ψ Γ2,g >≡ µ AL (Ψ Γ1,f [A]Ψ Γ2,g [A]) = = N Γ 12 i=1 dh i f (h γ1 , • • • , h γ N (Γ 12 ) )g(h γ1 , • • • , h γ N (Γ 12 ) ) ( 1 8 )
where dh i corresponds to the invariant SU (2)-Haar measure, Γ 12 ⊂ Σ is a graph containing both Γ 1 and Γ 2 , and we have used the same notation f (resp. g) to denote the extension of the function f (resp. g) on the graph Γ 12 . The auxiliary Hilbert space H aux is defined as the Cauchy completion of Cyl under (18). The (generalized) connection is quantized by promoting the holonomy (16) to an operator acting by multiplication in H aux as follows:

h γ [A]Ψ[A] = h γ [A]Ψ[A] . ( 19
)
The triad is associated with operators in H aux defining the flux of electric field across one dimensional lines. Namely, for a one dimensional path η and a smearing field α : Σ → su(2) we define

E(η, α) ≡ α i Êa i dx b ab . (20)
The associated quantum operator in H aux can be defined from its action on holonomies. More precisely one has

E(η, α) h γ [A] = i 2 p o ηγ αh γ [A] (for η target of γ) o ηγ h γ [A]α (for η source of γ) , (21)
where the curve γ is assumed to have one of its endpoints at η, o ηγ = ±1 is the sign of the orientation of the pair of oriented curves in the order (η, γ), and where p = G is the Planck length in three dimensions (the action vanishes if the curves are tangential to each other). In terms of the triad operator we can construct geometric operators corresponding to the area of regions in Σ or the length of curves. The operators (19), and (21) are the basic extended variables in terms of which we shall regularize and quantize the constraints [START_REF] Reisenberger | A left-handed simplicial action for euclidean general relativity[END_REF] and [START_REF] Rovelli | The projector on physical states in loop quantum gravity[END_REF]. So far we have not specified the space of graphs that we are considering as this will also be part of the regularization procedure studied in the following section.

IV. DISCRETIZATION PRESCRIPTION

According to the canonical quantization program, we now need to introduce a quantization of the smeared constraints [START_REF] Reisenberger | A left-handed simplicial action for euclidean general relativity[END_REF], and (12) preserving the gauge symmetry algebra at the quantum level. As we mentioned in previous sections, before quantizing it is necessary to translate the classical variables entering the definition of the constraints in terms of holonomies of the connection and fluxes of the electric field. In order to do that we need to define a discrete structure on top of which we can construct these extended variables. We do so by introducing an arbitrary finite cellular decomposition C Σ of Σ. We denote n the number of plaquettes (2-cells) which from now on will be denoted by the index p ∈ C Σ . We assume the plaquettes to be squares with edges (1-cells denoted e ∈ C Σ ) of length ε in a local coordinate system. It will also be necessary to use the dual complex C Σ * with its dual plaquettes p * ∈ C Σ * and edges e * ∈ C Σ * (see Fig. 2). Both cellular decompositions inherit the orientation from the orientation of Σ. The cellular decomposition defines the regulating structure. We now need to write the classical constraints in terms of extended variables in such a way that the naive continuum limit is satisfied. Namely, it is necessary that (for smooth field configurations) the regulated classical constraints become the classical constraints in the limit → 0 (or equivalently n → ∞).

Consequently, the phase space variables E a i and A i a are discretized as follows: the local connection A i a field is now replaced by the assignment of group elements h(e) = P exp(e A) ∈ SU (2) to the set of edges e ∈ C Σ . We discretize the triad field E a i by assigning to each dual 1-cell e * the su( 2) element E i (e * ) ≡ e * ab E b i (x)dx a , i.e. the flux of electric field across the dual edge e * . With this decomposition of Σ we can write the regularized versions of the constraints ( 11) and ( 12) as:

G R [α] = p * ∈CΣ * tr[α p * G p * ] = 0 (22) and C R [N ] = p∈CΣ tr[N p C p ], (23)
where G p * and C p are explicitly defined below. Finally, we must define the set of states to be considered when studying the regularized constraint algebra. The allowed states will be a subset Cyl(C Σ ) ⊂ Cyl consisting of all cylindrical functions whose underlying graph is contained in the one-skeleton of C Σ . In other words, the allowed graphs must consist of collections of 1-cells e ∈ C Σ .

A. Regularized constraints algebra

With the prescription introduced in the previous section we are now ready to compute the discrete version of the algebra [START_REF] Noui | Three dimensional loop quantum gravity: Physial scalar produt and spin foam models[END_REF]. Let us start with the sub-algebra of the Gauss constraint. Due to the simplicity of the action of SU (2) gauge transformations on holonomies it is straightforward to obtain expressions for the regularized constraints that are anomaly free as far as it concerns the subalgebra of SU (2) gauge transformations. Thus the Gauss constraints are more simply quantized by concentrating on their exponentiated versions. Namely, instead of writing the infinitesimal generator of SU (2) gauge transformations as a self-adjoint operator acting on H aux it is simpler to directly construct an unitary operator generating finite SU (2) gauge transformations. This follows from the fact that under a gauge transformation g : Σ → SU (2) the generalized connection transforms according to

h γ [A] -→ g -1 t h γ [A]g s ( 24
)
where g s = g(x s ), g t = g(x t ) ∈ SU (2) are the value of the gauge transformation at the source and target points of γ respectively. Finite SU (2) gauge transformations are then represented by the unitary operator U G (g) whose action on Ψ Γ,f ∈ Cyl is

U G (g) Ψ Γ,f ({h γi }) ≡ Ψ Γ,f ({g -1 t i h γi g s i }). ( 25
)
The requirement that the quantization of the constraints satisfy the quantum counterpart of (13) translates into the following equations for the unitary generators

U G (g 1 )U G (g 2 ) = U G (g 2 g 1 ). ( 26
)
The previous equations is satisfied by our definition and hence the canonical quantization of the Gauss constraint presented here is anomaly free. Given α : Σ → su(2) one can compute the infinitesimal generator G(α) from the previous line as

G(α) = -i d dt U G (exp(tα))| t=0 . (27)
Now the previous definition translates into the graphical action shown below which can be directly obtained by concentrating on a single graph node and formally expanding to first order in α equation ( 25) where one has replaced g = 1 + α. This allows us to define the action of Tr[α p * G p * ] appearing in (22) where we have regularized the Gauss constraint with a sum over dual plaquettes. In fact, since each dual plaquette correspond to a single node of the cellular decomposition, we can define the action of Tr[α p * G p * ] on the node (to which the dual plaquette p * corresponds) which is target of four holonomies using the following graphical notation: where we have omitted the index p * for the smearing field in the figure. Notice that from (21) and the last line, the action of the Gauss constraint on the node can be interpreted as the flux operator E(C, α) across an infinitesimal circle C (a zero area circle) centered at the node which is reminiscent of the well known geometric interpretation of the Gauss law in the Abelian case such as electromagnetism.

Tr[α p * G p * ] h 1 ⊗ h 2 ⊗ h 3 ⊗ h 4 = ( 2
The other cases corresponding to all possible orientations of the four holonomies can be obtained from (21) or from (27) in a similar fashion. The action of the Gauss constraint on arbitrary elements of Cyl(C Σ ) can be obtained from the previous equations by the standard rules of differentiation of functions of finitely many copies of SU (2).

From the action of the Gauss constraint in a given dual plaquette above it is immediate to see how the Gauss constraint sub-algebra is preserved also at the quantum level without anomalies, as we expect from equations ( 26) and (27). Infact, due to the local action of the Gauss constraint it is sufficient to concentrate on the action of the commutator [G R (α), G R (β)] on a single node. At the regularized level this means that only the dual plaquette around the given node is really relevant and so, among all the terms in the commutator due to the sum over dual plaquettes in the definition of G R , only one gives a non vanishing contribution, namely:

[G R (α), G R (β)] Ψ = [T r[α p * G p * ], T r[β p * G p * ]] h 1 ⊗ h 2 ⊗ h 3 ⊗ h 4 = = ([α p * , β p * ] h 1 ) ⊗ h 2 ⊗ h 3 ⊗ h 4 + h 1 ⊗ ([α p * , β p * ] h 2 ) ⊗ h 3 ⊗ h 4 + + h 1 ⊗ h 2 ⊗ ([α p * , β p * ] h 3 ) ⊗ h 4 + h 1 ⊗ h 2 ⊗ h 3 ⊗ ([α p * , β p * ] h 4 ) = = G R ([α, β]) Ψ . (29)
Hence, the quantum Gauss constraints reproduce the correct commutator algebra and are thus anomaly free as far as the SU (2) sub algebra is concerned. This property is related to the well known, and extremely useful fact in the context of lattice gauge theories, that discretization does not break the Yang-Mills gauge symmetry.

We shift now the attention to the definition of the regulated constraints associated with (23). According to ( 14) and ( 15), we can write the regulated constraint as that the latter expression converges to the local curvature of the connection A (for smooth configurations of A) in the limit → 0. To this end we define the holonomies g e ∈ SU (2) associated to each oriented edge e ∈ C Σ . In terms of this definition the holonomy around a single plaquette

C R [N ] = F R [N ] + E R [ΛN ] ( 3
p ∈ C Σ becomes W p = g e p 1 • • • g e p 4
where here e p i for i = 1, • • • , 4 are the corresponding edges bounding the plaquette of interest. Finally using that (for smooth configurations)

W p [A] = 1 + ε 2 F p [A] + O(ε 2 ) a natural candidate for regulated F [N ] is F R [N ] = p F [N p ] = p tr[N p (W p )]. ( 31
)
Notice that the previous regularization corresponds to the choice n = 1 at each plaquette in the general formula given in equation [START_REF] Roberts | Skein theory and Turaev-Viro invariants[END_REF]. At the end of this paper we will discuss the generalization of our analysis to arbitrary regularizations. We will argue that the main result of this work is indeed generic and thus independent of the choice made here for the sake of simplicity. The regularization of E[ΛN ] is more subtle as one needs to replace the classical smooth field E a i by extended flux operators which, according to (21), act as 'grasping' operators. An important observation, that follows from the action of the regulated Gauss constraint, is that the grasping operators must act at the endpoints of the edge holonomies in order to avoid inconsistency. Therefore, instead of smearing the E-field along the edges of the usual dual cellular decomposition η ∈ C Σ we need to work with the flux of E-field associated with the shifted edges depicted on the right of Figure 2. By abuse of notation we shall keep denoting the shifted edges η ∈ C Σ . We write the regulated quantity corresponding to E[ΛN ] as

E R [ΛN ] = p E[ΛN p ], (32)
where (concentrating on the shadowed plaquette in Figure 2)

E[ΛN p ] = Λ ijk N i p (E(η 1 , τ j )E(η 2 , τ k )) + Λ ijk N i p (E(η 2 , τ j )E(η 3 , τ k )) + Λ ijk N i p (E(η 3 , τ j )E(η 4 , τ k )) + Λ ijk N i p (E(η 4 , τ j )E(η 1 , τ k )) , (33)
where η i ∈ C Σ are the four shifted edges shown in the figure that are dual to the shadowed plaquette p ∈ C Σ , and the operators E(η, α) are defined in (21). Let us start with the quantum version of

{C[N ], C[M ]} = ΛG({N, M }) . ( 34
)
In order to compute the discrete analog of (34) it is sufficient to derive the action of the r.h.s and l.h.s of this commutation relation around a single vertex, more precisely around a generic non SU (2)-invariant state at a vertex of the cellular decomposition, since both sides of the commutator (34) vanish for N p , M p belonging to two plaquettes p and p that do not share a common vertex. Therefore the quantum version of eq. ( 34) has to be computed for a state of the form shown in Fig. 3 and the sums over plaquettes inside the expressions of C R [N ] and C R [M ] give not vanishing contributions only for a finite number of plaquettes, precisely the ones around the given vertex.

Let us call this non SU (2) invariant state Ψ and write the discrete action of the l.h.s of (34) on Ψ:

[C R [N ], C R [M ]] Ψ = = [[C(N 1 ), C(M 1 )] + [C(N 1 ), C(M 2 )] + [C(N 1 ), C(M 3 )] + [C(N 1 ), C(M 4 )] + +[C(N 2 ), C(M 2 )] + [C(N 2 ), C(M 1 )] + [C(N 2 ), C(M 3 )] + [C(N 2 ), C(M 4 )] + +[C(N 3 ), C(M 3 )] + [C(N 3 ), C(M 1 )] + [C(N 3 ), C(M 2 )] + [C(N 3 ), C(M 4 )] + +[C(N 4 ), C(M 4 )] + [C(N 4 ), C(M 1 )] + [C(N 4 ), C(M 2 )] + [C(N 4 ), C(M 3 )]] Ψ. (35)
We have now to choose also a convention of signs for the action of each E field on holonomies, we pick the one shown in Fig. 4, where the dashed arrows represent the E field and the black ones holonomies. With these conventions we are now ready to compute the action (35). Let us start with the terms involving the commutator of regulated quantities at the same plaquette. For the plaquette 1 we have:

[C(N 1 ), C(M 1 )] Ψ = [ F [N 1 ], E[ΛM 1 ] + E[ΛN 1 ], F [M 1 ] Ψ = = 1 2 M 1 1 N + 1 4 M 1 1 N + 1 4 M 1 1 N + + 1 2 M 1 1 N + 1 2 M 1 1 N + - 1 2 M 1 1 N - 1 2 M 1 1 N -(same diagrams switching N 1 ↔ M 1 ) ,
where we have used the Leibnitz rule when the flux operators act on products of holonomies, and we have symmetrized the local action of the graspings at the node. More precisely, when the product of E-fields appearing in E R (ΛN ) act at a given point there is, on the one hand, a factor ordering ambiguity associated to the non-commutativity of the flux operators, and, on the other hand, an ambiguity associated to the place at which the smearing field N is contracted with the Wilson line appearing in the regularization of F R (N ). By symmetrization we mean that every time we find this ambiguity we simply sum over all ordering possibilities with equal weights and divide by their number. This is the origin of the weight factors in the previous equation.

It is easy to see that the first three diagrams in the previous equation give an amplitude that is symmetric under N 1 ↔ M 1 and so are canceled. This is a consequence of the symmetrized action of graspings acting at the same point discussed above. It is important to notice that if we did not make this choice of (symmetrized) action we would immediately get contributions to the commutator that do not vanish when acting on gauge invariant states! The avoidance of this obvious anomaly justifies that above choice which is in addition very natural. Thus only the contributions of the last four diagrams remain. The result is:

[C(N 1 ), C(M 1 )] Ψ = = Λ (M 1 α N 1 β -N 1 α M 1 β ) 8 (tr[{τ j , τ β }W 1 ] h 1 αjk τ k h 2 ⊗ h 3 ⊗ h 4 -tr[{τ k , τ β }W 1 ] αjk τ j h 1 ⊗ h 2 ⊗ h 3 ⊗ h 4 ) = Λ 8 tr[W 1 ] (([N 1 , M 1 ]h 1 ) ⊗ h 2 ⊗ h 3 ⊗ h 4 + h 1 ⊗ ([N 1 , M 1 ]h 2 ) ⊗ h 3 ⊗ h 4 ) , (36)
where we have used {τ i , τ j } = -1 2 δ ij in the last line. An analogous calculation shows that:

[C(N 2 ), C(M 2 )] Ψ = Λ 8 tr[W 2 ] (h 1 ⊗ ([N 2 , M 2 ]h 2 ) ⊗ h 3 ⊗ h 4 + h 1 ⊗ h 2 ⊗ ([N 2 , M 2 ]h 3 ) ⊗ h 4 ) , (37)
[C(N 3 ), C(M 3 )] Ψ = Λ 8 tr[W 3 ] (h 1 ⊗ h 2 ⊗ ([N 3 , M 3 ]h 3 ) ⊗ h 4 + h 1 ⊗ h 2 ⊗ h 3 ⊗ ([N 3 , M 3 ]h 4 )) , (38)
[C(N 4 ), C(M 4 )] Ψ = Λ 8 tr[W 4 ] (([N 4 , M 4 ]h 1 ) ⊗ h 2 ⊗ h 3 ⊗ h 4 + h 1 ⊗ h 2 ⊗ h 3 ⊗ ([N 4 , M 4 ]h 4 )) . (39)
We have now to compute the commutators among constraints in neighboring plaquettes. Let us start with [C(N 1 ), C(M 2 )]:

[C(N 1 ), C(M 2 )] Ψ = [ F [N 1 ], E[ΛM 2 ] + E[ΛN 1 ], F [M 2 ] Ψ = - 2 1 M N - 2 1 M N - 2 M N 1 - M N 2 1 = - Λ 8 (N 1 α M 2 β tr[{τ k , τ α }W 1 ] h 1 ⊗ h 2 ⊗ βjk τ j h 3 ⊗ h 4 + M 2 α N 1 β tr[{τ j , τ α }W 2 ] βjk τ k h 1 ⊗ h 2 ⊗ h 3 ⊗ h 4 ) = Λ 16 tr[W 1 ] h 1 ⊗ h 2 ⊗ [N 1 , M 2 ]h 3 ⊗ h 4 + Λ 16 tr[W 2 ] [N 1 , M 2 ]h 1 ⊗ h 2 ⊗ h 3 ⊗ h 4 ).
An analogous computation shows that the commutators among constraints in plaquettes which just share a vertex ((1, 3) and (2, 4)) vanish, while for the other contributions we have:

[C(N 1 ), C(M 4 )] Ψ = Λ 16 tr[W 1 ] h 1 ⊗ h 2 ⊗ h 3 ⊗ [N 1 , M 4 ]h 4 + Λ 16 tr[W 4 ] h 1 ⊗ [N 1 , M 4 ]h 2 ⊗ h 3 ⊗ h 4 , [C(N 2 ), C(M 1 )] Ψ = Λ 16 tr[W 1 ]h 1 ⊗ h 2 ⊗ [N 2 , M 1 ] h 3 ⊗ h 4 + Λ 16 tr[W 2 ] [N 2 , M 1 ] h 1 ⊗ h 2 ⊗ h 3 ⊗ h 4 , [C(N 2 ), C(M 3 )] Ψ = Λ 16 tr[W 3 ] h 1 ⊗ [N 2 , M 3 ] h 2 ⊗ h 3 ⊗ h 4 + Λ 16 tr[W 2 ] h 1 ⊗ h 2 ⊗ h 3 ⊗ [N 2 , M 3 ] h 4 , [C(N 3 ), C(M 2 )] Ψ = Λ 16 tr[W 3 ] h 1 ⊗ [N 3 , M 2 ] h 2 ⊗ h 3 ⊗ h 4 + Λ 16 tr[W 2 ] h 1 ⊗ h 2 ⊗ h 3 ⊗ [N 3 , M 2 ] h 4 , [C(N 3 ), C(M 4 )] Ψ = Λ 16 tr[W 3 ] [N 3 , M 4 ] h 1 ⊗ h 2 ⊗ h 3 ⊗ h 4 + Λ 16 tr[W 4 ] h 1 ⊗ h 2 ⊗ [N 3 , M 4 ] h 3 ⊗ h 4 , [C(N 4 ), C(M 3 )] Ψ = Λ 16 tr[W 3 ] [N 4 , M 3 ] h 1 ⊗ h 2 ⊗ h 3 ⊗ h 4 + Λ 16 tr[W 4 ] h 1 ⊗ h 2 ⊗ [N 4 , M 3 ] h 3 ⊗ h 4 , [C(N 4 ), C(M 1 )] Ψ = Λ 16 tr[W 1 ] h 1 ⊗ h 2 ⊗ h 3 ⊗ [N 4 , M 1 ] h 4 + Λ 16 tr[W 4 ] h 1 ⊗ [N 4 , M 1 ] h 2 ⊗ h 3 ⊗ h 4 .
The fact that four different values of the smearing field enter the action of the regularized constraint at a single point is an artifact of the discretization since the values of the smearing field N , being discretized, changes from plaquette to plaquete in the four plaquettes surrounding the node of interest. The 'multi-valuedness' of smearing fields created by the discretization makes more obscure the meaning of the previous equations and should not affect the final result.

A more transparent interpretation of the equations is obtained by coarse graining the discrete smearing fields (the equalities that are produced by this means will be denoted by the symbol = c.g. , where c.g. stands here for coarse graining). Therefore we can evaluate the action (35) by the appropriate coarse-graining procedure 4 ≡ M and see that at the quantum level the algebra of the curvature constraint with itself presents an anomaly, namely:

N 1 = N 2 = N 3 = N 4 ≡ N , M 1 = M 2 = M 3 = M
[C R [N ], C R [M ]] Ψ = c.g. = c.g. Λ 8 (tr[W 1 ] + tr[W 2 ] + tr[W 3 ] + tr[W 4 ]) [([N, M ]h 1 ) ⊗ h 2 ⊗ h 3 ⊗ h 4 + + h 1 ⊗ ([N, M ]h 2 ) ⊗ h 3 ⊗ h 4 + h 1 ⊗ h 2 ⊗ ([N, M ]h 3 ) ⊗ h 4 + h 1 ⊗ h 2 ⊗ h 3 ⊗ ([N, M ]h 4 )] = = Λ G R ( tr[W ] 2 [N, M ]) Ψ , (40)
where

G R ( tr[W ] 2 [N, M ]) = v ( p∈v tr[W p ]/8)( p∈v G p [N, M ]).
Notice that the classical analog reduces-once the regulator is removed ( → 0) for smooth field configurations and due to the fact that Tr[W ]/2 → 1-to the correct algebra:

{C[N ], C[M ]} = Λ G({N, M }).
The presence of tr[W ] in the regulated algebra will not disappear in the quantum case: this is a genuine quantization anomaly introduced by our regularization method.

We can now compute the action on the state Ψ of the commutator between the scalar and the Gauss constraints and verify that the relation

[C R (N ), G R (M )] = C R ([N, M ]) (41)
holds. In this case when acting with the lhs of (41) on a single node, for the curvature constraint there will be four relevant terms in the sum over plaquettes entering the definition of C R (N ) (the terms related to the four plaquettes around the given node), each with a different value of the smearing field N p , while for the Gauss constraint there is only one relevant term (the one related to the dual plaquette around the given node) and we will denote M p * the value of the smearing field associated to this term. Thereby, for the lhs of (41) there are four commutators to compute and for each of them there are two terms: one related to the F R part of C R and one to the E R part.

Since the F (N p ), for all the four plaquettes (p = 1, 2, 3, 4) around the given node, depend only on holonomies the Leibnitz rule applied to the their commutators with the Gauss constraint in the dual plaquette around the given node

G[M p * ] implies directly [F R (N ), G R (M )] Ψ = F R ([N, M ]) Ψ . ( 42
)
This implies that we can simply concentrate on the commutator of E R (ΛN ) with the Gauss constraint.

Starting with the plaquette 1, we have, omitting the index p * for the smearing field M p * :

[E(ΛN 1 ), G(M )] Ψ = = Λ N 1 M -Λ N 1 M + +Λ N 1 M -Λ N 1 M (43)
Simple algebra yields

[E(ΛN 1 ), G(M )] Ψ = Λ 4 N 1 h 1 ⊗ Mh 2 ⊗ h 3 ⊗ h 4 - Λ 4 Mh 1 ⊗ N 1 h 2 ⊗ h 3 ⊗ h 4 , (44)
In an analogous way one can compute:

[E(ΛN 2 ), G(M )] Ψ = Λ 4 h 1 ⊗ N 2 h 2 ⊗ Mh 3 ⊗ h 4 - Λ 4 h 1 ⊗ Mh 2 ⊗ N 2 h 3 ⊗ h 4 , (45) 13
[E(ΛN 3 ), G(M )] Ψ = Λ 4 h 1 ⊗ h 2 ⊗ N 3 h 3 ⊗ Mh 4 - Λ 4 h 1 ⊗ h 2 ⊗ Mh 3 ⊗ N 3 h 4 , (46)
[E(ΛN 4 ), G(M )] Ψ = Λ 4 Mh 1 ⊗ h 2 ⊗ h 3 ⊗ N 4 h 4 - Λ 4 N 4 h 1 ⊗ h 2 ⊗ h 3 ⊗ Mh 4 . (47)
We can now sum up all the contribution and we get:

[E R (ΛN ), G R (M )] Ψ = + Λ 4 N 1 h 1 ⊗ Mh 2 ⊗ h 3 ⊗ h 4 - Λ 4 Mh 1 ⊗ N 1 h 2 ⊗ h 3 ⊗ h 4 + + Λ 4 h 1 ⊗ N 2 h 2 ⊗ Mh 3 ⊗ h 4 - Λ 4 h 1 ⊗ Mh 2 ⊗ N 2 h 3 ⊗ h 4 + + Λ 4 h 1 ⊗ h 2 ⊗ N 3 h 3 ⊗ Mh 4 - Λ 4 h 1 ⊗ h 2 ⊗ Mh 3 ⊗ N 3 h 4 + + Λ 4 Mh 1 ⊗ h 2 ⊗ h 3 ⊗ N 4 h 4 - Λ 4 N 4 h 1 ⊗ h 2 ⊗ h 3 ⊗ Mh 4 . (48)
We have now to compute the r.h.s of (41). Starting again from the plaquette 1,we get:

E(Λ[N 1 , M]) Ψ = Λ 4 N 1 i M j ijk k rs τ r h 1 ⊗ τ s h 2 ⊗ h 3 ⊗ h 4 = = Λ 4 N 1 h 1 ⊗ Mh 2 ⊗ h 3 ⊗ h 4 - Λ 4 Mh 1 ⊗ N 1 h 2 ⊗ h 3 ⊗ h 4 . (49)
An analogous computation shows that:

E(Λ[N 2 , M]) Ψ = Λ 4 h 1 ⊗ N 2 h 2 ⊗ Mh 3 ⊗ h 4 - Λ 4 h 1 ⊗ Mh 2 ⊗ N 2 h 3 ⊗ h 4 , (50)
E(Λ[N 3 , M]) Ψ = Λ 4 h 1 ⊗ h 2 ⊗ N 3 h 3 ⊗ Mh 4 - Λ 4 h 1 ⊗ h 2 ⊗ Mh 3 ⊗ N 3 h 4 , (51)
E(Λ[N 4 , M]) Ψ = Λ 4 Mh 1 ⊗ h 2 ⊗ h 3 ⊗ N 4 h 4 - Λ 4 N 4 h 1 ⊗ h 2 ⊗ h 3 ⊗ Mh 4 . (52)
Thus, summing all the four contributions to the action of the r.h.s of (41) on Ψ, we get:

E R (Λ[N, M ]) Ψ = + Λ 4 N 1 h 1 ⊗ Mh 2 ⊗ h 3 ⊗ h 4 - Λ 4 Mh 1 ⊗ N 1 h 2 ⊗ h 3 ⊗ h 4 + + Λ 4 h 1 ⊗ N 2 h 2 ⊗ Mh 3 ⊗ h 4 - Λ 4 h 1 ⊗ Mh 2 ⊗ N 2 h 3 ⊗ h 4 + + Λ 4 h 1 ⊗ h 2 ⊗ N 3 h 3 ⊗ Mh 4 - Λ 4 h 1 ⊗ h 2 ⊗ Mh 3 ⊗ N 3 h 4 + + Λ 4 Mh 1 ⊗ h 2 ⊗ h 3 ⊗ N 4 h 4 - Λ 4 N 4 h 1 ⊗ h 2 ⊗ h 3 ⊗ Mh 4 . (53)
From eq.( 48) and ( 53)

we conclude that [E R (ΛN ), G R (M )] = E R (Λ[N, M ]) which combined with (42) yields [C R (N ), G R (M )] = C R ([N, M ]), (54)
as expected.

To summarize, we have seen that the quantum version of the constraints algebra (13) of gravity in 2 + 1 dimensions with non-vanishing cosmological constant reads:

[C R (N ), C R (M )] = c.g. Λ G R ( tr[W ] 2 [N, M ]) [G R (N ), G R (M )] = G R ([N, M ]) [C R (N ), G R (M )] = C R ([N, M ]) (55)
Relations (55) show that just the commutators among the scalar constraints present an anomaly due to the presence of the factor tr[W ] 2 in the smearing of the Gauss law on the r.h.s. of the first equation. We see that the regularization does not break the internal gauge group SU (2); however, it does break the part of the gauge symmetry group related to spacetime diffeomorphisms. Notice also that the anomaly is a genuine quantum effect. If we had computed the Poisson algebra of regularized constraints instead we would have found basically the same result (where commutators are replaced by Poisson brackets). However, in that case the problematic factor disappears in the continuum limit as 4). It is useful to rewrite the result in terms of a constraint reparametrization that exhibits more clearly the compact character of the gauge symmetry group of Riemannian 2+1 gravity with positive cosmological constant. At the classical level we know that the algebra (13) generates a local su(2) ⊗ su( 2) symmetry, as one can see immediately by defining F ± (N ) = C(N ) ± √ Λ G(N ) and computing:

tr[W ] 2 = 1 + O(
{F ± (N ), F ± (M )} = ±2 √ Λ F ± ([N, M ]) {F ± (N ), F ∓ (M )} = 0 . (56)
In order to see what modification to this local symmetry appears at the quantum level we define the operators

F R± (N ) = C R (N ) ± √ Λ G R (N ) and then compute the commutators [F R± (N ), F R± (M )] and [F R± (N ), F R∓ (M )]. The result is [F R± (N ), F R± (M )] = ±2 √ Λ F R± ([N, M ]) + Λ 2 G R ((tr[W ] -2)[N, M ]) (57)
and

[F R± (N ), F R∓ (M )] = Λ 2 G R ((tr[W ] -2)[N, M ]). (58)
V. DISCUSSION We have precisely computed the regulated quantum constraint algebra for Riemannian 2+1 gravity in the connection formulation of LQG. The nature of the kinematical Hilbert space of LQG imposes the need of a regularization in the definition of the quantum constraints. This is so, due to the fact that only the holonomy (and not the local connection) and conjugate fluxes (instead of the local triad) can be quantized in the kinematical LQG representation: the fundamental operators representing phase space variables are of extended nature. We have studied a simple regularization of the constraints that leads to the correct naive continuum limit of both the constraints and their Poisson algebra. However, when these regulated constraints are quantized the regulated quantum constraint algebra becomes anomalous due to the presence of plaquette loop operators that do not go away in the refinement limit of the regulating lattice 5 .

There is a large freedom in the choice of regularization of the constraints. One source of ambiguity comes from the fact that the flux operators corresponding to the local triad do not commute in the quantum theory. As the classical constraints of the theory considered here are non linear in the triad field-due to the presence of a non vanishing cosmological constant-this non-commutativity introduces factor ordering ambiguities. We have shown that the requirement that the quantum constraint algebra be satisfied completely eliminates this source of ambiguity. More precisely, the caculations leading to eq. ( 36) and finally to eq. ( 40) show that if we do not symmetrize over all possible orderings then [C R (N ), C R (M )] Ψ = 0 when acting on states Ψ annihilated by the quantum Gauss constraints.

Another source of ambiguity can be found in the choice of paths along which the fluxes and holonomies used in the regularization are defined. For simplicity we have chosen here a square lattice C Σ and its dual. However, it should be clear from our calculation that the presence of the anomaly found here is independent of this choice. It cannot be removed by playing with this freedom. Finally, there is the ambiguity in the choice of representation used in the regularization of the curvature constraint. Instead of regularizing the local curvature using the fundamental representation as in (31) we could have used the more general function

f [NW ] = n/2 j c j χj (NW ) j(j+1)(2j+1) n/2 k c k W =1+ 2 F = 2 N i F i + O( 2 ), (59)
for arbitrary coefficients c j and some positive integer n. Calculations along the lines presented here show that it is not possible to correct the anomaly found here by playing with the parameters of this space of functions. Let us mention that the anomaly found here seem to be related to the results obtain in reference [22] by a different approach. If we write the constraint algebra as in (57) and (58) the anomaly is parametrized by the local quantity (Tr[W ] -2). If for a moment we think of this factor as a classical quantity, and we evaluate it on a smooth connection field configuartion, then this factor is proportional to the local curvature squared of the connection. This is in direct correspondence with the results of [22].

The anomaly found represents an unexpected difficulty for the implementation of the standard Dirac quantization in the LQG representation. In particular, the group averaging techniques used in the context of 2 + 1 gravity without cosmological constant to solve the first class constraints at the quantum level are not viable. Nevertheless, from the point of view of 3 + 1 gravity, the anomaly appearing in the constraints algebra is a mild one. In fact, in the four dimensional case one has to deal with structure functions in the constraints algebra already at the classical level. From this perspective the problem is well known and studied. One might hope that the techniques developed in this context might turn out useful to contour this obstacle (e.g. Thiemann et al. master constraint program [23]).

At this point we can only speculate with the possibility that the anomaly found here may be at the end related with the deformation of the classical symmetry of gravity leading to the quantum group structure underlying the quantization of 2+1 gravity with non-vanishing cosmological constant found by other methods.

FIG. 1 : 4 FIG. 2 :

 142 FIG. 1: Orientation of plaquette holonomies chosen for the regularization of F [N ].

FIG. 3 :FIG. 4 :

 34 FIG.3: On the left: a generic non SU (2) invariant state Ψ in our cellular decomposition. On the right: explicit illustration of the orientation of the paths used in the regularization of the constraints.

Even though this problem is a feature of quantum systems with local degrees of freedom it manifests itself also in loop quantum cosmology due to the peculiar nature of the model whose definition uses the regulating-structures of the full theory[START_REF] Chiou | Loop quantum cosmology with higher order holonomy corrections[END_REF].

In [18] one obtains a constraint algebra which mimics the constraint algebra of

4d gravity by using phase space dependent smearing functions. This choice considerably complicates the quantization of the (so defined) new constraints as their algebraic structure is no longer a Lie algebra.

A generalized connection is a map from the set of paths γ ⊂ Σ to SU[START_REF] Turaev | sum invariants of 3-manifolds and quantum 6j-symbols[END_REF]. It corresponds to an extension of the notion of holonomy hγ [A] introduced above.

We would like to point to the existence of results along the lines of this work but in 3+1 dimensions by R. Loll[21] where the 3d diffeomorphism constraint sub-algebra is tested in a lattice regularization of quantum gravity.

Acknowledgements

We would like to thank Bianca Dittrich, Antonino Marcianó, and Michael Reisenberger for many very useful discussions on issues related to the subject of this paper and Eugenio Bianchi, Carlo rovelli, and Simone Speziale for suggesting a way to improve the regularization of the Gauss law. This work was supported in part by the Agence Nationale de la Recherche; grant ANR-06-BLAN-0050 and from l'Institut Universitaire de France.