STEFANO BERTINICACCIATORI, SERGIO LUIGIBIANCA L. CERCHIAImostra contributor esterni nascondi contributor esterni 

Stefano Bertini, Sergio L. Cacciatori, and Bianca L. Cerchiai

Citation: Journal of Mathematical Physics 47, 043510 (2006); doi: 10.1063/1.2190898 View online: http://dx.doi.org/10.1063/1.2190898

View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/47/4?ver=pdfcov Published by the AIP Publishing

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On the Euler angles for SU „N…

Stefano Bertini^a兲

Dipartimento di Fisica dell’Università di Milano, Via Celoria 16, I-20133 Milano, Italy Sergio L. Cacciatori^b兲

Dipartimento di Matematica dell’Università di Milano, Via Saldini 50, I-20133 Milano, Italy and INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy

Bianca L. Cerchiai^c兲

Lawrence Berkeley National Laboratory, Theory Group, Bldg 50A5104, 1 Cyclotron Rd, Berkeley, California 94720-8162 and Department of Physics, University of California, Berkeley, California 94720

共Received 9 January 2006; accepted 6 March 2006; published online 25 April 2006兲

In this paper we reconsider the problem of the Euler parametrization for the unitary groups. After constructing the generic group element in terms of generalized angles, we compute the invariant measure on SU共N兲 and then we determine the full range of the parameters, using both topological and geometrical methods. In par- ticular, we show that the given parametrization realizes the group SU共N+1兲 as a fibration of U共N兲 over the complex projective space CPⁿ. This justifies the inter- pretation of the parameters as generalized Euler angles. © 2006 American Institute of Physics. 关DOI:10.1063/1.2190898兴

I. INTRODUCTION

The importance of group theory in all branches of physics is a well-known fact. Explicit realizations of group representations are often necessary technical tools. Often it is finite dimen- sional and compact Lie groups and then the knowledge of the associated algebra, which describes the group in a neighborhood of the identity, is enough for this purpose.

There are however cases where an explicit expression of the full global group structure is needed, as, for example, when nonperturbative computations come into play. In most of these cases, the main objectives are two: First, one would like to find a relative simple parametrization, making all the computations manageable. Second, one needs to determine the full range of the parameters, in order to be able to handle global questions.

If both such points can seem unnecessary at an abstract level, they become essential at a most concrete level, e.g., in instantonic calculus or in nonperturbative lattice gauge theory computa- tions. The necessary computer memory for simulations is in fact drastically diminished.

The case of SU共N兲 was first considered and solved by Tilma and Sudarshan, in Ref. 1. There, they provide a parametrization, in terms of angular parameters, for the unitary groups. In particu- lar, in the first paper they consider special groups, SU共N兲, together with some applications to qubit and qutrit configurations. In the second paper, they give an extension to U共N兲 groups, using the fibration structure of SU共N+1兲 as U共N兲 fiber over the complex projective space CPⁿ.

In this paper we reconsider the problem of finding a generalized Euler parametrization for special unitary groups. The intent is to provide a fully explicit and elementary共which does not mean short兲 proof of the beautiful results of Ref. 1. Our motivation is that the determination of the range of the parameters is a quite difficult task, so that disagreements are present in the literature

a兲Electronic mail: bertiste@tiscalinet.it

b兲Electronic mail: sergio.cacciatori@mi.infn.it

c兲Electronic mail: blcerchiai@lbl.gov

47, 043510-1

0022-2488/2006/47共4兲/043510/13/$23.00 © 2006 American Institute of Physics

even for SU共3兲 共for example, in Ref. 2兲. Therefore, we think that a careful deduction is necessary in order to corroborate the results of Tilma and Sudarshan. Also, all our proofs based essentially on inductive procedures, and they are explicit, in order to be easily accessible to anyone who needs them.

Our construction is quite different from Ref. 1, and as a result our parametrization differs slightly from theirs. However, this does not affect the final expression of the invariant measure.

To illustrate the spirit of our construction, let us start by taking a look at the Euler parametri- zation for SU共2兲.

Starting from the Pauli matrices

␴

1=冉^{0 1}1 0冊^,

^␴

²⁼冉^{0 − ii 0} 冊^,

^␴

³⁼冉¹0 − 1⁰ 冊^{, 共1.1兲}

it is known that the generic element of SU共2兲 can be written as

g = e^i␾␴3e^i␪␴2e^i␺␴3. 共1.2兲 Here

␾

苸关0,

␲

兴,

␪

苸关0,

␲

^{/ 2}兴,

␺

苸关0,2

␲

兴 are the so-called Euler angles for SU共2兲. They are related to the well-known Euler angles traditionally used in classical mechanics to describe the motion of a spin. From the point of view of the structure of the representation,共1.2兲 is obtained starting from a one parameter subgroup exp共i

␪␴

2兲 and then acting on it both from the left and from the right with a maximal subgroup of SU共2兲 which does not contain the first subgroup. We can rewrite it in the schematic form g = U共1兲exp共i

␪␴

2兲U共1兲. On the other hand, the group SU共2兲 is topologically equivalent to the three-sphere S³, and admits a Hopf fibration structure with fiber S¹ over the base S²⯝CP¹.

To recognize this fibration structure in共1.2兲, we can apply the methods used in Refs. 3 and 4.

After introducing the metric 具A兩B典=¹₂Tr共AB兲 on the algebra, the metric on the group can be computed as ds²=¹₂Tr J丢J, where J = −ig⁻¹dg are the left-invariant currents. Following Ref. 3, it is possible to separate the fiber from the base by writing g = hU共1兲, where h=e^i␾␴3e^i␪␴2 and U共1兲=e^i␺␴3. To find the metric on the fiber, let us fix the point on the base and compute the currents along the fiber, J_F= −iU共1兲⁻¹dU共1兲=d

␺␴

3. The metric on the fiber is then simply given by ds_F²= d

␺

². To determine the metric on the base, we first must project out from the current J_B= −ih⁻¹dh the component along the fiber, in order to be left with the reduced current on the basis J˜

B= d

␺␴

2+ sin共2

␺

兲d

␺ ␴

1, which then in turn provides the metric

ds_B²=¹₄关d共2

␺

兲²+ sin²共2

␺

兲d共2

␾

兲²兴. 共1.3兲 This corresponds in fact with the metric of a sphere of radius ¹₂. It is easy to see that, introducing the complex coordinates z = tan

␺

^ei␾and their complex conjugates, the metric ds_B² reduces to the standard Fubini-Study metric forCP¹.

This shows that the Euler parametrization captures the Hopf fibration structure of SU共2兲, which is the starting point for our construction. Mimicking what we said about SU共2兲, let us write the generic element of SU共N+1兲 as g=U共N兲e共

␪

兲U共N兲, where e共

␪

兲 is a one parameter subgroup not contained in U共N兲. The first difficulty we must face here is that this expression for a generic SU共N+1兲 group has redundancies, which must be eliminated. After this problem is solved, we then must show that the parametrization respects the Hopf fibration structure of SU共N+1兲.

II. THE SU„N… ALGEBRA

The generators of su共N兲 are all the N⫻N traceless Hermitian matrices. A convenient choice for a base are the generalized Gell-Mann matrices as explained in Ref. 1. Let us remind how they can be constructed using an inductive procedure. Let 兵␭i其_i=1^N2−1 be the Gell-Mann base for su共N兲:

They are N⫻N matrices which can be embedded in su共N+1兲 adding a null column and a null row

␭˜i=

冉

^␭0

^ជ

^{i 00}

^ជ 冊

^{. 共2.1兲}

We will omit the tilde from now on. The dimension of SU共N兲 being 共N+1兲²− 1, we must add 2N + 1 matrices to obtain a Gell-Mann base for su共N+1兲. This can be done as follows: Set

兵␭N²+2a−2其_␣␤=

␦

␣,a

␦

␤,N+1+

␦

␣,N+1

␦

␤,a,

共2.2兲 兵␭N²+2a−1其_␣␤= i共−

␦

␣,a

␦

␤,N+1+

␦

␣,N+1

␦

␤,a兲,

for a = 1 , . . . , N. The last matrix we need is diagonal and traceless so that we can take␭_{共N + 1兲}²−1

=

⑀

N+1diag兵1, ... ,1,−N其.

One can easily verify that the base of matrices兵␭I其_I=1^{共N + 1兲2−1} so obtained satisfies the normal- ization condition Trace兵␭I␭J其=2

␦

IJif we choose

⑀

N+1=冑^{2 /关共N+1兲2−共N+1兲兴.}

These are exactly the matrices we need to generate the group elements.

III. THE EULER PARAMETRIZATION FOR SU„N+1…: INDUCTIVE CONSTRUCTION It is a well-known fact that special unitary groups SU共N+1兲 can be geometrically understood as U共N兲 fibration over the complex projective space CP^N. Now U共N兲 is generated by the first N²− 1 generalized Gell-Mann matrices plus the last one ␭_{共N + 1兲}²−1. Using the fact that all the remaining generators of SU共N+1兲 can be obtained from the commutators of these matrices with

␭N²+1, one is tempted to write the general element of SU共N+1兲 in the form

SU共N + 1兲 = U共N兲e^ix␭N2+1U共N兲. 共3.1兲 However to describe SU共N+1兲 we need 共N+1兲²− 1 parameters, while in the rhs they are 2N²+ 1: There are共N−1兲² redundancies. Inspired at first by dimensional arguments, we propose that an U共N−1兲 subgroup can be subtracted from the left U共N兲 in the following way. Let us write U共N兲 in the form U共N兲=SU共N兲e^{i␺␭共N + 1兲2−1}. Inductively, we can think that also SU共N兲 can be recovered from U共N−1兲e^{i␾␭共N − 1兲2+1}U共N−1兲 eliminating the redundant parameters, so that it will have the form SU共N兲=he^{i␾␭共N − 1兲2+1}SU共N−1兲e^{i␪␭N2−1}. We then choose to eliminate the appearing SU共N−1兲 together with the phase e^{i␺␭共N + 1兲2−1}. In this way the SU共N+1兲 group element can be written in the form SU共N+1兲=he^{i␾␭共N − 1兲2+1}e^{i␪␭N2−1}e^ix␭N2+1U共N兲. By induction, assuming N艌2 we arrive to the final form of our Ansatz about the parametrization of the general element g 苸SU共N+1兲,

g = e^i␪1␭3e^i␾1␭2兿

a=2 N

关e^{i共␪a/⑀a兲␭a2−1}e^i␾a␭a2+1兴U共N兲关

␣

1, . . . ,

␣

N²兴, 共3.2兲

where U共N兲关

␣

1, . . . ,

␣

N²兴 is a parametrization of U共N兲 which in turn can be obtained inductively using the fact

U共N兲 = 关SU共N兲 ⫻ U共1兲兴/ZN. 共3.3兲

The Ansatz共3.2兲 contains the correct number of parameters. However, we need to show that it is a good Ansatz, meaning that at least locally it must generate the whole tangent space to the identity. Using the Backer-Campbell-Hausdorff formula and some properties of the Gell-Mann matrices,共essentially the fact that the commutators of ␭_{共k − 1兲}²+1 with the first共k−1兲²− 1 matrices generate all the remaining matrices of the su共k兲 algebra but the last one兲 it is easy to show that

e^i␪1␭1e^i␾1␭3兿

a=2 N

关e^{i共␪a/⑀a兲␭a2−1}e^i␾a␭a2+1兴 = e^{i兺j=1共N + 1兲2−2aj␭j}, 共3.4兲

where ajare all nonvanishing functions of the 2N parameters

␪

a,

␾

a. Thus in a change of coordi- nates共from the

␪

a,

␾

a to the aj兲 only 2N of the ajcan be chosen as independent parameters. We could choose the last ones, corresponding to the coefficients of the matrices兵␭k其_k=N2

N²+2N−1

. In this way, the N²free parameters for the remaining matrices come out exactly from the U共N兲 factors in 共3.2兲.

We have not entered into details here because a second simple proof of the validity of this parametrization will be given by constructing a nonsingular invariant measure from our Ansatz.

IV. INVARIANT MEASURE AND THE RANGE OF THE PARAMETERS

A. The invariant measure

To construct the invariant measure for the group starting from共3.2兲, we will adopt the same method used in Ref. 3, with UªU共N兲 as the fiber group. Let us then write 共3.2兲 as

g = h · U. 共4.1兲

Starting from the computation of the left invariant currents jh= −ih⁻¹dh, we can define the one forms

e^lª¹2Tr关jh·␭N²+l−1兴, l = 1, ... ,2N, 共4.2兲 which turns out to give the Vielbein one forms of the base space of the fibration. If eគ denotes the corresponding Vielbein matrix, the invariant measure for SU共N+1兲 will then take the form

d

␮

_SU共N+1兲^{= det e}គ · d

␮

U共N兲, 共4.3兲

d

␮

U共N兲 being the invariant measure for U共N兲. Using 共3.3兲 with U共1兲=e^{i共␻/⑀N+1兲␭共N + 1兲2−1}we obtain the recursion relation共note that here

␻

is allowed to vary in the range关0,2

␲

^{/ N}兴兲

d

␮

SU共N+1兲= det eគ · d

␮

SU共N兲

d

␻

⑀

N+1

. 共4.4兲

Then we will concentrate on the det eគ term. To this end let us write 共3.2兲 in the form

g = h_N+1关

␪

a,

␾

a兴 · U关

␣

i兴. 共4.5兲 Here we will consider N艌3 so that the relation

h_N+1= hNe^{i共␪N/⑀N兲␭N2−1}e^i␾N␭N2+1, 共4.6兲 is true. If we introduce the right currents Jh_N+1= −ih_N+1⁻¹ dh_N+1then the Vielbein共4.2兲 takes the form

e_l^兵N其= 1

2Tr兵Jh_N+1␭l其 = d

␾

N

␦

l,N²+1+ 1 2

⑀

N

d

␪

NTr兵e^{−i␾N␭N2+1}␭N²−1e^i␾N␭N2+1␭N²+l其

+1

2Tr兵e^{−i共␪N/⑀N兲␭N2−1}Jh_Ne^{i共␪N/⑀N兲␭N2−1}e^i␾N␭N2+1␭N²+le^{−i␾N␭N2+1}其, 共4.7兲 and using the relations in Appendix A we find

eគ^兵N其=

冢

^d

^␾

^{00N sin}

^␾

^Ncos00

^␾

^Nd

^␪

^{N 12sinsin}

^{␾ ␾}

^{NNTr关ecos−i}

^␾

^{共␪NN12/⑀Na=2兺N兲␭0NTr2−1}

^冋

^Jh

^␣

^{N1eaiJ共␪hNN␭/⑀Na兲␭2−1N2}

^册

^{−1Mគ 兴}

冣

^{, 共4.8兲}

where we introduced the兵N其 index to remember that this is a 2N⫻2N matrix associated to the group SU共N+1兲. Here Mគ is a column of matrices, M2j−1=␭j²+1, M_2j=␭j², j = 1 , 2 , . . . , N − 1. For- mula共4.8兲 then reads as follows: Jh_Nis a 1-form with components Jh_N,c, c = 1 , 2 , . . . , 2n − 2, with respects to the coordinates X^c, defined as X^2j−1=

␪

j, X^2j=

␾

j. To find the component共r,c兲 of 共4.8兲 one must then take the cth component of Jh_N,cand the rth component of Mគ before to compute the trace.

The invariant measure is then

det eគ^兵N其= d

␾

Nd

␪

Ncos

␪

Nsin^2N−1

␾

Ndet冉^{12Tr关e−i共␪N/⑀N兲␭N2−1JhNei共␪N/⑀N兲␭N2−1Mគ 兴}冊^{. 共4.9兲}

We now use the recurrence relation

J_h

N=␭_{共N − 1兲}²+1d

␾

N−1+ 1

⑀

N−1

e^{−i␾N−1␭共N − 1兲2+1}␭_{共N − 1兲}²−1e^{i␾N−1␭共N − 1兲2+1}d

␪

N−1

+ e^{−i␾N−1␭共N − 1兲2+1}e^{−i共␪N−1/⑀N−1兲␭共N − 1兲2−1}Jh_N−1e^{i共␪N−1/⑀N−1兲␭共N − 1兲2−1}e^{i␾N−1␭共N − 1兲2+1}. 共4.10兲 Computing the traces different cases arise depending on whether j = N − 1 or j⬍N−1; using again the relations in Appendix A it is not too difficult to show that the last determinant is equal to

det

冢

^dd

^{␾ ␾}

^{N−1N−1cos共Nsin共N}

^{␪ ␪}

^{NN兲兲 −1212cos共Nsin共N}

^{␪ ␪}

^{NN兲sin共2兲sin共2}

^{␾ ␾}

^{N−1N−1兲d兲d}

^{␪ ␪}

^N−1N−1

冣

⫻ det冉^12cos

^␾

^{N−1Tr关e−i共␪N−1/⑀N−1兲␭共N − 1兲2−1JhN−1ei共␪N−1/⑀N−1兲␭共N − 1兲2−1Mគ 兴}冊^,

which set into共4.9兲 in turn yields the recurrence relation

det eគ^兵N其= d

␾

Nd

␪

N

sin^2N−1

␾

N

tan^2N−4

␾

N−1

det eគ^兵N−1其, 共4.11兲

which can be solved to give

det eគ^兵N其= 2 d

␪

Nd

␾

Ncos

␾

Nsin^2N−1

␾

N兿

a=1 N−1

关sin

␾

acos^2a−1

␾

ad

␪

ad

␾

a兴. 共4.12兲

This is the same result as found in Ref. 1.

B. The range of the parameters

At this point we are able to determine the range of the parameters in such a way as to cover the whole group. We will do this only for the base space: The remaining ranges for the fiber can be determined recursively, as discussed above, remembering in particular that the U共1兲 phase in U共k兲 can be taken in 关0,2

␲

^{/ k}兴.

We then proceed as in Ref. 3. We first choose the ranges so as to generate a closed 关共N+1兲²− 1兴-dimensional closed manifold which then must wrap around the group manifold of

SU共N+1兲 an integer number of times. This can be done by looking at the measure 共4.12兲 on the base manifold and noticing that it is nonsingular when 0⬍

␾

a⬍

␲

/ 2, whereas

␪

acan take all the period values

␪

a苸关0,2

␲

兴, for all a=1, ... ,N. However, note that the angles

␪

1,

␾

1,

␪

2generate the whole SU共2兲 group when 0艋

␪

1艋

␲

, 0⬍

␾

a⬍

␲

/ 2 and 0艋

␪

2艋2

␲

. We can then restrict

␪

1

苸关0,

␲

兴. The rest of the variety is generated by the remaining U共N兲 part.

If we call V_N+1the manifold obtained this way we then find

Vol共VN+1/U共N兲兲 =冕0

␲

d

␪

兿

a=2 N 冕0

2␲

d

␪

a兿

b=1 N 冕0

␲/2

d

␾

b

再

^cos

^␾

^Nsin2N−1

^␾

^{NN−1兿c=1关sin}

^␾

^ccos2c−1

^␾

^c兴

冎

=

␲

^N

N!, 共4.13兲

or equivalently

Vol共VN+1兲 = Vol共U共N兲兲

␲

^N

N!. 共4.14兲

This is exactly the recursion relation found in Appendix B. Therefore, it is the correct range of the parameters for every N艌2, if we have V₃= SU共3兲, as can be easily checked directly or by com- parison with the results given in Appendix A of Ref. 2共see also Appendix B of Ref. 4兲. The next step is to determine the parametrization of SU共N+1兲 for every value of N. It is given by 共3.2兲 with

0艋

␪

1艋

␲

^{, 0}艋

␪

a艋 2

␲

,a = 2, . . . ,N,

0艋

␻

艋2

␲

N , 0艋

␾

a艋

␲

2,a = 1, . . . ,N, 共4.15兲

and the remaining parameters which cover SU共N兲 共determined inductively兲.

To prove that our parametrization is well defined we can do more: We are in fact able to show that the induced metric on the base manifold is exactly the Fubini-Study metric overCP^N. V. THE GEOMETRIC ANALYSIS OF THE FIBRATION

We will now show that the metric induced on the base space takes exactly the form of the Fubini-Study metric in trigonometric coordinates as given in Appendix C. To do so we will again use inductive arguments.

The metric on the base is ds_B²=关eគ^兵N其兴^T丢eគ^兵N其, where T indicates transposition and eគ^兵N其is given in共4.8兲. Using the relations in Appendix A and defining

XN=1 2兺

a=2 N

Tr关Jh_N

⑀

a␭a²−1兴 共5.1兲

the metric takes the form

ds_B²= d²

␾

N+ sin²

␾

N

再

^关d

^␪

^{N+ XN兴2+N−1兺j=1}

^冋

^{12Tr共e−i共␪N/⑀N兲␭N2−1JhNei共␪N/⑀N兲␭N2−1兲␭j2}

^册

²

+兺

j=1

N−1

冋

^{12Tr共e−i共␪N/⑀N兲␭N2−1JhNei共␪N/⑀N兲␭N2−1兲␭j2+1}

册

²

冎

^{− sin4}

^␾

^N关d

^␪

^{N+ XN兴2. 共5.2兲}

This is an encouraging form, which upon comparison with 共C3兲 suggests the identification

␰

=

␾

N. With this identification in mind, let us first remark that the following recursion relation holds:

XN= cos²

␾

N−1共d

␪

N−1+ X_N−1兲, 共5.3兲 which can be shown by inserting 共4.10兲 in 共5.1兲 and then applying 共A6兲 and 共A11兲. A direct computation yields

X₃= cos²

␾

2共d

␪

2+ cos共2

␾

1兲d

␪

1兲, 共5.4兲 from which, through repeated application of the recurrence relation共5.3兲, we obtain

XN=兺

k=1

N−3

冋

^{兿i=1k cos2}

^␾

^N−i

册

^d

^␪

^N−k+

冋

^{N−2兿i=1 cos2}

^␾

^N−i

册

^共d

^␪

^{2+ cos共2}

^␾

^1兲d

^␪

^{1兲. 共5.5兲}

At this point we must to compare d

␪

N+ XNwith the coefficient of sin⁴

␰

ⁱⁿ共C3兲. In fact, to bring d

␪

N+ XNto the desired form兺_i=1^N 共R˜ⁱ兲²d

␺

i, one is tempted to just set

␪

i=

␺

iand

␾

␮=

␻

␮. However, this cannot be the case because the R˜ⁱdoes not satisfy the condition兺共R˜ⁱ兲²= 1.

These observations, together with explicit calculations for the case N = 4 and N = 5, suggest that we should simply take some linear combination

␺

i=

␺

i共

␪

j兲. This can be done as follows: Let us introduce new variables^˜

␪

K, k = 1 , . . . , N, such that

␪

˜N=

␪

N,

␪

N−k=^˜

␪

N−k−^˜

␪

N−k+1, k = 1, . . . ,N − 3,

共5.6兲

␪

1+

␪

2=

␪

^˜1−

␪

^˜3,

␪

1−

␪

2=^˜

␪

3−^˜

␪

2. In this way d

␪

N+ XNtakes the desired form

d

␪

N+ XN=兺

i=1 N

共Rⁱ共

␻

␮兲兲²d

␺

i 共5.7兲

with

␻

␮=

␾

␮,

␮

= 1 , . . . N − 1,

␺

i=^˜

␪

N−i+1, i = 1 , . . . , N and

R₁= sin

␾

N−1, Rk= sin

␾

N−k兿

i=1 k−1

cos

␾

N−i, k = 2, . . . ,N − 1,

共5.8兲 RN=兿

i=1 N−1

cos

␾

N−i.

These formulas agree with the expressions in Appendix C. As the last step, in Appendix D we finally show that, after performing the change of variables described above, the coefficients of sin²

␰

^{and sin2}

␾

N also agree. This proves that the metric induced on the base CP^N of the U共N兲 fibration by the invariant metric on SU共N+1兲 is nothing else but the natural Fubini-Study metric in trigonometric coordinates.

We can now use this result as a different method to fix the range of the parameters. In fact, 共R¹, . . . , R^N兲 parametrize the positive orthant of a sphere, if 0⬍

␾

i⬍

␲

/ 2, i = 1 , . . . , N − 1. More- over, the identification of

␾

N with

␰

^yields

␾

苸关0,

␲

^{/ 2}兴. Finally, it is easy to show that the conditions ^˜

␪

i苸关0,2

␲

兴 are equivalent to

␪

1苸关0,

␲

兴 and

␪

i苸关0,2

␲

兴, i=2, ... ,N. These are the same results obtained in共4.15兲.

VI. CONCLUSIONS

In this paper, we have reconsidered the problem of constructing a generalized Euler param- etrization for SU共N兲. The parametrization we find differs slightly from the one described by Tilma and Sudarshan. In fact, comparing our results with the expression共18兲 in Ref. 1, it is possible to

see that we have chosen␭_{共k − 1兲}²−1instead of␭3. Furthermore, we have computed the corresponding invariant measure, which turns out to coincide with the result in Ref. 1, despite the slight differ- ences in the choice of the parametrization.

To determine the range of the parameters, we have used two distinct methods, both yielding the same result. To better motivate the name “Euler angles,” we have carefully shown that the parametrization captures the Hopf fibration structure of the SU共N兲 groups. In particular the change of coordinate we found to evidentiate the fibrations, gives an explicit map between the Euler coordinates introduced starting from the generalized Gell-Mann matrices, and the ones introduced in Ref. 5 using geometrical considerations.

We have given a quite explicit proof of every assertion. Apart from corroborating the results of Tilma and Sudarshan, we think that our work is providing a complete toolbox of computation techniques useful in applied theoretical physics as well as for experimental physicists.

ACKNOWLEDGMENTS

The authors共S.B. and S.C.兲 would like to thank G. Berrino, Professor R. Ferrari and Dr. L.

Molinari for helpful conversations. One of the authors共B.L.C.兲 is supported by the INFN under Grant No. 8930/01. This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the U.S. Department of Energy under Contract No. DE- AC02-05CH11231 and in part by the U.S. Department of Energy, Grant No. DE-FG02- 03ER41262. The work of the author共S.C.兲 is supported by INFN, COFIN prot. 2003023852គ008 and the European Commission RTN program MRTN-CT-2004-005104, in which the author共S.C.兲 is associated to the University of Milano-Bicocca.

APPENDIX A: SOME COMMUTATORS

Using the explicit form of the generalized Gell-Mann matrices constructed with the conven- tions of Sec. II, we find the useful commutators

关␭N²+1,␭N²+2j兴 = − i␭j², 关␭N²+1,␭N²+2j+1兴 = i␭j²+1, 共A1兲

when j = 1 , . . . , N − 1.

Others interesting relations easy to check are

关␭N²+1,␭N²兴 = − i共N + 1兲

⑀

N+1␭_{共N + 1兲}²−1− i兺

a=2 N

⑀

a␭a²−1,

关␭N²+1,␭a²−1兴 = i

⑀

a␭N², 共A2兲

关␭N²+1,␭_{共N + 1兲}²−1兴 = i共N + 1兲

⑀

N+1␭N²,

where a = 1 , . . . , N, from which remembering that

⑀

k=冑^{2 / k}共k−1兲, one also finds

关␭N²+1,关␭N²+1,␭N²兴兴 = 4␭N². 共A3兲 From the first two commutators we find the very useful relations

e^ix␭N2+1␭j²+1e^−ix␭N2+1= 1

sin x␭N²+2j+1− 1

tan xe^ix␭N2+1␭N²+2j+1e^−ix␭N2+1,

e^ix␭N2+1␭j²e^−ix␭N2+1= − 1

sin x␭N²+2j+ 1

tan xe^ix␭N2+1␭N²+2je^−ix␭N2+1, 共A4兲 when j = 1 , . . . , N − 1.

Other useful relations easy to prove using the previous relations are

兺a=2 N

⑀

a

2+共N + 1兲²

⑀

N+1

2 = 4, 共A5兲

兺a=2 N

⑀

a

2+共N + 1兲

⑀

N+1

2 = 2, 共A6兲

Tr关e^−ix␭N2+1␭N²−1e^ix␭N2+1␭N²+I兴 =

⑀

N

␦

I0sin共2x兲, 共A7兲

1

2Tr关␭ae^ix␭N2+1␭N²+2ie^−ix␭N2+1兴 =

␦

a,i²sin x, a艋 N²− 1, i = 1, . . . ,N − 1, 共A8兲

1

2Tr关␭ae^ix␭N2+1␭N²+2i−1e^−ix␭N2+1兴 = −

␦

a,i²+1sin x, a艋 N²− 1, i = 1, . . . ,N − 1, 共A9兲

1

2Tr

冋

^{eix␭N2+1␭N2e−ix␭N2+1N兺a=12−1Ca␭a}

册

^{= sin共2x兲12b=2兺N Tr关Cb2−1}

^⑀

^{b2−1兴, 共A10兲}

兺a=2 N

Tr关Ae^{ix␭共N − 1兲2+1}␭a²−1e^{−ix␭共N − 1兲2+1}兴 = cos²x兺

a=2 N

Tr关A

⑀

a␭a²−1兴, 共A11兲

e^ix␭N2−1␭_{共N − 1兲}²e^{−ix␭N2−1}= cos共N

⑀

Nx兲␭_{共N − 1兲}²− sin共N

⑀

Nx兲␭_{共N − 1兲}²+1, 共A12兲

e^ix␭N2−1␭_{共N − 1兲}²+1e^{−ix␭N2−1}= cos共N

⑀

Nx兲␭_{共N − 1兲}²+1+ sin共N

⑀

Nx兲␭_{共N − 1兲}², 共A13兲 where we used Aª兺_i=1^{共N − 1兲2−1}Aⁱ␭i.

APPENDIX B: THE TOTAL VOLUME OF SU„k…

The total volume for the groups SU共k兲 can be found following Macdonald in Ref. 6. First remember that, in the sense of rational cohomology, SU共k兲 is equivalent to the product of odd dimensional spheres

SU共k + 1兲 ⬃兿

j=1 k

S²ⁱ⁺¹, 共B1兲

where we choosen k + 1 to obtain recursive relations. The total volume of the group is then uniquely determined when a metric is established on the Lie algebra. We chose the metric induced by the scalar product共A兩B兲=¹₂Tr共AB兲, for A,B苸su共k+1兲. In this way the Gell-Mann generators are orthonormal. The formula for the total volume is共Ref. 6兲

Vol共SU共k + 1兲兲 =兿

j=1 k

Vol共S²ⁱ⁺¹兲 · Vol共Tk兲兿

␣⬎0兩

␣

^Ú兩², 共B2兲

where

␣

^Ú are the coroots associated to positive roots and Vol共Tk兲 is the volume of the torus generated by the simple coroots.

For su共k+1兲 the simple coroots are si= Li− L_i+1, i = 1 , . . . , k where Li is the diagonal matrix with the only nonvanishing entry兵Li其ii= 1. After writing siin terms of ␭j, as

s_i=兺

a=1 k 1

2Tr兵si␭_{共a + 1兲}²−1其␭_{共a + 1兲}²−1, 共B3兲 it is easy to prove the recursive relation

Vol共Tk兲 =冑^{k + 1}2k Vol共Tk−1兲. 共B4兲 From this we find

Vol共SU共k + 1兲兲 = Vol共SU共k兲兲2

␲

^k+1

k! 冑^{k + 1}2k , 共B5兲

where we used the fact that all the positive coroots have unitary length. If we note that the phase e^{i共␪k/⑀k兲␭共k + 1兲2−1} generates a U共1兲 group of volume 2

␲

冑k共k+1兲/2 and that U共k兲=关SU共k兲

⫻U共1兲兴/Zk, we can finally write

Vol共SU共k + 1兲兲 = Vol共U共k兲兲

␲

^k

k!, 共B6兲

APPENDIX C: THE FUBINI-STUDY METRIC FOR CP^N

CP^Nis a Kähler manifold of complex dimension N. In a local chart, which uses holomorphic inhomogeneous coordinates兵zⁱ其_i=1^N 苸C, the Kähler potential is K共zⁱ, z¯^j兲=k/2 log共1+兺_i=1^N 兩zⁱ兩²兲 with k a constant. The associated Kähler metric gij¯=

⳵

^{2K /}

⳵

^zi

⳵

^{¯zj is then}

ds_CPN

2 = k

冉

^{1 +兺i=1N}兺^{dzi=1N i兩zdz¯i兩i2−}兺^{i,j=1N zidz¯i¯zjdzj}

共1 +兺^{i=1N 兩zi兩2兲2}

冊

^{. 共C1兲}

Notice that obviously it is not possible to cover the whole space with a single chart, but the set of points which cannot be covered has vanishing measure. For our purpose it is therefore enough to consider a single chart.

Let us now search for a trigonometric coordinatization. To this aim let us introduce the new real coordinates

␰

^,

␻

␮,

␺

i,

␮

= 1 , . . . , N − 1, i = 1 , . . . , N, such that

zⁱ= tan

␰

^Ri共

␻

␮兲e^i␺i. 共C2兲 Here Rⁱ共

␻

␮兲 is a parametrization of the unit sphere Sⁿ⁻¹, construced as an immersion inR^N, where 兺_i=1^N 共Rⁱ兲²= 1 and

␻

_␮ are the angles of the sphere. However, notice that we are restricted to the positive orthant only: R_i⬎0. If

␻

_␮ are the standard angles 共starting, for example, with the azi- muthal one

␻

1兲, then

␻

_␮苸关0,

␲

^{/ 2}兴,

␰

苸关0,

␲

^{/ 2}兴, and

␺

i苸关0,2

␲

兴. This choice of coordinates finally gives