U (r) g=

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(r p)~(r p)

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(1117)

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( p)~( p)

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Its action on the elements of the coset reads as a projective one:

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= g = (A + B)(C + D ) 1

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(1118)

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11311 Remark This action of the group on one of its cosets is not mysterious We already described an example of it in (836) On a general level, let G be a group and H be a subgroup of G The right coset G/H is the set of equivalence classes modulo H de ned by the relation g = g if and only if there exists h H such that g = g h Thus, G/H =

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{g H = {g h, h H}, g G} Now, let g 0 G and g 0 H be its equivalence class An element g G transports this class by left multiplication: g 0 H g ( g 0 H) = ( g g 0)H In the present case, it is enough to note that any element g= A C B D U (r)

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11 Fermionic Coherent States

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with nonsingular D can be factorized as g = (Z g )h g with g = BD 1 , (Z g ) like in (1116) and h g U ( p) U (r p) So, by simple matrix multiplication, we have g (Z ) = (Z )h = (Z ), where is like in (1118)

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The compact manifold U (r)/(U ( p) U (r p)) has a rich structure It is, like C or the sphere S 2 or the Poincar half-plane, a K hler manifold: its Riemannian symplectic structure is simply encoded by the K hler potential F = F ( , ) given by F ( , ) = ln det(I p +

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(1119)

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The metric element and the 2-form derive from it as follows: ds 2 =

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g d g d

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d , d

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g =

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2 F ( , ) ,

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(1120a) (1120b)

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The volume element is given by d ( , ) = dim V Vol U (r)/U (r p) det I p +

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r

d

(1121)

In the normalization factor, V denotes the nite-dimensional space carrying the unitary irreducible representation of U (r) The explicit form of the coherent states is established thanks to the existence of a Baker Campbell Hausdorff Zassenhaus factorization formula for the group U (r) This formula expresses the displacement operator appearing in (1113) in terms of the variables i j :

exp

i j ai a j 1u j u p p+1uiur

i j a j a i =

i j ai a j

exp

1uiu p p+1u j ur

i j a a j i (1122)

exp

1u j u p p+1uiur

1ui, ju p p+1ui, j ur

i j a a j exp i

Equation (1122) is proven through the use of the following factorization in the nite matrix representation of the operators involved: I r p Z Z Z Z I p Z Z = I r p 0 e 1 0 0 e 2 I r p 0 Ip ,

(1123) with e 1 = (I r p Z Z ) 1/2 and e 2 = (I p Z Z )1/2

113 Coherent States for Systems of Identical Fermions

By using the factorization (1122) and the fact that the two right factors stabilize the extremal state a a j |extr = 0 for i 1ui = j u p / or p + 1 u i, j u r ,

one can write the coherent states as

i j ai a j

1 exp ))1/2 (N ( ,

|extr ,

(1124)

1u ju p p+1uiur

with N ( , ) = det I p~ p The scalar product (overlap) between two coherent states is then given in terms of the K hler potential by | = eF ( , ) (N ( , ))1/2 (N ( , ))1/2

(1125)

These coherent states, as an overcomplete family of vectors in V , resolve the identity: d ( , ) |

U (r)/(U ( p) U (r p))

| = Id

(1126)

This implies that any state | in the Hermitian space of fermionic levels can be expanded in terms of those states, | = d ( , ) ( ) |

where the contravariant symbol of | is given by

def ( ) = | = (N ( , )) 1/2 f ( )

Here, f ( ) is an analytical function of , thus providing the space V with a realization of the Fock Bargmann type

1132 Fermionic Symmetry SO(2r)

A second relevant algebra, namely, so(2r), is constructed from the operators a and i a j Its generators are the r(2r 1) operators 1 a a j i j , i 2 ai a j 1 u i, j u r 1ui = j ur / (1127) (1128)