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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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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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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)