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The case of n identical bosons in a single s shell is very simple and will play later on an important role. Here only the isospin part needs symmetrization, and for isospin 1 particles all possible states are uniquely specified by the particle number and the total isospin quantum number T. This happens because each state (n,T) is generated at most by two states, namely the states (n-1, T-1) and (n-1, T+1) which yield, together with the normalization condition imposed on the two coefficients, a single possible state.

Thus in order to generate the successive (n,T) states let us consider

the recursion in isospin quantum numbers

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These four terms can be recoupled so as to bring out the symmetry character of the two last particles. It must be symmetric, namely the recoupled

terms where the two last particles have t = 1

must be zero which yields the

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We now get the coefficients of the state together with the normalization condition

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) a single

2n-1)

Let us first consider the case of Fermions and denote ( normalized Slater determinant for n-1 particles. The n particle antisymmetrized state in the R representation is

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where P is the appropriate permutation operator and phase. The normalization is i

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A correctly antisymmetrized normalized coupled state is a linear combination of the form

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where the single particle CFP ( ) has been introduced. In the occupation number representation the equivalent expression of Eq. (2.81) is

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with S a normalized product of n-1 creation operators. Since all indivi

n-1

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Comparing with the expression (2.82) in R space, the coupled state vector in the occupation number representation is thus seen to be

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where then factor corrects for the normalization of the usual CFP coefficients calculated in tables with the definitions of Eq. (2.82) and Eq. (2.83).

For Bosons the same result applies. This can be most easily seen from the fact that the norm is determined by the number of ways a particular coupling scheme appears. In that respect the only difference between Bosons and Fermions is the appearance of a minus sign in the contraction, i.e., then factor in Eq. (2.66), The number of permutations (and the corresponding coupling schemes) are the same for both cases. Hence the relations (2.81) and (2.85) hold also for Boson and consequently the result (2.86) remains true. Of course the CFP's are different for Bosons and Fermions. In particular an arbitrary number of Bosons can occupy a given Nej shell in contrast to Fermions for which the CFP's vanish for n > 2j+1.

II.5.3 Matrix elements of Fermion-Boson systems

The field operators which will be constructed in the following chapters are of the general invariant form (see Eq. (2.21)).

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A

X

where F(x,y; àμλ'μ'LT) is the invariant matrix element for a given point process of multipolarities & and ♬ in angular momentum and isospin spaces. B and [λμ] [λ'μ'l are respectively linear combination of products of Boson and Fermion y operators of multipolarities, ' and isospin μ, μ'. For simplicity we shall limit here our expression to a scalar operator, for example the energy, for which =0, 7=0 and λ =λ', μ=μ'.

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F(x,y; Au)

is an invariant matrix element whose calculation is given in Chapter III (free field energy), Chapter IV (center of mass energy), Chapter VI (field interactions).

We consider now an initial state |i, IT) made of many Fermions coupled

to JT and many Bosons coupled to

JT

:

i

i

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where Y and X represent linear combinations of Fermion and Boson operators which are properly orthonormalized as discussed above, defining completely a configuration i in a given coupling scheme. We have a similar expression for the final state f) and the matrix element is here,

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Thus after evaluation of the basic invariant matrix elements F, the many body problem appears as the usual evaluation of mean values of complicated products of creation and annihilation operators according to well known technique. A few simple examples will be worked out in the following chapters.

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K

We consider a complex field (r,t) of spin 0, isospin 1 and charge k. The Lagrangian is

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