Consistent chiral three-nucleon interactions in ... - Theory Center

3 Mathematical basics
states.
In addition, the states |α〉 are antisymmetrized with respect to particle exchange
1 ↔ 2 only. This is assured by taking only states that obey (−1) l12+sab+tab = −1. It is
not antisymmetrized for particle exchange 1 ↔ 3 or 2 ↔ 3.
Once the matrix representation of A is known, we can solve its eigenvalue prob-
lem to get access to the fully antisymmetrized states |EJTi〉 which are eigenstates
of the antisymmetrizer A. Because AA = A holds true due to the projection
property of A, we will find two eigenspaces. One corresponding to eigenvalue 1,
spanned by the fully antisymmetrized states, and one corresponding to eigenvalue
0, made up of spurious states that we are not interested in. At the end we have the
antisymmetric three-particle eigenstates |EJTi〉 as expansion in the basis states
|α〉
|EJTi〉 =
〈α|EJTi〉|α〉 =
cα,i|α〉 . (3.91)
α
α
The expansion coefficients cα,i are called coefficients of fractional parentage (CFPs).
The quantum number
E = 2n12 + l12 + 2n3 + l3
(3.92)
corresponds to the total harmonic oscillator energy of the three-nucleon state. The
i is no physical quantum number, it just labels states with same E, J and T in an
arbitrary ordering.
Having the fully antisymmetrized three-particle relative states |EJTi〉 we can
explicitly construct a projector on the antisymmetric relative Hilbert space by us-
ing the dyadic product
Arel =
E,J,MJ T,MT
|EJMJTMTi〉〈EJMJTMTi| . (3.93)
i
We can extend this to an antisymmetrizer of the complete Hilbert space simply by
multiplying with a dyadic product of center-of-mass states
A =
ncm,lcm
mcm
≡
ncm,lcm
mcm
|ncmlcmmcm〉〈ncmlcmmcm| ⊗
E,J,MJ T,MT
i
E,J,MJ T,MT
|EJMJTMTi〉〈EJMJTMTi|
i
|ncmlcmmcm〉 ⊗ |EJMJTMTi〉〈ncmlcmmcm| ⊗ 〈EJMJTMTi| ,
(3.94)
since the center-of-mass states are invariant with respect to particle permutations.
32
After explaining the basic idea of the antisymmetrization procedure, we now