 The consistency in these results became an important justification and motivated a very strong growth in theoretical physics in the late 50s and 60s.The RPA vacuum formula_1 for a bosonic system can be expressed in terms of non-correlated bosonic vacuum formula_2 and original boson excitations formula_3

formula_4

where "Z" is a symmetric matrix with formula_5 and

formula_6

The normalization can be calculated by

formula_7

where formula_8 is the singular value decomposition of formula_9.formula_10

formula_11

formula_12

formula_13

the connection between new and old excitations is given by

formula_14.