The quantum spin 1/2 X- Y model in dimensions d= 1, 2 and 3 is studied by means of the extended Migdal approximation for the scale factor of the transformation b = 2. It is found, with the imposed conditions to get the recursion equations, that an unstable fixed point between a T=O fixed point and that of T== exists for d>l. However, alternate imposed conditions lead to an unusual behavior that the T = 0 fixed point does not appear in any dimension and that a stable fixed point appears at a lower temperature than that of an unstable one for d > 2. The same model on the simple cubic and face-centered cubic lattices is also treated by the two-step decimation transformation. Recently much attention has been paid to the X- Y model to clarify the nature of a phase transition in two dimensions. As for the quantum X- Y model, it is believed with little doubt that the one-dimensional model remains paramagnetic at all temperatures and that there occurs a second order phase transition in three dimensions to an ordered phase with the long range order (transverse magnetization) at a finite temperature.])<l) The spin 1/2 X- Y model in two dimensions has no long range order at non-zero temperature,4) but shows evidence of a divergent susceptibility and a non-singular specific heat at a finite temperature.3).5),6) For the classical case in two dimensions, it is believed that the topological phase transition with a line of critical points occurs at a finite temperature.7)~l l) Several authorsl2)~16) have studied the X- Y model with spin 1/2 in two dimensions by means of the real-space nonlinear renormalization-group transformations. Nevertheless, their results were contradictory and in conclusive. Dekeyser et a1.17) suggested with a real-space linear renormalization transformation and with the imposition of marginality for temperature deviations from criticality that the same model shows a critical behavior of the same kind as that predicted for the classical case. Recently an extension of the Migdal approximation 18 ),19) to quantum spin systems has been made by Suzuki and Takano,20) (hereafter referred to as ST), and Barma et a1. 21l The approximation employed is to perform the quantum mechanical calculation only for the one-dimensional decimation transformation (DT) in the Migdal approximation: The bond-moving approximation could be