Abstract: By introducing the imaginary time, we show that Gor’kov’s Ginzburg–Landau equation at zero temperature can be extended to an exact relativistic form. Based on this equation, we propose a quantum field theory with the imaginary time, which is intended to describe the quantum critical phenomena in the zero-temperature overdoped cuprate. By using such a quantum field theory, we show that the anomalous two-class scaling between the transition temperature T_c and zero-temperature superfluid phase stiffness \;\rho _s\left( 0 \right) observed in the overdoped side of single-crystal \rmL\rma_2 - x\rmS\rmr_x\rmCu\rmO_4 films can be derived exactly. In this paper, we further theoretically show that, for 2-dimensional overdoped cuprate films, the transition temperature T_c and the zero-temperature coherence length \xi \left( 0 \right) will obey a two-class scaling as well. When the transition temperature T_c is less than a characteristic temperature scale T_Q, the transition temperature T_c and the zero-temperature coherence length \xi(0) obeys the scaling \xi \left( 0 \right) \propto T_c^ - 1.34 , which is a quantum critical behavior. Nevertheless, when the transition temperature is larger than another characteristic temperature scale T_M, the scaling relationship will yield \xi \left( 0 \right) \propto T_c^ - 1 , which is a mean-field behavior.