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We use the Becchi-Rouet-Stora-Tyutin (BRST) method to quantize the $t$-$J$model in the $U(1)$ gauge slave boson representation. While the temporalcomponent of the gauge field plays a role of a Lagrange multiplier to enforcethe no double occupancy constraint, the spatial components do that to enforcethe vanishing counterflow constraints of the spinon and holon currents. Theseconstraints on the ordered states of the $t$-$J$ model belong to Dirac'ssecond-class constraints. The BRST quantization theory which is consistent withthe second-class constraints was not built in literature. We successfullydevelop such a BRST quantization theory in which the gauge invariance isguaranteed and the redundant gauge degrees of freedom are removed by propergauge fixing conditions while the no double occupancy and vanishing counterflowconstraints are exactly retained. Furthermore, the gauge fixing conditionsendow the gauge field with dynamics. This turns the strongly correlatedelectron model into a weakly coupled slave boson model, most of whose physicalobservable can be calculated by the conventional quantum many-body perturbationtheory. We focus on the properties of the strange metal phase in the $t$-$J$model. The electron momentum distribution and the spectral function arecalculated, and their non-Fermi liquid behaviors agree with the angle resolvedphotoemission spectroscopy measurements for the cuprate materials. We alsostudy the responses of the strange metal state to the external electromagneticfields. The non-Fermi liquid anomalies observed in cuprates are captured by ourcalculations. Especially, we find that the Hall resistivity decreases astemperature raises and the sign of the Hall resistivity varies from negative topositive when the dopant concentration varies from the optimal doping one tounderdoping one for the temperature $T>T^*$.

Non-Fermi Liquid Behavior of the $t$-$J$ Model in the Strange Metal Phase: $U(1)$ Gauge Theory Consistent with Local Constraints