  In the slave particle representation with $U(1)$ gauge symmetry, local
constraints on physical states characterized by various mean field solutions
belong to Dirac's second-class ones. Although constrained systems are
extensively investigated, realistic methods to solve the gauge theory problem
with second-class constraints are yet to be developed. We formulate a
Becchi-Rouet-Stora-Tyutin (BRST) quantization theory, called consistent $U(1)$
gauge theory, that is consistent with both first- and second-class local
constraints for strongly correlated condensed matter systems. In our consistent
$U(1)$ gauge theory, the redundant gauge degrees of freedom are removed by
proper gauge fixing conditions while the constraints are exactly retained and
the gauge invariance is guaranteed by the BRST symmetry. Furthermore, the gauge
fixing conditions endow the gauge field with dynamics. This turns the strongly
correlated electron model into a weakly coupled slave boson model, so most of
the system's physical properties can be calculated by the conventional quantum
many-body perturbation method. We focus on the property of the strange metal
phase in the $t$-$J$ model. The electron momentum distribution and the spectral
function are calculated, and the non-Fermi liquid behavior agrees with the
angle-resolved photoemission spectroscopy measurements for cuprate materials.
We also study the electromagnetic responses of the strange metal state. The
observed non-Fermi liquid anomalies are captured by our calculations.
Especially, we find that the Hall resistivity decreases as temperature
increases, and the sign of the Hall resistivity varies from negative to
positive when the dopant concentration varies from optimal doping to
underdoping in the strange metal regime.
