  With the Becchi-Rouet-Stora-Tyutin (BRST) quantization of gauge theory, we
solve the long-standing difficult problem of the local constraint conditions,
i.e., the single occupation of a slave particle per site, in the slave particle
theory. This difficulty is actually caused by inconsistently dealing with the
local Lagrange multiplier $\lambda_i$ which ensures the constraint: In the
Hamiltonian formalism of the theory, $\lambda_i$ is time-independent and
commutes with the Hamiltonian while in the Lagrangian formalism, $\lambda_i(t)$
becomes time-dependent and plays a role of gauge field. This implies that the
redundant degrees of freedom of $\lambda_i(t)$ are introduced and must be
removed by the additional constraint, the gauge fixing condition $\partial_t
\lambda_i(t)=0$. In literature, this gauge fixing condition was missed. We add
this gauge fixing condition and use the BRST quantization of gauge theory for
Dirac's first-class constraints in the slave particle theory. This gauge fixing
condition endows $\lambda_i(t)$ with dynamics and leads to important physical
results. As an example, we study the Hubbard model at half-filling and find
that the spinon is gapped in the weak $U$ and the system is indeed a
conventional metal, which resolves the paradox that the weak coupling state is
a superconductor in the previous slave boson mean field theory. For the $t$-$J$
model, we find that the dynamic effect of $\lambda_i(t)$ substantially
suppresses the $d$-wave pairing gap and then the superconducting critical
temperature may be lowered at least a factor of one-fifth of the mean field
value which is of the order of 1000 K. The renormalized $T_c$ is then close to
that in cuprates.
