Why does quantum analogue of current density operator $mathbf j = rho mathbf v$ involve an anti-commutator?

I am reading through Luttinger’s "Theory of Thermal Transport Coefficients". In equation (1.14), the charge-density operator is defined as
$$rho(mathbf r) = esum_j delta(mathbf r-mathbf r_j).$$ In equation (1.16), the current-density operator is subsequently defined as $$mathbf j(mathbf r) = frac{e}{2}sum_j left( mathbf v_jdelta(mathbf r-mathbf r_j) + delta(mathbf r-mathbf r_j) mathbf v_j right)$$
where $mathbf v_j$ is the velocity operator of the $j^text{th}$ particle. My question pertains to why the current-density is defined in this way, with the anti-commutator

$$mathbf j(mathbf r) = esum_j frac{1}{2} {mathbf v_j,delta(mathbf r-mathbf r_j) }. $$ It is clear that the role of the anti-commutator is to make the current-density Hermitian, as the charge-density and velocity operators don’t commute themselves, but there are other ways to construct a Hermitian operator. For example, taking
$$mathbf j(mathbf r) = esum_j frac{i}{2} [mathbf v_j,delta(mathbf r-mathbf r_j) ] $$
where $[,]$ denotes the commutator is also Hermitian. Why is the anti-commutator considered the "correct" way to construct a Hermitian operator?