In the book "Quantum Field Theory", the author Srednicki has mentioned on page 44 that "if we adopt Weyl ordering, where the quntum hamiltonian $H(P,Q)$ is given in terms of the classical hamiltonian $H(p,q)$ by $$H(P,Q)\equiv \int \frac{dx}{2\pi}\frac{dk}{2\pi}e^{ixP+ikQ}\int dpdq e^{-ixp-ikq}H(p,q),\tag{6.6}$$ then $\cdots$". How does Weyl ordering define and where does the above expression come from?
And my second related question is: How does the expression $$\langle q'',t''|q',t'\rangle=\int \prod_{k=1}^{N}dq_k \prod_{j=0}^{N}\frac{dp_j}{2\pi}e^{ip_j(q_{j+1}-q_j)}e^{-iH(p_j,\bar{q_j})\delta t}\tag{6.7}$$ convert into $$\langle q'',t''|q',t'\rangle=\int\mathcal{D}q\mathcal{D}pe^{i\int_{t'}^{t''}dt\big(p(t)\dot{q}(t)-H(p(t),q(t))\big)}\tag{6.8}$$ when we take the formal limit $\delta t\to 0$?
