I am studying PDEs whose symbols satisfy
\begin{equation}
\partial^\alpha_\xi\partial^\beta_xp(x,\xi) \lesssim M(x,\xi)\Psi(x,\xi)^{\alpha}\Phi(x,\xi)^{\beta}
\end{equation}
for all multiindices $\alpha,\beta$ and $(x,\xi) \in \mathbb{R}^{2n}$.
It is said that we must look at the metric
\begin{equation}
g_{x,\xi}=\frac{dx^2}{\Phi(x,\xi)^2}+\frac{d\xi^2}{\Psi(x,\xi)^2}
\end{equation}
for the phase space to study such PDEs.
My question How do I relate the symbol to the metric? What is reason behind such a choice of metric?
You can in fact reformulate the conditions on the symbol $p$ by saying that $$ \vert p^{(k)}(X) T^k\vert\le C_k M(X)g_X(T)^{k/2}, $$ with $X=(x,\xi)$. Here $p^{(k)}(X) $ stands for the $k$th derivative (a $k$multilinear form) and $T$ is a vector in $\mathbb R^{2n}$. Using Lars Hörmander's notations in Chapter 18 of the third volume of ALPDO (SpringerGrundlehren 274), you have $$ p\in \mathcal S(M, g). $$ The asset of this presentation is that you can for instance give rather simple sufficient conditions on the metric $g$ to ensure that symbols in $\mathcal S(1, g)$ give rise to bounded operators on $L^2(\mathbb R^n)$. Also, you can prove that you have an algebra of operators and results of type $$ P_1P_2\quad \text{has a symbol in $\mathcal S(M_1M_2, g)$}, $$ whenever $P_j$ has a symbol in $\mathcal S(M_j, g)$, under some rather simple conditions on the weights $M_j$. You may have as well some asymptotic calculus and define a "Planck" function $ h(X) $ which in your case is $(\Phi(X)\Psi(X))^{1} $ (which is required to be $\le 1$, some type of Heisenberg Uncertainty principle) and show that, for $P_j$ as above with with symbol $p_j$, $$ P_1P_2\text{Op}(p_1p_2)\quad \text{has a symbol in $\mathcal S(M_1M_2 h, g)$}. $$ The first occurrence of this type of conditions was given by Richard Beals and Charles Fefferman in their solution of the NirenbergTreves conjecture for local solvability of principal type differential operators: they had to resort to a nonhomogeneous microlocalization scheme involving the metric in your question in order to obtain some factorization result which fails to hold with standard homogeneous microlocalization arguments.

$\begingroup$ The metric introduced on the phase space may not be canonical in relation to the symplectic structure of the space. $\endgroup$ – Rahul Raju Pattar Jun 7 at 4:12

1$\begingroup$ When you are given a positivedefinite quadratic form $Q$ on a symplectic vector space, you may define $Q^\sigma$ the symplectic inverse (you can check the definition in the Hörmander reference given above). With $Q=g_X$, Hörmander's conditions on the metric involve $g_X$ and $g_X^\sigma$. $\endgroup$ – Bazin Jun 7 at 8:10