Interpretation of a parameter in forming a pseudodifferential operator

Question

In Zworski's Semiclassical Analysis, he defines the following method of quantization: for a symbol $a = a(x,\xi) \in \mathscr{S}(\mathbb{R}^{2n})$ and $u \in \mathscr{S}(\mathbb{R}^n)$,

$$ Op_t(a)u(x) : = \frac{1}{(2\pi h)^n} \int_{\mathbb{R}^n \times \mathbb{R}^n} e^{\tfrac{i}{h}\langle x - y, \xi \rangle} a(tx + (1 - t)y, \xi) u(y) dy d\xi$$

where $t \in [0,1]$ is some parameter. The presence of $h$ is only because this form of quantization is motivated by quantum mechanics, and the Weyl Quantization is equal to $Op_{1/2}(a)$, with $t = 1/2$. It's also useful to notice that the above family of quantizations obeys the adjoint formula $$ Op_t(a)^* = Op_{1 - t}(\bar{a}),$$ from which we can see that Weyl quantization on real symbols give self-adjoint operators, as desired in quantum mechanics. However, for $t \neq 1/2$, we lose this self-adjointness, and the only other value of $t$ treated in the text (as far as I know) is $t = 1$, because the quantization formula is very simple, and it is also one of the traditional ways to form pseudodifferential operators. My question is:

What are contexts in which we might use a quantization of the above form with $t \neq \tfrac{1}{2}, 1$? Are there any natural situations in which they arise?

Furthermore, perhaps I'm really more interested in

How can we interpret the role of the parameter $t$ in forming the pseudodifferential operators?

Answer 1

Let me answer to your second query and make $h=1$. You have $$ Op_1(a(x) \xi)= a(x) D_x,\quad \text{with $D_x=-i\partial_x$}, $$ $$ Op_0(a(x) \xi)= D_x a(x), $$ $$ Op_{1/2}(a(x) \xi)= \frac 12D_x a(x)+\frac 12a(x)D_x. $$ With $t=1$, you start with the derivations and then you multiply by the coefficients (in the case of a differential operator).

With $t=0$, you start with the multiplications and then you take derivatives.

$t=1/2$ is a symmetric compromise between the two bad solutions above. Note that the most important property of Weyl quantization is its symplectic invariance and not only the fact that real-valued symbols (Hamiltonians) get quantized by (formally) selfadjoint operators.