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:



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> 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?
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Furthermore, perhaps I'm really more interested in



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> How can we interpret the role of the parameter $t$ in forming the pseudodifferential operators?
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Interpretation of a parameter in forming a pseudodifferential operator
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.


