Variations (proportional to unity operator) in Schwingers quantum action principle

At the moment I'm working with the quantum action principle of J. Schwinger. For this I read the following paper: http://arxiv.org/abs/1503.08091. The auther mentioned that Schwinger considered for the variation of the action operator $S$ only variations $\delta q_a$ which are proportional to the unity operator (see page 36 of the paper).

But why does it make senese to ristrict the possible variations only to c-numbers?

In the paper on p.13 (https://arxiv.org/pdf/hep-th/0204003.pdf) I read that not all variations are allowed.

And also Schwinger mentioned some restriction in his paper "The Theory of Quantized Fields. I" under equation (2.17):

This expression for $\delta_0 \mathcal{L}$ is to be understood symbolically, since the order of the operators in $\mathcal{L}$ must not be altered in the course of effecting the variation. Accordingly, the commutation properties of $\delta _0 \phi^a$ are involved in obtaining the consequences of the stationary requirement on the action integral. For simplicity, we shall introduce here the explicit assumption that the commutation properties of $\delta _0 \phi^a$ and the structure of the lagrange function must be so related that identical contributions are produced by terms that differ fundamentally only in the position of $\delta _0 \phi^a$ .

It doesn't make sense to restrict the possible variations only to c-numbers. Section 3.5 on page 80 of Schwinger's book "Quantum Kinematics and Dynamics" sets up the action operator, $$\hat{S}=\int \left(\frac{1}{4}\hat{x}^{a}A_{ab}\frac{d\hat{x}^{b}}{dt} -\frac{1}{4}\frac{d\hat{x}^{a}}{dt}A_{ab}\hat{x}^{b}-\hat{H}\right)dt$$ The dynamical variables $\hat{x}^{a}$ are Hermitian. They stand for all the dynamical variables of a system. So, if there are $m$ coordindates $\hat{q}^{i}$ for $i=1,\ldots ,m$ and $m$ momenta $\hat{p}^{i}$ the operators $\hat{x}^{a}$ contain the coords and the momenta and so label $a=1,\ldots ,2m$. The constant matrix $A_{ab}$ needs to be anti-Hermitian $\hat{A}=-A$ since the action $\hat{S}$ is Hermitian. Schwinger varies the action by allowing the dynamical variables to change by $\delta\hat{x}^{a}$. The operator variations $\delta\hat{x}^{a}$ do not commute with the dynamical variables so Schwinger moves the variations to the left or the right using commutation relations and defines left and right partial derivatives, $$\delta\hat{H}=\delta\hat{x}^{a}\frac{\partial_{l}\hat{H}}{\partial\hat{x}^{a}} =\frac{\partial_{r}\hat{H}}{\partial\hat{x}^{a}}\delta\hat{x}^{a}$$ The condition $\delta\hat{S}=0$ then implies Hamilton's equations in the form, $$A_{ab}\frac{d\hat{x}^{b}}{dt}=\frac{\partial_{l}\hat{H}}{\partial\hat{x}^{a}}$$ and an equivalent equation, $$\frac{d\hat{x}^{a}}{dt}A_{ab}=-\frac{\partial_{r}\hat{H}}{\partial\hat{x}^{b}}$$ The last two equations are (3.28) on page 83 of Quantum Kinematics and Dynamics.