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, \begin{equation} \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 \end{equation} 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, \begin{equation} \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} \end{equation} The condition $\delta\hat{S}=0$ then implies Hamilton's equations in the form, \begin{equation} A_{ab}\frac{d\hat{x}^{b}}{dt}=\frac{\partial_{l}\hat{H}}{\partial\hat{x}^{a}} \end{equation} and an equivalent equation, \begin{equation} \frac{d\hat{x}^{a}}{dt}A_{ab}=-\frac{\partial_{r}\hat{H}}{\partial\hat{x}^{b}} \end{equation} The last two equations are (3.28) on page 83 of Quantum Kinematics and Dynamics.

  • $\begingroup$ But is this really the Hamilton's equition as we know it for a quatum system (like Dirac equation)? I also added some information to my question. One paper I found says that there are restrictions to the variation. The other (by Schwinger) says that Schwingers assumes more or less the position-independece of the variaton terms. $\endgroup$
    – Alpha001
    Dec 4 '16 at 16:12
  • $\begingroup$ Or maybe I don't understand really what Schwinger's assumption with the identical contrubition to δSδS by different positions mean. Also how can we ensure that the commutators (which appear if we bring the variations to the left/right) do not depend on the position of variations? $\endgroup$
    – Alpha001
    Dec 4 '16 at 18:25

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