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At the moment I'm working with the quantum[/PLAIN] [Broken] action principle of J. Schwinger. For this I read several paper and books (like: Quantum kinematics and dynamics by J. Schwinger, Schwinger's Quantum action principle by K.A. Milton, http://arxiv.org/abs/1503.08091, Introduction to quantum field theory by P. Roman and The Theory of Quantized Fields. I by J. Schwinger and some more).

Some authors restricted the variations to be c-numbers. But I don't really understand why this restriction makes sense?

Schwinger itself does not mentioned any restriction for quantum mechanics in his book "Quantum kinematics and dynamics". He also didn't assume that the variations commute with the other operators.

But he made some restriction for QFT in his paper "The Theory of Quantized Fields. I" under equation (2.17):

This expression for [itex] \delta_0 \mathcal{L} [/itex] is to be understood symbolically, since the order of the operators in [itex]\mathcal{L}[/itex] must not be altered in the course of effecting the variation. Accordingly, the commutation properties of [itex]\delta _0 \phi^a[/itex] 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 [itex]\delta _0 \phi^a[/itex] 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 [itex]\delta _0 \phi^a[/itex].

Why did Schwinger restrict the variations only in the case of quantum field theory?

Some authors restricted the variations to be c-numbers. But I don't really understand why this restriction makes sense?

Schwinger itself does not mentioned any restriction for quantum mechanics in his book "Quantum kinematics and dynamics". He also didn't assume that the variations commute with the other operators.

But he made some restriction for QFT in his paper "The Theory of Quantized Fields. I" under equation (2.17):

This expression for [itex] \delta_0 \mathcal{L} [/itex] is to be understood symbolically, since the order of the operators in [itex]\mathcal{L}[/itex] must not be altered in the course of effecting the variation. Accordingly, the commutation properties of [itex]\delta _0 \phi^a[/itex] 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 [itex]\delta _0 \phi^a[/itex] 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 [itex]\delta _0 \phi^a[/itex].

Why did Schwinger restrict the variations only in the case of quantum field theory?

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