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At the moment I'm working with a paper from Kimball A. Milton (a student of Julian Schwinger) and he uses a notation which I can't find anywhere. He wrote (page 33 eq. (4.19) http://arxiv.org/abs/1503.08091):

$$p_a(t) \cdot [q_a(t+dt)-q_a(t)]$$

He says this $\cdot$ denotes the symmetric multiplication of the $p$ and $q$ operators. But what does this mean? I can't find a definition of this dot product.

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I would guess that he means, $$ a \cdot b = ab+ba = \{a,b\} $$ There maybe a factor of $\frac{1}{2}$ involved in the definition.

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It could be related to the operation, where, the order of operators does not matter e.g.

$$ \hat{\mathcal{O}}_1 \cdot \hat{\mathcal{O}}_2 \equiv \hat{\mathcal{O}}_2 \cdot \hat{\mathcal{O}}_1 $$

much like the commutation matrices $\mathbf{AB}=\mathbf{BA}$ in mathematics. Hence, it could mean that

$$p_a(t) \cdot [q_a(t+dt)-q_a(t)] \equiv [q_a(t+dt)-q_a(t)] \cdot p_a(t) $$

It depends upon what kind of binary operation ``$\cdot$'' is and what kind of operators are involved.

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