# What is the symmetric multiplication of operators?

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.

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.

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.