Proof for Negele and Orland equation (2.34) The equation (2.34) of Negele and Orland has
$$\mathcal H_\text{A}(\hat{\mathbf p},\hat{\mathbf x}) =
\frac{1}{2m}\left(\hat {\mathbf p} - \frac e c \mathbf A(\hat{\mathbf x})\right)^2.\tag{2.34a}$$
And it says in normal form,
$$\mathcal H_\text{A}(\hat{\mathbf p},\hat{\mathbf x}) =
\frac{1}{2m}\left(\hat {\mathbf p}^2 - 2\frac e c \hat{\mathbf p}\mathbf A(\hat {\mathbf x}) - \frac e c i \nabla\cdot\mathbf{A}(\hat{\mathbf x})+(\frac e c \mathbf A(\hat{\mathbf x}))^2) \right).\tag{2.34b}$$


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*The second term is particularly very odd for me. Since it is normal ordered, I understand I need to move $\mathbf p$ to left. But doesn't this act on $\mathbf A$ then? I also wonder why there is no dot product between $\mathbf p$ and $\mathbf A$ in the second term.

*Also the third term seems just $\frac e c \mathbf p \cdot \mathbf {A}(\hat{\mathbf x})$ and could be combined with the second term?

*Don't the sign for the third term need to be (+) given that $\hat{\mathbf p} = -i\nabla$?
Because I think this should be (in a non-normal ordered form),
$$\mathcal H_\text{A}(\hat{\mathbf p},\hat{\mathbf x}) =
\frac{1}{2m}\left(\hat {\mathbf p}^2 - 2\frac e c \mathbf A(\hat {\mathbf x}) \cdot \hat{\mathbf p} - \frac e c \mathbf p\cdot\mathbf{A}(\hat{\mathbf x})+(\frac e c \mathbf A(\hat{\mathbf x}))^2) \right).$$
Please let me know the answer. Any answer will be appreciated.
 A: *

*Yes, the missing dot in the dot product of the second term 
$$\tag{2} -2\frac{e}{c}\hat{\mathbf{p}} \cdot \mathbf{A}(\hat{\mathbf{x}}) $$
of eq. (2.34b) is a typo. The operators $\hat{\mathbf{p}}$ and $\mathbf{A}(\hat{\mathbf{x}})$ do not commute, due to the CCRs
$$\tag{CCR} [\hat{x}^i~,~ \hat{p}_j]_{-}~=~i\hbar\delta^i_j~{\bf 1}.$$

*The second and third term taken together are equal to the anticommutator
$$\tag{2+3} -\frac{e}{c} \{ \hat{\mathbf{p}} ~;~\mathbf{A}(\hat{\mathbf{x}}) \}_{+}.$$ 
This becomes clear when one tries to derive (2.34b) from (2.34a). In particular, it is implicitly meant that the derivative $\nabla$ in the third term 
$$\tag{3} \frac{e}{c} \frac{\hbar}{i} \left(\nabla\cdot\mathbf{A}(\hat{\mathbf{x}})\right)
~=~\frac{e}{c}\frac{\hbar}{i} [\nabla~;~\mathbf{A}(\hat{\mathbf{x}})]_{-} 
~=~\frac{e}{c}  [\hat{\mathbf{p}} ~;~\mathbf{A}(\hat{\mathbf{x}})]_{-} $$
of eq. (2.34b) only acts on $\mathbf{A}(\hat{\mathbf{x}})$ and not beyond to the right.  This notational issue is similar to my Phys.SE answer here. 

*The sign in the third term of eq. (2.34b) is correct.
