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Let $A=A^+ +A^-$ where $A^+,A^-$ denote the creation and annihilation portion of the field. Then in Eduardo Fradkin, Field Theories of Condensed Matter Physics, equation (5.284), it states that $$ :e^A::e^B: ~=~ e^{[A^+,B^-]}:e^{A+B}:\tag{5.284} $$ where $::$ denotes normal-ordering of $A^+,A^-$. I'm familiar with the regular Baker-Hausdorff formula, but I'm not sure why this identity is true.

EDIT: Here's my attempt. \begin{align} :A^n: &= \text{He}_n(A) \\ :e^A: &= \sum_{n=0}^\infty \frac{1}{n!}\text{He}_n(A) =e^{A-1/2}\\ :e^A::e^B: &= e^{-1} e^A e^B\\ &= e^{-1} e^{A+B} e^{\frac{1}{2} [A,B]} \\ &= :e^{A+B}:e^{-1/2} e^{i \Im[A^+,B^-]} \end{align} where I implicitly assumed that $[A,B]$ is a complex multiple of the identity. However, you can see that my result doesn't quite match the equation.

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Ref. 1 contains several$^1$ typos, e.g. the aforementioned eq. (5.284) if we use$^1$ the definition above eq. (5.262):

Let $\phi^+(x)$ ($\phi^-(x)$) denote the piece of $\phi(x)$ which depends on the creation (annihilation) operators only, $$\phi(x) ~=~\phi^+(x)+\phi^-(x).\tag{5.262}$$

The corrected eq. (5.284) is derived as follows:

$$ :e^A::e^B:~=~e^{A^+}e^{A^-}e^{B^+}e^{B^-}~=~e^{A^+}e^{[A^-,B^+]}e^{B^+}e^{A^-}e^{B^-}$$ $$~=~e^{[A^-,B^+]}e^{A^++B^+}e^{A^-+B^-}~=~e^{[A^-,B^+]}:e^{A+B}:\tag{5.284'}$$

References:

  1. E. Fradkin, Field Theories of Condensed Matter Physics, 2nd ed. (2013).

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$^1$ Independently, there is a wrong sign in the truncated BCH formula (5.269).

$^2$ Alternatively, eq. (5.284) is correct in its printed form if we use the opposite notation for creation & annihilation operators.

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