How to Derive the Relation for W_{\varepsilon}[J] Using the Exponential Map?

[parton]

I have the following relation:

$$W_{\varepsilon}[J] = \mathrm{exp} \left[ - \varepsilon \int \mathrm{d} x \, \left( \dfrac{\delta}{\delta J(x)} \right)^{n} \right] \mathrm{exp}(W[J])$$

where W is a functional of J.

Now I read in a textbook that it follows

$$W_{\varepsilon}[J] = W[J] - \varepsilon \mathrm{e}^{-W[J]} \int \mathrm{d} x \, \left( \dfrac{\delta}{\delta J(x)} \right)^{n} \mathrm{e}^{W[J]} + \mathcal{O}(\varepsilon^{2})$$.

Unfortunately I absolutely don't know how to obtain this result. Maybe some kind of Baker-Campbell-Hausdorff formula is used here or a relation such as
$$\mathrm{e}^{A} B \mathrm{e}^{-A} = B + [A,B] + ...$$
but nevertheless I don't know how to obtain the result above. Does anyone have an idea?

 

[mark726]

I can provide some insight into how this result is obtained. First, it is important to understand that this relation is known as the exponential map, which is commonly used in functional analysis and quantum field theory.

To start, let's simplify the notation by using a dummy variable t instead of x. Then, we can rewrite the original relation as:

W_{\varepsilon}[J] = \mathrm{exp} \left[ - \varepsilon \int \mathrm{d} t \, \left( \dfrac{\delta}{\delta J(t)} \right)^{n} \right] \mathrm{exp}(W[J])

Next, we can expand the exponential term using its Taylor series:

\mathrm{exp}(W[J]) = \sum_{k=0}^{\infty} \dfrac{1}{k!} \left( \dfrac{\delta}{\delta J(t)} \right)^{k} W[J]

Substituting this into the original relation, we get:

W_{\varepsilon}[J] = \mathrm{exp} \left[ - \varepsilon \int \mathrm{d} t \, \left( \dfrac{\delta}{\delta J(t)} \right)^{n} \right] \sum_{k=0}^{\infty} \dfrac{1}{k!} \left( \dfrac{\delta}{\delta J(t)} \right)^{k} W[J]

Using the properties of the exponential and derivative operators, we can rearrange this as:

W_{\varepsilon}[J] = \sum_{k=0}^{\infty} \dfrac{(-\varepsilon)^{k}}{k!} \int \mathrm{d} t \, \left( \dfrac{\delta}{\delta J(t)} \right)^{n+k} W[J]

Now, we can focus on the integral term and use integration by parts to move the derivative operator onto the exponential term:

\int \mathrm{d} t \, \left( \dfrac{\delta}{\delta J(t)} \right)^{n+k} W[J] = \left[ \left( \dfrac{\delta}{\delta J(t)} \right)^{n+k-1}