Consider the following self-interacting real scalar field theory $$ \mathcal{L} = \frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi) - \frac{1}{2}m^2\phi^2 - \frac{\lambda}{4!}\phi^4 $$ with $m^2 > 0$ and $\lambda > 0$. It is well-known that the generating functional of the full (i.e., interacting) theory can be written as $$ Z[J] = \exp\Biggl\{-i\frac{\lambda}{4!} \int dx\ \frac{\delta^4}{\delta J(x)^4}\Biggr\}Z_0[J] $$ where $$ Z_0[J] = \mathcal{N} \exp\Biggl\{-\frac{1}{2} \int dz \int dw\ J(z) \Delta_F(z - w) J(w)\Biggr\} $$ is the generating functional of the corresponding free theory. Here $$ \Delta_F(z - w) = \int \frac{dp}{(2\pi)^d}\ \frac{i}{p^2 - m^2} e^{ip(z - w)} $$ is Feynman's propagator of the scalar field $\phi$ and $\mathcal{N}$ is a normalization constant.
We can expand -in powers of $\lambda$- the exponential operator $$ \exp\Biggl\{-i\frac{\lambda}{4!} \int dx\ \frac{\delta^4}{\delta J(x)^4}\Biggr\} = \sum^\infty_{\ell=0} \frac{(-i)^\ell \lambda^\ell}{(4!)^\ell\ell!} \int dx_1 \cdots \int dx_\ell\ \frac{\delta^{4\ell}}{\delta J(x_1)^4 \cdots \delta J(x_\ell)^4}. $$ On the other hand, my problem is that:
I am not sure how to make a series expansion of the generating functional $Z_0[J]$ of the free theory.
I see two possibilities:
- According to equation (1.49) from this document (here, the author is working with the $\phi^3$-real scalar theory and the generating functional is called $W[J]$ instead of $Z[J]$), I should be able to make the expansion \begin{align} Z_0[J] &= \mathcal{N} \exp\Biggl\{-\frac{1}{2} \int dz \int dw\ J(z) \Delta_F(z - w) J(w)\Biggr\} \\ &= \mathcal{N} \sum^\infty_{k=0} \frac{1}{k!} \Biggl(-\frac{1}{2} \int dz \int dw\ J(z) \Delta_F(z - w) J(w)\Biggr)^k \\ &= \mathcal{N} \sum^\infty_{k=0} \frac{(-1)^k}{2^kk!} \int dz_1 \int dw_1 \cdots \int dz_k \int dw_k\ J(z_1)J(w_1) \cdots J(z_k)J(w_k) \Delta_F(z_1 - w_1) \cdots \Delta_F(z_k - w_k). \end{align} If this is correct, then we would have the functional expansion \begin{align} Z[J] &= \mathcal{N} \sum^\infty_{\ell=0} \sum^\infty_{k=0} \frac{(-1)^k(-i)^\ell \lambda^\ell}{(4!)^\ell\ell!2^kk!} \int dx_1 \cdots \int dx_\ell\ \int dz_1 \int dw_1 \cdots \int dz_k \int dw_k \\ &\quad \times \Delta_F(z_1 - w_1) \cdots \Delta_F(z_k - w_k) \frac{\delta^{4\ell}}{\delta J(x_1)^4 \cdots \delta J(x_\ell)^4} J(z_1)J(w_1) \cdots J(z_k)J(w_k). \end{align} Consequently, some lowest-order terms are \begin{align} \frac{Z[J]}{\mathcal{N}} &= \lambda^0\Biggl\{1 - \frac{1}{2} \int dz_1 \int dw_1\ \Delta_F(z_1 - w_1)J(z_1)J(w_1) + \cdots\Biggr\} \\ &\quad + \lambda\Biggl\{- \frac{i}{8} \int dx_1\ \Delta_F(x_1 - x_1)\Delta_F(x_1 - x_1) + \cdots\Biggr\} + \mathcal{O}(\lambda^2). \end{align}
- According to equation (92) from this document, the correct series expansion of $Z_0[J]$ is given by a Volterra series \begin{align} Z_0[J] &= \mathcal{N} \exp\Biggl\{-\frac{1}{2} \int dz \int dw\ J(z) \Delta_F(z - w) J(w)\Biggr\} \\ &= \sum^\infty_{k=0} \frac{1}{k!} \int dz_1 \cdots \int dz_k\ J(z_1) \cdots \delta J(z_k) \frac{\delta^kZ_0[J]}{\delta J(z_1) \cdots \delta J(z_k)}\Biggr|_{J=0}.\end{align} If this is correct, then we would have the functional expansion \begin{align} Z[J] &= \sum^\infty_{\ell=0} \sum^\infty_{k=0} \frac{(-i)^\ell \lambda^\ell}{(4!)^\ell\ell!k!} \int dx_1 \cdots \int dx_\ell\ \int dz_1 \cdots \int dz_k \\ &\quad \times \Biggl[\frac{\delta^{4\ell}}{\delta J(x_1)^4 \cdots \delta J(x_\ell)^4} J(z_1) \cdots J(z_k)\Biggr] \frac{\delta^kZ_0[J]}{\delta J(z_1) \cdots \delta J(z_k)}\Biggr|_{J=0}. \end{align} Consequently, some lowest-order terms are \begin{align} \frac{Z[J]}{\mathcal{N}} &= \lambda^0\Biggl\{1 - \frac{1}{2} \int dz_1 \int dz_2\ J(z_1)J(z_1)\Delta_F(z_1 - z_2) \\ &\quad - \frac{1}{4!}\int dz_1 \int dz_2 \int dz_3 \int dz_4\ J(z_1)J(z_2)J(z_3)J(z_4) \\ &\quad \times \Bigl[\Delta_F(z_1 - z_2)\Delta_F(z_3 - z_4) + \Delta_F(z_1 - z_3)\Delta_F(z_2 - z_4) + \Delta_F(z_1 - z_4)\Delta_F(z_2 - z_3)\Bigr] + \cdots\Biggr\} \\ &\quad + \lambda\Biggl\{\cdots\Biggr\} + \mathcal{O}(\lambda^2). \end{align} Thus, as far as I can see both expansion are not yielding the same result. I would like to know which one is the correct one and why is the other wrong; or maybe both are equivalent but I can't see it.
EDIT: if both expansion for the generating functional $Z_0[J]$ are equal, i.e. $$ \sum^\infty_{k=0} \frac{1}{k!} \int dz_1 \cdots \int dz_k\ J(z_1) \cdots \delta J(z_k) \frac{\delta^kZ_0[J]}{\delta J(z_1) \cdots \delta J(z_k)}\Biggr|_{J=0} = \mathcal{N} \sum^\infty_{k=0} \frac{(-1)^k}{2^kk!} \int dz_1 \int dw_1 \cdots \int dz_k \int dw_k\ J(z_1)J(w_1) \cdots J(z_k)J(w_k) \Delta_F(z_1 - w_1) \cdots \Delta_F(z_k - w_k) $$ then, could we set the following identification? $$ \frac{\delta^kZ_0[J]}{\delta J(z_1) \cdots \delta J(z_k)}\Biggr|_{J=0} = \mathcal{N} \frac{(-1)^k}{2^k} \int dw_1 \cdots \int dw_k\ J(w_1) \cdots J(w_k) \Delta_F(z_1 - w_1) \cdots \Delta_F(z_k - w_k) $$