# Correct way to expand the generating functional

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:

1. 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}
2. 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? $$\frac{\delta^kZ_0[J]}{\delta J(z_1) \cdots \delta J(z_k)}\Biggr|_{J=0} = \mathcal{N} \frac{(-1)^k}{2^k} \int dw_1 \cdots \int dw_k\ J(w_1) \cdots J(w_k) \Delta_F(z_1 - w_1) \cdots \Delta_F(z_k - w_k)$$

The two series for $$Z_0[J]$$ are equivalent. This generating functional is defined by : \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 \\ \end{align} Then, the second series expansion is Taylor's formula : \begin{align} Z_0[J]&= \sum^\infty_{k=0} \frac{1}{k!} \int dz_1 \cdots \int dz_k\ J(z_1) \cdots J(z_k) \frac{\delta^kZ_0[J]}{\delta J(z_1) \cdots \delta J(z_k)}\Biggr|_{J=0}\end{align}

This formula actually holds for any functional $$F[J]$$ (which admits a formal series expansion) :

\begin{align} F[J]&= \sum^\infty_{k=0} \frac{1}{k!} \int dz_1 \cdots \int dz_k\ J(z_1) \cdots J(z_k) \frac{\delta^kF[J]}{\delta J(z_1) \cdots \delta J(z_k)}\Biggr|_{J=0}\end{align}

To prove Taylor's formula, we assume that we can expand $$F$$ as : $$F[J] = \sum_{k=0}^\infty \int dz_1\ldots\int dz_kF_k(z_1,\ldots,z_k)J(z_1)\ldots J(z_k)$$

where the $$F_k$$ are integral kernels. The calculations should work even if the $$F_k$$ are higher order distributions, but this is not needed here.

Then, we compute its $$k$$th functional derivative at $$J = 0$$. This vanishes except on the term which contains exactly $$k$$ insertions of $$J$$. Therefore : $$\left.\frac{\delta^k F[J]}{\delta J(z_1)\ldots \delta J(z_k)}\right|_{J=0} = \sum_{\sigma \in \mathfrak S_k}F_k(z_{\sigma(1)},\ldots z_{\sigma(k)})$$ where the sum runs over all permutation of $$\{1,\ldots,k\}$$. When we integrate over $$z_1,\ldots,z_k$$, we can relabel the variables : \begin{align} \int dz_1 \cdots \int dz_k\ J(z_1) \cdots J(z_k) \frac{\delta^kF[J]}{\delta J(z_1) \cdots \delta J(z_k)}\Biggr|_{J=0} &=\sum_{\sigma \in \mathfrak S_k}\int dz_1 \cdots \int dz_k\ J(z_1) \cdots J(z_k)F_k(z_{\sigma(1)},\ldots z_{\sigma(k)}) \\ &= k! \int dz_1\ldots\int dz_kF_k(z_1,\ldots,z_k)J(z_1)\ldots J(z_k) \end{align} Dividing by $$k!$$ and summing over $$k$$, we see that Taylor's formula holds.

NB : the same calculations would have worked directly on $$Z_0[J]$$ but the precise expression for the kernels $$F_k(z_1,\ldots,z_k)$$ is hard to write down formally (Wick's theorem). Since it's precise form is not needed, it is easier to do the proof in a more general setting.

Edit : The expression for the functional derivatives of the free generating functional are given, for $$n$$ an even integer, by : $$\begin{gather} \left.\frac{\delta^{n} Z_0[J]}{\delta J(z_1) \ldots \delta J(z_{n})}\right|_{J=0} &= \left(-\frac 12\right)^n\sum_{\sigma \in\mathfrak S_{n}} \Delta_F(z_{\sigma(1)}-z_{\sigma(2)})\ldots \Delta_F(z_{\sigma(n-1)} -z_{\sigma(n)}) \end{gather}$$ For $$n$$ odd, the functional derivative vanishes.

• I understood your steps but unfortunately the only thing I see is a derivation of the Volterra series for $Z_0[J]$, i.e., how the kernels $F_k$ are related to de functional derivatives at $J=0$. I still can't see how the two series expansions of $Z_0[J]$ are equal. If this is true (which I'm also inclined to think so), then shouldn't we able to make the identification that I made in the edit of the question? Jun 30, 2022 at 20:04
• The equation in your edit cannot be true, since it contains some $J$ insertions in the RHS, which are set to zero. I edited in the correct formula. Jun 30, 2022 at 21:37
• Yes, absolutely. I thought I was comparing coefficients of series, but now I see I'm comparing two integrands which, of course, are not necessarily equal, i.e., $\int dx\ f(x) = \int dx\ g(x)\ \nRightarrow\ f(x) = g(x)$. Also, from your edit (Wick theorem) I was able to see that, after a change of variables, the Volterra series of $Z_0$ will reduce to its Taylor series. Thanks for your help! Jul 1, 2022 at 2:37