Functional Derivation of Perturbation Theory

The functional derivation replaces each field in the interaction by a source derivative acting on the free Gaussian generating functional. Expanding that differential operator produces interaction vertices; differentiating the Gaussian produces every free contraction with the same multiplicities as Wick’s theorem. Normalization by the zero-source result removes vacuum bubbles, and $-i\log Z$ selects connected diagrams. The method is equivalent to the operator derivation when the action, state, ordering, regulator, and boundary prescription match.

Required background. Momentum-Space Feynman Rules fixes the action-to-rule conventions, while The Generating Functional fixes source differentiation and normalization.

Helpful background. Gaussian Fields and Sources develops the regulated completion of the free functional integral.

The free Gaussian functional

For a canonically normalized real scalar with the Feynman boundary prescription, normalize the free functional by $Z_0[0]=1$:

$$Z_0[J] =\exp\!\left[ -\frac12\int\mathrm d^d x\,\mathrm d^d y\, J(x)D_F(x-y)J(y) \right],$$

where

$$D_F(x-y) =\int\frac{\mathrm d^d p}{(2\pi)^d} \frac{i\,e^{-ip\cdot(x-y)}}{p^2-m^2+i0}.$$

Since

$$\frac{\delta}{\delta J(x)} e^{i\int J\phi} =i\phi(x)e^{i\int J\phi},$$

a field inside the functional integral may be represented by

$$\phi(x)\longrightarrow \frac1i\frac{\delta}{\delta J(x)}.$$

The source definition, normalized derivatives, and interacting differential operator are derived in Schwartz 2014, § 14.3, pp. 261–264.

Repeated differentiation of $Z_0$ proves the pairing rule. For example,

$$(-i)^4 \left. \frac{\delta^4 Z_0} {\delta J_1\delta J_2\delta J_3\delta J_4} \right|_{J=0} =D_{12}D_{34}+D_{13}D_{24}+D_{14}D_{23}.$$

Odd derivatives vanish at $J=0$ because the Gaussian is centered. The result is the vacuum-correlator consequence of Wick’s theorem; it does not by itself reproduce the theorem’s normal-ordered operator terms.

Turn the interaction into a differential operator

Split $S=S_0+S_{\mathrm{int}}$. Then

$$Z[J] =\frac{ \exp\!\left[ iS_{\mathrm{int}}\!\left( \frac1i\frac{\delta}{\delta J} \right) \right]Z_0[J] }{ \left. \exp\!\left[ iS_{\mathrm{int}}\!\left( \frac1i\frac{\delta}{\delta J} \right) \right]Z_0[J] \right|_{J=0} }, \qquad Z[0]=1.$$

For

$$S_{\mathrm{int}} =-\int\mathrm d^d x\,\frac{\lambda}{4!}\phi^4,$$

this becomes

$$Z[J] =\frac{ \exp\!\left[ -\frac{i\lambda}{4!} \int\mathrm d^d x\, \left(\frac1i\frac{\delta}{\delta J(x)}\right)^4 \right]Z_0[J] }{ \left. \exp\!\left[ -\frac{i\lambda}{4!} \int\mathrm d^d x\, \left(\frac1i\frac{\delta}{\delta J(x)}\right)^4 \right]Z_0[J] \right|_{J=0} }.$$

Every power of $\lambda$ supplies an integrated four-valent vertex. Every derivative acting on the quadratic exponent either pairs with another derivative to produce $D_F$ or leaves a factor linear in $J$ that must be removed by later derivatives. Setting $J=0$ keeps only complete contractions. The $1/4!$ and Taylor $1/V!$ are therefore canceled in exactly the same way as in the operator expansion, leaving the graph symmetry factor.

This formula also keeps two expansions conceptually separate. Expanding the interaction operator counts powers of the coupling and creates vertices; differentiating with respect to the external source creates the requested correlator legs. One should perform both operations before setting $J=0$. Setting the source to zero too early retains only the vacuum functional, while treating an external derivative as another interaction insertion gives the wrong Taylor factors.

Recover correlators and connected graphs

The normalized time-ordered correlators are

$$G^{(n)}(x_1,\ldots,x_n) =\left. (-i)^n \frac{\delta^n Z[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0}.$$

Choose one ultraviolet regulator and use it consistently in the numerator and denominator; in the following formulas, $D_F(0)$ denotes the corresponding regulated coincidence limit. At first order in $\lambda$, differentiating twice reproduces the two classes found by operator Wick expansion:

$$-\frac{i\lambda}{2} \int\mathrm d^d z\, D_F(x-z)D_F(0)D_F(z-y)$$

and a factor proportional to

$$D_F(x-y) \left[-\frac{i\lambda}{8} \int\mathrm d^d z\,D_F(0)^2\right].$$

The second class also occurs in the denominator and cancels before the regulator is removed. The first remains connected to both external derivatives and still requires the usual ultraviolet renormalization. This provides a direct coefficient check independent of the operator manipulation.

Now define

$$W[J]=-i\log Z[J].$$

Then

$$G_c^{(n)}(x_1,\ldots,x_n) =\left. \frac{1}{i^{n-1}} \frac{\delta^n W[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0}$$

contains only graphs connected to all $n$ external insertions. Exponentiating $iW$ reconstructs the partitions of the full correlator into connected blocks. This is the functional form of the linked-cluster relation developed on Connected, Disconnected, and Vacuum Diagrams.

Momentum-space rules from the same operator

Fourier transforming after differentiation sends each free contraction to $i/(p^2-m^2+i0)$. The integral over each local interaction point produces $(2\pi)^d\delta^{(d)}(\sum p)$, and derivatives in $S_{\mathrm{int}}$ act as $-ip_\mu$ on their own fields. Thus the functional method reproduces the same propagators, vertices, and loop measures as the operator method.

This gives a useful round trip:

Step	Functional statement	Diagrammatic statement
free inverse	quadratic Gaussian kernel	propagator with boundary prescription
interaction expansion	powers of $iS_{\mathrm{int}}[(1/i)\delta_J]$	integrated vertices
source differentiation	pairings from the Gaussian	internal and external lines
divide by zero-source value	remove common vacuum factor	cancel vacuum bubbles
take $-i\log$	cumulant generator	select connected external graphs
Fourier transform	convolutions become products and deltas	momentum-space rules

Failure of any row to reproduce the operator calculation signals a convention mismatch, a missing measure factor, or an illegitimate manipulation of distributions.

Fermions and ordered Grassmann sources

For Dirac fields introduce independent odd sources $\eta$ and $\bar\eta$. The free functional is exponential in $\bar\eta S_F\eta$, and interactions are obtained by replacing $\psi$ and $\bar\psi$ with ordered left/right derivatives. Because these derivatives anticommute, their order generates external exchange signs and the minus associated with closed fermion loops.

The determinant check is especially useful. Integrating a quadratic Dirac field in a bosonic background gives $\det(M_0+V)$ rather than the inverse square root produced by a commuting Gaussian. Expanding $\operatorname{Tr}\log(1+M_0^{-1}V)$ yields cyclic traces, with the Grassmann Gaussian supplying the statistics sign of each closed loop. This check applies only after the source order and the normalization of the interaction operator have been matched to the operator calculation.

One must state the source term and derivative convention before quoting a rule. Reversing the source order changes intermediate signs but cannot change a consistently computed amplitude. Srednicki’s Dirac-field derivation displays the ordered functional operator and its propagators in Srednicki 2007, § 45, pp. 282–291.

Equivalence and its limits

The operator and functional derivations agree when they describe the same regulated theory and the same in–out vacuum boundary condition. The equivalence is checked by matching:

• the free two-point distribution including $+i0$;
• interaction coefficients and derivative placements;
• vacuum normalization;
• external and Grassmann ordering; and
• contact or measure terms for derivative and constrained systems.

The compact functional formula is formal before a regulator and integration contour are specified. It does not prove convergence of the functional measure or perturbative series. Nor does $W[J]$ here equal the quantum effective action: the latter is a Legendre transform that organizes one-particle-irreducible graphs, which lie beyond this page.

Common pitfalls

Using $\delta/\delta J$ instead of $(1/i)\delta/\delta J$. The source appears as $e^{iJ\phi}$, so every field replacement carries $1/i$. Missing it changes vertex phases.

Forgetting the normalization denominator. The unnormalized functional contains vacuum bubbles. Divide by the zero-source value before interpreting source derivatives as normalized correlators.

Calling $W$ the effective action. $W=-i\log Z$ generates connected correlators. The effective action is obtained only after a Legendre transform under additional assumptions.

Check your understanding

Apply four source derivatives to $Z_0[J]$ and set $J=0$. Obtain the three scalar pairings and account for their multiplicities.

Solution

Expand the Gaussian through fourth order in the source:

$$Z_0[J]=1-\frac12 JD_FJ+\frac{1}{2!}\left(-\frac12JD_FJ\right)^2+O(J^6).$$

Only the quadratic square survives four derivatives at $J=0$. For any selected pairing, the derivatives can be assigned to the two quadratic factors in $2!$ ways and to the two source slots inside each factor in $2^2$ ways. These $2^2 2!$ assignments cancel the denominator $2!\,2^2$, leaving each of $D_{12}D_{34}$, $D_{13}D_{24}$, and $D_{14}D_{23}$ once. The prefactor $(-i)^4$ converts the source derivatives to the stated correlator convention.

Where to continue

• Compare connected and full functionals: Connected, Disconnected, and Vacuum Diagrams derives the partition structure in detail.
• Apply ordered odd sources: Fermion Signs and Closed Loops checks the graded signs against operator permutations.
• Import a gauge-fixed action: Gauge-Fixed Perturbation Rules, Ghost Diagrams, and Identity Checks applies the same machinery only after gauge fixing and the Faddeev–Popov determinant are declared.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.