Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part III, Chapter 3

Path Integrals in QFT

Connected diagrams, W[J], the effective action, and Dyson-Schwinger equations

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3.8 Connected Generating Functional W[J]

The generating functional Z[J] produces all diagrams, connected and disconnected. It is far more efficient to work with the functional that generates only connected correlation functions. We define:

$$\boxed{Z[J] = e^{iW[J]} \quad \Longleftrightarrow \quad W[J] = -i\ln Z[J]}$$

The functional W[J] generates connected n-point functions:

$$\langle\phi(x_1)\cdots\phi(x_n)\rangle_c = \left(\frac{1}{i}\right)^{n-1}\frac{\delta^n W[J]}{\delta J(x_1)\cdots\delta J(x_n)}\Bigg|_{J=0}$$

The Linked Cluster Theorem

Theorem: The logarithm of the sum of all diagrams equals the sum of all connected diagrams.

Proof sketch: Suppose the set of all connected diagrams is$\{C_1, C_2, C_3, \ldots\}$. A general (possibly disconnected) diagram is a product of connected pieces. The sum of all diagrams is:

$$Z[J] = \sum_{\{n_k\}} \prod_k \frac{(C_k)^{n_k}}{n_k!} = \prod_k \sum_{n_k=0}^{\infty} \frac{(C_k)^{n_k}}{n_k!} = \prod_k e^{C_k} = \exp\left(\sum_k C_k\right)$$

The $1/n_k!$ is the symmetry factor for $n_k$ identical disconnected pieces. Therefore $iW[J] = \sum_k C_k$, confirming that W generates only connected diagrams.

💡Why Connected Diagrams Matter

Disconnected diagrams describe independent processes happening in different regions of spacetime. They carry no new information beyond what is already contained in the connected pieces. For S-matrix elements, only connected diagrams contribute to actual scattering events.

Example: Free Field W[J]

For the free field, $Z_0[J] = Z_0[0]\exp[\frac{i}{2}JD_FJ]$, so:

$$W_0[J] = -i\ln Z_0[0] + \frac{1}{2}\int d^4x\,d^4y\; J(x)\,D_F(x-y)\,J(y)$$

The connected 2-point function is simply $D_F(x-y)$, and all higher connected n-point functions vanish in the free theory (as expected - interactions are needed for nontrivial correlations).

3.9 The Effective Action $\Gamma[\varphi]$

We now define the classical field (vacuum expectation value in the presence of the source):

$$\varphi(x) \equiv \langle\phi(x)\rangle_J = \frac{\delta W[J]}{\delta J(x)}$$

The effective action $\Gamma[\varphi]$ is defined as the Legendre transform of W[J]:

$$\boxed{\Gamma[\varphi] = W[J] - \int d^4x\; J(x)\,\varphi(x)}$$

where J(x) on the right-hand side is understood as a functional of $\varphi$(by inverting the relation $\varphi = \delta W/\delta J$). The key property is:

$$\frac{\delta\Gamma[\varphi]}{\delta\varphi(x)} = -J(x)$$

When J = 0, the physical vacuum satisfies $\delta\Gamma/\delta\varphi = 0$. This is the quantum equation of motion, which includes all loop corrections!

1PI Diagrams and the Effective Action

The effective action generates one-particle irreducible (1PI) diagrams. A diagram is 1PI if it cannot be disconnected by cutting a single internal line. The n-th functional derivative gives the 1PI n-point vertex function:

$$\Gamma^{(n)}(x_1,\ldots,x_n) = \frac{\delta^n\Gamma[\varphi]}{\delta\varphi(x_1)\cdots\delta\varphi(x_n)}\Bigg|_{\varphi=\varphi_0}$$

💡Tree-Level Effective Action = Classical Action

At tree level (zeroth order in $\hbar$), the effective action equals the classical action: $\Gamma[\varphi] = S[\varphi] + O(\hbar)$. Loop corrections systematically dress the classical vertices. The full $\Gamma[\varphi]$ is the "quantum-corrected classical action" that encodes all the physics.

Inverse Propagator from $\Gamma$

The 2-point vertex function is the inverse propagator:

$$\int d^4z\;\Gamma^{(2)}(x,z)\,G_c(z,y) = \delta^4(x-y)$$

where $G_c(x,y)$ is the connected (full) propagator. In momentum space:

$$\Gamma^{(2)}(p) = G_c(p)^{-1} = p^2 - m^2 - \Sigma(p^2)$$

where $\Sigma(p^2)$ is the self-energy (sum of all 1PI two-point diagrams). The full propagator is thus:

$$\boxed{G_c(p) = \frac{i}{p^2 - m^2 - \Sigma(p^2) + i\epsilon}}$$

3.10 Dyson-Schwinger Equations

The Dyson-Schwinger equations are the quantum equations of motion derived from the path integral. They follow from the fact that the integral of a total derivative vanishes:

$$0 = \int \mathcal{D}\phi\;\frac{\delta}{\delta\phi(x)}\left[e^{i(S[\phi]+\int J\phi)}\right]$$

Carrying out the differentiation:

$$\left\langle\frac{\delta S[\phi]}{\delta\phi(x)} + J(x)\right\rangle_J = 0$$

For $\phi^4$ theory, this gives an exact (non-perturbative) relation:

$$(\Box_x + m^2)\langle\phi(x)\rangle_J + \frac{\lambda}{3!}\langle\phi^3(x)\rangle_J = J(x)$$

The Dyson-Schwinger equations relate n-point functions to (n+2)-point functions, forming an infinite tower of coupled equations. Truncating this tower at finite order gives non-perturbative approximation schemes.

Key Concepts (This Page)

  • $W[J] = -i\ln Z[J]$ generates connected diagrams only (linked cluster theorem)
  • Effective action $\Gamma[\varphi]$ is the Legendre transform of W[J]
  • $\Gamma[\varphi]$ generates 1PI vertex functions
  • $\Gamma^{(2)}$ is the inverse propagator: encodes the self-energy
  • Dyson-Schwinger equations are exact quantum equations of motion from the path integral
  • Hierarchy: Z[J] (all diagrams) $\to$ W[J] (connected) $\to$ $\Gamma[\varphi]$ (1PI)
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