Quantum Field Theory Course

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

Part III, Chapter 3

Path Integrals in QFT

The generating functional Z[J] and n-point correlation functions

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3.3 The Generating Functional Z[J]

To extract correlation functions from the path integral, we introduce an external source $J(x)$ coupled linearly to the field:

$$\boxed{Z[J] = \int \mathcal{D}\phi\, \exp\left\{i\int d^4x\left[\mathcal{L}(\phi) + J(x)\phi(x)\right]\right\}}$$

The source J(x) is an arbitrary function that we set to zero at the end of the calculation. Its role is purely a bookkeeping device: functional derivatives with respect to J pull down factors of the field $\phi$.

💡Sources as Probes

Think of J(x) as an external "knob" that you can turn at every spacetime point. By gently wiggling J and seeing how Z responds, you learn about the quantum field's correlation structure. This is analogous to tapping a drum and listening to the response to determine its vibrational modes.

n-Point Functions from Functional Derivatives

The n-point correlation function (Green's function) is obtained by differentiating Z[J] n times:

$$\boxed{\langle 0|T\{\phi(x_1)\cdots\phi(x_n)\}|0\rangle = \frac{1}{Z[0]}\left(\frac{1}{i}\right)^n \frac{\delta^n Z[J]}{\delta J(x_1)\cdots \delta J(x_n)}\Bigg|_{J=0}}$$

Each functional derivative $\delta/\delta J(x_k)$ "inserts" a field operator$\phi(x_k)$ into the path integral. The factor of 1/i per derivative comes from the $e^{i\int J\phi}$ coupling.

3.4 Completing the Square for Free Fields

For the free Klein-Gordon field, the generating functional with source is:

$$Z_0[J] = \int \mathcal{D}\phi\, \exp\left\{i\int d^4x\left[-\frac{1}{2}\phi(\Box + m^2 - i\epsilon)\phi + J\phi\right]\right\}$$

We complete the square by shifting the integration variable. Define the shifted field:

$$\phi(x) = \phi'(x) + \int d^4y\, D_F(x-y)\,J(y)$$

where $D_F(x-y)$ is the Feynman propagator satisfying$(\Box_x + m^2 - i\epsilon)D_F(x-y) = -\delta^4(x-y)$. Substituting:

$$-\frac{1}{2}\phi(\Box+m^2)\phi + J\phi = -\frac{1}{2}\phi'(\Box+m^2)\phi' + \frac{1}{2}\int d^4x\,d^4y\; J(x)\,D_F(x-y)\,J(y)$$

The cross terms cancel perfectly. Since $\mathcal{D}\phi = \mathcal{D}\phi'$ (the measure is translation-invariant), the $\phi'$ integral factors out as $Z_0[0]$:

$$\boxed{Z_0[J] = Z_0[0]\,\exp\left[\frac{i}{2}\int d^4x\,d^4y\; J(x)\,D_F(x-y)\,J(y)\right]}$$

💡Why This Result is Remarkable

The entire content of free field theory is encoded in a single exponential involving the propagator $D_F$. All n-point functions follow from differentiating this compact expression. The propagator is the only "building block" needed.

3.5 Explicit Calculation: 2-Point Function

Let us verify the formalism by computing the 2-point function. We need two functional derivatives of $Z_0[J]$:

$$\frac{1}{i}\frac{\delta Z_0[J]}{\delta J(x_1)} = Z_0[J]\int d^4y\, D_F(x_1-y)\,J(y)$$

Taking the second derivative and setting J = 0:

$$\frac{1}{Z_0[0]}\left(\frac{1}{i}\right)^2 \frac{\delta^2 Z_0[J]}{\delta J(x_1)\delta J(x_2)}\Bigg|_{J=0} = D_F(x_1 - x_2)$$

Therefore:

$$\boxed{\langle 0|T\{\phi(x_1)\phi(x_2)\}|0\rangle = D_F(x_1 - x_2) = \int\frac{d^4k}{(2\pi)^4}\frac{i}{k^2 - m^2 + i\epsilon}\,e^{-ik\cdot(x_1-x_2)}}$$

This reproduces the Feynman propagator obtained from canonical quantization and Wick's theorem.

3.6 The 4-Point Function and Wick's Theorem

For the free-field 4-point function, we need four derivatives:

$$\langle 0|T\{\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\}|0\rangle_0 = \frac{1}{Z_0[0]}\left(\frac{1}{i}\right)^4 \frac{\delta^4 Z_0[J]}{\delta J(x_1)\cdots\delta J(x_4)}\Bigg|_{J=0}$$

Since $Z_0[J] = Z_0[0]\exp[\frac{i}{2}JDJ]$, the fourth derivative produces all possible pairings:

$$\langle 0|T\{\phi_1\phi_2\phi_3\phi_4\}|0\rangle_0 = D_F(x_1-x_2)D_F(x_3-x_4) + D_F(x_1-x_3)D_F(x_2-x_4) + D_F(x_1-x_4)D_F(x_2-x_3)$$

This is exactly Wick's theorem! The free-field n-point function equals the sum over all possible pairings (contractions) of field operators. For 2n fields, there are $(2n-1)!! = (2n-1)(2n-3)\cdots 3\cdot 1$ such pairings.

General Pattern: Odd n Vanishes

For odd n, the n-point function of the free field vanishes identically:

$$\langle 0|T\{\phi(x_1)\cdots\phi(x_{2n+1})\}|0\rangle_0 = 0$$

This follows because $Z_0[J]$ is an even functional of J (the exponent is quadratic in J), so odd derivatives vanish at J = 0. Physically, this reflects the$\phi \to -\phi$ symmetry of the free theory.

3.7 Generating Functional for Interacting Theories

For the interacting theory $\mathcal{L} = \mathcal{L}_0 + \mathcal{L}_{\text{int}}$with $\mathcal{L}_{\text{int}} = -(\lambda/4!)\phi^4$, the key insight is:

$$Z[J] = \exp\left[i\int d^4x\;\mathcal{L}_{\text{int}}\left(\frac{1}{i}\frac{\delta}{\delta J(x)}\right)\right] Z_0[J]$$

This remarkable formula says: replace every $\phi(x)$ in the interaction Lagrangian by the functional derivative $(1/i)\delta/\delta J(x)$, then act on the free generating functional. Expanding in powers of $\lambda$:

$$Z[J] = \left[1 - \frac{i\lambda}{4!}\int d^4z\left(\frac{1}{i}\frac{\delta}{\delta J(z)}\right)^4 + O(\lambda^2)\right] Z_0[J]$$

Each term generates Feynman diagrams at a given order in $\lambda$. The functional derivatives act on $Z_0[J]$ and produce all possible Wick contractions, recovering the diagrammatic expansion systematically.

Key Concepts (This Page)

  • Z[J] generates all n-point functions via functional differentiation
  • Completing the square gives $Z_0[J] = Z_0[0]\exp[\frac{i}{2}JD_FJ]$
  • 2-point function reproduces the Feynman propagator $D_F(x-y)$
  • 4-point function yields Wick's theorem: sum over all contractions
  • Odd n-point functions vanish for the free field ($\phi\to-\phi$ symmetry)
  • Interactions enter via $\mathcal{L}_{\text{int}}(\delta/i\delta J)$ acting on $Z_0[J]$
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