Quantum Field Theory Course

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

Part III, Chapter 5 — Page 2 of 3

Feynman Rules Derivation

Systematically deriving position-space and momentum-space rules from the path integral

5.7 From the Path Integral to Feynman Rules

We now derive the Feynman rules systematically from the generating functional. For $\phi^4$ theory, the action is:

$$S[\phi] = \int d^4x \left[\frac{1}{2}(\partial_\mu\phi)^2 - \frac{1}{2}m^2\phi^2 - \frac{\lambda}{4!}\phi^4\right]$$

The generating functional is:

$$Z[J] = \int \mathcal{D}\phi \, \exp\left(iS_0[\phi] - i\frac{\lambda}{4!}\int d^4z\,\phi^4(z) + i\int d^4x\, J(x)\phi(x)\right)$$

The key insight is to replace $\phi(z)$ in the interaction by functional derivatives with respect to $J(z)$:

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

where $Z_0[J]$ is the free generating functional:

$$Z_0[J] = Z_0[0]\exp\left(-\frac{1}{2}\int d^4x\,d^4y\, J(x)\,D_F(x-y)\,J(y)\right)$$

💡The Master Formula

This is the central result: the full interacting generating functional is obtained by acting with a differential operator (the interaction) on the free generating functional. Every functional derivative brings down a propagator endpoint, and the exponential generates all possible contractions.

5.8 Position-Space Feynman Rules

Expanding the interaction exponential order-by-order in $\lambda$ and applying Wick's theorem, we obtain the position-space Feynman rules for $\phi^4$ theory.

The Propagator Rule

Each line (contraction) connecting spacetime points $x$ and $y$ contributes a Feynman propagator:

$$\boxed{\text{Propagator: } x \longrightarrow y \quad = \quad D_F(x-y) = \int \frac{d^4p}{(2\pi)^4}\frac{i}{p^2 - m^2 + i\epsilon}\,e^{-ip\cdot(x-y)}}$$

The Vertex Rule

Each interaction vertex where four lines meet at a point $z$ contributes:

$$\boxed{\text{Vertex at } z: \quad (-i\lambda)\int d^4z}$$

The factor $-i\lambda$ comes from $-i\frac{\lambda}{4!}$ in the action, multiplied by $4!$ from the number of ways to contract 4 identical fields at the vertex with the external propagators. We integrate over the vertex position $z$ because the interaction can occur anywhere in spacetime.

External Points

Each external point $x_i$ in an n-point function contributes a factor of 1. The propagator connects directly to the external point.

Putting It Together

For the n-point function at order $\lambda^V$ (V vertices):

$$G^{(n)}(x_1,\ldots,x_n)\Big|_{\lambda^V} = \sum_{\text{diagrams}} \frac{1}{S}\prod_{\text{lines}} D_F(x_i - x_j) \prod_{v=1}^{V}(-i\lambda)\int d^4z_v$$

where $S$ is the symmetry factor of the diagram.

5.9 Momentum-Space Feynman Rules

Fourier transforming to momentum space is far more practical. Insert the momentum representation of the propagator:

$$D_F(x-y) = \int \frac{d^4p}{(2\pi)^4}\frac{i}{p^2-m^2+i\epsilon}\,e^{-ip\cdot(x-y)}$$

At each internal vertex $z_v$, we integrate over $z_v$, which produces a delta function enforcing momentum conservation:

$$\int d^4z_v \, e^{-i(p_1+p_2+p_3+p_4)\cdot z_v} = (2\pi)^4\delta^4(p_1+p_2+p_3+p_4)$$

💡Conservation at Vertices

The spacetime integral over each vertex position enforces 4-momentum conservation at that vertex. This is a direct consequence of translation invariance of the action. Every Feynman vertex conserves energy and momentum individually, not just globally.

The complete momentum-space Feynman rules for $\phi^4$ theory are:

Complete Momentum-Space Rules

  1. Internal propagator carrying momentum $p$:
    $$\frac{i}{p^2 - m^2 + i\epsilon}$$
  2. Vertex: factor of $-i\lambda$
  3. Momentum conservation at each vertex: $(2\pi)^4\delta^4(\sum p_i)$
  4. Loop integration: For each undetermined loop momentum $k$:
    $$\int \frac{d^4k}{(2\pi)^4}$$
  5. External lines: factor of 1 for each external leg
  6. Symmetry factor: divide by $S$
  7. Overall factor: $(2\pi)^4\delta^4(\sum p_{\text{ext}})$ for overall momentum conservation

Counting Loops

The number of independent loop momenta is determined by:

$$\boxed{L = I - V + 1}$$

where $I$ is the number of internal propagators and $V$ is the number of vertices. This is because each vertex gives one delta function (constraint), but one overall delta function is for global momentum conservation, leaving $I - (V - 1) = I - V + 1$ undetermined loop momenta. For connected diagrams with $E$ external legs, we also have $L = I - V + 1$, since $E$ external momenta are fixed.

5.10 Symmetry Factors: Systematic Derivation

Symmetry factors arise from the mismatch between the combinatorial factors in the Wick expansion and the naive Feynman rule factor. Let us derive them carefully.

In the perturbative expansion, a term at order $\lambda^V$ comes with the prefactor:

$$\frac{1}{V!}\left(-\frac{i\lambda}{4!}\right)^V$$

The $1/V!$ is from the Taylor expansion of the exponential, and $(1/4!)^V$ is from the Lagrangian. When we perform Wick contractions, we get combinatorial factors from:

  • Permuting identical vertices: $V!$ ways (cancels the $1/V!$)
  • Permuting legs at each vertex: $(4!)^V$ for all vertices (partially cancels the $(1/4!)^V$)

The symmetry factor $S$ is what remains uncanceled:

$$\boxed{S = |\text{Aut}(\Gamma)| = \text{number of automorphisms of diagram } \Gamma}$$

💡What Are Automorphisms?

An automorphism of a Feynman diagram is a permutation of internal lines and vertices that leaves the diagram topologically unchanged. The more symmetric a diagram, the larger its symmetry factor, and the more "overcounted" it is in the naive expansion.

Common Symmetry Factors in $\phi^4$ Theory

  • Tadpole (self-loop at vertex): $S = 2$ — can exchange the two ends of the loop
  • Bubble diagram (two propagators between same vertices): $S = 2$ — can swap the two propagators
  • Sunset diagram (three propagators between two vertices): $S = 3! = 6$ — can permute three propagators
  • Figure-eight vacuum diagram: $S = 2^3 = 8$ — two self-loops plus vertex exchange

5.11 Connected Diagrams and Vacuum Bubbles

An important simplification: the denominator $Z[0]$ exactly cancels all vacuum bubble diagrams (diagrams with no external legs). This is the content of the linked-cluster theorem:

$$\frac{\langle 0|T\{\phi(x_1)\cdots\phi(x_n)\,e^{-i\int H_{\text{int}}}\}|0\rangle}{\langle 0|T\{e^{-i\int H_{\text{int}}}\}|0\rangle} = \sum_{\text{connected diagrams with } x_1,\ldots,x_n \text{ attached}}$$

This means we only need to sum connected diagrams when computing correlation functions. The generating functional for connected diagrams is:

$$W[J] = -i\ln Z[J] = \sum_{\text{connected diagrams}}$$

Furthermore, the 1PI (one-particle irreducible) effective action $\Gamma[\phi_c]$, obtained by Legendre transform of $W[J]$, generates only diagrams that cannot be disconnected by cutting a single propagator. This hierarchical organization — full diagrams, connected diagrams, 1PI diagrams — is central to the renormalization program.

Key Concepts — Feynman Rules

  • Feynman rules are derived by expanding the path integral in the coupling and applying Wick's theorem
  • Position-space: propagator $D_F(x-y)$, vertex $(-i\lambda)\int d^4z$
  • Momentum-space: propagator $i/(p^2-m^2+i\epsilon)$, vertex $-i\lambda$, loop integral $\int d^4k/(2\pi)^4$
  • Momentum conservation holds at every vertex from translation invariance
  • Loop count: $L = I - V + 1$
  • Symmetry factor $S$ = number of automorphisms of the diagram
  • Vacuum bubbles cancel between numerator and denominator
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