Quantum Field Theory Course

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

Part IV, Chapter 3 | Page 2 of 3

QED Feynman Rules from First Principles

Deriving propagators, vertices, and external line factors from the QED Lagrangian

3.6 The Electron Propagator

The Feynman propagator for the Dirac field is the time-ordered two-point function:

$$S_F^{ab}(x-y) = \langle 0|T[\psi^a(x)\bar{\psi}^b(y)]|0\rangle$$

where a, b are spinor indices. To derive this, we start from the free Dirac equation$(i\gamma^\mu\partial_\mu - m)\psi = 0$. The propagator is the Green's function satisfying:

$$(i\gamma^\mu\partial_\mu - m)S_F(x-y) = i\delta^{(4)}(x-y)$$

Fourier transforming with $S_F(x-y) = \int\frac{d^4p}{(2\pi)^4}e^{-ip\cdot(x-y)}\tilde{S}_F(p)$:

$$(\gamma^\mu p_\mu - m)\tilde{S}_F(p) = i \qquad \Longrightarrow \qquad \tilde{S}_F(p) = \frac{i}{\not{p} - m}$$

To make sense of the pole at $p^2 = m^2$, we use the Feynman $i\epsilon$ prescription (positive-frequency modes propagate forward in time, negative-frequency backward):

$$\boxed{\tilde{S}_F(p) = \frac{i(\not{p} + m)}{p^2 - m^2 + i\epsilon}}$$

where we used $(\not{p} - m)(\not{p} + m) = p^2 - m^2$ to rationalize the denominator. The numerator $\not{p} + m$ is a $4 \times 4$ matrix in spinor space. This is the Feynman rule for an internal electron line carrying momentum p.

3.7 The Photon Propagator

The photon propagator is derived from the Maxwell Lagrangian. In momentum space, the equation of motion for $A_\mu$ (before gauge fixing) reads:

$$(-k^2 g^{\mu\nu} + k^\mu k^\nu)\tilde{A}_\nu(k) = J^\mu(k)$$

The operator $(-k^2 g^{\mu\nu} + k^\mu k^\nu)$ is not invertible because it has a zero eigenvalue (acting on $k^\nu$ gives zero). This reflects gauge invariance — physically equivalent configurations are counted multiple times in the path integral. We must fix a gauge.

Adding the gauge-fixing term $-\frac{1}{2\xi}(\partial_\mu A^\mu)^2$ to the Lagrangian, the equation becomes:

$$\Big[-k^2 g^{\mu\nu} + \Big(1 - \frac{1}{\xi}\Big)k^\mu k^\nu\Big]\tilde{A}_\nu = J^\mu$$

This is now invertible. The photon propagator (Green's function) in a general $R_\xi$ gauge is:

$$\boxed{\tilde{D}_F^{\mu\nu}(k) = \frac{-i}{k^2 + i\epsilon}\bigg(g^{\mu\nu} - (1-\xi)\frac{k^\mu k^\nu}{k^2}\bigg)}$$

In Feynman gauge ($\xi = 1$), this simplifies dramatically:

$$\tilde{D}_F^{\mu\nu}(k)\Big|_{\xi=1} = \frac{-ig^{\mu\nu}}{k^2 + i\epsilon}$$

Feynman gauge is almost always used in practice because of its simplicity. Physical results (S-matrix elements) are independent of the gauge parameter $\xi$, as guaranteed by the Ward identity (Page 3).

💡Why Does the Propagator Carry Lorentz Indices?

The photon is a spin-1 particle with polarization. The $g^{\mu\nu}$ in the numerator sums over all four polarization states, including the unphysical longitudinal and timelike polarizations. These unphysical modes cancel in any gauge-invariant amplitude — this cancellation is the content of the Ward identity. The $k^\mu k^\nu$ terms in the general-$\xi$ propagator project onto the longitudinal mode and drop out of physical amplitudes.

3.8 The Interaction Vertex

The vertex factor comes directly from the interaction Lagrangian. Recall:

$$\mathcal{L}_{\text{int}} = -e\bar{\psi}\gamma^\mu\psi A_\mu$$

In the S-matrix expansion, the first-order term is:

$$S^{(1)} = -ie\int d^4x\,\bar{\psi}_I(x)\gamma^\mu\psi_I(x)A_{\mu,I}(x)$$

Applying Wick's theorem and contracting each field with external particles, the spacetime integral over x produces a momentum-conserving delta function. The remaining algebraic factor at each vertex is:

$$\boxed{\text{QED vertex factor} = -ie\gamma^\mu}$$

accompanied by momentum conservation: $p_1 + p_2 + p_3 = 0$ (all momenta directed inward). The $\gamma^\mu$ matrix acts on the spinor indices and the Lorentz index $\mu$ is contracted with the photon propagator or polarization vector.

3.9 External Line Factors

External particles correspond to contracting a field operator with an asymptotic state. For example, contracting $\psi_I(x)$ with an incoming electron state $|e^-(p,s)\rangle$produces the spinor $u^s(p)e^{-ip\cdot x}$. The plane wave factor is absorbed into the momentum-conserving delta function, leaving the spinor as the external line factor.

$$\text{External fermion lines:}$$$$\begin{array}{ll} \text{Incoming } e^- : & u^s(p) \\ \text{Outgoing } e^- : & \bar{u}^s(p) \\ \text{Incoming } e^+ : & \bar{v}^s(p) \\ \text{Outgoing } e^+ : & v^s(p) \end{array}$$

The logic for positrons follows from the fact that $\bar{\psi}$ creates a positron (or annihilates an electron), while $\psi$ creates an electron (or annihilates a positron). External photon lines contribute their polarization vector:

$$\text{Incoming photon: } \epsilon_\mu(k), \qquad \text{Outgoing photon: } \epsilon_\mu^*(k)$$

3.10 Complete QED Feynman Rules

Assembling the results above, the complete set of momentum-space Feynman rules for QED is:

$$\text{1. Electron propagator: } \frac{i(\not{p}+m)}{p^2-m^2+i\epsilon}$$
$$\text{2. Photon propagator (Feynman gauge): } \frac{-ig_{\mu\nu}}{k^2+i\epsilon}$$
$$\text{3. Vertex: } -ie\gamma^\mu \text{ with 4-momentum conservation at each vertex}$$

Procedural rules for constructing amplitudes:

  • Draw all topologically distinct diagrams connecting the given external particles
  • Assign momenta to all lines, enforcing conservation at each vertex
  • Write the amplitude by reading each diagram: follow fermion lines backward (against the arrow), writing spinors, vertices, and propagators from left to right
  • Integrate over all undetermined loop momenta: $\int\frac{d^4\ell}{(2\pi)^4}$
  • Include a factor of $(-1)$ for each closed fermion loop (from anti-commuting Grassmann fields)
  • Divide by the symmetry factor of the diagram

💡Reading Feynman Diagrams

The key rule for QED amplitudes: trace backward along each fermion line. An electron entering on the left and exiting on the right gives a factor like$\bar{u}(p')(-ie\gamma^\mu)\frac{i(\not{q}+m)}{q^2-m^2+i\epsilon}(-ie\gamma^\nu)u(p)$. The overall expression reads naturally as: "electron with spinor u(p) interacts at the first vertex, propagates with momentum q, interacts at the second vertex, and exits with spinor $\bar{u}(p')$." Each photon line connecting two vertices contributes a propagator whose Lorentz indices contract with the $\gamma^\mu$ at each end.

Key Concepts

  • The electron propagator $i(\not{p}+m)/(p^2-m^2+i\epsilon)$ is the Green's function of the Dirac operator
  • The photon propagator requires gauge fixing; in Feynman gauge it is $-ig_{\mu\nu}/(k^2+i\epsilon)$
  • The vertex factor $-ie\gamma^\mu$ comes directly from the interaction Lagrangian $-e\bar{\psi}\gamma^\mu\psi A_\mu$
  • External lines contribute Dirac spinors u, v (and their conjugates) or polarization vectors $\epsilon_\mu$
  • Fermion lines are read backward (against the momentum arrow) when writing down the amplitude
  • Closed fermion loops carry an extra factor of $(-1)$ from Grassmann anti-commutation
  • Physical results are gauge-independent (guaranteed by the Ward identity)
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