Feynman Propagator for the Dirac Field

Hello! Welcome back to our study of Quantum Field Theory.

In our last lesson, we constructed the Fock space for the Dirac field. We saw how imposing anti-commutation relations on the field operators naturally leads to Fermi-Dirac statistics and the Pauli exclusion principle, providing a complete description of the state space for non-interacting fermions.

Now that we have the field operators and the states they act upon, our next task is to describe how a free fermion propagates from one point in spacetime to another. This is the central role of the propagator. Today's lesson is dedicated to deriving the Feynman propagator for the Dirac field. This is a crucial step, as the propagator is not only a Green's function for the free theory's equation of motion but also the fundamental building block for describing particle interactions in Feynman diagrams.

1. The Dirac Field in the Heisenberg Picture

To discuss propagation between two spacetime points, $x$ and $y$ , we need to work in the Heisenberg picture where the operators evolve in time. The time-dependent Dirac field operator $ψ (x)$ and its Dirac adjoint $ψ ˉ (x)$ are given by the same plane-wave expansions we used before, but now with the full spacetime dependence in the exponents:

ψ (x) = s = 1, 2 \sum \int \frac{d ^{3} p}{( 2 π ) ^{3}} \frac{1}{2 E _{p}} (b_{p}^{s} u^{s} (p) e^{- i p \cdot x} + c_{p}^{s †} v^{s} (p) e^{+ i p \cdot x})

ψ ˉ (x) = s = 1, 2 \sum \int \frac{d ^{3} p}{( 2 π ) ^{3}} \frac{1}{2 E _{p}} (b_{p}^{s †} u ˉ^{s} (p) e^{+ i p \cdot x} + c_{p}^{s} v ˉ^{s} (p) e^{- i p \cdot x})

Here, $p^{0} = E_{p} = ∣ p ∣^{2} + m^{2}$ , and the creation and annihilation operators $b, b^{†}, c, c^{†}$ are now fixed, time-independent operators.

2. Causality and Field Anti-Commutators

For the scalar field, we found that the commutator $[ϕ (x), ϕ (y)]$ vanishes for spacelike separations, ensuring that measurements at $x$ and $y$ cannot affect each other. Let's compute the analogous quantity for the Dirac field, which is the anti-commutator ${ψ_{α} (x), ψ ˉ_{β} (y)}$ . This object is a 4x4 matrix in spinor space.

The detailed calculation involves substituting the field expansions and using the anti-commutation relations for the creation/annihilation operators, along with the spinor completeness relations you've seen before ( $\sum_{s} u^{s} (p) u ˉ^{s} (p) = \neq p + m$ and $\sum_{s} v^{s} (p) v ˉ^{s} (p) = \neq p - m$ ). The result is non-zero and can be related to the scalar field propagator.

Crucially, for a spacelike separation $(x - y)^{2} < 0$ , the anti-commutator vanishes:

{ψ_{α} (x), ψ ˉ_{β} (y)} = 0 for (x - y)^{2} < 0

This might seem weaker than the bosonic case, as the fields themselves don't commute. However, all physical observables in a theory of fermions are constructed from an even number of fermion fields (e.g., the charge current $j^{μ} = ψ ˉ γ^{μ} ψ$ ). You can verify that any such bilinear will commute with another bilinear at spacelike separation. This ensures that causality is respected.

The following reading provides the derivation of this anti-commutator and the associated discussion on causality.

5. Quantizing the Dirac Field

To see the explicit derivation of the general fermionic propagator and the causality argument, please read the designated section from David Tong's QFT notes.

Please read Section 5.4, "Propagators" (pages 113-114). Focus on the derivation of Equation (5.27) for the anti-commutator iS(x-y) and the subsequent discussion on causality related to Equation (5.31).

3. Defining the Feynman Propagator

While the anti-commutator is an important theoretical object, the quantity that appears in perturbation theory and Feynman diagrams is the Feynman propagator, defined as the time-ordered vacuum expectation value (VEV) of the fields.

For the Dirac field, the Feynman propagator $S_{F} (x - y)$ is defined as:

S_{F, α β} (x - y) \equiv ⟨ 0∣ T {ψ_{α} (x) ψ ˉ_{β} (y)} ∣0 ⟩

The time-ordering symbol $T$ for fermions has a critical difference from its bosonic counterpart. Due to their anti-commuting nature, swapping the order of two fermionic operators introduces a minus sign.

T {ψ (x) ψ ˉ (y)} = {ψ (x) ψ ˉ (y) - ψ ˉ (y) ψ (x) if x^{0} > y^{0} if y^{0} > x^{0}

This minus sign is essential. For spacelike separated points, the time-ordering is frame-dependent. The minus sign ensures that the definition of the propagator remains consistent and Lorentz invariant, because in this regime $ψ (x) ψ ˉ (y) = - ψ ˉ (y) ψ (x)$ .

4. Deriving the Propagator in Momentum Space

Our main task is to evaluate the definition of $S_{F} (x - y)$ and find its integral representation. We do this by calculating the two time-ordered cases separately.

Case 1: $x^{0} > y^{0}$
We compute $⟨ 0∣ ψ (x) ψ ˉ (y) ∣0 ⟩$ . When we substitute the field expansions, only the term containing $b_{p}^{s}$ from $ψ (x)$ and $b_{q}^{s^{'} †}$ from $ψ ˉ (y)$ gives a non-zero VEV, as $⟨ 0∣ b b^{†} ∣0 ⟩ \neq = 0$ . All other combinations result in an annihilation operator acting on the vacuum on the right or left. The calculation yields:

⟨ 0∣ ψ (x) ψ ˉ (y) ∣0 ⟩ = \int \frac{d ^{3} p}{( 2 π ) ^{3}} \frac{1}{2 E _{p}} (\neq p + m) e^{- i p \cdot (x - y)}

Case 2: $y^{0} > x^{0}$
We compute $⟨ 0∣ - ψ ˉ (y) ψ (x) ∣0 ⟩$ . This time, the non-zero contribution comes from the $c_{p}^{s}$ term in $ψ ˉ (y)$ and the $c_{q}^{s^{'} †}$ term in $ψ (x)$ . This corresponds to an antiparticle created at $x$ and annihilated at $y$ , which is equivalent to a particle propagating "backward in time" from $y$ to $x$ . The calculation gives:

⟨ 0∣ - ψ ˉ (y) ψ (x) ∣0 ⟩ = \int \frac{d ^{3} p}{( 2 π ) ^{3}} \frac{1}{2 E _{p}} (\neq p + m) e^{- i p \cdot (x - y)} (after a change of variables p \to - p)

It turns out that these two expressions, valid for different time orderings, can be combined into a single, elegant 4-momentum integral using the same contour integration techniques used for the scalar propagator. This gives the final expression for the Feynman propagator:

S_{F} (x - y) = \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{i ( \neq p + m )}{p ^{2} - m ^{2} + i ϵ} e^{- i p \cdot (x - y)}

The term multiplying the exponential is the momentum-space propagator:

S ~_{F} (p) = \frac{i ( \neq p + m )}{p ^{2} - m ^{2} + i ϵ}

Notice that since $p^{2} - m^{2} = (\neq p - m) (\neq p + m)$ , we can write this more compactly as:

S ~_{F} (p) = \frac{i}{\neq p - m + i ϵ}

This is the form you will most often see and use in calculations.

The following reading walks through this derivation.

5. Quantizing the Dirac Field

The following section in Tong's notes details the derivation of the Feynman propagator, from its time-ordered definition to its final momentum-space representation.

Please read Section 5.5, "The Feynman Propagator" (pages 114-115). Pay close attention to the definition (5.34) with the crucial minus sign, and follow how the two time-ordered cases combine to give the final integral representation (5.35).

Test your understanding!

Show that the Feynman propagator $S_{F} (x - y)$ is a Green's function for the Dirac operator. Specifically, prove the relation:

(i \neq \partial_{x} - m) S_{F} (x - y) = i δ^{(4)} (x - y)

Hint: Apply the operator $(i \neq \partial_{x} - m)$ to the integral representation of $S_{F} (x - y)$ . What does the derivative operator $\partial_{x}$ do to the exponential $e^{- i p \cdot (x - y)}$ ?

Show answer

We start with the integral representation of the propagator:

S_{F} (x - y) = \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{i ( \neq p + m )}{p ^{2} - m ^{2} + i ϵ} e^{- i p \cdot (x - y)}

Now we apply the Dirac operator $(i \neq \partial_{x} - m)$ . The derivative $i \neq \partial_{x} = i γ^{μ} \partial_{x^{μ}}$ acts on the exponential, bringing down a factor of $i γ^{μ} (- i p_{μ}) = γ^{μ} p_{μ} = \neq p$ .

(i \neq \partial_{x} - m) S_{F} (x - y) = \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{i ( \neq p + m )}{p ^{2} - m ^{2} + i ϵ} (\neq p - m) e^{- i p \cdot (x - y)}

Using the identity $(\neq p + m) (\neq p - m) = p^{2} - m^{2}$ , the expression inside the integral simplifies significantly:

= \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{i ( p ^{2} - m ^{2} )}{p ^{2} - m ^{2} + i ϵ} e^{- i p \cdot (x - y)}

In the limit $ϵ \to 0$ , the fraction $\frac{p ^{2} - m ^{2}}{p ^{2} - m ^{2} + i ϵ}$ is simply 1 for all $p$ . So we are left with:

= i \int \frac{d ^{4} p}{( 2 π ) ^{4}} e^{- i p \cdot (x - y)}

The remaining integral is precisely the definition of the four-dimensional Dirac delta function, $δ^{(4)} (x - y)$ . Therefore, we have shown:

(i \neq \partial_{x} - m) S_{F} (x - y) = i δ^{(4)} (x - y)

This confirms that the Feynman propagator is the Green's function for the Dirac operator, sourced by a point impulse at $y$ .

Conclusion

In this lesson, we have derived one of the most important quantities in QFT: the Feynman propagator for the Dirac field. This result encapsulates the dynamics of a free fermion propagating through spacetime.

Key Takeaways:

The Feynman propagator for the Dirac field, $S_{F} (x - y)$ , is the time-ordered vacuum expectation value of $ψ (x) ψ ˉ (y)$ .
Time-ordering for fermions includes a crucial minus sign for odd permutations, i.e., $T {ψ (x) ψ ˉ (y)} = - ψ ˉ (y) ψ (x)$ for $y^{0} > x^{0}$ .
In momentum space, the propagator takes the compact form $S ~_{F} (p) = \frac{i}{\neq p - m + i ϵ}$ , which is the inverse of the Dirac operator in momentum space.
The propagator is the Green's function for the Dirac equation, satisfying $(i \neq \partial_{x} - m) S_{F} (x - y) = i δ^{(4)} (x - y)$ .

Preview of the Next Lesson:
We have now developed the full theory of the free Dirac field. The next logical step is to introduce interactions. We will see how demanding that the Dirac Lagrangian be invariant under a local U(1) phase transformation forces us to introduce a new field—the electromagnetic field. This principle of local gauge invariance is the cornerstone of modern particle physics, and it will lead us directly to the interaction term of Quantum Electrodynamics (QED).

Recommended resources

1. The Dirac Field in the Heisenberg Picture

2. Causality and Field Anti-Commutators

3. Defining the Feynman Propagator

4. Deriving the Propagator in Momentum Space

Conclusion