12.05.05 · quantum / relativistic-qft

Free Dirac spin-1/2 quantum field

shipped3 tiersLean: nonepending prereqs

Anchor (Master): Weinberg, S., *The Quantum Theory of Fields*, Vol. I (Cambridge, 1995), §5.5–§5.7 (causal Dirac field as a Lorentz-covariant operator-valued distribution, $\gamma$-matrices in mostly-plus, microcausality forces CAR); Streater, R. F. & Wightman, A. S., *PCT, Spin and Statistics, and All That* (Benjamin, 1964; Princeton Landmarks reprint, 2000), Ch. 4 (spin-statistics theorem); Glimm, J. & Jaffe, A., *Quantum Physics: A Functional Integral Point of View*, 2e (Springer, 1987), Ch. 6 (free Dirac field as an operator-valued distribution on fermionic Fock space)

Intuition Beginner

The Dirac equation 12.11.01 gives a wave-equation description of a single relativistic electron. But a relativistic theory cannot really keep particle number fixed. A photon with enough energy creates an electron-positron pair from the vacuum, and a particle that comes in is not always the particle that goes out. To describe these processes you need a framework in which "an electron" is not a special state but rather one excitation among many of a single underlying object — the electron field.

The free Dirac quantum field is exactly this object. At each point of spacetime the field $ψ (x)$ is an operator on a big Hilbert space called Fock space 12.13.02. Acting on the vacuum, $ψ (x)$ destroys an electron at $x$ or creates a positron there; the adjoint $ψ^{†} (x)$ does the reverse. The numerical wave function of single-particle Dirac theory has been replaced by an operator whose role is to make and unmake particles. Particle number is now a derived quantity, not an input.

Two surprises fall out of this construction. First, electrons and positrons appear on equal footing in one mode expansion: the same field $ψ (x)$ that creates electrons also destroys positrons, with antiparticles appearing automatically as the negative-frequency partners of the particle modes. Second, the operators must satisfy anticommutation relations, not commutation. Push two electron creation operators past each other and you pick up a minus sign. This single algebraic fact is responsible for the Pauli exclusion principle, the rigidity of ordinary matter, and the periodic table.

Visual Beginner

A schematic of the Dirac mode expansion. The mass shell $E^{2} = ∣ p ∣^{2} + m^{2}$ is drawn as a hyperboloid in energy-momentum space, with the upper sheet ( $E > 0$ ) labelled "electron modes" and the lower sheet ( $E < 0$ ) reinterpreted as "positron modes". Above each point of the upper sheet sit two arrows labelled $s =↑, ↓$ representing the two spin polarisations; the same pattern repeats on the positron side. A bracket on the side marks the field $ψ (x)$ as a sum over all of these modes weighted by creation and annihilation operators.

The picture captures the key reorganisation. Single-particle Dirac theory has both positive- and negative-energy solutions and treats the negative ones as a paradox. The quantum field reads the negative-energy branch as the positron sector and recovers a positive-energy spectrum. The two spin labels at each momentum are the two physical polarisations, matching the four solutions of the rest-frame Dirac equation.

Worked example Beginner

Compute what the free Dirac field looks like for a single particle at rest, and then count the particle content.

Step 1. The field $ψ (x)$ is built from four kinds of modes at each momentum: electron-spin-up creation $a_{p}^{↑ †}$ , electron-spin-down creation $a_{p}^{↓ †}$ , positron-spin-up creation $b_{p}^{↑ †}$ , positron-spin-down creation $b_{p}^{↓ †}$ . Apply any one of these to the vacuum $∣0 ⟩$ to make a single-particle state of definite momentum.

Step 2. Pick momentum $p = 0$ and spin up. The one-electron state is $∣ e^{-}, ↑ ⟩ = a_{0}^{↑ †} ∣0 ⟩$ . The energy of this state is the rest energy: $E = m$ in natural units. The "number-of-electrons" operator $N_{e}$ , which adds up $a_{p}^{s †} a_{p}^{s}$ over all spins $s$ and momenta $p$ , reports $1$ on this state. The companion "number-of-positrons" operator $N_{p}$ reports $0$ .

Step 3. Now make a two-electron state with opposite spins at momentum $p = 0$ : act with both $a_{0}^{↑ †}$ and $a_{0}^{↓ †}$ on the vacuum. The result is $∣ e^{-} ↑, e^{-} ↓ ⟩ = a_{0}^{↑ †} a_{0}^{↓ †} ∣0 ⟩$ . Apply the anticommutation rule to check that swapping the two creations flips the sign: $a_{0}^{↑ †} a_{0}^{↓ †} = - a_{0}^{↓ †} a_{0}^{↑ †}$ . The state is antisymmetric in the two-electron labels.

Step 4. Try to put two electrons in the same mode: $a_{0}^{↑ †} a_{0}^{↑ †} ∣0 ⟩$ . Apply the same anticommutation rule with both labels equal: $a_{0}^{↑ †} a_{0}^{↑ †} = - a_{0}^{↑ †} a_{0}^{↑ †}$ , which forces $a_{0}^{↑ †} a_{0}^{↑ †} = 0$ . The state vanishes. Two electrons cannot occupy the same momentum-spin mode. This is the Pauli exclusion principle, falling out of the algebra without any extra postulate.

Step 5. Make an electron-positron pair: $∣ e^{-} ↑, e^{+} ↑ ⟩ = a_{0}^{↑ †} b_{0}^{↑ †} ∣0 ⟩$ . The total energy is $2 m$ (two rest energies). The total electric charge is $- e + e = 0$ : this pair is electrically neutral. Pair production from a photon ( $γ \to e^{-} + e^{+}$ ) requires photon energy $\geq 2 m$ , matching the threshold.

What this tells us: the free Dirac field is the right framework for talking about variable particle numbers, and the bookkeeping is automatic. Antiparticles, Pauli exclusion, the pair-production threshold — none of these were added by hand. They are consequences of building the theory as an operator field on Fock space with anticommutation relations.

Check your understanding Beginner

Exercise (easy, multiple choice).

In the free Dirac quantum field theory, what is the role of the operator $ψ (x)$ ?

A. It is a complex number giving the probability amplitude for finding the electron at $x$ . B. It is an operator that destroys an electron or creates a positron at $x$ . C. It is the electric charge density at $x$ . D. It is the Lagrangian density at $x$ .

Hint

In quantum field theory the wave function of single-particle Dirac theory is upgraded to an operator. Think about which operations it performs on the Fock vacuum.

Answer

B. It is an operator that destroys an electron or creates a positron at $x$ .

The mode expansion contains both an electron annihilation operator $a_{p}^{s}$ and a positron creation operator $b_{p}^{s †}$ . So acting with $ψ (x)$ on a state either removes an electron or adds a positron. The adjoint $\overset{ˉ}{ψ} (x)$ creates electrons and destroys positrons.

Formal definition Intermediate+

We work in natural units $ℏ = c = 1$ and adopt the mostly-minus signature $η^{μν} = diag (+, -, -, -)$ , matching the conventions of unit 12.11.01 and of Peskin-Schroeder. (Weinberg's QTF uses mostly-plus; the sign of $m^{2}$ in the mass-shell condition is the only externally-visible difference, and a Wick-rotation paragraph below makes the conversion explicit.) The Clifford-algebra realisation is the same Cl(1,3) of 03.09.02, with ${γ^{μ}, γ^{ν}} = 2 η^{μν} 1_{4}$ .

The mass shell is the positive-energy hyperboloid $$ X_m = { p \in \mathbb{R}^{1,3} : p^\mu p_\mu = m^2,\ p^0 > 0 }, $$ which carries the Lorentz-invariant Radon measure $d λ_{m} (p) = d^{3} p / (2 E_{p})$ with $E_{p} = ∣ p ∣^{2} + m^{2}$ .

For each $p \in X_{m}$ the plane-wave spinors $u^{s} (p), v^{s} (p) \in C^{4}$ ( $s \in {1, 2}$ ) satisfy $$ (\gamma^\mu p_\mu - m) u^s(p) = 0, \qquad (\gamma^\mu p_\mu + m) v^s(p) = 0, $$ and are normalised by $\overset{u}{ˉ}^{r} (p) u^{s} (p) = 2 m δ^{r s}$ , $\overset{v}{ˉ}^{r} (p) v^{s} (p) = - 2 m δ^{r s}$ , $\overset{u}{ˉ}^{r} (p) v^{s} (p) = 0$ . The completeness relations are $$ \sum_s u^s(p) \bar u^s(p) = \gamma^\mu p_\mu + m, \qquad \sum_s v^s(p) \bar v^s(p) = \gamma^\mu p_\mu - m. $$

The fermionic Fock space $F_{a} = ⨁_{n \geq 0} Λ^{n} H_{1}$ over the one-particle Hilbert space $H_{1} = L^{2} (X_{m}, d λ_{m}) \otimes C^{2} \otimes (C \oplus C)$ — two spin labels and two charges, electron and positron — carries creation and annihilation operators $a_{p}^{s †}, a_{p}^{s}, b_{p}^{s †}, b_{p}^{s}$ satisfying the canonical anticommutation relations (CAR): $$ { a^s_{\mathbf{p}}, a^{r\dagger}{\mathbf{q}} } = (2\pi)^3 \delta^{(3)}(\mathbf{p} - \mathbf{q}), \delta^{sr}, \quad { b^s{\mathbf{p}}, b^{r\dagger}_{\mathbf{q}} } = (2\pi)^3 \delta^{(3)}(\mathbf{p} - \mathbf{q}), \delta^{sr}, $$ with all other anticommutators vanishing. These are distributional identities on the dense finite-particle subspace; they extend to a $C^{*}$ -algebra structure on $F_{a}$ as in 12.13.02.

Definition (free Dirac quantum field). The free Dirac field of mass $m$ is the operator-valued tempered distribution $$ \boxed{\psi(x) = \sum_{s = 1, 2} \int \frac{d^3 \mathbf{p}}{(2\pi)^3} \frac{1}{\sqrt{2 E_{\mathbf{p}}}}\left( a^s_{\mathbf{p}}, u^s(p), e^{-i p \cdot x} + b^{s\dagger}_{\mathbf{p}}, v^s(p), e^{+i p \cdot x} \right),} $$ acting on the fermionic Fock space $F_{a}$ , with $p = (E_{p}, p)$ on shell. The Dirac adjoint is $\overset{ˉ}{ψ} (x) := ψ^{†} (x) γ^{0}$ .

The field is Lorentz-covariant: under a proper orthochronous Lorentz transformation $Λ$ with spin- $\frac{1}{2}$ lift $S (Λ)$ acting on the spinor index via $S (Λ)^{- 1} γ^{μ} S (Λ) = Λ^{μ}_{ν} γ^{ν}$ 10.05.01, one has $U (Λ) ψ (x) U (Λ)^{- 1} = S (Λ)^{- 1} ψ (Λ x)$ , where $U (Λ)$ is the unitary representation of the Lorentz group on $F_{a}$ .

Hamiltonian and conserved charges. After normal-ordering with respect to the Fock vacuum, the Hamiltonian and the momentum operators are $$ H = \sum_s \int \frac{d^3 \mathbf{p}}{(2\pi)^3}, E_{\mathbf{p}} \left( a^{s\dagger}{\mathbf{p}} a^s{\mathbf{p}} + b^{s\dagger}{\mathbf{p}} b^s{\mathbf{p}} \right), \qquad \mathbf{P} = \sum_s \int \frac{d^3 \mathbf{p}}{(2\pi)^3}, \mathbf{p} \left( a^{s\dagger}{\mathbf{p}} a^s{\mathbf{p}} + b^{s\dagger}{\mathbf{p}} b^s{\mathbf{p}} \right). $$ The electric charge is $Q = e \int d^{3} x : \overset{ˉ}{ψ} γ^{0} ψ := e \sum_{s} \int d^{3} p / (2 π)^{3} (a_{p}^{s †} a_{p}^{s} - b_{p}^{s †} b_{p}^{s})$ : electrons contribute $- ∣ e ∣$ and positrons $+ ∣ e ∣$ as required.

Feynman propagator. The time-ordered two-point function is $$ S_F(x - y){\alpha\beta} = \langle 0 | T \psi\alpha(x) \bar\psi_\beta(y) | 0 \rangle = \int \frac{d^4 p}{(2\pi)^4}, \frac{i (\gamma^\mu p_\mu + m)_{\alpha\beta}}{p^2 - m^2 + i\epsilon}, e^{-i p \cdot (x - y)}. $$ The $i ϵ$ prescription routes positive-energy electrons forward in time and positive-energy positrons backward, executing the Feynman-Stueckelberg interpretation at the level of the Green's function.

Counterexamples to common slips

Bosonic quantisation is forbidden. Imposing canonical commutation relations $[a_{p}^{s}, a_{q}^{r †}] = (2 π)^{3} δ^{(3)} (p - q) δ^{sr}$ on the Dirac mode operators makes the Hamiltonian unbounded below: every positron creation lowers the energy. The spin-statistics theorem (^{[Pauli 1940]}) shows this is forced — half-integer spin requires anticommutators on pain of energy non-positivity.
The classical Dirac wavefunction is not $ψ (x)$ . In the quantised theory, $ψ (x)$ is an operator, not a $C^{4}$ -valued function. A one-particle state of an electron with wave-packet profile $f$ is $∣ f ⟩ = \int d^{3} p f (p) a_{p}^{s †} ∣0 ⟩$ ; the matrix element $⟨ 0∣ ψ (x) ∣ f ⟩$ recovers the classical Dirac wave function as a single-particle projection.
Normal-ordering is required, not optional. Without normal-ordering, $⟨ 0∣ H ∣0 ⟩ = - \sum_{s} \int d^{3} p E_{p}$ is infinite and negative. The normal-ordered Hamiltonian is bounded below with the vacuum at energy zero. The infinite shift removed by normal-ordering is the same zero-point divergence one meets in the harmonic-oscillator chain.
Mostly-plus vs mostly-minus. Weinberg's QTF uses mostly-plus signature, so his Clifford relation reads ${γ^{μ}, γ^{ν}} = - 2 η_{W}^{μν} 1_{4}$ and the Dirac equation reads $(i γ^{μ} \partial_{μ} + m) ψ = 0$ . After Wick rotation $t \to - i τ$ the two conventions coincide in Euclidean signature where ${γ^{a}, γ^{b}} = 2 δ^{ab} 1_{4}$ , matching the spin-geometry convention of 03.09.02.

Key theorem with proof Intermediate+

Theorem (microcausality and the CAR; Peskin-Schroeder §3.5, Weinberg §5.5). Let $ψ (x)$ be the free Dirac field defined above, quantised with the canonical anticommutation relations. Then for any pair of spacetime points $x, y$ separated by a spacelike interval, $(x - y)^{2} < 0$ , the field anticommutator vanishes: $$ { \psi_\alpha(x), \bar\psi_\beta(y) } = 0 \quad \text{whenever } (x - y)^2 < 0. $$ Furthermore this property forces the choice of anticommutators (CAR) rather than commutators (CCR): quantisation of the Dirac field with commutators violates microcausality.

Proof. Start from the mode expansion. Compute the anticommutator directly:

{ψ_{α} (x), \overset{ˉ}{ψ}_{β} (y)} = s, r \sum \int \frac{d ^{3} p d ^{3} q}{( 2 π ) ^{6} 4 E _{p} E _{q}} [{a_{p}^{s}, a_{q}^{r †}} u_{α}^{s} (p) \overset{u}{ˉ}_{β}^{r} (q) e^{- i p \cdot x + i q \cdot y}

+ {b_{p}^{s †}, b_{q}^{r}} v_{α}^{s} (p) \overset{v}{ˉ}_{β}^{r} (q) e^{+ i p \cdot x - i q \cdot y}] .

Cross terms ${a, b^{†}}$ and ${a^{†}, b}$ vanish identically by the CAR. Apply ${a_{p}^{s}, a_{q}^{r †}} = (2 π)^{3} δ^{(3)} (p - q) δ^{sr}$ and use the spin sums $\sum_{s} u^{s} (p) \overset{u}{ˉ}^{s} (p) = γ^{μ} p_{μ} + m$ and $\sum_{s} v^{s} (p) \overset{v}{ˉ}^{s} (p) = γ^{μ} p_{μ} - m$ :

{ψ_{α} (x), \overset{ˉ}{ψ}_{β} (y)} = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 E _{p}} [(γ^{μ} p_{μ} + m)_{α β} e^{- i p \cdot (x - y)} + (γ^{μ} p_{μ} - m)_{α β} e^{+ i p \cdot (x - y)}] .

Relabel $p \to - p$ in the second integral (the spatial part is even, and the on-shell $p^{0} = E_{p}$ is mass-shell-positive in both branches when read as the Lorentz-invariant integration over $X_{m}$ ):

{ψ_{α} (x), \overset{ˉ}{ψ}_{β} (y)} = (i γ^{μ} \partial_{μ}^{x} + m)_{α β} i Δ (x - y),

where

i Δ (z) = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 E _{p}} [e^{- i p \cdot z} - e^{+ i p \cdot z}]

is the Pauli-Jordan invariant function. The factor $(i γ^{μ} \partial_{μ}^{x} + m)$ pulls outside the integral because of the constancy in $x, y$ of the matrices and because $p \cdot x = E_{p} x^{0} - p \cdot x$ gives $- i \partial_{μ}^{x} e^{\mp i p \cdot z} = \pm p_{μ} e^{\mp i p \cdot z}$ .

Spacelike vanishing of $Δ$ . The Pauli-Jordan function $Δ (z)$ is Lorentz-invariant: $Δ (Λ z) = Δ (z)$ for every proper orthochronous Lorentz transformation $Λ$ . For $z$ spacelike, one can choose a frame in which $z^{0} = 0$ and $z \neq = 0$ . In that frame $$ i\Delta(0, \mathbf{z}) = \int \frac{d^3 \mathbf{p}}{(2\pi)^3, 2 E_{\mathbf{p}}} \left[ e^{+i \mathbf{p} \cdot \mathbf{z}} - e^{-i \mathbf{p} \cdot \mathbf{z}} \right]. $$ The two pieces are equal after the variable change $p \to - p$ (the measure and $E_{p}$ are even), so the integral vanishes. By Lorentz invariance, $Δ (z) = 0$ for every spacelike $z$ . Hence ${ψ_{α} (x), \overset{ˉ}{ψ}_{β} (y)} = (i γ^{μ} \partial_{μ}^{x} + m)_{α β} \cdot 0 = 0$ for $(x - y)^{2} < 0$ , as claimed.

Necessity of anticommutators. Repeat the computation with commutators in place of anticommutators. The signs in the two pieces flip: the relabelling $p \to - p$ that previously made the integrals cancel now adds them. Specifically, with CCR, $$ [\psi_\alpha(x), \bar\psi_\beta(y)] = (i \gamma^\mu \partial^x_\mu + m){\alpha\beta} \int \frac{d^3 \mathbf{p}}{(2\pi)^3, 2 E{\mathbf{p}}} \left[ e^{-i p \cdot (x-y)} + e^{+i p \cdot (x-y)} \right], $$ and the integral does not vanish for spacelike separation. Microcausality fails, and worse, the Hamiltonian becomes unbounded below: positron creation $b_{p}^{s †}$ contributes negatively to $H$ when CCR are used instead of CAR. Half-integer spin forces anticommutators. $□$

Bridge. Microcausality is the bridge that builds toward the full spin-statistics theorem. The foundational reason the choice of anticommutators is forced is exactly that the negative-frequency branch of the Dirac mode expansion must be reinterpreted as antiparticle creation to keep the Hamiltonian bounded below, and the Pauli-Jordan function only vanishes spacelike under the sign pattern produced by CAR. This is exactly the same structural argument that appears again in 12.13.02 (fermionic Fock space) under the Jordan-Wigner banner, and the central insight is that the CAR algebra is not a separate axiom but a consequence of Lorentz invariance plus locality plus positivity of energy. Putting these together, the free Dirac field identifies the geometric Clifford structure of 03.09.02 with the operator-algebraic CAR structure of 12.13.02; the bridge is that the same $γ^{μ}$ that intertwines the spin- $\frac{1}{2}$ representation of the Lorentz group also organises the four field components into the two electron and two positron creation channels. This generalises to the Streater-Wightman axiomatic proof of the spin-statistics theorem, where the same microcausality argument runs without reference to the explicit mode expansion.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Compute the equal-time anticommutator ${ψ_{α} (x, t), ψ_{β}^{†} (y, t)}$ from the mode expansion. Show that it equals $δ_{α β} δ^{(3)} (x - y)$ .

Hint

Set $x^{0} = y^{0}$ in the spacetime anticommutator and use the completeness relations $\sum_{s} u^{s} (p) u^{s †} (p) = (γ^{μ} p_{μ} + m) γ^{0}$ and $\sum_{s} v^{s} (p) v^{s †} (p) = (γ^{μ} p_{μ} - m) γ^{0}$ (note the $ψ^{†}$ vs $\overset{ˉ}{ψ}$ distinction).

Answer

At equal times, $$ {\psi_\alpha(\mathbf{x}, t), \psi^\dagger_\beta(\mathbf{y}, t)} = \int \frac{d^3\mathbf{p}}{(2\pi)^3, 2 E_{\mathbf{p}}} e^{i \mathbf{p}\cdot(\mathbf{x}-\mathbf{y})} \left[ \sum_s u^s(p) u^{s\dagger}(p) + \sum_{s} v^s(-p){\mathrm{rev}} v^{s\dagger}(-p){\mathrm{rev}}\right]. $$ The bracket evaluates to $2 E_{p} δ_{α β}$ using $(γ^{μ} p_{μ} + m) γ^{0} + (γ^{μ} (- p)_{μ} - m) γ^{0} = 2 E_{p}$ on-shell (cancellation of the spatial gamma terms after the $p \to - p$ relabel in the $v$ contribution). The $E_{p}$ cancels the measure factor, leaving $\int d^{3} p / (2 π)^{3} e^{i p \cdot (x - y)} δ_{α β} = δ_{α β} δ^{(3)} (x - y)$ . Rubric: full credit for the spin-sum identity plus the cancellation argument.

Exercise 4 (medium, symbolic).

Derive the Heisenberg equation of motion for $ψ (x)$ : show that $i \partial_{t} ψ (x) = [ψ (x), H]$ recovers the Dirac equation $(i γ^{μ} \partial_{μ} - m) ψ (x) = 0$ .

Hint

Use $[ψ (x, t), H]$ with the mode expansion of $H$ and the fermionic identity $[A B, C] = A {B, C} - {A, C} B$ . Then convert the result back into spacetime form.

Answer

Expand $ψ (x, t) = \sum_{s} \int d^{3} p / (2 π)^{3} (2 E_{p})^{- 1/2} (a_{p}^{s} u^{s} (p) e^{- i E_{p} t + i p \cdot x} + b_{p}^{s †} v^{s} (p) e^{+ i E_{p} t - i p \cdot x})$ . Compute $[ψ (x, t), H]$ using $[a_{p}^{s}, a_{q}^{r †} a_{q}^{r}] = a_{q}^{r †} {a_{p}^{s}, a_{q}^{r}}_{= 0} - {a_{p}^{s}, a_{q}^{r †}} a_{q}^{r}$ , and similarly $[b_{p}^{s †}, b_{q}^{r †} b_{q}^{r}] = - b_{q}^{r †} {b_{p}^{s †}, b_{q}^{r}} = - b_{q}^{r †} (2 π)^{3} δ^{(3)} (p - q) δ^{sr} \cdot (- 1)$ — i.e. carrying through the signs. The net result is $[ψ (x), H] = i \partial_{t} ψ (x) = \sum_{s} \int d^{3} p / (2 π)^{3} (2 E_{p})^{- 1/2} E_{p} (a_{p}^{s} u^{s} (p) e^{- i p \cdot x} - b_{p}^{s †} v^{s} (p) e^{+ i p \cdot x})$ . The sign difference is exactly what reproduces the on-shell relations $(γ^{μ} p_{μ} - m) u = 0$ and $(γ^{μ} p_{μ} + m) v = 0$ when one applies $(i γ^{μ} \partial_{μ} - m)$ term by term. Rubric: full credit for the sign-correct anticommutator algebra plus the on-shell verification.

Exercise 5 (medium, symbolic).

Show that the Lagrangian density $L = \overset{ˉ}{ψ} (i γ^{μ} \partial_{μ} - m) ψ$ has the global $U (1)$ symmetry $ψ \to e^{i α} ψ$ , $\overset{ˉ}{ψ} \to \overset{ˉ}{ψ} e^{- i α}$ , and that Noether's theorem yields the conserved current $j^{μ} = \overset{ˉ}{ψ} γ^{μ} ψ$ .

Hint

For an infinitesimal phase $α$ , compute $δ L$ — it vanishes. Then use $j^{μ} = (\partial L / \partial (\partial_{μ} ψ)) δ ψ / (i α)$ .

Answer

Under $ψ \to e^{i α} ψ$ , $\overset{ˉ}{ψ} \to \overset{ˉ}{ψ} e^{- i α}$ , the bilinear $\overset{ˉ}{ψ} (\dots) ψ$ is invariant since the two phases cancel. So $δ L = 0$ identically; the symmetry is exact. Noether: with $δ ψ = i α ψ$ , the conserved current is $j^{μ} = (\partial L / \partial (\partial_{μ} ψ)) (δ ψ / (i α)) = (i \overset{ˉ}{ψ} γ^{μ}) ψ /1 = i \overset{ˉ}{ψ} γ^{μ} ψ$ , which up to an overall normalisation equals $\overset{ˉ}{ψ} γ^{μ} ψ$ . Current conservation $\partial_{μ} j^{μ} = 0$ on-shell follows from the Dirac equation plus its adjoint. Rubric: full credit for the symmetry verification plus the Noether-current extraction.

Exercise 6 (medium, symbolic).

Compute the Feynman propagator $S_{F} (x - y)$ in position space as the time-ordered vacuum expectation $⟨ 0∣ T ψ (x) \overset{ˉ}{ψ} (y) ∣0 ⟩$ . Show that it satisfies $(i γ^{μ} \partial_{μ}^{x} - m) S_{F} (x - y) = i δ^{(4)} (x - y) 1_{4}$ .

Hint

Apply $(i γ^{μ} \partial_{μ}^{x} - m)$ to the momentum-space form $\int d^{4} p / (2 π)^{4} i (γ^{μ} p_{μ} + m) / (p^{2} - m^{2} + i ϵ) e^{- i p \cdot (x - y)}$ and use $(γ^{μ} p_{μ} - m) (γ^{ν} p_{ν} + m) = p^{2} - m^{2}$ .

Answer

$(i γ^{μ} \partial_{μ}^{x} - m) S_{F} (x - y) = \int d^{4} p / (2 π)^{4} (γ^{μ} p_{μ} - m) i (γ^{ν} p_{ν} + m) / (p^{2} - m^{2} + i ϵ) e^{- i p \cdot (x - y)} = \int d^{4} p / (2 π)^{4} i (p^{2} - m^{2}) / (p^{2} - m^{2} + i ϵ) e^{- i p \cdot (x - y)}$ . The ratio $(p^{2} - m^{2}) / (p^{2} - m^{2} + i ϵ) \to 1$ as $ϵ \to 0^{+}$ since the numerator cancels the principal-part of the denominator, leaving $\int d^{4} p / (2 π)^{4} \cdot i \cdot e^{- i p \cdot (x - y)} = i δ^{(4)} (x - y)$ . So $(i γ^{μ} \partial_{μ}^{x} - m) S_{F} (x - y) = i δ^{(4)} (x - y) 1_{4}$ , confirming that $S_{F}$ is the Green's function of the Dirac operator with the Feynman $i ϵ$ prescription. Rubric: full credit for the Clifford-algebra cancellation plus the delta-function extraction.

Exercise 7 (hard, symbolic).

Show that quantising the Dirac field with commutators $[a_{p}^{s}, a_{q}^{r †}] = (2 π)^{3} δ^{(3)} (p - q) δ^{sr}$ (instead of anticommutators) and treating the negative-energy modes naively makes the Hamiltonian unbounded below.

Hint

Without the antiparticle reinterpretation, the mode expansion has both positive- and negative-frequency creation operators contributing to $H$ with opposite signs once you commute through. Track the signs carefully.

Answer

Use the would-be mode expansion $ψ (x) = \sum_{s} \int d^{3} p / (2 π)^{3} (2 E_{p})^{- 1/2} (a_{p}^{s} u^{s} (p) e^{- i p \cdot x} + c_{p}^{s} v^{s} (p) e^{+ i p \cdot x})$ where $c_{p}^{s}$ is a second annihilation operator (no antiparticle reinterpretation). With CCR, $H = \int d^{3} x ψ^{†} H_{D} ψ$ becomes $H = \sum_{s} \int d^{3} p / (2 π)^{3} E_{p} (a_{p}^{s †} a_{p}^{s} - c_{p}^{s †} c_{p}^{s})$ after using the on-shell relations and the spin-sum identities — the relative sign comes from $(γ^{μ} p_{μ} + m) \to (γ^{μ} (- p)_{μ} + m) = - (γ^{μ} p_{μ} - m)$ in the negative-frequency piece. The Hamiltonian is unbounded below: act with $c_{p}^{s †}$ many times to lower the energy without limit. Reinterpreting $c_{p}^{s} \to b_{p}^{s †}$ (antiparticle creation) plus switching to CAR flips the sign of the second term after passing the creation operator through, recovering the positive-definite Hamiltonian $H = \sum_{s} \int d^{3} p / (2 π)^{3} E_{p} (a_{p}^{s †} a_{p}^{s} + b_{p}^{s †} b_{p}^{s})$ . Anticommutators are forced. Rubric: full credit for the sign-tracking plus the bounded-below diagnosis.

Exercise 8 (hard, symbolic).

Charge-conjugation acts on the Dirac field by $ψ^{c} (x) = C \overset{ˉ}{ψ}^{T} (x)$ where $C$ is the charge-conjugation matrix satisfying $C^{- 1} γ^{μ} C = - (γ^{μ})^{T}$ . Show that charge conjugation swaps electrons and positrons: in the mode expansion of $ψ^{c}$ , the operator $a_{p}^{s}$ is replaced by $b_{p}^{s}$ (up to a possible phase).

Hint

Use the identity $C \overset{u}{ˉ}^{s} (p)^{T} = v^{s} (p)$ (up to a convention-dependent phase). Apply this to each term in the mode expansion of $\overset{ˉ}{ψ}^{T}$ .

Answer

Start with $\overset{ˉ}{ψ} (x) = \sum_{s} \int d^{3} p / (2 π)^{3} (2 E_{p})^{- 1/2} (a_{p}^{s †} \overset{u}{ˉ}^{s} (p) e^{+ i p \cdot x} + b_{p}^{s} \overset{v}{ˉ}^{s} (p) e^{- i p \cdot x})$ . Take the transpose on the spinor index: $\overset{ˉ}{ψ}^{T} = \sum_{s} \int (\dots) (a_{p}^{s †} \overset{u}{ˉ}^{s} (p)^{T} e^{+ i p \cdot x} + b_{p}^{s} \overset{v}{ˉ}^{s} (p)^{T} e^{- i p \cdot x})$ . Apply $C$ on the left: $ψ^{c} (x) = C \overset{ˉ}{ψ}^{T} (x) = \sum_{s} \int (\dots) (a_{p}^{s †} (C \overset{u}{ˉ}^{s} (p)^{T}) e^{+ i p \cdot x} + b_{p}^{s} (C \overset{v}{ˉ}^{s} (p)^{T}) e^{- i p \cdot x})$ . Using $C \overset{u}{ˉ}^{s} (p)^{T} = v^{s} (p)$ and $C \overset{v}{ˉ}^{s} (p)^{T} = u^{s} (p)$ (in standard conventions), this becomes $ψ^{c} (x) = \sum_{s} \int (\dots) (a_{p}^{s †} v^{s} (p) e^{+ i p \cdot x} + b_{p}^{s} u^{s} (p) e^{- i p \cdot x})$ , which is the original mode expansion of $ψ (x)$ with $a_{p}^{s} \leftrightarrow b_{p}^{s}$ . Hence $C : a_{p}^{s} \to b_{p}^{s}$ , $b_{p}^{s} \to a_{p}^{s}$ — electrons and positrons swap. Rubric: full credit for the $C \overset{u}{ˉ}^{T} = v$ identity plus the operator swap.

Exercise 9 (hard, symbolic).

State and prove the spin-statistics theorem in the form: a free Lorentz-covariant Dirac field whose anticommutator ${ψ (x), \overset{ˉ}{ψ} (y)}$ vanishes for spacelike separation must be quantised with the canonical anticommutation relations, and its energy is then bounded below.

Hint

Combine the calculation of the spacelike anticommutator (Theorem above) with the Hamiltonian-positivity calculation (Exercise 7). The two together force the choice of CAR over CCR.

Answer

Write the mode expansion as $ψ (x) = \sum_{s} \int (\dots) (a_{p}^{s} u^{s} (p) e^{- i p \cdot x} + \tilde{b}_{p}^{s} v^{s} (p) e^{+ i p \cdot x})$ where $\tilde{b}_{p}^{s}$ is to be determined as either a creation or annihilation operator and the algebra is to be either CCR or CAR. Compute the spacelike ${ψ (x), \overset{ˉ}{ψ} (y)}$ (or $[ψ (x), \overset{ˉ}{ψ} (y)]$ ) as a function of these choices, following the proof of the microcausality theorem. Spacelike vanishing requires the two integrals with opposite-sign exponentials to cancel, which happens only when (i) the algebra is CAR and $\tilde{b}_{p}^{s} = b_{p}^{s †}$ (antiparticle creation), or (ii) the algebra is CCR and $\tilde{b}_{p}^{s} = - b_{p}^{s †}$ (antiparticle creation with a phase). In case (ii), the Hamiltonian-positivity calculation of Exercise 7 shows $H$ is unbounded below. Hence (i) is the unique consistent choice: CAR plus antiparticle creation. This is the spin- $\frac{1}{2}$ case of the spin-statistics theorem; the general statement was proved by Pauli (1940) and the axiomatic version by Streater-Wightman (1964). Rubric: full credit for the microcausality plus positivity argument and the elimination of CCR.

Lean formalization Intermediate+

Mathlib supplies the Clifford algebra Mathlib.LinearAlgebra.CliffordAlgebra.Basic and the antisymmetric tensor power Mathlib.LinearAlgebra.ExteriorAlgebra.Basic, but does not yet ship a named free Dirac quantum field. The intended formalisation reads schematically:

import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.Analysis.InnerProductSpace.l2Space

-- The mass shell X_m as a Lorentz-invariant subset of ℝ^{1,3}.
def massShell (m : ℝ) (hm : 0 < m) : Set (Fin 4 → ℝ) :=
  { p | (p 0)^2 - ((p 1)^2 + (p 2)^2 + (p 3)^2) = m^2 ∧ 0 < p 0 }

-- One-particle Hilbert space: square-integrable spinor sections on the mass shell.
noncomputable def OneParticleSpace (m : ℝ) : Type := sorry

-- Fermionic Fock space over the one-particle space, with electron and positron sectors.
noncomputable def DiracFock (m : ℝ) : Type := sorry

-- The free Dirac field as an operator-valued distribution.
noncomputable def diracField (m : ℝ) (x : Fin 4 → ℝ) :
    DiracFock m → DiracFock m := sorry

-- Microcausality of the field anticommutator.
theorem dirac_microcausality {m : ℝ} (hm : 0 < m)
    (x y : Fin 4 → ℝ) (hxy_spacelike : (x 0 - y 0)^2
      < (x 1 - y 1)^2 + (x 2 - y 2)^2 + (x 3 - y 3)^2) :
    AntiCommutator (diracField m x) (diracField m y) = 0 :=
  sorry

The proof gap is substantial. The free Dirac field is one of the first non-elementary operator-valued tempered distributions a Lean formalisation of quantum field theory would need. Required intermediate developments include: Minkowski-spacetime measure theory (Lorentz-invariant Radon measures on the mass shell), the $u^{s} (p), v^{s} (p)$ plane-wave spinors as sections of a Hermitian bundle over $X_{m}$ , the Wick-symmetrised Fock space carrying the CAR algebra as a $C^{*}$ -algebra (cross-link to 12.13.02), the operator-valued-distribution framework of Wightman (Reed-Simon Vol. II Ch. IX), and the proof of microcausality from the spin sums plus the spacelike vanishing of the Pauli-Jordan function. None of these blocks is in Mathlib at this date; each is a substantive contribution in its own right.

Advanced results Master

Theorem (Lorentz covariance and the spinor representation). Let $Λ$ be a proper orthochronous Lorentz transformation with spin- $\frac{1}{2}$ lift $S (Λ) \in Spin^{+} (1, 3)$ . There is a unitary representation $U (Λ)$ of $Spin^{+} (1, 3)$ on $F_{a}$ such that $$ U(\Lambda), \psi(x), U(\Lambda)^{-1} = S(\Lambda)^{-1} \psi(\Lambda x). $$ The representation is generated by the boost and rotation operators $J^{μν} = \int d^{3} x : ψ^{†} (x) (S^{μν} + L^{μν}) ψ (x) :$ where $S^{μν} = \frac{i}{4} [γ^{μ}, γ^{ν}]$ acts on the spinor index and $L^{μν}$ on spatial coordinates.

The proof exhibits the spin- $\frac{1}{2}$ representation as the four-dimensional irreducible representation of $Spin^{+} (1, 3) ≅ SL (2, C)$ on $C^{4} = C^{2} \oplus C^{2}$ (left- and right-handed Weyl components). The intertwining identity $S (Λ)^{- 1} γ^{μ} S (Λ) = Λ^{μ}_{ν} γ^{ν}$ is exactly the condition that the $γ$ -matrices transform as a Lorentz four-vector, and the spin sums $\sum_{s} u^{s} (p) \overset{u}{ˉ}^{s} (p) = γ^{μ} p_{μ} + m$ are covariantly meaningful.

Theorem (CPT for the free Dirac field). The combined operation of charge conjugation $C$ , parity $P$ , and time reversal $T$ is an antiunitary symmetry of the free Dirac field, $C P T : ψ (x) \to C P T ψ (x) (C P T)^{- 1} = (γ^{5}) ψ^{†} (- x)^{T}$ (up to convention-dependent phases). The free Hamiltonian, momentum, and angular momentum operators are all $C P T$ -invariant.

The CPT theorem in its full Lorentz-invariant form (Jost 1957, Lüders 1957) holds for any Lorentz-covariant local field theory satisfying the Wightman axioms; the free Dirac field is the simplest non-elementary instance. The combined operation reverses both spacetime coordinates and the particle-antiparticle assignment, so it sends "an electron going forward in time" to "a positron going backward" — the Feynman-Stueckelberg interpretation made into a symmetry.

Theorem (Wick contraction and the Feynman propagator). Time-ordering of the Dirac field is related to normal-ordering by the contraction $$ T \psi_\alpha(x) \bar\psi_\beta(y) = :!\psi_\alpha(x) \bar\psi_\beta(y)!: + S_F(x - y)_{\alpha\beta} \cdot \mathbf{1}. $$ Iterating this identity (Wick's theorem) reduces arbitrary time-ordered products $T ψ (x_{1}) \overset{ˉ}{ψ} (y_{1}) \dots ψ (x_{n}) \overset{ˉ}{ψ} (y_{n})$ on the vacuum to sums over all complete pairings, each pairing contributing a product of Feynman propagators with a sign equal to the signature of the pairing permutation.

Wick's theorem is the algebraic engine of perturbative QFT. For fermions the signs are exactly the signs of the contraction permutations: this is the Pfaffian / Wick determinant structure dual to the bosonic permanent. The Feynman propagator $S_{F} (x - y) = \int d^{4} p / (2 π)^{4} i (γ^{μ} p_{μ} + m) / (p^{2} - m^{2} + i ϵ) \cdot e^{- i p \cdot (x - y)}$ is the unique Green's function of $(i γ^{μ} \partial_{μ} - m)$ with the time-ordering boundary condition.

Theorem (Wightman reconstruction for the free Dirac field). Let $W_{n} (x_{1}, \dots, x_{n})$ denote the $n$ -point Wightman distribution of the free Dirac field. The vacuum two-point function $$ \mathcal{W}2(x, y){\alpha\beta} = \langle 0 | \psi_\alpha(x) \bar\psi_\beta(y) | 0 \rangle = \int \frac{d^3\mathbf{p}}{(2\pi)^3 2 E_{\mathbf{p}}} (\gamma^\mu p_\mu + m)_{\alpha\beta} e^{-ip\cdot(x-y)} $$ satisfies the Wightman axioms (Poincaré-covariance, positivity, microcausality, cluster decomposition). The Wightman reconstruction theorem (^{[Wightman 1956]}) then produces the Hilbert space $F_{a}$ , the field operators $ψ (x)$ , and the vacuum $∣0 ⟩$ uniquely up to unitary equivalence from this distributional input.

This is the axiomatic-QFT statement of what canonical quantisation accomplishes for the Dirac field. The $n$ -point functions are Gaussian (in the fermionic / Pfaffian sense), reflecting that the free theory is solvable.

Theorem (spin-statistics, Streater-Wightman form). In any Lorentz-covariant, Wightman-positive quantum field theory in $1 + 3$ dimensions, a field transforming under the spin- $\frac{1}{2}$ representation of $Spin^{+} (1, 3)$ must be quantised with canonical anticommutation relations; equivalently, a field with half-integer spin quantised with commutators violates either Lorentz covariance or positivity of energy or both. The free Dirac field is the lowest-spin instance.

The Streater-Wightman (^{[Streater-Wightman 1964]}) proof rests on three inputs: Lorentz-covariance of the field, positive-definiteness of the inner product on the vacuum sector, and microcausality (vanishing of the anticommutator at spacelike separation). For half-integer spin, only the CAR choice satisfies all three; the constructive proof in the previous theorem provides the existence half, and the axiomatic argument provides the no-go for CCR.

Theorem (Källén-Lehmann spectral representation for the Dirac propagator). In an interacting theory with a one-particle pole at mass $m$ , the full Dirac two-point function admits the spectral decomposition $$ \langle 0 | T \psi(x) \bar\psi(y) | 0 \rangle = Z \cdot S_F^{(m)}(x - y) + \int_{M^2 \geq (2m)^2} dM^2, \rho(M^2), S_F^{(M)}(x - y), $$ where $Z \in (0, 1]$ is the wavefunction renormalisation, $S_{F}^{(M)}$ is the free Dirac propagator at mass $M$ , and $ρ (M^{2})$ is a positive spectral density supported above the multi-particle threshold. For the free theory $Z = 1$ and $ρ \equiv 0$ .

The Källén-Lehmann representation is the spectral analogue of the Plancherel decomposition for a self-adjoint operator: the propagator is a positive measure on the spectrum of $P^{2} = H^{2} - ∣ P ∣^{2}$ , and the free theory's pure-pole structure becomes the simplest possible spectral measure. Positivity of $ρ$ is the Wightman-positivity input, and the propagator's ultraviolet behaviour is constrained by the convergence of the spectral integral.

Theorem (LSZ reduction for spin- $\frac{1}{2}$ ). In a Lorentz-covariant theory with the Dirac field as an asymptotic field, the $S$ -matrix element between an in-state of electrons and an out-state of electrons (and positrons) is given by the amputated time-ordered Green's function multiplied by the on-shell $u, v$ spinors: $$ \langle \text{out} | S | \text{in} \rangle = \prod_{i} (Z_2)^{1/2} \bar u(p_i^{out}) \cdots G_{\mathrm{amp}}(p_i^{out}, p_j^{in}) \cdots u(p_j^{in}) (Z_2)^{1/2}. $$ The factors $Z_{2}$ are the wavefunction renormalisations of the Dirac field, the $\overset{u}{ˉ}, u$ project onto the one-particle pole, and amputation strips the external propagators.

The LSZ formula reduces the calculation of scattering amplitudes to the calculation of amputated $n$ -point Green's functions, computable via Feynman diagrams once an interaction Lagrangian is specified. For the free Dirac field $S = 1$ , so LSZ is the consistency check that the free field reproduces the identity $S$ -matrix, but the framework is set up for the interacting case.

Synthesis. The free Dirac field is the foundational reason quantum field theory and the relativistic spin- $\frac{1}{2}$ particle become two views of one object. The central insight is that promoting the classical Dirac equation 12.11.01 to an operator equation forces simultaneously the existence of antiparticles, the canonical anticommutation relations, Lorentz covariance via the spin- $\frac{1}{2}$ representation of $Spin^{+} (1, 3)$ , and the spin-statistics theorem. This is exactly the unification that builds toward 12.13.02 (fermionic Fock space) as the algebraic substrate, and identifies the geometric Clifford algebra of 03.09.02 with the operator-algebraic CAR. Putting these together, the mode expansion $ψ (x) = \sum_{s} \int (\dots) (a_{p}^{s} u^{s} (p) e^{- i p \cdot x} + b_{p}^{s †} v^{s} (p) e^{+ i p \cdot x})$ does double duty: read as a sum of one-particle wave functions it recovers the classical Dirac theory; read as an operator equation it generates the entire $S$ -matrix machinery via Wick's theorem and the LSZ formula. The bridge between these readings is the Wightman two-point function $W_{2} (x, y)$ , which on the one hand is the classical Green's function of the Dirac operator with a specific boundary prescription, and on the other hand encodes the full operator-algebra structure of the quantum field via the reconstruction theorem.

Several pairings that look distinct at first inspection turn out to be the same. Charge-conjugation $C$ that swaps $a \leftrightarrow b$ in the mode expansion is dual to parity-times-time-reversal $P T$ in the sense of the CPT theorem, and the three together form a discrete antiunitary symmetry of any Lorentz-covariant Wightman field theory. The Feynman propagator $S_{F} (x - y)$ and the Pauli-Jordan function $Δ (x - y)$ are two slices of the same Wightman two-point distribution, differing only in their analytic continuation prescriptions; the bridge is the spectral representation, where each is a contour-deformation of the other in the complex $p^{0}$ -plane. The Pfaffian sign structure of fermionic Wick contractions and the CAR algebra are two presentations of the same exterior-algebra structure, with the CAR being the Hilbert-space realisation of the Pfaffian's algebraic identities. Finally, the spin-statistics theorem identifies the half-integer-spin transformation property under $Spin^{+} (1, 3)$ with the requirement of CAR quantisation, identifying $X$ with $Y$ where $X$ is the topological double-cover structure of the Lorentz group and $Y$ is the antisymmetric tensor structure of fermionic Fock space — a deep identification that recurs throughout the construction of physically reasonable quantum field theories.

Full proof set Master

Proposition (positivity of the normal-ordered Hamiltonian). Let $ψ (x)$ be the free Dirac field defined above, and let $H$ be the normal-ordered Hamiltonian $$ H = \sum_s \int \frac{d^3\mathbf{p}}{(2\pi)^3} E_{\mathbf{p}} \left( a^{s\dagger}{\mathbf{p}} a^s{\mathbf{p}} + b^{s\dagger}{\mathbf{p}} b^s{\mathbf{p}} \right). $$ Then $H \geq 0$ on the Fock space $F_{a}$ , with $H ∣0 ⟩ = 0$ on the vacuum, and the spectrum of $H$ is ${0} \cup [m, \infty)$ .

Proof. Each operator $N_{p, e}^{s} := a_{p}^{s †} a_{p}^{s}$ is the number operator for electrons of momentum $p$ and spin $s$ . Since $a_{p}^{s}$ satisfies ${a_{p}^{s}, a_{q}^{r †}} = (2 π)^{3} δ^{(3)} (p - q) δ^{sr}$ , the operator $N_{p, e}^{s}$ on a smeared mode $a^{s †} (f) = \int d^{3} p / (2 π)^{3} f (p) a_{p}^{s †}$ has spectrum ${0, 1}$ — this is the Pauli-exclusion eigenvalue structure of 12.13.02. The integrand $E_{p} \geq m > 0$ is strictly positive on the mass shell. Each individual term $E_{p} N_{p, e}^{s}$ is therefore a positive self-adjoint operator. The same is true for the positron sector $b_{p}^{s †} b_{p}^{s}$ .

$H$ is the integral of a positive density, hence $⟨ χ ∣ H ∣ χ ⟩ \geq 0$ for every $χ$ in the dense subspace of finite-particle states. The vacuum eigenvalue is $H ∣0 ⟩ = 0$ since each annihilation operator kills $∣0 ⟩$ . The lowest excited state is a single electron or positron at rest, with energy $E = m$ . Multi-particle states have energy $\geq 2 m$ (two particles, each at least at rest) up to a continuum that fills $[m, \infty)$ once one allows momentum smearing. The spectrum is ${0} \cup [m, \infty)$ , with isolated eigenvalue $0$ and continuous spectrum starting at $m$ . $□$

Proposition (Lorentz invariance of $i Δ$ ). The Pauli-Jordan function $$ i\Delta(z) = \int \frac{d^3\mathbf{p}}{(2\pi)^3 2 E_{\mathbf{p}}} \left[ e^{-ip\cdot z} - e^{+ip\cdot z} \right]{p^0 = E{\mathbf{p}}} $$ is invariant under proper orthochronous Lorentz transformations: $Δ (Λ z) = Δ (z)$ for every $Λ \in SO^{+} (1, 3)$ .

Proof. The integration measure $d^{3} p / (2 E_{p})$ is the Lorentz-invariant measure on the upper mass shell $X_{m}$ , restricted to the spatial slice. Equivalently it can be written as $\int d^{4} p δ (p^{2} - m^{2}) θ (p^{0})$ , which is manifestly Lorentz-invariant under proper orthochronous transformations. The exponentials $e^{\pm i p \cdot z}$ are Lorentz-scalars in $z$ once $p$ is integrated against an invariant measure. Hence the change of variables $p \to Λ^{- 1} p$ in the integral defining $Δ (Λ z)$ produces $Δ (z)$ . $□$

Proposition (microcausality from spacelike vanishing of $Δ$ ). For any spacelike-separated $x, y$ , the field anticommutator vanishes: ${ψ_{α} (x), \overset{ˉ}{ψ}_{β} (y)} = 0$ .

Proof. From the proof of the key theorem, ${ψ (x), \overset{ˉ}{ψ} (y)}_{α β} = (i γ^{μ} \partial_{μ}^{x} + m)_{α β} i Δ (x - y)$ . It suffices to show $Δ (z) = 0$ for spacelike $z$ . By the previous Proposition, $Δ$ is Lorentz-invariant. For spacelike $z$ choose a frame in which $z^{0} = 0$ and $z \neq = 0$ . Then $$ i\Delta(0, \mathbf{z}) = \int \frac{d^3\mathbf{p}}{(2\pi)^3 2 E_{\mathbf{p}}} \left[ e^{+i \mathbf{p}\cdot\mathbf{z}} - e^{-i \mathbf{p}\cdot\mathbf{z}} \right]. $$ The two pieces are equal after $p \to - p$ (the measure and $E_{p}$ are even in $p$ ), so the integrand cancels pointwise. Hence $Δ (0, z) = 0$ , and by Lorentz invariance $Δ (z) = 0$ for every spacelike $z$ . $□$

Proposition (Feynman propagator as Green's function). $(i γ^{μ} \partial_{μ}^{x} - m) S_{F} (x - y) = i δ^{(4)} (x - y) 1_{4}$ where $S_{F} (x - y) = \int d^{4} p / (2 π)^{4} i (γ^{μ} p_{μ} + m) / (p^{2} - m^{2} + i ϵ) e^{- i p \cdot (x - y)}$ .

Proof. Apply the operator under the integral. Use $(γ^{μ} p_{μ} - m) (γ^{ν} p_{ν} + m) = p^{2} - m^{2}$ from the Clifford algebra plus the on-shell identities: $$ (i\gamma^\mu\partial^x_\mu - m) S_F(x-y) = \int \frac{d^4 p}{(2\pi)^4} \frac{i(\gamma^\mu p_\mu - m)(\gamma^\nu p_\nu + m)}{p^2 - m^2 + i\epsilon} e^{-ip\cdot(x-y)} = i \int \frac{d^4 p}{(2\pi)^4} \frac{p^2 - m^2}{p^2 - m^2 + i\epsilon} e^{-ip\cdot(x-y)}. $$ As $ϵ \to 0^{+}$ , the ratio $(p^{2} - m^{2}) / (p^{2} - m^{2} + i ϵ) \to 1$ pointwise off the mass shell, and the integral converges to the distributional identity $i \int d^{4} p / (2 π)^{4} e^{- i p \cdot (x - y)} = i δ^{(4)} (x - y)$ . $□$

Proposition (CAR forced by microcausality plus positivity). Assume $ψ (x) = \sum_{s} \int (\dots) (a_{p}^{s} u^{s} (p) e^{- i p \cdot x} + \tilde{b}_{p}^{s} v^{s} (p) e^{+ i p \cdot x})$ with $a$ and $\tilde{b}$ satisfying either CCR or CAR. If ${ψ (x), \overset{ˉ}{ψ} (y)} = 0$ (or $[ψ (x), \overset{ˉ}{ψ} (y)] = 0$ ) for spacelike $x, y$ and $H \geq 0$ , then $\tilde{b}_{p}^{s} = b_{p}^{s †}$ and the algebra is CAR.

Proof. Direct computation (Exercises 7 and 9) shows that spacelike vanishing of the (anti)commutator requires the two integrals $\int e^{- i p \cdot z}$ and $\int e^{+ i p \cdot z}$ to cancel. With CAR plus $\tilde{b} = b^{†}$ , the signs work out as in the microcausality theorem. With CCR plus $\tilde{b} = b^{†}$ , the signs do not cancel and microcausality fails. With CCR plus $\tilde{b} = b$ (annihilation), one would recover spacelike vanishing of the commutator but the Hamiltonian $H = \sum_{s} \int (\dots) E_{p} (a_{p}^{s †} a_{p}^{s} - b_{p}^{s †} b_{p}^{s})$ would be unbounded below (Exercise 7). Hence CAR plus antiparticle creation is the unique choice satisfying all three constraints. This is the constructive spin- $\frac{1}{2}$ piece of the spin-statistics theorem. $□$

Proposition (charge as integral of the Noether current). The electric charge operator $Q = e \int d^{3} x : j^{0} (x) :$ with $j^{μ} = \overset{ˉ}{ψ} γ^{μ} ψ$ acts on the Fock space as $Q = e \sum_{s} \int d^{3} p / (2 π)^{3} (a_{p}^{s †} a_{p}^{s} - b_{p}^{s †} b_{p}^{s})$ . In particular $Q ∣0 ⟩ = 0$ , $Q a_{p}^{s †} ∣0 ⟩ = e a_{p}^{s †} ∣0 ⟩$ , and $Q b_{p}^{s †} ∣0 ⟩ = - e b_{p}^{s †} ∣0 ⟩$ , so the electron has charge $+ e$ and the positron has charge $- e$ in the convention $e < 0$ for the physical electron, or with the opposite sign for the symmetric convention.

Proof. Insert the mode expansion of $ψ$ and $\overset{ˉ}{ψ}$ into $j^{0} = ψ^{†} ψ$ (note $\overset{ˉ}{ψ} γ^{0} = ψ^{†}$ ), integrate over $x$ to localise to equal momenta via $\int d^{3} x e^{i (p - q) \cdot x} = (2 π)^{3} δ^{(3)} (p - q)$ , and apply the spin-sum identities $\sum_{s} u^{s} (p) u^{s †} (p) / (2 E_{p}) = 1_{4}$ (and similarly for $v$ ) on shell. The cross terms $a u^{s †} \cdot v^{r} b^{†}$ pick up oscillating exponentials $e^{\pm 2 i E_{p} t}$ which average out in the equal-time normal-ordered expression. The diagonal terms produce $\sum_{s} \int d^{3} p / (2 π)^{3} (a_{p}^{s †} a_{p}^{s} - b_{p}^{s †} b_{p}^{s})$ . Multiplying by $e$ gives the claimed result. The eigenvalues on the one-particle states follow from $[Q, a_{p}^{s †}] = e a_{p}^{s †}$ and $[Q, b_{p}^{s †}] = - e b_{p}^{s †}$ , again by the standard fermionic-commutator algebra. $□$

Theorem (spin-statistics, full Streater-Wightman form), stated without proof — see Streater-Wightman 1964 Ch. 4 ^{[Streater-Wightman 1964]}. The full theorem applies to any local Wightman field with finite-dimensional Lorentz representation. The half-integer-spin case requires CAR (the constructive case proved above is its lowest-spin instance); the integer-spin case requires CCR. The route to the general statement uses the analyticity of Wightman functions in the complexified Lorentz group (Bargmann-Hall-Wightman theorem) to convert the spacelike-vanishing constraint into a constraint on the Schwinger functions, where the spin-statistics connection appears as a sign in the Euclidean two-point function.

Connections Master

Dirac equation and relativistic spin 12.11.01. The free Dirac quantum field promotes the classical Dirac equation to an operator-valued tempered distribution. The plane-wave solutions $u^{s} (p), v^{s} (p)$ developed there are reused here as the spinor coefficients in the mode expansion. The relation is: classical Dirac theory is the one-particle sector of the quantum-field theory, recovered by taking the matrix element $⟨ 0∣ ψ (x) ∣ f ⟩$ on a wave-packet one-electron state $∣ f ⟩$ .
Fermionic Fock space and Pauli exclusion 12.13.02. The free Dirac field is canonically quantised on the antisymmetric Fock space $F_{a} = ⨁_{n} Λ^{n} H_{1}$ . The CAR algebra developed there is the algebraic substrate: every mode $p, s$ contributes a copy of the two-dimensional CAR generators ${a, a^{†}}$ , and the Dirac field is built by tensoring these copies together over the mass-shell measure. The Pauli exclusion principle and the antisymmetry of multi-electron states are direct consequences of this Fock structure.
Clifford algebra 03.09.02. The spinor index $α \in {1, 2, 3, 4}$ of the Dirac field is a representation of the Clifford algebra Cl(1, 3) generated by the $γ$ -matrices. The intertwining identity $S (Λ)^{- 1} γ^{μ} S (Λ) = Λ^{μ}_{ν} γ^{ν}$ identifies the spin- $\frac{1}{2}$ representation of the Lorentz group with the Clifford module structure of $C^{4}$ , providing the algebraic bridge between the geometric spin-structure machinery of 03.09.02 and the operator-field machinery of this unit.
Special relativity and Lorentz transformations 10.05.01. The Lorentz covariance of the Dirac field is the input that forces the spin-statistics theorem and the mass-shell structure. The unitary representation $U (Λ)$ on Fock space lifts the Lorentz action on Minkowski spacetime to an action on quantum states, and the intertwining condition $U (Λ) ψ (x) U (Λ)^{- 1} = S (Λ)^{- 1} ψ (Λ x)$ is what makes the field Lorentz-covariant rather than merely Lorentz-invariant.
Angular momentum operators / SU(2) 12.05.01. The non-relativistic limit of the Dirac field recovers the Pauli theory: the upper two components of $ψ (x)$ become the two-component Pauli spinor, and the SU(2) angular-momentum generators of 12.05.01 are recovered as the rotation generators of the Lorentz subgroup acting on those upper components. The spin- $\frac{1}{2}$ algebra of 12.05.01 is the non-relativistic remnant of the full spin- $\frac{1}{2}$ representation of $Spin^{+} (1, 3)$ developed here.
Canonical quantum field theory 12.12.01. The free Dirac field is one of the three textbook quantum field theories — alongside the free Klein-Gordon scalar field and the free Maxwell / Proca vector field — that are simultaneously exactly solvable and physically relevant. The canonical quantisation procedure of 12.12.01 specialises here to the fermionic case where the equal-time commutators are replaced by equal-time anticommutators and the symmetric Fock space is replaced by the antisymmetric one.

Historical & philosophical context Master

The free Dirac quantum field condenses a sequence of insights between 1927 and 1940. Dirac's 1927 paper The Quantum Theory of the Emission and Absorption of Radiation (Proc. Roy. Soc. A 114, 243–265) ^{[Dirac 1927]} introduced the idea of a quantised field of photons, with creation and annihilation operators acting on a Hilbert space of variable photon number. Jordan and Wigner's 1928 paper Über das Paulische Äquivalenzverbot (Z. Phys. 47, 631–651) ^{[Jordan-Wigner 1928]} introduced anticommutation relations as the appropriate quantisation rule for matter fields, demonstrating that the Pauli exclusion principle was a consequence of the algebra rather than a separate postulate. Heisenberg and Pauli's two 1929–30 papers Zur Quantendynamik der Wellenfelder (Z. Phys. 56, 1 and 59, 168) ^{[Heisenberg-Pauli 1929]} gave the first Lagrangian-based covariant quantisation of the Dirac field as an operator-valued field, with canonical anticommutators specified at equal times and Lorentz invariance verified at the level of the action. The negative-energy solutions, originally interpreted by Dirac through the hole-theory of A Theory of Electrons and Protons (Proc. Roy. Soc. A 126, 360–365, 1930), were translated into the modern antiparticle interpretation by the late 1930s, with the experimental confirmation provided by Anderson's 1933 The Positive Electron (Phys. Rev. 43, 491–494) ^{[Anderson 1933]}.

The spin-statistics theorem was crystallised by Fierz's 1939 Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin (Helv. Phys. Acta 12, 3) ^{[Fierz 1939]} and Pauli's 1940 The Connection Between Spin and Statistics (Phys. Rev. 58, 716–722) ^{[Pauli 1940]}. Pauli's argument used the explicit form of the commutator / anticommutator of free fields and showed that half-integer spin combined with bosonic quantisation produces either a Hamiltonian unbounded below or a violation of microcausality. The axiomatic version, due to Lüders (1958) and packaged by Streater and Wightman in their 1964 monograph PCT, Spin and Statistics, and All That ^{[Streater-Wightman 1964]}, deduces the same conclusion from the Wightman axioms: positivity, Poincaré-covariance, and local commutativity together force the spin-statistics correspondence on any Lorentz-covariant quantum field. The constructive proof presented here is the spin- $\frac{1}{2}$ case of the same theorem.

Weinberg's 1995 reformulation in The Quantum Theory of Fields, Vol. I §5 ^{[Weinberg 1995]}, inverts the historical order: the Dirac field is derived from the Wigner classification of irreducible unitary representations of the Poincaré group, with the spinor structure appearing as the lowest-mass-shell half-integer-spin representation. The mode expansion, antiparticle interpretation, and CAR all fall out of the requirement that the field generate a Lorentz-covariant operator algebra with local commutators / anticommutators. This route makes manifest that the Dirac field is uniquely determined by symmetry plus locality plus positivity. The Glimm-Jaffe 1987 functional-integral approach ^{[Glimm-Jaffe 1987]} presents the same object from the Euclidean side, where the Dirac field becomes a Grassmann-valued random field with covariance equal to the Wick-rotated Feynman propagator, identifying the Wightman-Schwinger correspondence as a Wick rotation in the spin- $\frac{1}{2}$ setting.

Bibliography Master

@article{Dirac1927Radiation,
  author  = {Dirac, P. A. M.},
  title   = {The Quantum Theory of the Emission and Absorption of Radiation},
  journal = {Proc. Roy. Soc. London A},
  volume  = {114},
  year    = {1927},
  pages   = {243--265}
}

@article{JordanWigner1928,
  author  = {Jordan, P. and Wigner, E.},
  title   = {{\"U}ber das Paulische {\"A}quivalenzverbot},
  journal = {Z. Phys.},
  volume  = {47},
  year    = {1928},
  pages   = {631--651}
}

@article{HeisenbergPauli1929,
  author  = {Heisenberg, W. and Pauli, W.},
  title   = {Zur Quantendynamik der Wellenfelder},
  journal = {Z. Phys.},
  volume  = {56},
  year    = {1929},
  pages   = {1--61}
}

@article{HeisenbergPauli1930,
  author  = {Heisenberg, W. and Pauli, W.},
  title   = {Zur Quantendynamik der Wellenfelder II},
  journal = {Z. Phys.},
  volume  = {59},
  year    = {1930},
  pages   = {168--190}
}

@article{Dirac1930HoleTheory,
  author  = {Dirac, P. A. M.},
  title   = {A Theory of Electrons and Protons},
  journal = {Proc. Roy. Soc. London A},
  volume  = {126},
  year    = {1930},
  pages   = {360--365}
}

@article{Anderson1933Positron,
  author  = {Anderson, C. D.},
  title   = {The Positive Electron},
  journal = {Phys. Rev.},
  volume  = {43},
  year    = {1933},
  pages   = {491--494}
}

@article{Fierz1939SpinStatistics,
  author  = {Fierz, M.},
  title   = {{\"U}ber die relativistische Theorie kr{\"a}ftefreier Teilchen mit beliebigem Spin},
  journal = {Helv. Phys. Acta},
  volume  = {12},
  year    = {1939},
  pages   = {3--37}
}

@article{Pauli1940SpinStatistics,
  author  = {Pauli, W.},
  title   = {The Connection Between Spin and Statistics},
  journal = {Phys. Rev.},
  volume  = {58},
  year    = {1940},
  pages   = {716--722}
}

@article{Wightman1956VEVs,
  author  = {Wightman, A. S.},
  title   = {Quantum Field Theory in Terms of Vacuum Expectation Values},
  journal = {Phys. Rev.},
  volume  = {101},
  year    = {1956},
  pages   = {860--866}
}

@book{StreaterWightman1964,
  author    = {Streater, R. F. and Wightman, A. S.},
  title     = {PCT, Spin and Statistics, and All That},
  publisher = {W. A. Benjamin (reprinted Princeton Univ. Press 2000)},
  year      = {1964}
}

@book{BjorkenDrellRQF,
  author    = {Bjorken, J. D. and Drell, S. D.},
  title     = {Relativistic Quantum Fields},
  publisher = {McGraw-Hill},
  year      = {1965}
}

@book{ItzyksonZuber1980,
  author    = {Itzykson, C. and Zuber, J.-B.},
  title     = {Quantum Field Theory},
  publisher = {McGraw-Hill},
  year      = {1980}
}

@book{GlimmJaffe1987,
  author    = {Glimm, J. and Jaffe, A.},
  title     = {Quantum Physics: A Functional Integral Point of View},
  publisher = {Springer-Verlag},
  edition   = {2},
  year      = {1987}
}

@book{PeskinSchroeder1995,
  author    = {Peskin, M. E. and Schroeder, D. V.},
  title     = {An Introduction to Quantum Field Theory},
  publisher = {Westview Press},
  year      = {1995}
}

@book{Weinberg1995QTFv1,
  author    = {Weinberg, S.},
  title     = {The Quantum Theory of Fields, Vol. I: Foundations},
  publisher = {Cambridge University Press},
  year      = {1995}
}

@book{Srednicki2007QFT,
  author    = {Srednicki, M.},
  title     = {Quantum Field Theory},
  publisher = {Cambridge University Press},
  year      = {2007}
}

@misc{TongQFTNotes,
  author       = {Tong, D.},
  title        = {Quantum Field Theory (DAMTP Cambridge lecture notes)},
  year         = {2007},
  howpublished = {Lecture notes, University of Cambridge}
}

Prerequisites

03.09.02 pending
10.05.01 pending
12.05.01 pending
12.11.01 pending
12.13.02 pending

Tier anchors

beginner: Zee, *Quantum Field Theory in a Nutshell*, 2e (Princeton, 2010), Ch. II.1 (The Dirac field, sketch); Aitchison & Hey, *Gauge Theories in Particle Physics*, 4e (CRC, 2013), Vol. I Ch. 7 (Dirac quanta and antiparticles, physics-style)
intermediate: Peskin & Schroeder, *An Introduction to Quantum Field Theory* (Westview, 1995), Ch. 3 §3.5 (quantisation of the Dirac field, CAR, mode expansion, antiparticle interpretation); Srednicki, *Quantum Field Theory* (Cambridge, 2007), Chs. 36–43 (Dirac field, Weinberg-convention $\gamma$-matrices, $u, v$ spinors); Tong, *Quantum Field Theory*, DAMTP Cambridge lecture notes, §5 (Quantising the Dirac field)
master: Weinberg, S., *The Quantum Theory of Fields*, Vol. I (Cambridge, 1995), §5.5–§5.7 (causal Dirac field as a Lorentz-covariant operator-valued distribution, $\gamma$-matrices in mostly-plus, microcausality forces CAR); Streater, R. F. & Wightman, A. S., *PCT, Spin and Statistics, and All That* (Benjamin, 1964; Princeton Landmarks reprint, 2000), Ch. 4 (spin-statistics theorem); Glimm, J. & Jaffe, A., *Quantum Physics: A Functional Integral Point of View*, 2e (Springer, 1987), Ch. 6 (free Dirac field as an operator-valued distribution on fermionic Fock space)

References

raw/pdfs/qft/qft.pdf · Tong's QFT lecture notes — §5 Quantising the Dirac field; mode expansion, CAR, Hamiltonian and momentum, microcausality of $\{\psi(x), \bar\psi(y)\}$
Weinberg, S. — The Quantum Theory of Fields, Vol. I · Cambridge University Press (1995), §5.5–§5.7 (causal Dirac field, $\gamma$-matrix conventions, antiparticles from microscopic causality, spin-statistics)
Dirac, P. A. M. — The Quantum Theory of the Emission and Absorption of Radiation · Proc. Roy. Soc. A 114, 243–265 (1927) — first quantum-field paper, photon emission/absorption operators; the seed for what becomes the quantised Dirac field
Jordan, P. & Wigner, E. — Über das Paulische Äquivalenzverbot · Z. Phys. 47, 631–651 (1928) — introduction of fermionic anticommutation relations; the canonical quantisation of the Dirac field requires the CAR rather than the CCR
Heisenberg, W. & Pauli, W. — Zur Quantendynamik der Wellenfelder · Z. Phys. 56, 1–61 (1929); Z. Phys. 59, 168–190 (1930) — Lagrangian formulation of QFT and quantisation of the Dirac field as an operator-valued field
Pauli, W. — The Connection Between Spin and Statistics · Phys. Rev. 58, 716–722 (1940) — spin-statistics theorem; the Dirac field must be quantised with anticommutators on pain of energy unboundedness
Fierz, M. — Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin · Helv. Phys. Acta 12, 3–37 (1939) — precursor to Pauli 1940 on spin-statistics
Anderson, C. D. — The Positive Electron · Phys. Rev. 43, 491–494 (1933) — experimental discovery of the positron, the antiparticle predicted by the negative-frequency branch of the Dirac field
Peskin, M. E. & Schroeder, D. V. — An Introduction to Quantum Field Theory · Westview Press (1995), Ch. 3 §3.5 (quantisation of the Dirac field; mode expansion, CAR, Hamiltonian, normal-ordering, antiparticles)
Srednicki, M. — Quantum Field Theory · Cambridge University Press (2007), Chs. 36–45 (Dirac field, $u^s(p), v^s(p)$ spinors, propagator, LSZ for spin-1/2)
Streater, R. F. & Wightman, A. S. — PCT, Spin and Statistics, and All That · Benjamin (1964); Princeton Landmarks reprint (2000), Ch. 4 (spin-statistics theorem, axiomatic proof from Wightman positivity plus locality)
Glimm, J. & Jaffe, A. — Quantum Physics: A Functional Integral Point of View · Springer (2e, 1987), Ch. 6 (Dirac field on fermionic Fock space; canonical anticommutation distributions $\{\psi(x), \psi^\dagger(y)\}$; Wightman reconstruction)
Itzykson, C. & Zuber, J.-B. — Quantum Field Theory · McGraw-Hill (1980), Ch. 3 (free Dirac field; Feynman propagator $S_F(x-y)$; Wick rotation; charge-conjugation symmetry)
Bjorken, J. D. & Drell, S. D. — Relativistic Quantum Fields · McGraw-Hill (1965), Ch. 13 (quantisation of the Dirac field, hole theory's translation into the operator language)
Wightman, A. S. — Quantum Field Theory in Terms of Vacuum Expectation Values · Phys. Rev. 101, 860–866 (1956) — axiomatic framework in which the Dirac field is realised as an operator-valued tempered distribution

Estimated time

beginner: 18m
intermediate: 45m
master: 75m