Part I, Chapter 5

The Dirac Field

Relativistic quantum theory of spin-1/2 particles

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Video Lecture

Lecture 13: Introducing the Dirac Equation - MIT 8.323

Classical Dirac field theory and spinor structure (MIT QFT Course)

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Video Lecture

What is the Dirac Equation — Dirac Equation Explained

Accessible overview of the Dirac equation in relativistic quantum mechanics

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📚 Related Topics

• Lagrangian Formalism Page 7 - Original location of this content
• Dirac Quantization (Part II) - Quantization with anticommutators
• Spin 1/2 (QM) - Prerequisite spinor concepts

5.1 The Dirac Equation

To describe spin-1/2 particles (electrons, quarks, neutrinos), we need a spinor field ψ(x). The Dirac field is a 4-component complex object:

$$\psi(x) = \begin{pmatrix} \psi_1(x) \\ \psi_2(x) \\ \psi_3(x) \\ \psi_4(x) \end{pmatrix}$$

The Dirac Lagrangian is:

$$\boxed{\mathcal{L} = \bar{\psi}(i\gamma^\mu \partial_\mu - m)\psi}$$

where:

$\bar{\psi} = \psi^\dagger \gamma^0$ is the Dirac adjoint
γ^μ are the Dirac gamma matrices (4×4 matrices)
m is the fermion mass
We use natural units with ℏ = c = 1

Gamma Matrices

The gamma matrices satisfy the Clifford algebra:

$$\boxed{\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}\mathbb{I}}$$

where { , } denotes the anticommutator. In the Dirac (standard) representation:

$$\gamma^0 = \begin{pmatrix} \mathbb{I} & 0 \\ 0 & -\mathbb{I} \end{pmatrix}, \quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix}$$

where σⁱ are the Pauli matrices.

5.2 Deriving the Dirac Equation

Treating ψ and $\bar{\psi}$ as independent fields, the Euler-Lagrange equation for $\bar{\psi}$ gives:

$$\boxed{(i\gamma^\mu \partial_\mu - m)\psi = 0}$$

This is the Dirac equation. Multiplying by (iγ^ν∂_ν + m):

$$(i\gamma^\nu \partial_\nu + m)(i\gamma^\mu \partial_\mu - m)\psi = 0$$

Using the Clifford algebra:

$$-\gamma^\nu \gamma^\mu \partial_\nu \partial_\mu - m^2 = -\frac{1}{2}\{\gamma^\nu, \gamma^\mu\}\partial_\nu \partial_\mu - m^2 = -\partial_\mu \partial^\mu - m^2$$

Therefore each component satisfies the Klein-Gordon equation:

$$(\Box + m^2)\psi = 0$$

But the Dirac equation is first-order in time, unlike Klein-Gordon. This allows for a positive-definite probability density.

5.3 Dirac Adjoint and Conserved Current

Varying with respect to ψ gives the adjoint Dirac equation:

$$\bar{\psi}(i\overleftarrow{\partial}_\mu \gamma^\mu + m) = 0$$

or equivalently:

$$i\partial_\mu \bar{\psi} \gamma^\mu + m\bar{\psi} = 0$$

Probability Current

Multiplying the Dirac equation by $\bar{\psi}$ from the left and the adjoint equation by ψ from the right, then adding:

$$\partial_\mu(\bar{\psi}\gamma^\mu \psi) = 0$$

Therefore the Dirac current:

$$\boxed{j^\mu = \bar{\psi}\gamma^\mu \psi}$$

is conserved: ∂_μj^μ = 0. The time component:

$$j^0 = \psi^\dagger \psi \geq 0$$

is positive definite, solving the negative probability problem of the Klein-Gordon equation!

5.4 Global U(1) Symmetry

The Dirac Lagrangian is invariant under the global phase transformation:

$$\psi \to e^{i\alpha}\psi, \quad \bar{\psi} \to e^{-i\alpha}\bar{\psi}$$

where α is a constant. This U(1) symmetry leads to the conserved current j^μ = $\bar{\psi}\gamma^\mu \psi$ by Noether's theorem.

The conserved charge:

$$Q = \int d^3x \, j^0 = \int d^3x \, \psi^\dagger \psi$$

is the total fermion number (electric charge for electrons).

Lorentz Invariance

Under a Lorentz transformation Λ, the Dirac field transforms as:

$$\psi(x) \to S(\Lambda)\psi(\Lambda^{-1}x)$$

where S(Λ) is a 4×4 matrix representation of the Lorentz group satisfying:

$$S(\Lambda)^{-1}\gamma^\mu S(\Lambda) = \Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu$$

This ensures the Dirac Lagrangian is Lorentz invariant. Spinors transform under the (1/2, 0) ⊕ (0, 1/2) representation of the Lorentz group.

5.5 Energy-Momentum Tensor for Dirac Field

The canonical energy-momentum tensor is:

$$T^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \psi)}\partial^\nu \psi + \frac{\partial \mathcal{L}}{\partial(\partial_\mu \bar{\psi})}\partial^\nu \bar{\psi} - g^{\mu\nu}\mathcal{L}$$

For the Dirac Lagrangian:

$$\frac{\partial \mathcal{L}}{\partial(\partial_\mu \psi)} = i\bar{\psi}\gamma^\mu$$

Therefore:

$$\boxed{T^{\mu\nu}_{\text{can}} = i\bar{\psi}\gamma^\mu \partial^\nu \psi}$$

This is not symmetric. The symmetric (Belinfante) tensor is:

$$\boxed{T^{\mu\nu} = \frac{i}{4}\bar{\psi}\gamma^\mu \overleftrightarrow{\partial}^\nu \psi}$$

where $\overleftrightarrow{\partial}^\nu = \overrightarrow{\partial}^\nu - \overleftarrow{\partial}^\nu$.

Hamiltonian

The Hamiltonian density is:

$$\mathcal{H} = T^{00} = i\bar{\psi}\gamma^0 \partial_0 \psi = i\psi^\dagger \dot{\psi}$$

Using the Dirac equation, this can be rewritten as:

$$\mathcal{H} = \bar{\psi}(-i\gamma^i \partial_i + m)\psi$$

The energy density is:

$$T^{00} = i\psi^\dagger \dot{\psi} = \bar{\psi}(-i\vec{\gamma} \cdot \nabla + m)\psi$$

Total energy:

$$E = \int d^3x \, T^{00}$$

🎯 Key Concepts: The Dirac Field

• Dirac Lagrangian: $\mathcal{L} = \bar{\psi}(i\gamma^\mu \partial_\mu - m)\psi$
• Dirac equation: (iγ^μ∂_μ - m)ψ = 0 (first-order in time)
• Clifford algebra: {γ^μ, γ^ν} = 2g^μν
• Conserved current: j^μ = $\bar{\psi}\gamma^\mu \psi$
• Positive probability: ρ = ψ^†ψ ≥ 0 (solves Klein-Gordon problem)
• U(1) symmetry: Global phase invariance → charge conservation
• Spinor representation: (1/2, 0) ⊕ (0, 1/2) of Lorentz group
• Energy-momentum tensor: T^μν = i ψ̄γ^μ∂^νψ (not symmetric)

📖 What's Next?

Now that you understand the classical Dirac field, you're ready for:

• Part II: Dirac Quantization - Anticommutators and fermion creation operators
• Spin-Statistics Theorem - Why fermions need anticommutators
• Energy-Momentum Tensor Chapter - General theory and other field examples

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Chapter 5

The Dirac Field

Part I Overview →

Runnable Simulations

Dirac Gamma Matrix Algebra Verification

Python

script.py39 lines

import numpy as np

# Dirac gamma matrices in the Dirac (standard) representation
I2 = np.eye(2, dtype=complex)
zero2 = np.zeros((2, 2), dtype=complex)
sigma_x = np.array([[0, 1], [1, 0]], dtype=complex)
sigma_y = np.array([[0, -1j], [1j, 0]], dtype=complex)
sigma_z = np.array([[1, 0], [0, -1]], dtype=complex)
sigmas = [sigma_x, sigma_y, sigma_z]

# gamma^0 = diag(I, -I), gamma^i = [[0, sigma^i], [-sigma^i, 0]]
gamma = [None] * 4
gamma[0] = np.block([[I2, zero2], [zero2, -I2]])
for i in range(3):
    gamma[i+1] = np.block([[zero2, sigmas[i]], [-sigmas[i], zero2]])

# Minkowski metric
eta = np.diag([1.0, -1.0, -1.0, -1.0])

print("Verification of Clifford Algebra: {gamma^mu, gamma^nu} = 2 eta^{mu,nu} I_4")
print("=" * 60)

for mu in range(4):
    for nu in range(mu, 4):
        anticomm = gamma[mu] @ gamma[nu] + gamma[nu] @ gamma[mu]
        expected = 2.0 * eta[mu, nu] * np.eye(4)
        match = np.allclose(anticomm, expected)
        print(f"  {{gamma^{mu}, gamma^{nu}}} = {2*eta[mu,nu]:+.0f} * I_4  -->  {'PASS' if match else 'FAIL'}")

# gamma^5
gamma5 = 1j * gamma[0] @ gamma[1] @ gamma[2] @ gamma[3]
print(f"\ngamma^5 = i gamma^0 gamma^1 gamma^2 gamma^3")
print(f"  (gamma^5)^2 = I_4:  {'PASS' if np.allclose(gamma5 @ gamma5, np.eye(4)) else 'FAIL'}")
print(f"  Tr(gamma^5) = {np.trace(gamma5).real:.0f}  (should be 0)")
for mu in range(4):
    ac = gamma5 @ gamma[mu] + gamma[mu] @ gamma5
    print(f"  {{gamma^5, gamma^{mu}}} = 0:  {'PASS' if np.allclose(ac, 0) else 'FAIL'}")

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Code will be executed with Python 3 on the server

Dirac Equation Energy-Momentum Relation

Fortran

program.f9035 lines

program dirac_energy
  implicit none
  double precision :: m, p, E_pos, E_neg
  double precision :: p_values(10)
  integer :: i

m = 0.511d0  ! electron mass in MeV

print *, "Dirac Equation: Energy-Momentum Relation"
  print *, "E = +/- sqrt(p^2 + m^2), m_e = 0.511 MeV"
  print *, "=========================================="
  print *, ""
  print '(A10, A15, A15, A15)', "p (MeV)", "E+ (MeV)", "E- (MeV)", "v/c = p/E"
  print *, "----------------------------------------------------------"

p_values = (/0.0d0, 0.1d0, 0.25d0, 0.511d0, 1.0d0, &
               2.0d0, 5.0d0, 10.0d0, 100.0d0, 1000.0d0/)

do i = 1, 10
    p = p_values(i)
    E_pos = sqrt(p**2 + m**2)
    E_neg = -E_pos
    print '(F10.3, F15.6, F15.6, F15.6)', p, E_pos, E_neg, p/E_pos
  end do

print *, ""
  print *, "Key observations:"
  print *, "  - Negative energy solutions led Dirac to predict antimatter"
  print *, "  - At p >> m, E approaches p (ultrarelativistic limit)"
  print *, "  - At p = 0, E = m (rest mass energy)"
  print *, "  - v/c = p/E always < 1 for massive particles"

end program dirac_energy

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Code will be compiled with gfortran and executed on the server