advanced33 min studyLesson 655 in Physics

Dirac spinors and completeness relations

Fix gamma-matrix and spinor conventions before using on-shell spin sums as matrices in a relativistic field expansion.

Before you begin

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The question for today

What you’ll understand

Distinguish the Dirac adjoint from the ordinary positive norm.
Use positive-energy on-shell u and v equations with correct mass signs.
Check completeness using rest-frame rank, trace, and normalization.

The Dirac equation is first order in spacetime derivatives but still describes relativistic particles with a positive mass shell. Its plane-wave solutions carry four spinor components and come in two positive-frequency and two negative-frequency basis solutions for a massive field. Before quantization, these are classical mode functions. Their inner products and outer products fix the normalization of the quantum expansion. A completeness relation is a matrix identity over spin labels, rather than a scalar probability or a sum over every momentum.

The adjoint follows from Lorentz covariance

Keep natural units and the plus, minus, minus, minus metric. Gamma matrices satisfy the corresponding Clifford algebra. Their numerical entries depend on representation, but changing the spinor basis consistently leaves bilinears and completeness identities unchanged. The Dirac adjoint multiplies the Hermitian conjugate by gamma zero. It is chosen because the resulting scalar bilinear transforms covariantly, not because it is a positive-definite probability norm for every four-component vector.

The ordinary Hermitian product of an on-shell spinor with itself is positive and proportional to its energy in the chosen frame. The Dirac scalar product is instead proportional to mass and has opposite signs for u and v solutions. A negative value of the latter does not signal a negative-norm antiparticle state. Positivity of the physical Hilbert-space norm is established by the quantum ladder algebra and state construction, which should not be conflated with the indefinite spinor bilinear.

\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\mathbf1,\qquad \bar u=u^{\dagger}\gamma^0,\quad u_r^{\dagger}u_s=2E\delta_{rs},\quad \bar u_ru_s=2m\delta_{rs}

η = diag(1,−1,−1,−1), ℏ = c = 1, m > 0, and E > 0. u and v in this normalization have energy dimension one-half. For v, v†rvₛ = 2Eδrₛ while v̄rvₛ = −2mδrₛ.↔ Scroll wide equations horizontally.

Both frequency families use positive on-shell momentum

A positive-frequency mode multiplies a u spinor by the negative imaginary momentum phase. A negative-frequency mode multiplies a v spinor by the opposite phase, while the label p still has positive energy. Inserting those phases into the same Dirac equation produces opposite mass signs in the algebraic spinor equations. Changing the time phase without changing its equation is a common source of erroneous completeness signs.

Each massive family has two independent spin states at a fixed on-shell momentum. Squaring the first-order operator gives the Klein–Gordon dispersion, but that second-order equation alone does not select the spinor subspace. The spinors also transform nontrivially under Lorentz transformations. In the massless limit a rest frame disappears and helicity becomes a useful label; formulas dividing by mass cannot simply be carried into that limit, although the usual spin-sum expressions have well-defined massless counterparts.

(\not p-m)u_s(p)=0,\qquad (\not p+m)v_s(p)=0,\qquad \not p=\gamma^\mu p_\mu,\quad p^0=E=\sqrt{\mathbf p^2+m^2}

The associated modes are uₛ(p)e⁻ⁱᵖ·ˣ and vₛ(p)e⁺ⁱᵖ·ˣ with p·x = Et−p·x. Spin label s runs over two states. These are classical on-shell solution identities, before assigning fermionic quantum operators.↔ Scroll wide equations horizontally.

Outer products resolve the appropriate solution subspaces

Summing u times its Dirac adjoint over both spin labels produces the slash-momentum matrix plus mass; the corresponding v sum has minus mass. Each matrix has rank two on the massive shell. Acting on its appropriate spinor verifies the eigenvalue, while acting on the opposite family gives zero. The relative minus sign in the v scalar normalization is needed for this algebra and does not authorize a negative physical excitation energy.

Spin sums are useful because unobserved spin labels in amplitudes can be replaced by matrix expressions, provided conjugation and normalization remain consistent. A sum using Hermitian adjoints instead has an additional gamma zero on the right. Factors from a field's inverse square root of twice energy must still be included separately. The identity does not itself establish canonical anticommutation, interactions, or a fully normalized scattering probability; those require additional algebra, flux conventions, and dynamics.

\sum_{s=1}^2u_s(p)\bar u_s(p)=\not p+m,\qquad \sum_{s=1}^2v_s(p)\bar v_s(p)=\not p-m

Each result is a 4×4 matrix of energy dimension one. In a Dirac basis at rest, γ⁰ = diag(1,1,−1,−1). The u sum becomes diag(2m,2m,0,0), while the v sum becomes diag(0,0,−2m,−2m).↔ Scroll wide equations horizontally.

Work it through

Check a rest-frame spin sum

Use a Dirac representation with γ⁰ = diag(1,1,−1,−1), rest mass m = 2, and normalized u spinors (2,0,0,0)ᵀ and (0,2,0,0)ᵀ in the declared units. Compute each scalar normalization and their outer-product sum.

At rest E = m = 2. For either spinor, multiplying by γ⁰ leaves its nonzero upper component unchanged, so $\bar u_su_s=u_s^{\dagger}u_s=4=2m$ .
The outer products have a single 4 in the first or second diagonal position. Summing gives diag(4,4,0,0).
The covariant expression is $\not p+m=m(\gamma^0+\mathbf1)$ , exactly the same matrix. Its rank is two and its trace is 8 = 4m.

The result

Both scalar normalizations equal 4, and the summed matrix is diag(4,4,0,0), matching slash p plus m.

The living notebook

Explore the model

Interactive lab

Interactive lab · quantum

A basis label is not the whole relativistic spinor

Reset and set BOTH Mixing angle θ to 0 degrees and Relative phase φ to 0 degrees; predict basis-zero probability 1. Then Reset and set BOTH θ to 180 degrees and φ to 0 degrees; predict basis-one probability 1. This pure two-state geometry illustrates two orthogonal spin labels only. It has no four-component Dirac spinor, Lorentz boost, gamma matrix, or spinor completeness solver.

Bloch vector / P(0)P(1)Phase latitude

Probability |0⟩: 71.1%
Probability |1⟩: 28.9%
Bloch (x, y, z): 0.78, 0.45, 0.42

Mixing angle θ65°Relative phase φ30°

t = 0 s

A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA two-dimensional spin basis is part of a Dirac mode label, not a replacement for its relativistic four-component structure.

Pause & reason

Check your understanding

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Read the checkpoint explanation

The negative-frequency spinors satisfy (slash p + m)v = 0 and v̄v = −2m. Together these give Σv v̄ = slash p − m, rather than the positive-frequency u sum.

Your turn

Try it yourself

A massive on-shell momentum has E = 5 and spatial magnitude 4. Find m, u†u, ūu, v†v, and v̄v for one spin label. Also find the trace of the two-spin u completeness matrix.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

First use the mass shell; distinguish the frame-dependent Hermitian norm from the scalar bilinear.

Compare with the solution

The mass is $m=\sqrt{25-16}=3$ . The positive Hermitian products are u†u = v†v = 2E = 10. The scalar bilinears are ūu = 2m = 6 and v̄v = −2m = −6. Since every gamma matrix has zero trace, the u spin sum has trace 4m = 12. The negative v scalar is a spinor-bilinear convention, not a negative Hilbert-space norm.

Keep the idea

Take it with you

The Dirac adjoint and the ordinary Hermitian product encode different transformation properties.
Both frequency families are labeled using positive on-shell energy.
On-shell spin sums are rank-two matrix identities with normalization-dependent factors.

Read further

Original sources and deeper explanations to follow at your own pace.

David Tong, Quantum Field Theory, 4.1–4.7.3: Clifford algebra, Dirac solutions and inner/outer products

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