Spinor Conventions

A spinor formula is usable only together with its phase, momentum label, adjoint, normalization, and spin basis. This reference collects one coherent package and the most common normalization conversions. The solution and completeness derivations remain at Free Dirac Spinors.

Required background. Free Dirac Spinors supplies the modes; Gamma-Matrix Conventions fixes their matrix and adjoint definitions. Helpful background. Helicity and Chirality explains the different spin labels.

Spinor labels and mode phases

Use $\hbar=c=1$, $\eta=(+---)$, $p=(E,\mathbf p)$ with $E=\sqrt{\mathbf p^2+m^2}>0$, and $p\!\!\!/=\gamma^0E-\boldsymbol\gamma\cdot\mathbf p$. The label $p$ is future directed for both families:

Mode	Algebraic equation	Canonical four-momentum of the wave
$u_s(p)e^{-ip\cdot x}$	$(p\!\!\!/-m)u_s(p)=0$	$p^\mu$
$v_s(p)e^{+ip\cdot x}$	$(p\!\!\!/+m)v_s(p)=0$	$-p^\mu$

For $m>0$, choose orthonormal two-spinor bases $\chi_s,\eta_s$. In the Dirac basis, the covariantly normalized columns are

$$u_s(p)=\sqrt{E+m} \begin{pmatrix} \chi_s\\ \dfrac{\boldsymbol\sigma\cdot\mathbf p}{E+m}\chi_s \end{pmatrix},$$ $$v_s(p)=\sqrt{E+m} \begin{pmatrix} \dfrac{\boldsymbol\sigma\cdot\mathbf p}{E+m}\eta_s\\ \eta_s \end{pmatrix}.$$

The bases can be chosen independently for these solution and completeness identities. A specified charge-conjugation pairing can subsequently relate their labels and phases; it is additional information, not an implicit consequence of the displayed notation.

At rest, $u_s=\sqrt{2m}(\chi_s,0)^T$ and $v_s=\sqrt{2m}(0,\eta_s)^T$. The negative-frequency label does not make the Hilbert density negative.

Normalizations and spin sums

Use $\bar w=w^\dagger\gamma^0$ for either family.

Identity	Positive-frequency columns	Negative-frequency columns
Covariant scalar product	$\bar u_r u_s=2m\delta_{rs}$	$\bar v_r v_s=-2m\delta_{rs}$
Hilbert scalar product	$u_r^\dagger u_s=2E\delta_{rs}$	$v_r^\dagger v_s=2E\delta_{rs}$
Covariant spin sum	$\sum_su_s\bar u_s=p\!\!\!/+m$	$\sum_sv_s\bar v_s=p\!\!\!/-m$

For equal future labels, $\bar u_r(p)v_s(p)=0$, but in general

$$u_r(p)^\dagger v_s(p) =2\chi_r^\dagger (\boldsymbol\sigma\cdot\mathbf p)\eta_s \ne0.$$

Fixed-Hamiltonian orthogonality instead uses $v_s(E,-\mathbf p)$:

$$u_r(E,\mathbf p)^\dagger v_s(E,-\mathbf p)=0.$$

For $H(\mathbf p)=\boldsymbol\alpha\cdot \mathbf p+\beta m$, the Hilbert projectors are

$$\begin{aligned} P_+(\mathbf p) &=\frac1{2E}\sum_su_s(p)u_s(p)^\dagger,\\ P_-(\mathbf p) &=\frac1{2E}\sum_sv_s(E,-\mathbf p) v_s(E,-\mathbf p)^\dagger. \end{aligned}$$

They equal $(I\pm H/E)/2$ and sum to $I$. The covariant spin sums in the table are not themselves these unit-normalized orthogonal projectors. See The Dirac Hamiltonian for the spectral construction.

For another real positive normalization, rescale every accompanying formula:

Column convention	Hilbert norm	Covariant scalar norm
$u,v$ as above	$2E$	$+2m,-2m$
$u/\sqrt{2E},v/\sqrt{2E}$	$1$	$+m/E,-m/E$
$u/\sqrt{2m},v/\sqrt{2m}$, $m>0$	$E/m$	$+1,-1$

The unit-Hilbert normalization depends on $E$ and hence on the chosen frame. The spin sums divide by the same $2E$ or $2m$ factor as the corresponding outer products. If a spinor in a mode expansion is multiplied by a factor, its coefficient must be divided by that factor to retain the same field or wavefunction. Do not change the columns while leaving the expansion’s measure and coefficients unexplained. For state and amplitude normalizations, consult Relativistic Normalization. A four-component column alone is not a normalized momentum eigenstate of the one-particle Hilbert space.

Dirac and chiral components

For the accepted chiral ordering $\psi_C=(\psi_L,\psi_R)^T$, one explicit constant unitary map from the Dirac basis is

$$\begin{aligned} \psi_C&=V\psi_D,\\ V&=\frac1{\sqrt2} \begin{pmatrix} I_2&-I_2\\ I_2&I_2 \end{pmatrix},\\ \gamma_C^\mu&=V\gamma_D^\mu V^\dagger. \end{aligned}$$

It gives $\gamma_C^5=\operatorname{diag}(-I_2,I_2)$. If $\psi_D=(\xi,\zeta)^T$, then $\psi_L=(\xi-\zeta)/\sqrt2$ and $\psi_R=(\xi+\zeta)/\sqrt2$. The upper and lower Dirac components are therefore not the left and right chiral components. The full matrices and trace conventions are at Gamma-Matrix Conventions.

Fixed rest-spin labels define canonical spin columns for $m>0$. Helicity columns instead use eigenvectors of $\boldsymbol\sigma\cdot\widehat{\mathbf p}$. For a massless positive-frequency helicity eigenmode at nonzero momentum, its $\gamma^5$ eigenvalue equals twice its helicity in the stated convention. The relation between negative-frequency labels and physical antiparticle helicity also uses the conjugation and field-expansion conventions; do not infer it from the upper block of a $v$ column.

As $m\to0$ at fixed nonzero $\mathbf p$, the displayed component solutions and $2E$ Hilbert normalization remain well defined, while $\bar uu=\bar vv=0$. Dividing by $\sqrt{2m}$ has no such massless continuation. At $m=\mathbf p=0$, the energy projectors involving $H/E$ are undefined.

Charge conjugation uses two related matrices

In the Dirac basis, distinguish

$$B=i\gamma^2,\qquad C_D=i\gamma^2\gamma^0.$$

For a commuting solution column,

$$\psi^c=B\psi^*=C_D\bar\psi^{\,T}.$$

The two matrices act on different arguments. Substituting $C_D$ for $B$ in a formula that already conjugates $\psi$ introduces an extra matrix. Under a constant complex unitary basis change, the matrix of the antilinear map transforms as $B'=VBV^T$, rather than by similarity. The $C_D$ matrix likewise transforms by congruence in the barred-spinor formula.

Use Symmetry Conventions for the complete coordinate arguments, phases, charge/background transformations, and distinction between commuting columns and fermion field operators. The Spinor Algebra notebook provides explicit tests in a complex basis where the wrong transformation rule is detectable.

Checks before using a copied spinor

Check the free equation with the written phase and differential momentum, then compute both $w^\dagger w$ and $\bar w w$. Test a spin sum with the actual normalization, and test the energy projector at fixed canonical momentum. These four checks distinguish a harmless normalization change from a sign or momentum-label error.

If constructing a quantum field, creation and annihilation operators, their algebra, and the state are additional data. Their placement next to $u$ and $v$ is given at From Spinors to Fermion Fields; the numerical spinor column does not by itself determine particle counting.

References

• Bjorken, James D., and Sidney D. Drell. Relativistic Quantum Mechanics. McGraw–Hill, 1964.
• Dreiner, Herbi K., Howard E. Haber, and Stephen P. Martin. “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry.” Physics Reports 494, 1–196 (2010). doi:10.1016/j.physrep.2010.05.002; corrected arXiv version, appendix G.
• Thaller, Bernd. The Dirac Equation. Springer, 1992. doi:10.1007/978-3-662-02753-0.