Dirac Bilinear Table

The sixteen independent Dirac matrices organize spinor products into scalar, pseudoscalar, vector, axial-vector, and antisymmetric-tensor structures. This table fixes their definitions before listing transformation signs. The construction and parity derivation belong to Bilinear Covariants.

Required background. Bilinear Covariants supplies the adjoint and Lorentz transformation law. Helpful background. Symmetry Conventions specifies the conjugation operations and their statistics dependence.

Sixteen matrix structures form the bilinear basis

Use $\hbar=c=1$, $\eta=(+---)$, $\bar\psi=\psi^\dagger\gamma^0$, and the accepted gamma-matrix definitions:

$$\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3, \qquad \sigma^{\mu\nu}=\frac{i}{2} [\gamma^\mu,\gamma^\nu].$$

The following definitions apply in any consistently transformed gamma basis. “Lorentz type” in this first table refers to proper Lorentz transformations; parity distinguishes the scalar from the pseudoscalar and the vector from the axial vector.

Name	Definition	Matrix structures	Proper-Lorentz type
Scalar	$S=\bar\psi\psi$	$1$	Scalar
Pseudoscalar	$P_5=i\bar\psi\gamma^5\psi$	$1$	Scalar
Vector	$V^\mu=\bar\psi\gamma^\mu\psi$	$4$	Four-vector
Axial vector	$A_5^\mu=\bar\psi\gamma^\mu\gamma^5\psi$	$4$	Four-vector
Tensor	$T^{\mu\nu}=\bar\psi\sigma^{\mu\nu}\psi$	$6$	Antisymmetric rank-two tensor

The count $1+1+4+4+6=16$ counts independent matrices. It does not make the sixteen bilinear values of a single spinor freely specifiable: quadratic identities constrain them. In particular, $T^{\nu\mu}=-T^{\mu\nu}$ and $T^{\mu\mu}=0$ without summing.

The symbols $P_5,A_5$ avoid confusing these bilinears with a parity operator or electromagnetic potential. Some sources omit the $i$ from the pseudoscalar or write $\gamma^5\gamma^\mu$ for the axial matrix. Those conventions respectively change the reality assignment or reverse the axial bilinear’s sign.

Dirac Hermiticity and reality

For each matrix $\Gamma$ in the table,

$$\gamma^0\Gamma^\dagger\gamma^0=\Gamma.$$

Consequently a diagonal bilinear of an ordinary commuting complex column is real: $(\bar\psi\Gamma\psi)^*=\bar\psi\Gamma\psi$. The useful transition identity is instead

$$(\bar\chi\Gamma\psi)^* =\bar\psi\Gamma\chi.$$

A transition bilinear with $\chi\ne\psi$ need not be real. Nor does this criterion say that every $\Gamma$ is Hermitian under the ordinary matrix adjoint: it is $\gamma^0\Gamma$ that is Hermitian. The bare $\gamma^5$ is Dirac anti-Hermitian, which explains the $i$ in $P_5$.

For operator-valued fields the corresponding statement concerns Hermitian, properly defined local composite operators, not real numerical wavefunctions. Products at coincident points require the field theory’s prescription.

Parity and charge-conjugation lookup

Under parity compare the transformed bilinear at $(t,\mathbf x)$ with the original at $(t,-\mathbf x)$. The parity matrix is $\gamma^0$ up to an irrelevant phase for these bilinears.

Component	Parity sign
$S$	$+$
$P_5$	$-$
$V^0,\ V^i$	$+,\ -$
$A_5^0,\ A_5^i$	$-,\ +$
$T^{0i},\ T^{ij}$	$-,\ +$

For charge conjugation use the declared charge-conjugation map. Its action on a commuting wavefunction and on a fermionic field product gives different signs:

Bilinear	Commuting columns	Reordered fermionic bilinear
$S$	$-$	$+$
$P_5$	$-$	$+$
$V^\mu$	$+$	$-$
$A_5^\mu$	$-$	$+$
$T^{\mu\nu}$	$+$	$-$

The second column uses $\psi^c=B\psi^*$. The last column includes the extra minus sign from interchanging anticommuting fields. For quantum local products, use an appropriate definition, such as normal ordering for the free-field current. The commuting density $V^0=\psi^\dagger\psi$ remains nonnegative under conjugation; the normal-ordered quantum charge current is C-odd. These are statements about different objects.

Time reversal additionally conjugates $i$ and changes the time argument. Use the complete P/T/C table in Symmetry Conventions when that operation is needed; a Lorentz type alone does not specify it.

On-shell normalization checks

For a massive positive-energy spinor with $\bar u_s(p)u_s(p)=2m$, the free-spinor normalization gives, for the same spin and momentum,

$$S=2m,\qquad P_5=0,\qquad V^\mu=2p^\mu.$$

For a pure polarization with spin four-vector $s^\mu$ satisfying $p\cdot s=0$ and $s^2=-1$,

$$A_5^\mu=2m s^\mu.$$

See Bilinear Covariants for the polarization convention. These diagonal formulas do not replace transition matrix elements between different momenta.

A quick rest-frame sign check is $u=\sqrt{2m}(1,0,0,0)^{\mathsf T}$ in the Dirac basis. Then $V^0=2m$, $A_5^3=2m$, $T^{12}=2m$, and $T^{0i}=0$. Rescaling $u$ by a complex number $a$ multiplies every diagonal bilinear by $|a|^2$. Thus unit-Hilbert columns give $V^0=1$, not $2E$. The massive rest-frame spin vector is not a prescription for taking a massless rest-frame limit.

References

• Bjorken, James D., and Sidney D. Drell. Relativistic Quantum Mechanics. McGraw–Hill, 1964. Spinor currents and covariant matrix elements.
• 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. See also the corrected arXiv version, especially its four-component convention appendix, for fermionic identities and convention comparisons.