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The antiparticle of a fermion. In a Dirac field, its external wavefunctions are the negative-frequency $v$ spinors, while its physical energy is positive. Fermion-number arrows on Feynman diagrams run opposite an antifermion's physical momentum.

In a Dirac field mode expansion, the antiparticle contribution to $\bar{\psi }\Gamma \psi$ first appears with operator order $d{d}^{†}$. Normal ordering changes it to $-d^{†}d$ using the fermionic anticommutation relations. Thus its scattering coefficient is $-{\bar{v}}_{\mathrm{in}}\Gamma v_{\mathrm{out}}$. Pair-annihilation and creation terms have the operator orders $db$ and $b^{†}{d}^{†}$. Keeping these orders and a fixed external-state convention determines interference signs, especially between exchange and annihilation graphs in a Yukawa interaction.

The antiproton is the antiparticle of the proton: it has the same rest mass and spin $1/2$, with opposite electric charge and baryon number. A negative electric charge alone cannot identify an antiproton, because a negative pion also bends in that direction in a magnetic field. Combining magnetic rigidity with velocity determines a rest mass; agreement with the proton rest mass then makes the identification much more discriminating.

The mode expansion of a Dirac field explains how independent particle and antiparticle excitations enter a relativistic field description. It supplies a theoretical language for the antiproton, while the Chamberlain-Segrè antiproton identification supplies an experimental test. The proton is a composite baryon, so a free Dirac field is not a complete account of its internal dynamics or of antiproton production in a target.

An axial charge is the spatial integral of the time component of an axial current. For a Dirac field and an internal generator $t$, it is $\int {\psi }^{†}{\gamma }_{5}t\psi d^{3}x$. Its canonical action rotates left- and right-handed fields oppositely. It is conserved only when the current divergence and flux at infinity vanish; Dirac masses and a chiral anomaly can obstruct conservation. The nonsinglet quark generators have no color anomaly and obey the nonsinglet axial charge algebra of quark densities.

For a Dirac field, the charge-conjugation matrix implements ${\psi }^c=C{\bar{\psi }}^T$. In the standard gamma matrix representation one can take $C=i{\gamma }^2{\gamma }^0$. An overall phase is conventional. The transpose intertwining relation is different from the entrywise complex conjugation involved in an antiunitary operator.

The charge-conjugate Dirac field is ${\psi }^c=C{\bar{\psi }}^T$, where $C$ is the charge-conjugation matrix satisfying $C{\gamma }^{\mu T}{C}^{-1}=-{\gamma }^{\mu }$ in a common convention. Transposition acts on spinor indices, while the Dirac adjoint is $\bar{\psi }={\psi }^{†}{\gamma }^0$.

Conjugating a minimally coupled Dirac equation produces the equation with opposite electric charge. Thus charge conjugation exchanges particle and antiparticle excitations and reverses additive charges, while leaving the mass unchanged. For quantum electrodynamics, the gauge field must transform as well for the full coupled theory to be invariant. A relation between field expressions does not by itself prove that every interaction involving that field has charge conjugation symmetry.

The chiral anomaly is the quantum nonconservation of a classically conserved axial current. For massless charged Dirac fields, the classical axial rotation is a symmetry, but a quantization preserving the vector gauge symmetry gives an axial-current divergence proportional to $F_{\mu \nu }{}^{\star }{F}^{\mu \nu }$.

This is a physical effect: the electromagnetic chiral anomaly helps determine the neutral pion's decay amplitude into two photons. A global axial symmetry may fail in this way while the gauge theory remains consistent. A gauge anomaly is different because the lost gauge redundancy is needed to remove unphysical states.^[ref]

The free kinetic terms of left- and right-handed Weyl fields are separately invariant under unitary group rotations in flavour space. For $N$ massless Dirac fields, the subgroup preserving common fermion number is $U(N{)}_L\times U(N{)}_R$. A common nonzero Dirac mass term couples the two chirality sectors and leaves its diagonal $U(N)$ subgroup. Gauge interactions can obstruct some of the classical axial currents through a chiral anomaly.

The projectors $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$ select the two chirality eigenspaces of a Dirac field. They satisfy $P_L+P_R=I$, $P_L{P}_R=0$, and $P_L^2=P_L$, $P_R^2=P_R$.

Without a symmetry excluding odd scalar powers, a real-scalar Yukawa interaction with a massive Dirac field requires a general scalar potential through degree four. For a constant background, the fermion mass becomes $M+y\varphi$, and its divergent local determinant contribution contains $(M+y\varphi {)}^4$. Linear and cubic terms are therefore generated when $M\ne 0$. A zero renormalized tadpole is a renormalization condition, not a prohibition of its counterterm. An exact discrete chiral symmetry in a massless fermion theory can instead enforce an even scalar potential.

For a Hermitian fermion bilinear, an antiunitary operator implementing CPT conjugates numerical coefficients and exchanges particle and antiparticle fields at $-x$. Reordering the fermions then gives $\Theta :\overline{\psi }\Gamma \psi (x):{\Theta }^{-1}=:\overline{\psi }{\gamma }_5\Gamma {\gamma }_5\psi (-x):$ when ${\Gamma }^{†}={\gamma }^0\Gamma {\gamma }^0$. Thus scalar, pseudoscalar and tensor bilinears are even, while vector and axial-vector bilinears are odd. Each derivative contributes another minus sign. The graded reordering is essential; treating Dirac fields as commuting quantities gives the wrong transformation.

The global phase symmetry of a Dirac field gives the conserved Dirac current $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$.

The Dirac equation is the relativistic equation of motion for a spin-one-half Dirac field, $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$, in units $\hslash =c=1$. The gamma matrices obey the Clifford algebra, ensuring that each component also obeys the free Klein-Gordon equation.

Being first order in time allows an initial spinor to determine its subsequent free evolution. The components encode spin as well as particle and antiparticle solutions; quantization interprets both kinds of excitation with positive physical energy. Coupling through the gauge covariant derivative describes an electron interacting with an electromagnetic field. The nonrelativistic limit contains the coupling of spin to a magnetic field, explaining why a scalar relativistic wave equation alone cannot describe an electron.

A Dirac field theory invariant under $\psi \mapsto e^{i\alpha }\psi$, $\bar{\psi }\mapsto e^{-i\alpha }\bar{\psi }$ has a conserved global charge, subject to the quantum theory preserving the symmetry. In particle language this charge counts particles minus antiparticles. When the S-matrix commutes with it, a scattering amplitude between states of different charge vanishes. A neutral scalar and a Yukawa interaction preserve the charge, so a process with one incoming Dirac particle and only one outgoing Dirac antiparticle plus neutral particles is forbidden. This net charge is distinct from a sum of positive occupation numbers for all particle and antiparticle modes.

The Dirac Hamiltonian is the first-order time-evolution operator obtained from the Dirac equation. For a free massive particle,where the Dirac matrices satisfy $\{{\alpha }_i,{\alpha }_j\}=2{\delta }_{ij}I$, $\{{\alpha }_i,\beta \}=0$ and ${\beta }^2=I$. These identities give $H^2=c^2{P}^2+m^2{c}^4$, recovering the relativistic energy-momentum relation.

Minimal electromagnetic coupling replaces the canonical momentum by $\mathbf{P}-q\mathbf{A}$ and adds $q\varphi$, in SI units. The wavefunction has spinor components, and the free operator has positive- and negative-energy branches. The Dirac field interprets particle and antiparticle excitations with a stable vacuum; it is more complete than treating the negative branch as an ordinary one-particle spectrum with no lower bound.

For gamma matrices ${\gamma }^0={\sigma }_3$, ${\gamma }^1=i{\sigma }_1$, ${\gamma }^2=i{\sigma }_2$ and Minkowski metric $\mathrm{diag}(1,-1,-1)$, a one-axis spatial reflection acts with ${\gamma }^1$ on the two-component Dirac field. It preserves the kinetic term and reverses the Dirac mass term. Time reversal acts with ${\gamma }^2$ and includes conjugation of the operator coefficients; it has the same mass reversal. Their composition preserves a real mass. Charge conjugation can use ${\psi }_C={\gamma }^1{\psi }^{\ast }$, because ${\gamma }^{\mu }{\gamma }^1=-{\gamma }^1{\gamma }^{\mu \ast }$. The mass bilinear is charge-conjugation even when Grassmann field exchange signs are included, and the kinetic action is invariant up to a boundary term. These statements concern the classical free action.

An independent vierbein and metric-compatible spin connection allow spin density to source a torsion tensor. The connection equation from the Einstein-Hilbert action is algebraic in torsion. In the absence of connection-dependent matter it fixes the Levi-Civita connection. For a minimally coupled Dirac field, eliminating the connection generates a local four-fermion interaction. Thus its equivalent torsion-free second-order action includes this contact term; simply dropping it defines a different matter theory.

Without fixed gauge charges or interactions distinguishing particles from antiparticles, $N$ free Dirac fields give $2N$ left-handed Weyl fields after charge conjugation of the right-handed fields. Their massless kinetic term has unitary group $U(2N)$. Equal nonzero Dirac masses have equal positive singular values and can be written as $2N$ equal Majorana mass terms; their preserving group is the orthogonal group $O(2N)$. Explicitly, in a basis with symmetric mass matrix $m1$, invariance requires $V^{†}V=V^{T}V=1$, hence real $V$. Restricting to transformations preserving a chosen common fermion number gives the usual $U(N{)}_L\times U(N{)}_R$ at zero mass and $U(N)$ at nonzero mass.

For constant internal matrices acting on Dirac fields, the equal-time canonical anticommutation relations givewith the corresponding spatial delta functions for fields. The quartic terms cancel after fermion reordering. Integrating proves the displayed algebra for well-defined global Noether charges, with normal ordering and a regulator that preserves the nonanomalous internal symmetry. Orthogonal chiral projectors make the left/right mixed commutators zero. This reduces current-charge closure to a finite matrix calculation.

For a charged Dirac field excitation of mass $m$, multiplicity $N$, electric coupling $eQ$ and scalar Yukawa interaction $-yh\bar{\psi }\psi$, two orientations of a fermion loop induce a Higgs boson coupling to two photons. With metric $(+---)$ and the usual $S=1+iT$ convention, their physical on-shell scattering amplitude isThe Ward identity holds on both photon legs. At low external energy, the integral tends to $1/(3m^2)$. Thus fixed $y$ gives an inverse-mass coefficient, whereas $y=m/v$ gives a constant coefficient as $m$ increases at fixed $v$. The distinction is essential when applying heavy-field decoupling.