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A relativistic quantum field is a quantum field whose transformation and dynamics are compatible with special relativity. A scalar field, a Dirac field and a gauge field transform differently under Lorentz transformations, reflecting their different spin content.

Relativistic consistency also constrains locality. Microcausality requires local physical observables at spacelike separation to commute, so an operation in one region cannot immediately change measurement statistics in the other. Vacuum correlations across spacelike separation do not contradict that requirement. Field excitations can describe particles and antiparticles, allowing creation and annihilation processes when conserved quantities and available energy permit them. An interacting theory's observables therefore cannot generally be organized using one permanently fixed particle number.

For ${\mathcal{L}}_{\mathrm{int}}=g^2(\bar{\psi }\psi {)}^2/2$ with one species of Dirac field, the tree-level Feynman diagrams for $f(p_1)\bar{f}(p_2)\to f(p_3)\bar{f}(p_4)$ giveThe fermionic sign is the antisymmetry of the two external contractions. The initial spin average is one quarter of the spin sum. The two diagonal contributions are $g^4(t-4m^2{)}^2$ and $g^4(s-4m^2{)}^2$; the interference Dirac trace is $R=(s-4m^2{)}^2+(t-4m^2{)}^2-(u-4m^2{)}^2$, giving the displayed result after subtracting $g^{4}R/2$. Here the Mandelstam variables satisfy $s+t+u=4m^2$. At threshold this gives $16g^4{m}^4$; for zero mass it gives $g^4(s^2+t^2+u^2)/2$. The formula describes the leading scattering amplitude of an effective interaction and is not a statement of perturbative renormalizability.

For one unit-charge Dirac field in $d=4-2ϵ$, define the insertion as $i{\Pi }_{\mu \nu }=i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu })\Pi$. The one-loop ultraviolet pole is ${\Pi }_{\mathrm{UV}}=-e^2/(12{\pi }^2ϵ)$. The Ward identity cancels any apparent divergent vector mass term. With photon counterterm insertion $-i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu }){\delta }_3$, this gives ${\delta }_3=-e^2/(12{\pi }^2ϵ)$ in the minimal subtraction scheme.

For $N$ Dirac fields with common mass, $\mathcal{L}=\bar{\Psi }(i/\partial -m{1}_N)\Psi$ is invariant under constant complex-linear rotations $\Psi \mapsto U\Psi$ with $U\in U(N)$. A Hermitian flavour generator $T^A$ has Noether current $j_A^{\mu }=\bar{\Psi }{\gamma }^{\mu }{T}^A\Psi$. A nondegenerate fermion mass matrix restricts the symmetry to unitary matrices commuting with it. If the theory is entirely free and rotations into charge-conjugate fields are allowed, its enhanced internal symmetry of free Dirac fields can be larger than this subgroup.

A fermionic mode with canonical anticommutation relations $\{b,b^{†}\}=1$ has the symmetrically ordered Hamiltonian operator $H=\frac{E}{2}(b^{†}b-b{b}^{†})=E(b^{†}b-1/2)$. Its empty-state zero-point contribution is $-E/2$. A quantum harmonic oscillator with bosonic canonical commutation relations contributes $+E/2$ instead. For a Dirac field, sum over both particle and antiparticle modes and over spin: there are four independent oscillators per spatial momentum in four spacetime dimensions. Comparing bosonic and fermionic vacuum energies requires a common regularization in quantum field theory, matching degrees of freedom and a stated vacuum-energy subtraction convention.

A two-field vacuum expectation value. Its ordered, time-ordered and anticommutator versions are different distributions. For a free Dirac field, fermionic time ordering combines the positive-energy particle and antiparticle contributions into a Dirac propagator. Its nonzero value at spacelike separation does not violate microcausality, which constrains the field anticommutator.

A fermion bilinear transforming as a Lorentz four-vector. For a single Dirac field this is the Dirac current; distinct flavours can give a charged or flavour-changing current.

Three scalar Yukawa interactions on a closed Dirac field loop give two cyclic orientations. Odd gamma matrix traces vanish, while $\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu })=4g^{\mu \nu }$ gives the displayed numerator. For external invariant mass squares $M^2,{\mu }^2,{\mu }^2$, write $D_a=a^2-m^2$ and similarly for the other two lines. The numerator becomes $4m[D_a+D_b+D_c+4m^2-{\mu }^2-M^2/2]$, reducing the fermion loop to three scalar bubbles and a scalar triangle Feynman integral. The closed fermion loop sign must be included. A massive fermion produces a logarithmic local divergence.

For one four-component Dirac field with scalar interaction $-y\bar{\psi }\psi \varphi$ and the mostly-plus propagator convention, the fermion loop numerator is $4[M^2-k\cdot (k-p)]$. Reduce it to a bubble and tadpole: $-4y^2[(2M^2+p^2/2)J-I]$. Dimensional regularization gives pole parts $J=2/(\epsilon 16{\pi }^2)$ and $I=-2M^2/(\epsilon 16{\pi }^2)$. The local pole then requires both wavefunction renormalization and a scalar mass counterterm.