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A background gauge field is a nondynamical source for a conserved current, rather than a field integrated over in the path integral. For a Dirac current with source term $+B_{\mu }{j}^{\mu }$, the combined transformations $\psi \mapsto e^{i\alpha }\psi$ and $B_{\mu }\mapsto B_{\mu }+{\partial }_{\mu }\alpha$ preserve the action. Whether this also preserves the generating functional depends on the absence of the corresponding gauge anomaly.

For a nonanomalous Dirac current $j^{\mu }$, introduce $Z[B]=\int \mathcal{D}\bar{\psi }\mathcal{D}\psi e^{iS+i\int B_{\mu }{j}^{\mu }}$. The local phase change of variables has unit functional Jacobian, while its kinetic variation cancels the shift of the source. Thus $Z[B+\partial \alpha ]=Z[B]$. At first order, integration by parts yields ${\partial }_{\mu }\delta Z/\delta B_{\mu }=0$. Charged sources transform too when deriving the full Ward-Takahashi identity with Ward identity contact terms.

For a commuting spinor in mostly-plus signature, orient its Dirac current to the future and write $V^i={ϵ}^{†}{\alpha }^iϵ$, with Hermitian ${\alpha }^i=-{\gamma }^0{\gamma }^i$. For every spatial unit vector $u$, $(u\cdot \alpha {)}^2=1$ implies $|u\cdot V|\le {ϵ}^{†}ϵ=V^0$. Choosing $u$ along the spatial current proves $V^2\le 0$. Thus spinor bilinears supply causal vectors suitable for energy-condition estimates. A Grassmann-valued spinor does not have this ordinary positivity interpretation.

For normal ordering of a fermion bilinear, charge conjugation sends $\bar{\psi }\Gamma \psi$ to $\bar{\psi }(C^{-1}\Gamma C{)}^T\psi$. In particular the Dirac current is odd and the axial current is even under charge conjugation.

An interaction $-\lambda j^{\mu }{\partial }_{\mu }\varphi$ with an exactly conserved current is redundant: integration by parts gives $\lambda \varphi {\partial }_{\mu }{j}^{\mu }$ up to a boundary term. For the Dirac current, the local phase redefinition $\psi =e^{-i\lambda \varphi }\chi$ removes the coupling, so its on-shell scattering amplitudes vanish.

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

The Dirac electromagnetic current is the electric charge times the Dirac current. Its coupling $-e{j}^{\mu }{A}_{\mu }$ to the electromagnetic four-potential is the matter interaction of quantum electrodynamics.

A Dirac field is a quantum field transforming as a Dirac spinor. Its free Lagrangian density is $\mathcal{L}=i\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -m\bar{\psi }\psi$, giving the Dirac equation. Unlike a single-particle wave function, a quantized Dirac field can create and remove particles and antiparticles.

Its mode expansion of a Dirac field contains fermionic annihilation operators for particles and fermionic creation operators for antiparticles, with canonical anticommutation relations enforcing Fermi-Dirac statistics. For the electron field, these excitations are electrons and Positrons. A global phase symmetry gives the conserved Dirac current; coupling that current to the electromagnetic field produces the matter interaction in quantum electrodynamics.

A local expression containing two fermion fields and a matrix acting on Dirac spinor indices. The choices $\Gamma ={\gamma }^{\mu }$ and $\Gamma ={\gamma }^{\mu }{\gamma }^5$ give the Dirac current and axial current.

For a Dirac field with a global phase symmetry, fermion number is the vacuum-subtracted charge of the Dirac current. In a mode expansion it counts particles minus antiparticles, with the overall sign fixed by the chosen generator convention. It is distinct from the total number of fermionic excitations and from fermion parity, which can survive even when interactions violate the continuous charge.

A Killing spinor is a nonzero supersymmetry parameter that makes the fermionic transformations vanish on a bosonic background. In undeformed minimal four-dimensional supergravity it is parallel, ${\nabla }_{\mu }ϵ=0$; in its cosmological deformation the equation uses ${\mathcal{D}}_{\mu }={\nabla }_{\mu }+(m/2){\gamma }_{\mu }$. Its commuting Dirac current is a Killing vector, since the spinor equation makes the symmetric covariant derivative of the current vanish. General supergravity theories also impose the vanishing of matter-fermion transformations.

The equal-time canonical anticommutation relations implyApply $-i{\alpha }_{bc}^k{\partial }_{z^k}+m{\beta }_{bc}$, set $\mathbf{z}=\mathbf{y}$, and integrate over $\mathbf{y}$. The mass term cancels, leaving $i\int d^{3}y{\psi }^{†}{\alpha }^k\psi {\partial }_{y^k}{\delta }^{(3)}(\mathbf{x}-\mathbf{y})$. Integration by parts proves the title formula as a distribution identity. The Heisenberg equation of motion then yields ${\partial }_{t}n=-\nabla \cdot \mathbf{j}$ for the physical Dirac current $j^k=\bar{\psi }{\gamma }^k\psi$. The factor $-i$ is required because the commutator of Hermitian operators is anti-Hermitian.

In four dimensions a single commuting Majorana spinor is specified by a two-component complex spinor $\xi$. Its future Dirac current is proportional to $({\xi }^{†}\xi ,{\xi }^{†}\mathbf{\sigma }\xi )$, and the Pauli matrix multiplication law implies $|{\xi }^{†}\mathbf{\sigma }\xi {|}^2=({\xi }^{†}\xi {)}^2$. Thus it is null. A Dirac spinor or a sum of independent Majorana currents can be timelike. This distinction matters when applying a timelike-current classification to minimal real supersymmetry.

An operator equation in a path integral $\mathcal{O}(x)=0$ asserts that insertions of $\mathcal{O}(x)$ vanish in every time-ordered correlation function with the other insertion points separated from $x$. Derivatives of time ordering can produce Ward identity contact terms at coincidence. For example, conservation of a Dirac current gives a vanishing separated-point divergence, while a charged insertion contributes a delta-function contact term fixed by its transformation. A vanishing one-point expectation value alone is insufficient.

A parallel commuting Dirac current is causal. If timelike, the geometry locally splits as a time line times a three-dimensional Ricci-flat metric; three-dimensional Ricci flatness implies flatness. If null, the stabilizer of a parallel spinor in $\mathrm{Spin}(1,3)$ contains only null rotations, and the local vacuum metric is $d{s}^2=-2dudv+d{x}^2+d{y}^2+H(u,x,y)d{u}^2$ with $H_{xx}+H_{yy}=0$. Constant spinors obeying ${\gamma }^{+}ϵ=0$ are parallel. These are local statements; global quotients must preserve a compatible spin structure.

A nonzero spacetime parallel spinor makes the Nester two-form vanish, so its ADM boundary charge vanishes. For complete regular spin initial data in an asymptotically flat spacetime under the dominant energy condition, the ADM momentum is future causal. Its contraction with the spinor's nonzero future causal Dirac current cannot vanish if the momentum is timelike. Thus the invariant ADM mass is zero; in a rest frame this gives zero energy. Local parallel-spinor existence without asymptotic/global hypotheses does not establish this mass conclusion.

For the local variation $\delta \psi =i\alpha (x)\psi$, $\delta \bar{\psi }=-i\alpha (x)\bar{\psi }$, the terms proportional to $\alpha$ cancel and the differentiated phase givesThus, for a compactly supported $\alpha$, integration by parts givesThis Dirac current is the Noether current in the convention $\delta S=-\int j^{\mu }{\partial }_{\mu }\alpha$. Using the convention with a plus sign reverses the Noether current and the conserved charge together. There is one independent global symmetry in field theory, with four spacetime components of its Noether current. The associated conserved charge is $Q=\int d^{3}x{j}^0$, with vacuum subtraction when required.

Couple a nondynamical background gauge field $B_{\mu }$ to the Dirac current, choosing a plus sign for the source term:There is no integration over $B_{\mu }$. The gauge covariant derivative is ${\partial }_{\mu }-i{B}_{\mu }$, so the simultaneous transformations areThe kinetic term changes by $-j^{\mu }{\partial }_{\mu }\alpha$ and the source term changes by $+j^{\mu }{\partial }_{\mu }\alpha$. The action is therefore invariant, exactly for finite $\alpha$ as well as infinitesimally. Changing integration variables and using the invariant functional measure from part c proves the background gauge invariance of a current generating functional:The allowed transformations preserve the vacuum boundary conditions. Its infinitesimal version is ${\partial }_{\mu }\delta Z/\delta B_{\mu }=0$. Because $B_{\mu }$ is only a source, this statement needs no gauge fixing or photon kinetic term. The absence of a vector gauge anomaly is essential to the change-of-variables argument.

Collect the two Dirac fields into $\Psi =({\psi }_1,{\psi }_2{)}^T$, with a common nonzero mass $m$. ThenAmong complex-linear rotations of the Dirac fields, the internal symmetry of a classical field theory is $\Psi \mapsto U\Psi$, $U\in U(2)$, with $\bar{\Psi }\mapsto \bar{\Psi }{U}^{†}$. Unitarity preserves the kinetic term, and the mass matrix commutes with every $U$. This unitary flavour symmetry of degenerate Dirac fields has group $(SU(2)\times U(1))/{\mathbb{Z}}_2$, where the quotient avoids counting the common central element twice.

Choose the four Hermitian matrices $T^0=1_2/2$, $T^a={\sigma }^a/2$ for $a=1,2,3$, with ${\sigma }^a$ the Pauli matrices. Promoting the four parameters to functions gives $\delta S=-\int d^{4}x({\partial }_{\mu }{\alpha }^A)\bar{\Psi }{\gamma }^{\mu }{T}^A\Psi$. The four Noether currents are thereforeTheir divergences vanish by the equal-mass Dirac equations. The singlet is half the total Dirac current; the other three currents rotate the two species.

Free-theory qualification. If “global internal symmetry” allows mixing the Dirac fields with charge-conjugate fields, the completely free theory written here has a larger enhanced internal symmetry of free Dirac fields. Write ${\psi }_i=({\lambda }_{2i-1}+i{\lambda }_{2i})/\sqrt{2}$ using four Majorana fermions. Up to a total derivative,The full orthogonal group $O(4)$ acts on the four real species; its connected part is $SO(4)$. Its six Noether currents can be chosen as $J_{AB}^{\mu }=i{\bar{\lambda }}_A{\gamma }^{\mu }{\lambda }_B$ for $A<B$, up to the common sign convention. The antisymmetry of this Majorana fermion vector bilinear makes these the six independent rotations. A concrete rotation outside the stated complex-linear $U(2)$ is ${\lambda }_2\mapsto \cos \theta {\lambda }_2+\sin \theta {\lambda }_3$, ${\lambda }_3\mapsto -\sin \theta {\lambda }_2+\cos \theta {\lambda }_3$: it preserves the displayed Lagrangian density while mixing ${\psi }_i$ with their charge-conjugate fields. The usual $U(2)$ answer specifies the subgroup preserving the chosen common fermion number.

A relativistic theory must accommodate the energy-momentum relation $E^2={\mathbf{p}}^2+m^2$, causal interactions and the possibility of particle creation. Promoting the squared relation to a wave equation gives the Klein-Gordon field equationA plane wave $\varphi =N{e}^{-iEt+i\mathbf{p}\cdot \mathbf{x}}$ hasThe positive and negative branches both solve the local differential equation. Its being second order in time means that specifying only a wavefunction, as in the nonrelativistic Schrödinger equation, is not the complete initial datum.

For a complex scalar, the conserved Klein-Gordon inner product comes from the currentFor the plane wave, $j^0=2E|N{|}^2$, which is negative on the negative-frequency branch. Thus this conserved density cannot be a positive single-particle probability density for arbitrary Klein-Gordon solutions. Merely replacing it by $|\varphi {|}^2$ does not supply the missing conserved relativistic probability law. The indefinite Klein-Gordon charge density is an obstacle to that naive one-particle interpretation, not evidence that a consistent scalar field theory has negative probabilities.

The Dirac equation is first order,The gamma-matrix anticommutation relations imply $H_D^2=({\mathbf{p}}^2+m^2)I$, so its spectrum again contains $\pm E_{\mathbf{p}}$. But its conserved Dirac current has density $j^0={\psi }^{†}\psi \ge 0$. The Dirac equation solves the density-sign difficulty while leaving both energy branches. Treating every negative-energy solution as an ordinary accessible one-electron state would leave the energy unbounded below and permit radiative descent without a stable ground state.

The historical Dirac sea filled the negative-energy electron states and interpreted holes as positive-energy antiparticles. It motivates the Positron but is not the general modern explanation: bosons do not obey an exclusion principle and cannot be stabilized by filling a sea. Quantum field theory instead promotes the fields to operators and interprets negative-frequency modes using antiparticle creation operators.

For a complex scalar field, schematically,Bosonic commutation relations give a normal-ordered Hamiltonian and conserved charge of the formBoth particle species have positive excitation energies. The sign-changing scalar current now represents charge, rather than particle probability; opposite charges are physically allowed. The ordinary positive Fock space norm gives nonnegative probabilities. A real scalar field is its own antiparticle and has no corresponding complex-field U(1) charge.

For the Dirac field, the analogous expansion contains electron annihilation operators and positron creation operators. Fermionic anticommutation relations turn the negative-frequency contribution into a positive antiparticle excitation energy after the vacuum constant is removed. The Pauli exclusion principle and relativistic locality are thereby incorporated in a field description, rather than a postulated infinite population of physical negative-energy electrons. The normal-ordering step removes an additive vacuum constant; it does not by itself settle the separate gravitational vacuum-energy problem.

Interactions can change particle number while conserving total energy, momentum and charge. Trying to localize a particle on its Compton wavelength scale requires momenta of order its rest mass, where accessible interactions can involve pair creation. A fixed-particle wavefunction is consequently not a general description of such processes. Creation and annihilation operators act between different particle-number sectors, and field interactions describe scattering and production within one framework.

Finally, relativistic causality concerns observables: spacelike separated local bosonic observables commute, while fermionic fields satisfy the appropriate anticommutation conditions. This microcausality prevents a controllable influence through spacelike separated local operations. Free relativistic one-particle theories remain useful, and low-energy fixed-number approximations can be excellent. What fails is the naive insistence that unrestricted relativistic interactions, positive probabilities, stability and arbitrary particle production all fit an unchanged one-particle Schrödinger picture.^[ref]