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In the Weyl representation of the gamma matrices, write each Dirac field as ${\psi }_i=({\ell }_{i\alpha },r_i^{\dot{\alpha }}{)}^T$, where ${\ell }_i$ and $r_i$ have opposite chirality. With the Weyl sigma matrices ${\sigma }^{\mu }=(1,\mathbf{\sigma })$ and ${\bar{\sigma }}^{\mu }=(1,-\mathbf{\sigma })$,The Dirac mass term couples the two Weyl fields. Under $\ell \mapsto U_L\ell$, $r\mapsto U_{R}r$, its first term acquires $U_L^{†}{U}_R$, so a common nonzero mass requires $U_L=U_R$. At zero mass there is no such constraint:This is the chiral flavour symmetry of massless Dirac fields. Its eight Noether currents are $j_{L,A}^{\mu }={\ell }^{†}{\bar{\sigma }}^{\mu }{T}^A\ell$ and $j_{R,A}^{\mu }=r^{†}{\sigma }^{\mu }{T}^{A}r$. The sum $j_R+j_L$ is the vector current, and $j_R-j_L$ is the axial current, for each flavour generator. This statement concerns the free theory in flat spacetime; an interacting theory can have a chiral anomaly.

For completeness, allowing the extra transformations described in part e gives four left-handed Weyl fields, $q=({\ell }_1,{\ell }_2,r_1^c,r_2^c{)}^T$, where $r_i^c$ is the left-handed charge-conjugate field of $r_i$. After integration by parts their kinetic term is $i{q}^{†}{\bar{\sigma }}^{\mu }{\partial }_{\mu }q$, whose maximal linear unitary group is $U(4)$. The symmetric mass matrix for equal Dirac masses iswith the antisymmetric spinor contraction implicit. A unitary change of basis makes $M=m{1}_4$; preserving both the kinetic term and $M$ requires $V^{†}V=V^{T}V=1_4$, hence real $V\in O(4)$. Thus the unrestricted free-theory enhancement is $U(4)$ at zero mass and $O(4)$ at equal nonzero mass; the familiar $U(2{)}_L\times U(2{)}_R$ and diagonal $U(2)$ are their common-fermion number-preserving subgroups.

Combine the two equal-mass Dirac fields into $\Psi =({\psi }^1,{\psi }^2{)}^T$. A constant unitary matrix $U$ acting on this flavour index commutes with the common Dirac Hamiltonian, so $\Psi \mapsto U\Psi$ gives the unitary flavour symmetry of degenerate Dirac fields $U(2)$. Its real Lie algebra has dimension four. A convenient Hermitian generator basis is $T^0=1_2/2$, $T^a={\tau }^a/2$, $a=1,2,3$, where ${\tau }^a$ are the Pauli matrices.

For each generator the canonical anticommutation relations giveThe calculation is exactly the one in b, since $T^A$ commutes with both the Dirac gamma matrices and the common fermion mass matrix. ThusExplicitly their densities areThe two off-diagonal conserved currents are why counting only the individual fermion numbers misses two generators. The singlet generates $U(1)$; the other three generate special unitary group $SU(2)$. Unequal masses would break the off-diagonal conservation laws.

Scope of the count. Four is the count of currents of the displayed ${\bar{\psi }}^A{\gamma }^{\mu }{T}_{AB}{\psi }^B$ form. The completely free system also has an enhanced internal symmetry of free Dirac fields if mixing with charge-conjugate fields is admitted. For $m\ne 0$, write its two Dirac fields as four equal-mass Majorana spinors ${\chi }^I$. Their orthogonal group $O(4)$ rotations have six continuous generators, with conserved currents proportional to $i{\bar{\chi }}^I{\gamma }^{\mu }{\chi }^J$ for $I<J$; the four $U(2)$ currents form a subgroup. This larger count is relevant if “similar” means all local internal currents of the free system, without preserving a chosen fermion number. If $m=0$, the usual number-preserving chiral flavour symmetry of massless Dirac fields is $U(2{)}_L\times U(2{)}_R$ with eight currents; allowing charge conjugation mixing enlarges it to $U(4)$ with sixteen. These counts concern internal currents, excluding spacetime Noether currents.