= Enhanced internal symmetry of free Dirac fields
{title2=$m=0:\ U(2N);\qquad m\ne0:\ O(2N)$}
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> $m\mathbf1$, invariance requires $V^\dagger V=V^TV=\mathbf1$, 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>.
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