Dirac fields and fermionic excitations

Use natural units and metric $\eta=(+---)$. Gamma matrices obey the Clifford algebra, while $\bar\psi=\psi^{\dagger}\gamma^0$ is the Dirac adjoint. The free Lagrangian is $\bar\psi(i\gamma^\mu\partial_\mu-m)\psi$. Its equation is first order in time and space. The matrices are not arbitrary decoration: their anticommutation is exactly what makes the squared differential operator reproduce the scalar relativistic mass-shell condition.

Multiply the Dirac equation by the conjugate differential operator. Since ordinary partial derivatives commute, the antisymmetric part of the gamma-matrix product drops out and every spinor component satisfies the Klein–Gordon equation. This demonstrates the same energy-momentum dispersion, not an equivalence of spin-zero and spin-one-half theories. The allowed spinor solutions still satisfy the additional first-order constraints and transform differently under rotations and boosts.

The Dirac field is expanded in positive-frequency spinors multiplying particle annihilation operators and negative-frequency spinors multiplying antiparticle creation operators. Each massive on-shell particle has two spin states, with a corresponding antiparticle sector. The four components of the field organize this relativistically; they are not four simultaneous outcomes of measuring one spin component. Antiparticles appear as positive-energy excitations with opposite conserved charge, not as accessible negative-energy states.

For a discrete normalized fermionic mode, $\{b,b^{\dagger}\}=1$ and $\{b^{\dagger},b^{\dagger}\}=0$ imply $(b^{\dagger})^2=0$. One mode can be empty or occupied once. Two fermions may still share a spatial region when their full one-particle modes differ, for example through spin. Pauli exclusion is a constraint on states, not a classical force that forbids any spatial overlap.

Exchanging two fermionic creation operators reverses the sign of the state. That minus sign generates antisymmetric multiparticle amplitudes and exchange interference. In time-ordered fermion correlators, swapping two fermion operators also inserts a minus sign. The momentum-space free propagator is $i(\gamma^\mu p_\mu+m)/(p^2-m^2+i\epsilon)$, a matrix in spinor indices. Replacing it by the scalar propagator discards the numerator that carries spin information.

The spin-statistics connection is a structural theorem with assumptions including relativistic locality and a physically suitable Hilbert space and spectrum. It is not derived here merely by naming a field “spinor.” A useful practical check is that the correctly quantized free theory has positive particle and antiparticle excitation energies. The detailed proof and extensions in lower dimensions demand more care than an oscillator analogy supplies.