A. Zee, Quantum Field Theory in a Nutshell · Problem II.4.1

Field Theory Redux

Physics — QFT

Problem

Show that we would also get into trouble if we quantize the Dirac field with commutation instead of anticommutation rules. Calculate the commutator [J0(x⃗,0),J0(0⃗)][J^0(\vec{x}, 0), J^0(\vec{0})].
Intuitive explanation

The Dirac field's spin-statistics connection hinges on causality and energy positivity—commutation rules would violate both. By computing the equal-time commutator of the charge density J0J^0 with itself, you'll see that it fails to vanish at spacelike separation, allowing faster-than-light signaling and revealing negative-energy modes that destabilize the vacuum. This calculation exposes why fermions must obey anticommutation relations.

Intuitive explanation

The Dirac field's spin-statistics connection hinges on causality and energy positivity—commutation rules would violate both. By computing the equal-time commutator of the charge density J0J^0 with itself, you'll see that it fails to vanish at spacelike separation, allowing faster-than-light signaling and revealing negative-energy modes that destabilize the vacuum. This calculation exposes why fermions must obey anticommutation relations.

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