If you quantize the Dirac field using commutation relations (like a scalar field) instead of anticommutation relations, the resulting theory has a fatal problem. What is it?
AThe energy spectrum becomes continuous instead of discrete
BThe theory has no Lorentz invariance
CThe Hamiltonian is unbounded below — there is no ground state, and the theory is unstable
DThe propagator diverges for all momenta
The Dirac Hamiltonian has both positive and negative frequency solutions. If you use commutation relations, the negative-frequency modes contribute -E_p to the energy for each excitation. With bosonic statistics, you could put arbitrarily many quanta in these modes, driving the energy to negative infinity. Anticommutation relations solve this: the antiparticle creation operators (associated with negative-frequency solutions) contribute +E_p to the energy, and the Pauli exclusion principle prevents unlimited occupation. This is a deep connection between spin and statistics — spin-1/2 fields must be quantized with anticommutation relations for the theory to be stable.
Question 2 Multiple Choice
The Dirac field operator psi(x) destroys a particle or creates an antiparticle; psi-bar(x) creates a particle or destroys an antiparticle. Why does a single field operator perform two seemingly different actions?
AThis is a mathematical artifact with no physical significance
BBecause the Dirac equation has both positive-frequency and negative-frequency solutions, and the field operator is a sum over both — positive-frequency components destroy particles, while negative-frequency components create antiparticles
CBecause particles and antiparticles are the same object moving in different directions in time
DBecause the field operator is not Hermitian
The Dirac field is expanded as psi(x) = sum_s integral [b_{p,s} u_s(p) e^{-ipx} + d-dagger_{p,s} v_s(p) e^{+ipx}] d^3p / (2pi)^3 (2E_p), where u_s and v_s are positive- and negative-frequency spinors. The operator b_{p,s} destroys an electron with momentum p and spin s; d-dagger_{p,s} creates a positron. Both are present because the Dirac equation requires both types of solutions for completeness. The field operator is the object that appears in the Lagrangian and in interaction vertices — its dual role is what makes charge-changing processes possible.
Question 3 True / False
The quantized Dirac field has a vacuum with infinite negative energy (a filled Dirac sea of negative-energy states).
TTrue
FFalse
Answer: False
The Dirac sea was Dirac's original interpretation, in which all negative-energy states are filled and a 'hole' in the sea appears as a positron. Modern quantum field theory does not use the Dirac sea. Instead, anticommutation relations and the reinterpretation of negative-frequency modes as antiparticle creation operators ensure that the vacuum has zero energy (after normal ordering) and no particles. The positron is created by d-dagger, not by removing an electron from a filled sea. The two pictures give the same physical predictions, but the field-theoretic approach is cleaner and generalizes to all particle types.
Question 4 Short Answer
Explain the role of the spin-statistics connection in the quantization of the Dirac field: why must spin-1/2 fields be quantized with anticommutators?
Think about your answer, then reveal below.
Model answer: The spin-statistics theorem states that fields with half-integer spin must be quantized with anticommutation relations, and fields with integer spin with commutation relations. For the Dirac field specifically, the argument is that the Hamiltonian contains both positive- and negative-frequency contributions. With commutation relations, the negative-frequency sector would have an energy spectrum unbounded below, giving an unstable vacuum. Anticommutation relations flip the sign so that both particle and antiparticle excitations contribute positive energy. Additionally, microcausality (the requirement that field operators at spacelike separations commute or anticommute) requires anticommutators for half-integer spin fields to maintain Lorentz invariance.
The spin-statistics connection is not optional — it is a theorem provable from the axioms of relativistic quantum field theory (Lorentz invariance, locality, positive energy). Violating it leads to either negative-energy states or violations of causality. The fact that electrons are fermions is a consequence of their spin-1/2 nature combined with the requirements of a consistent relativistic quantum theory.