Past exam of the gauge field theory course of the University of Cambridge / 2014 / Paper 1 / 3 / a

Solution

A relativistic quantum field describes both creation and annihilation of excitations; a one-particle wavefunction alone is insufficient when interactions can change particle number. For a free complex scalar field, one conventional mode expansion of a free field is

ϕ (x) = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 E _{p}} [a_{p} e^{- i p \cdot x} + b_{p}^{†} e^{i p \cdot x}], E_{p} = p^{2} + m^{2} .

The apparently negative-frequency term creates a positive-energy antiparticle; it is not a physical particle of unbounded negative energy. The normal-ordered Hamiltonian and conserved global phase charge have the forms

H = \int \frac{d ^{3} p}{( 2 π ) ^{3}} E_{p} (a_{p}^{†} a_{p} + b_{p}^{†} b_{p}), Q = q \int \frac{d ^{3} p}{( 2 π ) ^{3}} (a_{p}^{†} a_{p} - b_{p}^{†} b_{p}) .

Particles and antiparticles have equal mass and opposite additive internal charges. Including both frequency sectors is also essential to the local field commutator: the positive- and negative-frequency terms combine so that spacelike-separated scalar observables commute, respecting microcausality.

A Dirac field similarly contains particle annihilation operators multiplying

u

v

spinors. Its canonical anticommutation relations implement Fermi-Dirac statistics, whereas scalar modes use commutators and Bose-Einstein statistics. These assignments are tied to the Spin-statistics theorem under the usual relativistic locality and positivity assumptions. Conserved charge constrains particle creation and annihilation, but particle and antiparticle numbers need not be separately conserved in an interacting theory.

An antiparticle need not be a distinct species. A neutral real scalar field is self-conjugate; the photon is also its own antiparticle. A Majorana fermion obeys a self-conjugacy condition and cannot carry an unbroken additive charge that distinguishes it from its antiparticle. Electric neutrality alone does not force self-conjugacy when other conserved quantum numbers distinguish the states.

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