= 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
$$
\phi(x)=\int\frac{d^3p}{(2\pi)^3\sqrt{2E_{\mathbf p}}}
\left[a_{\mathbf p}e^{-ip\cdot x}+b_{\mathbf p}^\dagger e^{ip\cdot x}\right],
\qquad E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}.
$$
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
$$
H=\int\frac{d^3p}{(2\pi)^3}E_{\mathbf p}
(a_{\mathbf p}^\dagger a_{\mathbf p}+b_{\mathbf p}^\dagger b_{\mathbf p}),\qquad
Q=q\int\frac{d^3p}{(2\pi)^3}
(a_{\mathbf p}^\dagger a_{\mathbf p}-b_{\mathbf p}^\dagger b_{\mathbf p}).
$$
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 $u$ spinors and antiparticle <creation operators> multiplying $v$ 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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