= Solution
A relativistic theory must accommodate the energy-momentum relation $E^2=\mathbf p^2+m^2$, causal interactions and the possibility of particle creation. Promoting the squared relation to a wave equation gives the <Klein-Gordon field> equation
$$
(\partial_t^2-\nabla^2+m^2)\phi=0.
$$
A plane wave $\phi=N e^{-iEt+i\mathbf p\cdot\mathbf x}$ has
$$
\boxed{E=\pm E_{\mathbf p},\qquad E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}.}
$$
The positive and negative branches both solve the local differential equation. Its being second order in time means that specifying only a wavefunction, as in the nonrelativistic Schrödinger equation, is not the complete initial datum.
For a complex scalar, the conserved <Klein-Gordon inner product> comes from the current
$$
j^\mu=i\big(\phi^*\partial^\mu\phi-\phi\partial^\mu\phi^*\big),\qquad
\partial_\mu j^\mu=0.
$$
For the plane wave, \b[$j^0=2E|N|^2$], which is negative on the negative-frequency branch. Thus this conserved density cannot be a positive single-particle probability density for arbitrary Klein-Gordon solutions. Merely replacing it by $|\phi|^2$ does not supply the missing conserved relativistic probability law. The <indefinite Klein-Gordon charge density> is an obstacle to that naive one-particle interpretation, not evidence that a consistent scalar field theory has negative probabilities.
The <Dirac equation> is first order,
$$
i\partial_t\psi=H_D\psi,\qquad H_D=\boldsymbol\alpha\cdot\mathbf p+\beta m.
$$
The gamma-matrix anticommutation relations imply $H_D^2=(\mathbf p^2+m^2)I$, so its spectrum again contains $\pm E_{\mathbf p}$. But its conserved <Dirac current> has density $j^0=\psi^\dagger\psi\ge0$. The Dirac equation solves the density-sign difficulty while leaving both energy branches. Treating every negative-energy solution as an ordinary accessible one-electron state would leave the energy unbounded below and permit radiative descent without a stable ground state.
The historical <Dirac sea> filled the negative-energy electron states and interpreted holes as positive-energy antiparticles. It motivates the <Positron> but is not the general modern explanation: bosons do not obey an exclusion principle and cannot be stabilized by filling a sea. <Quantum field theory> instead promotes the fields to operators and interprets negative-frequency modes using antiparticle creation operators.
For a complex scalar field, schematically,
$$
\widehat\phi(x)=\int\frac{d^3p}{(2\pi)^3\sqrt{2E_{\mathbf p}}}
\left(a_{\mathbf p}e^{-ipx}+b_{\mathbf p}^\dagger e^{ipx}\right).
$$
Bosonic commutation relations give a normal-ordered Hamiltonian and conserved charge of the form
$$
:H:=\int\frac{d^3p}{(2\pi)^3}E_{\mathbf p}(a^\dagger_{\mathbf p}a_{\mathbf p}+b^\dagger_{\mathbf p}b_{\mathbf p}),
\qquad Q\propto N_a-N_b.
$$
Both particle species have positive excitation energies. The sign-changing scalar current now represents charge, rather than particle probability; opposite charges are physically allowed. The ordinary positive <Fock space> norm gives nonnegative probabilities. A real scalar field is its own antiparticle and has no corresponding complex-field U(1) charge.
For the <Dirac field>, the analogous expansion contains electron annihilation operators and positron creation operators. Fermionic anticommutation relations turn the negative-frequency contribution into a positive antiparticle excitation energy after the vacuum constant is removed. The <Pauli exclusion principle> and relativistic locality are thereby incorporated in a field description, rather than a postulated infinite population of physical negative-energy electrons. The normal-ordering step removes an additive vacuum constant; it does not by itself settle the separate gravitational vacuum-energy problem.
Interactions can change particle number while conserving total energy, momentum and charge. Trying to localize a particle on its <Compton wavelength> scale requires momenta of order its rest mass, where accessible interactions can involve pair creation. A fixed-particle wavefunction is consequently not a general description of such processes. <Creation and annihilation operators> act between different particle-number sectors, and field interactions describe scattering and production within one framework.
Finally, relativistic causality concerns observables: spacelike separated local bosonic observables commute, while fermionic fields satisfy the appropriate anticommutation conditions. This <microcausality> prevents a controllable influence through spacelike separated local operations. Free relativistic one-particle theories remain useful, and low-energy fixed-number approximations can be excellent. What fails is the naive insistence that unrestricted relativistic interactions, positive probabilities, stability and arbitrary particle production all fit an unchanged one-particle Schrödinger picture.<Quantum Field Theory by David Tong>[§2.5 complex scalar fields; §5.1–5.3 fermionic quantization and the hole interpretation]{ref}
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