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Schrödinger equation is not Lorentz covariant. The Klein-Gordon (spin-0) and Dirac (spin-½) equations are relativistically invariant. Negative-energy solutions interpret as antiparticles, naturally leading to quantum field theory.
The Schrödinger equation treats time and space asymmetrically: it has a first derivative in time but second derivatives in space. Under a Lorentz boost — a change to a moving reference frame — this asymmetry breaks the equation's form, which means it cannot describe particles moving at relativistic speeds. The requirement of Lorentz covariance demands that the equation take the same form in all inertial frames, so the derivatives in time and space must appear symmetrically. The simplest relativistic equation uses the energy-momentum relation E² = (pc)² + (mc²)² and replaces E → iℏ∂/∂t and p → −iℏ∇ to get an equation with second derivatives in both space and time.
The result is the Klein-Gordon equation: (□ + m²c²/ℏ²)φ = 0, where □ is the d'Alembert operator. This works for spin-0 particles like pions. But it has a problem: the conserved current density can be negative, which would mean negative probability — physically nonsensical. More deeply, the Klein-Gordon equation has solutions with both positive and negative energy, and there is no simple way to discard the negative-energy solutions while keeping a complete set of states.
Dirac approached the problem differently. He required an equation linear in both space and time derivatives, of the form (iℏγ^μ ∂_μ − mc)ψ = 0. For this to be consistent with the relativistic energy-momentum relation, the coefficients γ^μ cannot be ordinary numbers — they must be 4×4 matrices, now called Dirac matrices. The wave function ψ becomes a four-component spinor. Two components represent spin-up and spin-down for positive energy, and two components represent spin-up and spin-down for negative energy. This doubling is not a defect; it is a prediction: for every particle, there exists an antiparticle with the same mass but opposite charge. The positron, discovered in 1932, confirmed this prediction.
The negative-energy sea interpretation (filled Dirac sea) was Dirac's original resolution, but it only works for fermions and becomes unwieldy. The modern understanding is that the correct framework is quantum field theory, where both positive- and negative-energy solutions are reinterpreted: the positive-energy modes are particle creation operators and the negative-energy modes are antiparticle annihilation operators. Relativistic quantum mechanics — Klein-Gordon and Dirac equations — works as a one-particle approximation when particle creation is suppressed, but the deeper truth is a field theory. This is where your Dirac notation and operator formalism become indispensable: the full machinery carries over directly into quantum field theory.