Dirac's transformation theory
Dirac's transformation theory is the unifying framework, developed by Paul Dirac in the late 1920s and laid out in his 1930 book, that brought the rival versions of quantum mechanics under one roof. In the mid-1920s there were two seemingly different theories: Heisenberg's matrix mechanics, all tables of numbers and abstract algebra, and Schrödinger's wave mechanics, all smooth wavefunctions and differential equations. They gave the same answers, but no one could see clearly why.
Dirac's insight was to step back from both pictures and treat the quantum state as an abstract vector, with matrix mechanics and wave mechanics revealed as nothing more than two different choices of basis for the same underlying object. Writing the wavefunction in the position basis recovers Schrödinger; writing it in the energy basis recovers Heisenberg. The two formulations are related by a change of basis, a 'transformation', which is where the theory gets its name.
To make this work elegantly, Dirac introduced the bra-ket notation and the abstract operator language that physicists still use today. His transformation theory turned quantum mechanics from a pair of competing recipes into a single coherent structure built on states, operators, and the freedom to switch between representations — the formalism that every modern textbook now teaches.
Heisenberg's matrices and Schrödinger's waves are the same theory written in two different bases.
Dirac's framework is a reformulation, not a new physical theory: it makes no prediction that matrix or wave mechanics did not already make. Its value is conceptual clarity and unifying power, which is why his bra-ket language became the standard.