The term transformation theory refers to a procedure used by P. A. M. Dirac in his early formulation of quantum theory, from around 1927.

The term is related to the famous wave-particle duality, according to which a particle (a "small" physical object) may display either particle or wave aspects, depending on the observational situation. Or, indeed, a variety of intermediate aspects, as the situation demands.

This "transformation" idea also refers to the changes a physical object may undergo in the course of time, whereby it may "move" between "positions" in its Hilbert "space".

Remaining in full use today, it would be regarded as a topic in the mathematics of Hilbert space, although technically speaking it is somewhat more general in scope). While the terminology is (with good reason) reminiscent of motion in ordinary space, the Hilbert space of a quantum object is more general, and holds its entire quantum state.