Dirac’s Form of Relativistic Quantum Mechanics

We give a physical interpretation to the covariant harmonic oscillator formalism by incorporating Dirac’s ideas on relativistic quantum mechanics. His papers from 1927 discuss the time-energy uncertainty relation and how it differs from Heisenberg’s. Dirac observed that time is a c-number. In 1945, Dirac attempted to use the four-dimensional harmonic oscillator with normalizable time-like wave functions. This use of the covariant oscillator formalism is consistent with Dirac’s overall plan to construct a relativistic quantum mechanics. In his 1949 paper, Dirac emphasizes that the task of constructing a relativistic dynamics is equivalent to constructing a representation of the Poincaré group. We discuss the front, instant, and point forms of quantum mechanics which Dirac proposes along with the problems Dirac had. Dirac’s light-cone coordinate system in which the longitudinal and time-like coordinate variables undergo only scale changes under Lorentz boosts is discussed. This can be helpful in describing Lorentz deformation of relativistic extended hadrons. We show that the constraint applicable to the point form is useful in describing the conservation of probability under Lorentz transformations. This allows us to construct Lorentz-transformed relativistic bound-state wave functions, needed in understanding relativistic hadrons as bound states of quarks and/or antiquarks.