Dirac membrane

In quantum mechanics, a Dirac membrane is a model of a charged membrane introduced by Paul Dirac in 1962. Dirac's original motivation was to explain the mass of the muon as an excitation of the ground state corresponding to an electron.^[1] Anticipating the birth of string theory by almost a decade, he was the first to introduce what is now called a type of Nambu–Goto action for membranes.^[2][3]

In the Dirac membrane model the repulsive electromagnetic forces on the membrane are balanced by the contracting ones coming from the positive tension. In the case of the spherical membrane, classical equations of motion imply that the balance is met for the radius ${\displaystyle 0.75r_{e}}$, where ${\displaystyle r_{e}}$ is the classical electron radius. Using Bohr–Sommerfeld quantisation condition for the Hamiltonian of the spherically symmetric membrane, Dirac finds the approximation of the mass corresponding to the first excitation as ${\displaystyle 53m_{e}}$, where ${\displaystyle m_{e}}$ is the mass of the electron, which is about a quarter of the observed muon mass.