On the Dirac Theory of Spin 1/2 Particles and Its NonRelativistic Limit
Abstract
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by twocomponent wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between c and c. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the nonrelativistic PauliHamiltonian for particles of spin 1/2 , one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the nonrelativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper nonrelativistic Hamiltonian (essentially the PauliHamiltonian) is obtained in the nonrelativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper nonrelativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.
 Publication:

Physical Review
 Pub Date:
 April 1950
 DOI:
 10.1103/PhysRev.78.29
 Bibcode:
 1950PhRv...78...29F