The Second-Quantized Theory of Spin-1/2 Particles in the Nonrelativistic Limit

Abstract

The second-quantized Dirac Hamiltonian for free electrons is transformed by a canonical transformation to a representation in which the positive and negative energy wave operators are separately represented by two-component operators. The transformation employed is the second-quantized analog of the one derived by Foldy and Wouthuysen in their discussion of the one-particle Dirac theory and its nonrelativistic limit. This transformation is then applied to the wave operators and the Hamiltonian in the second-quantized, charge-conjugate formalism for Dirac particles. The wave operators for positrons and electrons become linearly-independent two-component operators, and the Hamiltonian separates into an electron and a positron part, each of which contains only the corresponding two-component wave operators. It is also shown that by means of an appropriate, readily determinable sequence of canonical transformations, Hamiltonians for fields of spin-_ particles interacting via intermediary fields can also be reduced to nonrelativistic form. This is accomplished by transforming the Hamiltonian to a representation in which it is exhibited effectively as a series expansion in powers of the Compton wavelength of the spin-_ particle. Illustration of the method is provided by detailed examination of the case of nucleons interacting via the pseudoscalar meson field.