Editing Fermi's interaction (section)

===Hamiltonian===

The Hamiltonian is composed of three parts: <math>H_\text{h.p.}</math>, representing the energy of the free heavy particles, <math>H_\text{l.p.}</math>, representing the energy of the free light particles, and a part giving the interaction <math>H_\text{int.}</math>.

:<math>H_\text{h.p.} = \frac{1}{2}(1 + \rho)N + \frac{1}{2}(1 - \rho)P,</math>

where <math>N</math> and <math>P</math> are the energy operators of the neutron and proton respectively, so that if <math>\rho = 1</math>, <math>H_\text{h.p.} = N</math>, and if <math>\rho = -1</math>, <math>H_\text{h.p.} = P</math>.

:<math>H_\text{l.p.} = \sum_s H_s N_s + \sum_\sigma K_\sigma M_\sigma,</math>

where <math>H_s</math> is the energy of the electron in the <math>s^\text{th}</math> state in the nucleus's Coulomb field, and  <math>N_s</math> is the number of electrons in that state; <math>M_\sigma</math> is the number of neutrinos in the <math>\sigma^\text{th}</math> state, and <math>K_\sigma</math> energy of each such neutrino (assumed to be in a free, plane wave state).

The interaction part must contain a term representing the transformation of a proton into a neutron along with the emission of an electron and a neutrino (now known to be an antineutrino), as well as a term for the inverse process; the Coulomb force between the electron and proton is ignored as irrelevant to the <math>\beta</math>-decay process.

Fermi proposes two possible values for <math>H_\text{int.}</math>: first, a non-relativistic version which ignores spin:

:<math>H_\text{int.} = g \left[ Q \psi(x) \phi(x) + Q^* \psi^*(x) \phi^*(x) \right],</math>

and subsequently a version assuming that the light particles are four-component [[Dirac spinor]]s, but that speed of the heavy particles is small relative to <math>c</math> and that the interaction terms analogous to the electromagnetic vector potential can be ignored:

:<math>H_\text{int.} = g \left[ Q \tilde{\psi}^* \delta \phi + Q^* \tilde{\psi} \delta \phi^* \right],</math>

where <math>\psi</math> and <math>\phi</math> are now four-component Dirac spinors, <math>\tilde{\psi}</math> represents the Hermitian conjugate of <math>\psi</math>, and <math>\delta</math> is a matrix

:<math>\begin{pmatrix}
0 & -1 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & -1 & 0
\end{pmatrix}.</math>