Editing Fermi's interaction (section)

=== Definitions ===

The theory deals with three types of particles presumed to be in direct interaction: initially a “[[nucleon|heavy particle]]” in the “neutron state” (<math>\rho=+1</math>), which then transitions into its “proton state” (<math>\rho = -1</math>) with the emission of an electron and a neutrino.

====Electron state====
:<math>\psi = \sum_s \psi_s a_s,</math>

where <math>\psi</math> is the [[Coulomb wave function|single-electron wavefunction]], <math>\psi_s</math> are its [[stationary state]]s.

<math>a_s</math> is the [[Creation and annihilation operators|operator which annihilates an electron in state <math>s</math>]] which acts on the [[Fock space]] as

:<math>a_s \Psi(N_1, N_2, \ldots, N_s, \ldots) = (-1)^{N_1 + N_2 + \cdots + N_s - 1} (1 - N_s) \Psi(N_1, N_2, \ldots, 1 - N_s, \ldots).</math>

<math>a_s^*</math> is the creation operator for electron state <math>s:</math>

:<math>a_s^* \Psi(N_1, N_2, \ldots, N_s, \ldots) = (-1)^{N_1 + N_2 + \cdots + N_s - 1} N_s \Psi(N_1, N_2, \ldots, 1 - N_s, \ldots).</math>

====Neutrino state====
Similarly,

:<math>\phi = \sum_\sigma \phi_\sigma b_\sigma,</math>

where <math>\phi</math> is the single-neutrino wavefunction, and <math>\phi_\sigma</math> are its stationary states.

<math>b_\sigma</math> is the operator which annihilates a neutrino in state <math>\sigma</math> which acts on the Fock space as

:<math>b_\sigma \Phi(M_1, M_2, \ldots, M_\sigma, \ldots) = (-1)^{M_1 + M_2 + \cdots + M_\sigma - 1} (1 - M_\sigma) \Phi(M_1, M_2, \ldots, 1 - M_\sigma, \ldots).</math>

<math>b_\sigma^*</math> is the creation operator for neutrino state <math>\sigma</math>.

====Heavy particle state====
<math>\rho</math> is the operator introduced by Heisenberg (later generalized into [[isospin]]) that acts on a [[nucleon|heavy particle]] state, which has eigenvalue +1 when the particle is a neutron, and −1 if the particle is a proton. Therefore, heavy particle states will be represented by two-row column vectors, where

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

represents a neutron, and

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

represents a proton (in the representation where <math>\rho</math> is the usual <math>\sigma_z</math> [[Pauli matrices|spin matrix]]).

The operators that change a heavy particle from a proton into a neutron and vice versa are respectively represented by

:<math>Q = \sigma_x - i \sigma_y = \begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}</math>

and

:<math>Q^* = \sigma_x + i \sigma_y = \begin{pmatrix}0 & 0\\ 1 & 0\end{pmatrix}.</math>

<math>u_n</math> resp. <math>v_n</math> is an eigenfunction for a neutron resp. proton in the state <math>n</math>.