Editing Fermi's interaction (section)

==The "tentativo"==

=== 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>.

===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>

===Matrix elements===

The state of the system is taken to be given by the [[tuple]] <math>\rho, n, N_1, N_2, \ldots, M_1, M_2, \ldots,</math> where <math>\rho = \pm 1</math> specifies whether the heavy particle is a neutron or proton, <math>n</math> is the quantum state of the heavy particle, <math>N_s</math> is the number of electrons in state <math>s</math> and <math>M_\sigma</math> is the number of neutrinos in state <math>\sigma</math>.

Using the relativistic version of <math>H_\text{int.}</math>, Fermi gives the matrix element between the state with a neutron in state <math>n</math> and no electrons resp. neutrinos present in state <math>s</math> resp. <math>\sigma </math>, and the state with a proton in state <math>m</math> and an electron and a neutrino present in states <math>s</math> and <math>\sigma</math> as

:<math>H^{\rho=1, n, N_s=0, M_\sigma=0}_{\rho=-1,m,N_s=1,M_\sigma=1} = \pm g \int v_m^* u_n \tilde{\psi}_s \delta \phi^*_\sigma d\tau,</math>

where the integral is taken over the entire configuration space of the heavy particles (except for <math>\rho</math>). The <math>\pm</math> is determined by whether the total number of light particles is odd (−) or even (+).

===Transition probability===

{{See also|Fermi's golden rule}}

To calculate the lifetime of a neutron in a state <math>n</math> according to the usual [[quantum perturbation theory]], the above matrix elements must be summed over all unoccupied electron and neutrino states. This is simplified by assuming that the electron and neutrino eigenfunctions <math>\psi_s</math> and <math>\phi_\sigma</math> are constant within the nucleus (i.e., their [[Compton wavelength]] is much larger than the size of the nucleus). This leads to

:<math>H^{\rho=1, n, N_s=0, M_\sigma=0}_{\rho=-1,m,N_s=1,M_\sigma=1} = \pm g \tilde{\psi}_s \delta \phi_\sigma^* \int v_m^* u_n d\tau,</math>

where <math>\psi_s</math> and <math>\phi_\sigma</math> are now evaluated at the position of the nucleus.

According to [[Fermi's golden rule]]{{Elucidate|date=July 2016}}, the probability of this transition is

:<math>\begin{align}
\left|a^{\rho=1, n, N_s=0, M_\sigma=0}_{\rho=-1,m,N_s=1,M_\sigma=1}\right|^2 &= \left|H^{\rho=1, n, N_s=0, M_\sigma=0}_{\rho=-1,m,N_s=1,M_\sigma=1} \times \frac{\exp{\frac{2\pi i}{h} (-W + H_s + K_\sigma) t} - 1}{-W + H_s + K_\sigma}\right|^2 \\
&= 4 \left|H^{\rho=1, n, N_s=0, M_\sigma=0}_{\rho=-1,m,N_s=1,M_\sigma=1}\right|^2 \times \frac{\sin^2\left(\frac{\pi t}{h}(-W + H_s + K_\sigma)\right)}{(-W + H_s + K_\sigma)^2},
\end{align}</math>

where <math>W</math> is the difference in the energy of the proton and neutron states.

Averaging over all positive-energy neutrino spin / momentum directions (where <math>\Omega^{-1}</math> is the density of neutrino states, eventually taken to infinity), we obtain

:<math> \left\langle \left|H^{\rho=1, n, N_s=0, M_\sigma=0}_{\rho=-1,m,N_s=1,M_\sigma=1}\right|^2 \right \rangle_\text{avg} = \frac{g^2}{4\Omega} \left|\int v_m^* u_n d\tau\right|^2  \left( \tilde{\psi}_s \psi_s - \frac{\mu c^2}{K_\sigma} \tilde{\psi}_s \beta \psi_s\right),</math>

where <math>\mu</math> is the rest mass of the neutrino and <math>\beta</math> is the Dirac matrix.

Noting that the transition probability has a sharp maximum for values of <math>p_\sigma</math> for which <math>-W + H_s + K_\sigma = 0</math>, this simplifies to {{Elucidate|date=July 2016}}

:<math> t\frac{8\pi^3 g^2}{h^4} \times \left| \int v_m^* u_n d\tau \right|^2 \frac{p_\sigma^2}{v_\sigma}\left(\tilde{\psi}_s \psi_s - \frac{\mu c^2}{K_\sigma} \tilde{\psi}_s \beta \psi_s\right),</math>

where <math>p_\sigma</math> and <math>K_\sigma</math> is the values for which <math>-W + H_s + K_\sigma = 0</math>.

Fermi makes three remarks about this function:
 
* Since the neutrino states are considered to be free, <math>K_\sigma > \mu c^2</math> and thus the upper limit on the continuous <math>\beta</math>-spectrum is <math>H_s \leq W - \mu c^2</math>.
* Since for the electrons <math>H_s > mc^2</math>, in order for <math>\beta</math>-decay to occur, the proton–neutron energy difference must be <math>W \geq (m + \mu)c^2</math>
* The factor 
::<math>Q_{mn}^* = \int v_m^* u_n d\tau</math> 
:in the transition probability is normally of magnitude 1, but in special circumstances it vanishes; this leads to (approximate) [[selection rules]] for <math>\beta</math>-decay.

===Forbidden transitions===
{{Main article|Forbidden transition}}

As noted above, when the inner product <math>Q_{mn}^*</math> between the heavy particle states <math>u_n</math> and <math>v_m</math> vanishes, the associated transition is "forbidden" (or, rather, much less likely than in cases where it is closer to 1).

If the description of the nucleus in terms of the individual quantum states of the protons and neutrons is accurate to a good approximation, <math>Q_{mn}^*</math> vanishes unless the neutron state <math>u_n</math> and the proton state <math>v_m</math> have the same angular momentum; otherwise, the total angular momentum of the entire nucleus before and after the decay must be used.