In
particle physics, Fermi's interaction (also the Fermi theory of beta decay or the Fermi
four-fermion interaction) is an explanation of the
beta decay
In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar of that nuclide. For e ...
, proposed by
Enrico Fermi in 1933. The theory posits four
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s directly interacting with one another (at one vertex of the associated
Feynman diagram). This interaction explains beta decay of a
neutron by direct coupling of a neutron with an
electron, a
neutrino (later determined to be an
antineutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is ...
) and a
proton.
Fermi first introduced this coupling in his description of beta decay in 1933.
[
] The Fermi interaction was the precursor to the theory for the
weak interaction
In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
where the interaction between the proton–neutron and electron–antineutrino is mediated by a virtual
W− boson, of which the Fermi theory is the low-energy
effective field theory.
History of initial rejection and later publication
Fermi first submitted his "tentative" theory of beta decay to the prestigious science journal ''
Nature
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
'', which rejected it "because it contained speculations too remote from reality to be of interest to the reader.
" ''Nature'' later admitted the rejection to be one of the great editorial blunders in its history.
Fermi then submitted revised versions of the paper to
Italian
Italian(s) may refer to:
* Anything of, from, or related to the people of Italy over the centuries
** Italians, an ethnic group or simply a citizen of the Italian Republic or Italian Kingdom
** Italian language, a Romance language
*** Regional It ...
and
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
publications, which accepted and published them in those languages in 1933 and 1934.
[
Includes complete English translation of Fermi's 1934 paper in German] The paper did not appear at the time in a primary publication in English.
An English translation of the seminal paper was published in the
American Journal of Physics
The ''American Journal of Physics'' is a monthly, peer-reviewed scientific journal published by the American Association of Physics Teachers and the American Institute of Physics. The editor-in-chief is Beth Parks of Colgate University."Current Fr ...
in 1968.
Fermi found the initial rejection of the paper so troubling that he decided to take some time off from
theoretical physics, and do only experimental physics. This would lead shortly to his famous work with
activation of nuclei with slow neutrons.
The "tentativo"
Definitions
The theory deals with three types of particles presumed to be in direct interaction: initially a “
heavy particle” in the “neutron state” (
), which then transitions into its “proton state” (
) with the emission of an electron and a neutrino.
Electron state
:
where
is the
single-electron wavefunction,
are its
stationary state
A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, ener ...
s.
is the
operator which annihilates an electron in state which acts on the
Fock space
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intro ...
as
:
is the creation operator for electron state
:
:
Neutrino state
Similarly,
:
where
is the single-neutrino wavefunction, and
are its stationary states.
is the operator which annihilates a neutrino in state
which acts on the Fock space as
:
is the creation operator for neutrino state
.
Heavy particle state
is the operator introduced by Heisenberg (later generalized into
isospin) that acts on a
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
:
represents a neutron, and
:
represents a proton (in the representation where
is the usual
spin matrix).
The operators that change a heavy particle from a proton into a neutron and vice versa are respectively represented by
:
and
:
resp.
is an eigenfunction for a neutron resp. proton in the state
.
Hamiltonian
The Hamiltonian is composed of three parts:
, representing the energy of the free heavy particles,
, representing the energy of the free light particles, and a part giving the interaction
.
:
where
and
are the energy operators of the neutron and proton respectively, so that if
,
, and if
,
.
:
where
is the energy of the electron in the
state in the nucleus's Coulomb field, and
is the number of electrons in that state;
is the number of neutrinos in the
state, and
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
-decay process.
Fermi proposes two possible values for
: first, a non-relativistic version which ignores spin:
:
and subsequently a version assuming that the light particles are four-component
Dirac spinors, but that speed of the heavy particles is small relative to
and that the interaction terms analogous to the electromagnetic vector potential can be ignored:
:
where
and
are now four-component Dirac spinors,
represents the Hermitian conjugate of
, and
is a matrix
:
Matrix elements
The state of the system is taken to be given by the
tuple where
specifies whether the heavy particle is a neutron or proton,
is the quantum state of the heavy particle,
is the number of electrons in state
and
is the number of neutrinos in state
.
Using the relativistic version of
, Fermi gives the matrix element between the state with a neutron in state
and no electrons resp. neutrinos present in state
resp.
, and the state with a proton in state
and an electron and a neutrino present in states
and
as
:
where the integral is taken over the entire configuration space of the heavy particles (except for
). The
is determined by whether the total number of light particles is odd (−) or even (+).
Transition probability
To calculate the lifetime of a neutron in a state
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
and
are constant within the nucleus (i.e., their
Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It wa ...
is much smaller than the size of the nucleus). This leads to
:
where
and
are now evaluated at the position of the nucleus.
According to
Fermi's golden rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of ...
, the probability of this transition is
:
where
is the difference in the energy of the proton and neutron states.
Averaging over all positive-energy neutrino spin / momentum directions (where
is the density of neutrino states, eventually taken to infinity), we obtain
:
where
is the rest mass of the neutrino and
is the Dirac matrix.
Noting that the transition probability has a sharp maximum for values of
for which
, this simplifies to
:
where
and
is the values for which
.
Fermi makes three remarks about this function:
* Since the neutrino states are considered to be free,
and thus the upper limit on the continuous
-spectrum is
.
* Since for the electrons
, in order for
-decay to occur, the proton–neutron energy difference must be
* The factor
::
:in the transition probability is normally of magnitude 1, but in special circumstances it vanishes; this leads to (approximate)
selection rules for
-decay.
Forbidden transitions
As noted above, when the inner product
between the heavy particle states
and
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 good,
vanishes unless the neutron state
and the proton state
have the same angular momentum; otherwise, the angular momentum of the whole nucleus before and after the decay must be used.
Influence
Shortly after Fermi's paper appeared,
Werner Heisenberg
Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...
noted in a letter to
Wolfgang Pauli[
] that the emission and absorption of neutrinos and electrons in the nucleus should, at the second order of perturbation theory, lead to an attraction between protons and neutrons, analogously to how the emission and absorption of
photons
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
leads to the electromagnetic force. He found that the force would be of the form
, but that contemporary experimental data led to a value that was too small by a factor of a million.
The following year,
Hideki Yukawa
was a Japanese theoretical physicist and the first Japanese Nobel laureate for his prediction of the pi meson, or pion.
Biography
He was born as Hideki Ogawa in Tokyo and grew up in Kyoto with two older brothers, two older sisters, and two y ...
picked up on this idea,
[
] but in
his theory the neutrinos and electrons were replaced by a new hypothetical
particle with a rest mass approximately 200 times heavier than the electron.
[
]
Later developments
Fermi's four-fermion theory describes the
weak interaction
In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
remarkably well. Unfortunately, the calculated cross-section, or probability of interaction, grows as the square of the energy
. Since this cross section grows without bound, the theory is not valid at energies much higher than about 100 GeV. Here is the Fermi constant, which denotes the strength of the interaction. This eventually led to the replacement of the four-fermion contact interaction by a more complete theory (
UV completion)—an exchange of a
W or Z boson as explained in the
electroweak theory
In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...
.
The interaction could also explain
muon decay via a coupling of a muon, electron-antineutrino, muon-neutrino and electron, with the same fundamental strength of the interaction. This hypothesis was put forward by Gershtein and
Zeldovich
Yakov Borisovich Zeldovich ( be, Я́каў Бары́савіч Зяльдо́віч, russian: Я́ков Бори́сович Зельдо́вич; 8 March 1914 – 2 December 1987), also known as YaB, was a leading Soviet physicist of Bel ...
and is known as the Vector Current Conservation hypothesis.
In the original theory, Fermi assumed that the form of interaction is a contact coupling of two vector currents. Subsequently, it was pointed out by
Lee and
Yang that nothing prevented the appearance of an axial, parity violating current, and this was confirmed by
experiments carried out by
Chien-Shiung Wu
)
, spouse =
, residence =
, nationality = ChineseAmerican
, field = Physics
, work_institutions = Institute of Physics, Academia SinicaUniversity of California at BerkeleySmith CollegePrinceton UniversityColumbia UniversityZhejiang Unive ...
.
The inclusion of parity violation in Fermi's interaction was done by
George Gamow
George Gamow (March 4, 1904 – August 19, 1968), born Georgiy Antonovich Gamov ( uk, Георгій Антонович Гамов, russian: Георгий Антонович Гамов), was a Russian-born Soviet and American polymath, theoret ...
and
Edward Teller
Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the Teller–Ulam design), although he did not care fo ...
in the so-called
Gamow–Teller transitions which described Fermi's interaction in terms of parity-violating "allowed" decays and parity-conserving "superallowed" decays in terms of anti-parallel and parallel electron and neutrino spin states respectively. Before the advent of the electroweak theory and the
Standard Model,
George Sudarshan
Ennackal Chandy George Sudarshan (also known as E. C. G. Sudarshan; 16 September 1931 – 13 May 2018) was an Indian Americans, Indian American theoretical physicist and a professor at the University of Texas. Sudarshan has been credited wit ...
and
Robert Marshak
Robert Eugene Marshak (October 11, 1916 – December 23, 1992) was an American physicist, educator, and eighth president of the City College of New York.
Biography
Marshak was born in the Bronx, New York City. His parents, Harry and Rose Marshak ...
, and also independently
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
and
Murray Gell-Mann, were able to determine the correct
tensor structure (
vector minus
axial vector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its ...
, ) of the four-fermion interaction.
Fermi constant
The most precise experimental determination of the Fermi constant comes from measurements of the muon
lifetime
Lifetime may refer to:
* Life expectancy, the length of time a person is expected to remain alive
Arts, entertainment, and media
Music
* Lifetime (band), a rock band from New Jersey
* ''Life Time'' (Rollins Band album), by Rollins Band
* ...
, which is inversely proportional to the square of (when neglecting the muon mass against the mass of the W boson).
[
] In modern terms, the "reduced Fermi constant", that is, the constant in
natural units is
:
Here, is the
coupling constant
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
of the
weak interaction
In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
, and is the mass of the
W boson
In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , and ...
, which mediates the decay in question.
In the Standard Model, the Fermi constant is related to the
Higgs vacuum expectation value
:
.
[
]
More directly, approximately (tree level for the standard model),
:
This can be further simplified in terms of the
Weinberg angle
The weak mixing angle or Weinberg angle is a parameter in the Weinberg– Salam theory of the electroweak interaction, part of the Standard Model of particle physics, and is usually denoted as . It is the angle by which spontaneous symmetry bre ...
using the relation between the
W and Z bosons with
, so that
:
References
{{DEFAULTSORT:Fermi's Interaction
Interaction
Weak interaction