Fermi's interaction

In particle physics, Fermi's interaction (also the Fermi theory of beta decay) is an explanation of the beta decay, proposed by Enrico Fermi in 1933.[1] The theory posits four fermions 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) and a proton.[2]

Fermi first introduced this coupling in his description of beta decay in 1933.[3] The Fermi interaction was the precursor to the theory for the weak interaction where the interaction between the proton–neutron and electron–antineutrino is mediated by a virtual W boson.

Beta-minus Decay

decay in an atomic nucleus (the accompanying antineutrino is omitted). The inset shows beta decay of a free neutron. In both processes, the intermediate emission of a virtual
(which then decays to electron and antineutrino) is not shown.

History of initial rejection and later publication

Fermi first submitted his "tentative" theory of beta decay to the famous science journal Nature, which rejected it "because it contained speculations too remote from reality to be of interest to the reader.[4]" Nature later admitted the rejection to be one of the great editorial blunders in its history.[5] Fermi then submitted revised versions of the paper to Italian and German publications, which accepted and published them in those languages in 1933 and 1934.[6][7][8][9] The paper did not appear at the time in a primary publication in English.[5] An English translation of the seminal paper was published in the American Journal of Physics in 1968.[9]

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"


Electron state

where is the single-electron wavefunction, are its stationary states.

is the operator which annihilates an electron in state which acts on the Fock space as

is the creation operator for electron state :

Neutrino state


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


resp. is an eigenfunction for a neutron resp. proton in the state .


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 anti-neutrino), 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 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, 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.


Shortly after Fermi's paper appeared, Werner Heisenberg noted in a letter to Wolfgang Pauli[10] 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 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.[11]

The following year, Hideki Yukawa picked up on this idea,[12] 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.[13]

Later developments

Fermi's four-fermion theory describes the weak 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 GF is the Fermi coupling 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.

Fermi's interaction showing the 4-point fermion vector current, coupled under Fermi's Coupling Constant GF. Fermi's Theory was the first theoretical effort in describing nuclear decay rates for β decay.

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 and is known as the Vector Current Conservation hypothesis.[14]

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.[15][16]

The inclusion of parity violation in Fermi's interaction was done by George Gamow and Edward Teller 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 and Robert Marshak, and also independently Richard Feynman and Murray Gell-Mann, were able to determine the correct tensor structure (vector minus axial vector, VA) of the four-fermion interaction.[17][18]

Fermi coupling constant

The most precise experimental determination of the Fermi constant comes from measurements of the muon lifetime, which is inversely proportional to the square of GF (when neglecting the muon mass against the mass of the W boson).[19] In modern terms:[3][20]

Here g is the coupling constant of the weak interaction, and mW is the mass of the W boson which mediates the decay in question.

In the Standard Model, Fermi's 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 using the relation between the W and Z Bosons with , so that


  1. ^ Yang, C. N. (2012). "Fermi's β-decay Theory". Asia Pacific Physics Newsletter. 1 (01): 27–30. doi:10.1142/s2251158x12000045.
  2. ^ Feynman, R.P. (1962). Theory of Fundamental Processes. W. A. Benjamin. Chapters 6 & 7.
  3. ^ a b Griffiths, D. (2009). Introduction to Elementary Particles (2nd ed.). pp. 314–315. ISBN 978-3-527-40601-2.
  4. ^ Pais, Abraham (1986). Inward Bound. Oxford: Oxford University Press. p. 418. ISBN 0-19-851997-4.
  5. ^ a b Close, Frank (February 23, 2012). Neutrino. Oxford University Press. Retrieved May 5, 2017.
  6. ^ Fermi, E. (1933). "Tentativo di una teoria dei raggi β". La Ricerca Scientifica (in Italian). 2 (12).
  7. ^ Fermi, E. (1934). "Tentativo di una teoria dei raggi β". Il Nuovo Cimento (in Italian). 11 (1): 1–19. Bibcode:1934NCim...11....1F. doi:10.1007/BF02959820.
  8. ^ Fermi, E. (1934). "Versuch einer Theorie der beta-Strahlen. I". Zeitschrift für Physik (in German). 88: 161. Bibcode:1934ZPhy...88..161F. doi:10.1007/BF01351864.
  9. ^ a b Wilson, F. L. (1968). "Fermi's Theory of Beta Decay". American Journal of Physics. 36 (12): 1150. Bibcode:1968AmJPh..36.1150W. doi:10.1119/1.1974382. Includes complete English translation of Fermi's 1934 paper in German
  10. ^ Pauli, Wolfgang (1985). Scientific Correspondence with Bohr, Einstein, Heisenberg a.o. Volume II:1930–1939. Springer-Verlag Berlin Heidelberg GmbH. p. 250, letter #341, Heisenberg to Pauli, January 18th 1934.
  11. ^ Brown, Laurie M (1996). The Origin of the Concept of Nuclear Forces. Institute of Physics Publishing. Section 3.3.
  12. ^ Yukawa, H. (1935). "On the interaction of elementary particles. I.". Proceedings of the Physico-Mathematical Society of Japan. 17: 1.
  13. ^ Mehra, Jagdish (2001). The Historical Development of Quantum Theory, Volume 6 Part 2 (1932–1941). Springer. p. 832.
  14. ^ Gerstein, S. S.; Zeldovich, Ya. B. (1955). "Meson corrections in the theory of beta decay". Zh. Eksp. Teor. Fiz.: 698–699.
  15. ^ Lee, T. D.; Yang, C. N. (1956). "Question of Parity Conservation in Weak Interactions". Physical Review. 104 (1): 254–258. Bibcode:1956PhRv..104..254L. doi:10.1103/PhysRev.104.254.
  16. ^ Wu, C. S.; Ambler, E; Hayward, R. W.; Hoppes, D. D.; Hudson, R. P. (1957). "Experimental Test of Parity Conservation in Beta Decay". Physical Review. 105 (4): 1413–1415. Bibcode:1957PhRv..105.1413W. doi:10.1103/PhysRev.105.1413.
  17. ^ Feynman, R. P.; Gell-Mann, M. (1958). "Theory of the Fermi interaction" (PDF). Physical Review. 109 (1): 193. Bibcode:1958PhRv..109..193F. doi:10.1103/physrev.109.193.
  18. ^ Sudarshan, E. C.; Marshak, R. E. (1958). "Chirality invariance and the universal Fermi interaction". Physical Review. 109 (5): 1860. Bibcode:1958PhRv..109.1860S. doi:10.1103/physrev.109.1860.2.
  19. ^ Chitwood, D. B.; MuLan Collaboration; et al. (2007). "Improved Measurement of the Positive-Muon Lifetime and Determination of the Fermi Constant". Physical Review Letters. 99: 032001. arXiv:0704.1981. Bibcode:2007PhRvL..99c2001C. doi:10.1103/PhysRevLett.99.032001. PMID 17678280.
  20. ^ "CODATA Value: Fermi coupling constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved October 31, 2016.
  21. ^ Plehn, T.; Rauch, M. (2005). "Quartic Higgs coupling at hadron colliders". Physical Review D. 72: 053008. arXiv:hep-ph/0507321. Bibcode:2005PhRvD..72e3008P. doi:10.1103/PhysRevD.72.053008.
Beta decay transition

A Fermi transition or a Gamow–Teller transition are types of nuclear beta decay determined by changes in angular momentum or spin. In the Fermi transition, the spins of the emitted particles are antiparallel, coupling to , so the angular momentum of the initial and final angular momentum states of the nucleus are unchanged (). This is in contrast to a Gamow-Teller transition, where the spins of the emitted electron (positron) and antineutrino (neutrino) couple to total spin , leading to an angular momentum change between the initial and final angular momentum states of the nucleus.

Fermi and Gamow-Teller transitions correspond to two different forms of leading order behavior of the weak interaction Hamiltonian in the non-relativistic limit:

= isospin transition matrix which turn protons to neutrons and vice versa
= Pauli spin matrices, which lead to .
= identity operator in spin space, leaving unchanged.
= Weak vector coupling constant.
= Weak axial-vector coupling constant.

The theoretical work in describing these transitions was done between 1934 and 1936 by Nuclear Physicists George Gamow and Edward Teller at George Washington University.

Enrico Fermi

Enrico Fermi (Italian: [enˈriːko ˈfermi]; 29 September 1901 – 28 November 1954) was an Italian and naturalized-American physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and the "architect of the atomic bomb". He was one of very few physicists to excel in both theoretical physics and experimental physics. Fermi held several patents related to the use of nuclear power, and was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity by neutron bombardment and for the discovery of transuranium elements. He made significant contributions to the development of statistical mechanics, quantum theory, and nuclear and particle physics.

Fermi's first major contribution involved the field of statistical mechanics. After Wolfgang Pauli formulated his exclusion principle in 1925, Fermi followed with a paper in which he applied the principle to an ideal gas, employing a statistical formulation now known as Fermi–Dirac statistics. Today, particles that obey the exclusion principle are called "fermions". Pauli later postulated the existence of an uncharged invisible particle emitted along with an electron during beta decay, to satisfy the law of conservation of energy. Fermi took up this idea, developing a model that incorporated the postulated particle, which he named the "neutrino". His theory, later referred to as Fermi's interaction and now called weak interaction, described one of the four fundamental interactions in nature. Through experiments inducing radioactivity with the recently discovered neutron, Fermi discovered that slow neutrons were more easily captured by atomic nuclei than fast ones, and he developed the Fermi age equation to describe this. After bombarding thorium and uranium with slow neutrons, he concluded that he had created new elements. Although he was awarded the Nobel Prize for this discovery, the new elements were later revealed to be nuclear fission products.

Fermi left Italy in 1938 to escape new Italian racial laws that affected his Jewish wife, Laura Capon. He emigrated to the United States, where he worked on the Manhattan Project during World War II. Fermi led the team that designed and built Chicago Pile-1, which went critical on 2 December 1942, demonstrating the first human-created, self-sustaining nuclear chain reaction. He was on hand when the X-10 Graphite Reactor at Oak Ridge, Tennessee, went critical in 1943, and when the B Reactor at the Hanford Site did so the next year. At Los Alamos, he headed F Division, part of which worked on Edward Teller's thermonuclear "Super" bomb. He was present at the Trinity test on 16 July 1945, where he used his Fermi method to estimate the bomb's yield.

After the war, Fermi served under J. Robert Oppenheimer on the General Advisory Committee, which advised the Atomic Energy Commission on nuclear matters. After the detonation of the first Soviet fission bomb in August 1949, he strongly opposed the development of a hydrogen bomb on both moral and technical grounds. He was among the scientists who testified on Oppenheimer's behalf at the 1954 hearing that resulted in the denial of Oppenheimer's security clearance. Fermi did important work in particle physics, especially related to pions and muons, and he speculated that cosmic rays arose when material was accelerated by magnetic fields in interstellar space. Many awards, concepts, and institutions are named after Fermi, including the Enrico Fermi Award, the Enrico Fermi Institute, the Fermi National Accelerator Laboratory, the Fermi Gamma-ray Space Telescope, the Enrico Fermi Nuclear Generating Station, and the synthetic element fermium, making him one of 16 scientists who have elements named after them.

Index of physics articles (F)

The index of physics articles is split into multiple pages due to its size.

To navigate by individual letter use the table of contents below.

List of mathematical topics in quantum theory

This is a list of mathematical topics in quantum theory, by Wikipedia page. See also list of functional analysis topics, list of Lie group topics, list of quantum-mechanical systems with analytical solutions.

List of things named after Enrico Fermi

Enrico Fermi (1901–1954), an Italian-born, naturalized American physicist, is the eponym of the topics listed below.

Mathematical formulation of the Standard Model

This article describes the mathematics of the Standard Model of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group SU(3) × SU(2) × U(1). The theory is commonly viewed as containing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs particle.

The Standard Model is renormalizable and mathematically self-consistent, however despite having huge and continued successes in providing experimental predictions it does leave some unexplained phenomena. In particular, although the physics of special relativity is incorporated, general relativity is not, and the Standard Model will fail at energies or distances where the graviton is expected to emerge. Therefore, in a modern field theory context, it is seen as an effective field theory.

This article requires some background in physics and mathematics, but is designed as both an introduction and a reference.

Nuclear physics

Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied.

Nuclear physics should not be confused with atomic physics, which studies the atom as a whole, including its electrons.

Discoveries in nuclear physics have led to applications in many fields. This includes nuclear power, nuclear weapons, nuclear medicine and magnetic resonance imaging, industrial and agricultural isotopes, ion implantation in materials engineering, and radiocarbon dating in geology and archaeology. Such applications are studied in the field of nuclear engineering.

Particle physics evolved out of nuclear physics and the two fields are typically taught in close association. Nuclear astrophysics, the application of nuclear physics to astrophysics, is crucial in explaining the inner workings of stars and the origin of the chemical elements.

Quantum field theory

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics and is used to construct physical models of subatomic particles (in particle physics) and quasiparticles (in condensed matter physics).

QFT treats particles as excited states (also called quanta) of their underlying fields, which are—in a sense—more fundamental than the basic particles. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding fields. Each interaction can be visually represented by Feynman diagrams, which are formal computational tools, in the process of relativistic perturbation theory.

Rubby Sherr

Rubby Sherr (September 14, 1913 – July 8, 2013) was an American nuclear physicist who co-invented a key component of the first nuclear weapon while participating in the Manhattan Project during the Second World War. His academic career spanned nearly eight decades, including almost 40 years working at Princeton University.

UV completion

In theoretical physics, ultraviolet completion, or UV completion, of a quantum field theory is the passing from a lower energy quantum field theory to a more general quantum field theory above a threshold value known as the cutoff. In particular, the more general high energy theory must be well-defined at arbitrarily high energies.

The word "ultraviolet" in this so-called "ultraviolet regime" is only figurative, and refers to energies much higher than ultraviolet light per se. Rather, by analogy to the relationship between ultraviolet and visible light, it refers to energies higher than (and wavelengths shorter than) those "visible" to laboratory experiment.

The ultraviolet theory must be renormalizable; it can have no Landau poles; and most typically, it enjoys asymptotic freedom in the case that it is a quantum field theory (or at least has a nontrivial fixed point). However, it may also be a background of string theory whose ultraviolet behavior is at least as good as that of renormalizable quantum field theories. Besides these two known examples (QFT and string theory), it could be a completely different theory than string theory that behaves well at very high energies.

There is an analogous phrase "infrared completion", which applies to length scales longer than those "visible" to normal experiment, particularly cosmology distances.

Weak interaction

In particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is the mechanism of interaction between subatomic particles that is responsible for the radioactive decay of atoms. The weak interaction serves an essential role in nuclear fission, and the theory regarding it in terms of both its behavior and effects is sometimes called quantum flavordynamics (QFD). However, the term QFD is rarely used because the weak force is better understood in terms of electroweak theory (EWT). In addition to this, QFD is related to quantum chromodynamics (QCD), which deals with the strong interaction, and quantum electrodynamics (QED), which deals with the electromagnetic force.

The effective range of the weak force is limited to subatomic distances, and is less than the diameter of a proton. It is one of the four known force-related fundamental interactions of nature, alongside the strong interaction, electromagnetism, and gravitation.

Wu experiment

The Wu experiment was a nuclear physics experiment conducted in 1956 by the Chinese American physicist Chien-Shiung Wu in collaboration with the Low Temperature Group of the US National Bureau of Standards. The experiment's purpose was to establish whether or not conservation of parity (P-conservation), which was previously established in the electromagnetic and strong interactions, also applied to weak interactions. If P-conservation were true, a mirrored version of the world (where left is right and right is left) would behave as the mirror image of the current world. If P-conservation were violated, then it would be possible to distinguish between a mirrored version of the world and the mirror image of the current world.

The experiment established that conservation of parity was violated (P-violation) by the weak interaction. This result was not expected by the physics community, which had previously regarded parity as a conserved quantity. Tsung-Dao Lee and Chen-Ning Yang, the theoretical physicists who originated the idea of parity nonconservation and proposed the experiment, received the 1957 Nobel Prize in physics for this result. Chien-Shiung Wu’s role in the discovery was mentioned in the Nobel prize acceptance speech, but was not honored until 1978, when she was awarded the first Wolf Prize.

Yukawa potential

In particle, atomic and condensed matter physics, a Yukawa potential (also called a screened Coulomb potential) is a potential of the form

where g is a magnitude scaling constant, i.e. is the amplitude of potential, m is the mass of the particle, r is the radial distance to the particle, and α is another scaling constant, so that is the range. The potential is monotone increasing in r and it is negative, implying the force is attractive. In the SI system, the unit of the Yukawa potential is (1/meters).

The Coulomb potential of electromagnetism is an example of a Yukawa potential with equal to 1 everywhere. This can be interpreted as saying that the photon mass m is equal to 0.

In interactions between a meson field and a fermion field, the constant g is equal to the gauge coupling constant between those fields. In the case of the nuclear force, the fermions would be a proton and another proton or a neutron.

Beyond the
Standard Model

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