Isospin
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In
nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the ...
and
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the
flavour symmetry In particle physics, flavour or flavor refers to the ''species'' of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with ''flavour quantum numbers'' th ...
seen more broadly in the interactions of
baryon In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classif ...
s and
meson In particle physics, a meson ( or ) is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, ...
s. The name of the concept contains the term ''spin'' because its quantum mechanical description is mathematically similar to that of
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
(in particular, in the way it
couples Couple or couples may refer to : Basic meaning *Couple (app), a mobile app which provides a mobile messaging service for two people *Couple (mechanics), a system of forces with a resultant moment but no resultant force *Couple (relationship), tw ...
; for example, a proton–neutron pair can be coupled either in a state of total isospin 1 or in one of 0). But unlike angular momentum, it is a dimensionless quantity and is not actually any type of spin.
Etymologically Etymology ()The New Oxford Dictionary of English (1998) – p. 633 "Etymology /ˌɛtɪˈmɒlədʒi/ the study of the class in words and the way their meanings have changed throughout time". is the study of the history of the form of words an ...
, the term was derived from isotopic spin, a confusing term to which nuclear physicists prefer isobaric spin, which is more precise in meaning. Before the concept of quarks was introduced, particles that are affected equally by the strong force but had different charges (e.g. protons and neutrons) were considered different states of the same particle, but having isospin values related to the number of charge states.. A close examination of isospin symmetry ultimately led directly to the discovery and understanding of quarks and to the development of
Yang–Mills theory In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using ...
. Isospin symmetry remains an important concept in particle physics.


Quark content and isospin

In the modern formulation, isospin () is defined as a vector quantity in which up and down quarks have a value of  = 1/2, with the 3rd-component (3) being +1/2 for up quarks, and −1/2 for down quarks, while all other quarks have  = 0. Therefore, for hadrons in general, where u and d are the numbers of up and down quarks respectively, : I_3 = \frac(n_u - n_d). In any combination of quarks, the 3rd component of the isospin vector (3) could either be aligned between a pair of quarks, or face the opposite direction, giving different possible values for total isospin for any combination of quark flavours.
Hadrons In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the ele ...
with the same quark content but different total isospin can be distinguished experimentally, verifying that flavour is actually a vector quantity, not a scalar (up vs down simply being a projection in the quantum mechanical  axis of flavour space). For example, a strange quark can be combined with an up and a down quark to form a
baryon In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classif ...
, but there are two different ways the isospin values can combine either adding (due to being flavour-aligned) or cancelling out (due to being in opposite flavour directions). The isospin-1 state (the ) and the isospin-0 state (the ) have different experimentally detected masses and half-lives.


Isospin and symmetry

Isospin is regarded as a symmetry of the strong interaction under the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of the Lie group
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
, the two states being the up flavour and down flavour. In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, when a
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
has a symmetry, that symmetry manifests itself through a set of states that have the same energy (the states are described as being ''
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''). In simple terms, the energy operator for the strong interaction gives the same result when an up quark and an otherwise identical down quark are swapped around. Like the case for regular spin, the isospin operator I is
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
-valued: it has three components I''x'', I''y'', I''z'', which are coordinates in the same 3-dimensional vector space where the 3 representation acts. Note that this vector space has nothing to do with the physical space, except similar mathematical formalism. Isospin is described by two quantum numbers: the total isospin, and 3 an eigenvalue of the I''z''
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
for which flavor states are eigenstates, . In other words, each 3 state specifies certain flavor state of a
multiplet In physics and particularly in particle physics, a multiplet is the state space for 'internal' degrees of freedom of a particle, that is, degrees of freedom associated to a particle itself, as opposed to 'external' degrees of freedom such as the ...
. The third coordinate (), to which the "3" subscript refers, is chosen due to notational conventions that relate bases in 2 and 3 representation spaces. Namely, for the spin-1/2 case, components of I are equal to
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
divided by 2, and so I''z'' =  3, where : \tau_3 = \begin 1 & 0 \\ 0 & -1 \end. While the forms of these matrices are isomorphic to those of spin, ''these'' Pauli matrices only act within the Hilbert space of isospin, not that of spin, and therefore is common to denote them with τ rather than σ to avoid confusion. Although isospin symmetry is actually very slightly broken, SU(3) symmetry is more badly broken, due to the much higher mass of the strange quark compared to the up and down. The discovery of
charm Charm may refer to: Social science * Charisma, a person or thing's pronounced ability to attract others * Superficial charm, flattery, telling people what they want to hear Science and technology * Charm quark, a type of elementary particle * Ch ...
,
bottomness In physics, bottomness (symbol ''B''′ using a prime as plain ''B'' is used already for baryon number) or beauty is a flavour quantum number reflecting the difference between the number of bottom antiquarks (''n'') and the number of botto ...
and
topness Topness (''T'', also called truth), a flavour quantum number, represents the difference between the number of top quarks (t) and number of top antiquarks () that are present in a particle: :T = n_\text - n_\bar By convention, top quarks have a ...
could lead to further expansions up to SU(6) flavour symmetry, which would hold if all six quarks were identical. However, the very much larger masses of the charm, bottom, and top quarks means that SU(6) flavour symmetry is very badly broken in nature (at least at low energies), and assuming this symmetry leads to qualitatively and quantitatively incorrect predictions. In modern applications, such as
lattice QCD Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lat ...
, isospin symmetry is often treated as exact for the three light quarks (uds), while the three heavy quarks (cbt) must be treated separately.


Hadron nomenclature

Hadron nomenclature is based on isospin. * Particles of total isospin 3/2 are named
Delta baryon The Delta baryons (or baryons, also called Delta resonances) are a family of subatomic particle made of three up or down quarks (u or d quarks), the same constituent quarks that make up the more familiar protons and neutrons. Properties Four c ...
s and can be made by a combination of any three up or down quarks (but only up or down quarks). * Particles of total isospin 1 can be made from two up quarks, two down quarks, or one of each: ** certain
mesons In particle physics, a meson ( or ) is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, ...
further differentiated by total spin into
pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
s (total spin 0) and
rho meson Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
s (total spin 1) ** with an additional quark of higher flavour Sigma baryons * Particles of total isospin 1/2 can be made from: ** a single up or down quark together with an additional quark of higher flavour strange (
kaon KAON (Karlsruhe ontology) is an ontology infrastructure developed by the University of Karlsruhe and the Research Center for Information Technologies in Karlsruhe. Its first incarnation was developed in 2002 and supported an enhanced version of ...
s), charm ( D meson), or bottom ( B meson) ** a single up or down quark together with two additional quarks of higher flavour Xi baryon ** an up quark, a down quark, and either an up or a down quark
nucleons In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number). Until the 1960s, nucleons we ...
. Note that three identical quarks would be forbidden by the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulat ...
due to requirement of anti-symmetric wave function * Particles of total isospin 0 can be made from ** a neutral quark-antiquark pair: u\bar or d\bar eta mesons ** one up quark and one down quark, with an additional quark of higher flavour
Lambda baryons The lambda baryons (Λ) are a family of subatomic hadron particles containing one up quark, one down quark, and a third quark from a higher flavour generation, in a combination where the quantum wave function changes sign upon the flavour of an ...
** anything not involving any up or down quarks


History


Original motivation for isospin

Isospin was introduced as a concept in 1932, well before the 1960s development of the
quark model In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Ei ...
. The man who introduced it,
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 breakthrough paper. In the subsequent serie ...
, did so to explain symmetries of the then newly discovered
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
(symbol n): * The
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
of the neutron and the proton (symbol p) are almost identical: they are nearly degenerate, and both are thus often called
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number). Until the 1960s, nucleons were ...
s. Although the proton has a positive electric charge, and the neutron is neutral, they are almost identical in all other aspects. * The strength of the strong interaction between any pair of nucleons is the same, independent of whether they are interacting as protons or as neutrons. This behavior is not unlike the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
, where there are two possible states based on their spin. Other properties of the particle are conserved in this case. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937,
Eugene Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his co ...
introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated. Protons and neutrons were then grouped together as
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number). Until the 1960s, nucleons were ...
s because they both have nearly the same mass and interact in nearly the same way, if the (much weaker) electromagnetic interaction is neglected. In
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, the near mass-degeneracy of the neutron and proton points to an approximate symmetry of the Hamiltonian describing the strong interactions. It was thus convenient to treat them as being different states of the same particle. Heisenberg's particular contribution was to note that the mathematical formulation of this symmetry was in certain respects similar to the mathematical formulation of spin, whence the name "isospin" derives. The neutron and the proton are assigned to the
doublet Doublet is a word derived from the Latin ''duplus'', "twofold, twice as much",