Isospin

In nuclear physics and particle physics, isospin (I) is a quantum number related to the strong interaction. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions of baryons and mesons. In nuclear physics and particle physics, isospin (I) is a quantum number related to the strong interaction. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions of baryons and mesons. The name of the concept contains the term spin because its quantum mechanical description is mathematically similar to that of angular momentum (in particular, in the way it couples; for example, a proton-neutron pair can be coupled either in a state of total isospin 1 or in one of 0). Unlike angular momentum, however, it is a dimensionless quantity, and is not actually any type of spin. Etymologically, 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 were introduced, particles that are affected equally by the strong force but had different charges (e.g. protons and neutrons) were treated as being 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 of the development of Yang–Mills theory. Isospin symmetry remains an important concept in particle physics. In the modern formulation, isospin (I) is defined as a vector quantity in which up and down quarks have a value of I = ​1⁄2, with the 3rd-component (I3) being ​1⁄2 for up quarks, and −​1⁄2 for down quarks, while all other quarks have I = 0. Therefore, for hadrons in general, where nu and nd are the numbers of up and down quarks respectively. In any combination of quarks, the 3rd component of the isospin vector (I3) 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 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 z-axis of flavour-space). For example, a strange quark can be combined with an up and a down quark to form a baryon, 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 Σ0) and the isospin 0 state (the Λ0) have different experimentally detected masses and half-lives. Isospin is regarded as a symmetry of the strong interaction under the action of the Lie group SU(2), the two states being the up flavour and down flavour. In quantum mechanics, when a Hamiltonian has a symmetry, that symmetry manifests itself through a set of states that have the same energy (the states are described as being degenerate). In simple terms, that 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-valued: it has three components Ix, Iy, Iz which are coordinates in the same 3-dimensional vector space where the 3 representation acts. Note that it has nothing to do with the physical space, except similar mathematical formalism. Isospin is described by two quantum numbers: I, the total isospin, and I3, an eigenvalue of the Iz projection for which flavor states are eigenstates, not an arbitrary projection as in the case of spin. In other words, each I3 state specifies certain flavor state of a multiplet. The third coordinate (z), to which the '3' subscript refers, is chosen due to notational conventions which relate bases in 2 and 3 representation spaces. Namely, for the spin-​1⁄2 case, components of I are equal to Pauli matrices divided by 2, and so Iz = ​1⁄2 τ3, where