# Isospin

The (strong ) isospin in the theory of elementary particles a flavor quantum number that describes an internal symmetry of the strong interaction and is used for the classification of hadrons.

More generally, the concept is ( as well as in solid state physics ) is used to describe two-state systems. Here, the two quantum mechanical states are perceived as opposing orientations of the isospin ( ±). The system is a superposition of the two states, as represented by the other two components () will be described. It is then spoken of a isospin ( iso: " quantitatively equal ", from Ancient Greek ἴσος ), since the system appears the same as a Spin-1/2-Teilchen, although it is not necessarily to be a spin.

- 3.1 Weak hypercharge

## Formation

In scattering processes to mirror nuclei was found that the strong interaction does not distinguish between the neutral neutrons and positively charged protons, that is, that it works regardless of load. Regarding the nuclear neutron and proton are therefore identical particles. My slight mass difference is therefore due to their different electrical charge. Based on these facts and presumptions deduced Werner Heisenberg in 1932, that the proton and neutron two different charge states of one and the same particle, the nucleon.

For a further description he borrowed the quantum mechanical spin formalism from the analog behavior of the electrons. These come in two " forms " on (spin -up and spin-down ), which are not distinguishable by a certain force - here the purely electric power.

The name was coined in 1937 by isospin Eugene Wigner and was initially for isotopic spin. However, since this can be misconstrued as an indication of a change in the number of nucleons (see isotope), is now the term used isobaric spin.

The isospin was postulated long before the formulation of the quark model. Murray Gell-Mann joined the isospin with the flavor strangeness to the so called Eightfold Way, a direct precursor of the quark model and quantum chromodynamics.

## Formalism

As the normal spin of the fundamental fermions ( such as the electron) the isospin always has the value 1/2.

The third component canonically used (and often referred to ) of the isospin represents his attitude and has the two possible values 1 / 2 and -1 / 2. These are in the quark model for the two quarks

- U ( up, engl. above) and
- D (down, engl. below):.

The quarks s, c, b and t carry no isospin. For antiquarks, the sign of change.

Thus, as follows by the number of u and d quarks and the associated anti-quarks is given:

The difference between proton and neutron results from their composition:

- Proton p = UUD
- Neutron n = udd.

This assignment is in some books instead of the other way around and is just a convention, which is insignificant, as long as consistency is maintained.

### Hypercharge

Due to their isospin and their electrical charge to many particles using the Gell-Mann - Nishijima formula can assign a hypercharge:

The hypercharge is

- For up-and down - quark, respectively:
- For anti -up and anti -down quark respectively:
- For the nucleons ( proton p, neutron n ), respectively.

### Quantum field theory

In the framework of quantum field theory the isospin of the two-dimensional real vector space is allocated in the let the quarks u and d represent the basis vectors:

This makes it possible to describe the conversion of nucleons, as occurs in the radioactive decay. This is a transformation of SU (2) of symmetry, which is described in the theory of weak interaction.

Mathematically, these transformations mediated by ladder operators that are assigned to the gauge bosons of the field theory. For example, the transition is described by the matrix equation

In addition to strong isospin discussed above can also be a weak isospin of leptons and quarks introduce. According to him, the individual families are grouped into weak isospin doublets, within which the particles behave identically with respect to the weak interaction, and sometimes merge into one another:

Here, d ', s ', b', the eigenstates of the quarks with respect to the weak interaction, which are related by the CKM matrix with the eigenstates of the strong interaction. The index L indicates that there are only left-handed particles transitions within a doublet which are mediated by charged weak currents.

The weak isospin doublets of left-handed has the amount and the third component (synonym ):

- For the neutral leptons and the positively charged quarks:

- For the charged leptons and the negatively charged quarks:

Right-handed particles, however, occur only in singlets, as there are no right-handed neutrinos on the one hand ( Goldhaber experiment) and on the other hand, no neutral weak currents, which could change the quark species:

### Weak hypercharge

Because of their weak isospin and their electrical charge to all particles using a modified Gell-Mann - Nishijima formula can assign a weak hypercharge:

It is

- For left-handed leptons, whether loaded or not:
- For left-handed quarks, charged no matter whether positive or negative:

- For right-handed ( charged) leptons:
- For right-handed positively charged quarks:
- For right-handed negatively charged quarks.