# Isospin

The isotopic spin is in the theory of elementary a Flavor - quantum number , an inner symmetry of the strong interaction describes and for classifying the hadrons is used. The designation ( iso- : “quantitatively equal”, from ancient Greek ἴσος) indicates that the system appears like a spin 1/2 particle , although it is not a spin .

The concept is used more generally (also in solid-state physics ) to describe two-state systems . The two quantum mechanical states are understood as opposing orientations of the isospin (± ). If the system is in a superposition of the two states, this is described by the two other components ( ). ${\ displaystyle I_ {z}}$${\ displaystyle I_ {x}, I_ {y}}$

## discovery

In scattering processes on mirror cores it was found that the strong interaction does not differentiate between neutral neutrons and positively charged protons , i.e. This means that it acts independently of the charge. With regard to the nuclear force, the neutron and proton are identical, and their slight difference in mass is related to the electrical charge . Werner Heisenberg concluded from this in 1932 that the proton and the neutron are 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 corresponding behavior of the electrons . They also have two states ( spin-up and spin-down ) which cannot be distinguished by a certain force - here the purely electrical force.

The name Isospin was coined by Eugene Wigner in 1937 and initially stood for isotopic spin . However, since this can be misinterpreted as an indication of a change in the number of neutrons (cf. isotope ), the term isobaric spin is used today. Murray Gell-Mann combined the properties of isospin and strangeness in the Eightfold Way , a direct precursor of the quark model and quantum chromodynamics .

## formalism

up
Quark / antiquark u u
Isospin ${\ displaystyle I_ {z}}$ + ½
down
Quark / antiquark d d
Isospin ${\ displaystyle I_ {z}}$ + ½

Like the normal spin of the fundamental fermions (such as the electron), the quantum number of the isospin always has the value 1/2.

The canonically used third component (often also referred to as) of the isospin represents its setting and has the two possible values ​​+1/2 and −1/2. In the quark model, these stand for the two quarks${\ displaystyle I_ {z}}$${\ displaystyle I_ {3}}$

• u ( up ): and${\ displaystyle I_ {z} = + 1/2}$
• d ( down , Eng .: below): .${\ displaystyle I_ {z} = - 1/2}$

The quarks s , c , b and t do not have an isospin. For anti quarks the sign changes from . ${\ displaystyle I_ {z}}$

The number of u and d quarks and the associated antiquarks is thus given as follows : ${\ displaystyle I_ {z}}$

${\ displaystyle I_ {z} = {\ frac {1} {2}} {\ Big (} (n_ {u} -n _ {\ bar {u}}) - (n_ {d} -n _ {\ bar { d}}) {\ Big)}}$.

The difference between protons and neutrons results from their composition:

• Proton p = uud ${\ displaystyle \ Rightarrow I_ {z} = + 1/2}$
• Neutron n = udd .${\ displaystyle \ Rightarrow I_ {z} = - 1/2}$

This assignment is done the other way around in some books and is just a convention that doesn't matter as long as consistency is maintained.

### Hypercharge

Particle Components el. charge
${\ displaystyle Q}$
Isospin
${\ displaystyle I_ {z}}$
Hyperldg.
${\ displaystyle Y}$
Quarks Up u +2/3 + ½ +1/3
Anti-up u -2/3 -1/3
Down d -1/3 +1/3
Anti-down d +1/3 + ½ -1/3
Hadrons proton uud +1 + ½ +1
neutron udd 0 +1

Due to their isospin and their electrical charge , many particles can be assigned a hypercharge using the Gell-Mann-Nishijima formula : ${\ displaystyle Q}$ ${\ displaystyle Y}$

${\ displaystyle Y = 2 (Q-I_ {z}).}$

The hypercharge is

• for up and down quark respectively: ${\ displaystyle Y = + 1/3 \! \,}$
• for anti-up and anti-down quark: ${\ displaystyle Y = -1 / 3 \! \,}$
• for the nucleons ( protons p, neutron n) respectively: .${\ displaystyle Y = + 1 \! \,}$

### Quantum field theory

Within the framework of quantum field theory , the isospin is assigned the two-dimensional complex vector space in which the quarks u and d can be represented as basis vectors : ${\ displaystyle \ mathbb {C} ^ {2}}$

${\ displaystyle \ mathbf {u} = \ left ({\ begin {matrix} 1 \\ 0 \ end {matrix}} \ right), \ quad \ mathbf {d} = \ left ({\ begin {matrix} 0 \\ 1 \ end {matrix}} \ right).}$

This makes it possible to describe the transformation of nucleons as the radioactive decay takes place: . This is a transformation of SU (2) symmetry described in the weak interaction theory. ${\ displaystyle \ mathbf {n} \ to \ mathbf {p} + e ^ {-} + {\ bar {\ nu}}}$

Mathematically, these transformations are mediated by ladder operators , which are assigned to the gauge bosons of field theory. For example, the transition is described by the matrix equation ${\ displaystyle \ mathbf {d} \ rightarrow \ mathbf {u}}$

${\ displaystyle \ left ({\ begin {matrix} 0 & 1 \\ 0 & 0 \ end {matrix}} \ right) \ cdot \ left ({\ begin {matrix} 0 \\ 1 \ end {matrix}} \ right) = \ left ({\ begin {matrix} 1 \\ 0 \ end {matrix}} \ right).}$

## Individual evidence

1. ^ W. Heisenberg: About the construction of atomic nuclei . In: Journal of Physics . tape 77 , 1932, pp. 1–11 , doi : 10.1007 / BF01342433 , bibcode : 1932ZPhy ... 77 .... 1H .