   #copyright

Isospin

2007 Schools Wikipedia Selection. Related subjects: General Physics

   In physics, and specifically, particle physics, isospin (isotopic spin,
   isobaric spin) is a symmetry of the strong interaction as it applies to
   the interactions of the neutron and proton. Isospin symmetry is a
   subset of the flavour symmetry seen more broadly in the interactions of
   baryons and mesons. Isospin symmetry remains an important concept in
   particle physics, and a close examination of this symmetry historically
   led directly to the discovery and understanding of quarks and of the
   development of Yang-Mills theory.
   Flavour in particle physics
   Flavour quantum numbers
     * Lepton number: L
     * Baryon number: B
     * Electric charge: Q
     * Weak hypercharge: Y[W]
     * Weak isospin: T[z]
     * Isospin: I, I[z]
     * Hypercharge: Y
     * Strangeness: S
     * Charm: C
     * Bottomness: B'
     * Topness: T
     __________________________________________________________________

     * Y=B+S+C+B'+T
     * Q=I[z]+Y/2
     * Q=T[z]+Y[W]/2
     * B−L
     __________________________________________________________________

   Related topics:
     * CPT symmetry
     * CKM matrix
     * CP symmetry
     * Chirality

Symmetry

   Isospin was introduced by Werner Heisenberg to explain several related
   symmetries:
     * The mass of the neutron and the proton are almost identical: they
       are nearly degenerate, and are thus often called nucleons. Although
       the proton has a positive charge, and the neutron is neutral, they
       are almost identical in all other respects.
     * 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.
     * The mass of the pions which mediate the strong interaction between
       the nucleons are the same. In particular, the mass of the
       positively-charged pion is identical to that of the
       negatively-charged pion, and both have nearly the same mass as the
       neutral pion.

   In quantum mechanics, when a Hamiltonian has a symmetry, that symmetry
   manifests itself through a set of states that have (almost) the same
   energy; that is, the states are degenerate. In particle physics, mass
   is the same thing as energy (since E=mc²), and so the near
   mass-degeneracy of the neutron and proton points at a symmetry of the
   Hamiltonian describing the strong interactions. The neutron does have a
   slightly higher mass: the mass degeneracy is not exact. The proton is
   charged, the neutron is not. However, here, as the case would be in
   general for quantum mechanics, the appearance of a symmetry can be
   imperfect, as it is perturbed by other forces, which give rise to
   slight differences between states.

SU(2)

   Heisenberg's contribution was to note that the mathematical formulation
   of this symmetry was in certain respects similar to the mathematical
   formulation of spin, from whence the name "isospin" derives. To be
   precise, the isospin symmetry is given by the invariance of the
   Hamiltonian of the strong interactions under the action of the Lie
   group SU(2). The neutron and the proton are assigned to the doublet
   (the spin-1/2 or fundamental representation) of SU(2). The pions are
   assigned to the triplet (the spin-1 or adjoint representation) of
   SU(2).

   Just as is the case for regular spin, isospin is described by two
   numbers, I, the total isospin, and I[3], the component of the spin
   vector in a given direction. The proton and neutron both have I=1/2, as
   they belong to the doublet. The proton has I[3]=+1/2 or 'isospin-up'
   and the neutron has I[3]=−1/2 or 'isospin-down'. The pions, belonging
   to the triplet, have I=1, and π^+, π^0 and π^− have, respectively,
   I[3]=+1, 0, −1.

Yang-Mills

   Isospin symmetry was central to the original formulation of Yang-Mills
   theory. The pions were proposed to be the SU(2) gauge bosons of this
   theory. While it is now understood that isospin symmetry is not a true
   gauge symmetry, this initial confusion was historically important for
   the development of the overall ideas of gauge invariance.

Relationship to flavour

   The discovery and subsequent close analysis of additional particles,
   both mesons and baryons, made it clear that the concept of isospin
   symmetry could be broadened to an even larger symmetry group, now
   called flavour symmetry. Once the kaons and their property of
   strangeness became better understood, it started to become clear that
   these, too, seemed to be a part of an enlarged, more general symmetry
   that contained isospin as a subset. The larger symmetry was named the
   Eight-fold Way by Murray Gell-Mann, and was promptly recognized to
   correspond to the adjoint representation of SU(3). This immediately led
   to Gell-Mann's proposal of the existence of quarks. The quarks would
   belong to the fundamental representation of the flavour SU(3) symmetry,
   and it is from the fundamental rep and its conjugate (the quarks and
   the anti-quarks) that the higher representation (the mesons and
   baryons) could be assembled. In short, the theory of Lie groups and Lie
   algebras modelled the physical reality of particles in the most
   exceptional and unexpected way.

   The discovery of the J/ψ meson and charm led to the expansion of
   flavour symmetry to SU(4), and the discovery of the upsilon meson (and
   the corresponding top and bottom quarks) led to the current SU(6)
   flavour symmetry. Isospin symmetry is just one little corner of this
   broader symmetry. There are strong theoretical reasons, confirmed by
   experiment, that lead one to believe that things stop there, and that
   there are no further quarks to be found.

Isospin symmetry of quarks

   In the framework of the Standard Model, the isospin symmetry of the
   proton and neutron are reinterpreted as the isospin symmetry of the up
   and down quarks. Technically, the nucleon doublet is seen to be the
   product of a single quark (thus, a doublet) and a pair of quarks in a
   singlet state. That is, the proton wave function, in terms of quark
   flavour eigenstates, is described by

          \vert p\rangle = \vert u\rangle \frac{1}{\sqrt{2}} \left( \vert
          ud\rangle + \vert du\rangle \right) + \mbox {perms.}

   and the neutron by

          \vert n\rangle = \vert d\rangle \frac{1}{\sqrt{2}} \left( \vert
          ud\rangle + \vert du\rangle \right) + \mbox {perms.}

   where perms stands for permutations. Here, \vert u \rangle is the up
   quark flavour eigenstate, and \vert d \rangle is the down quark flavour
   eigenstate. Although the above is the technically correct way of
   denoting a proton and neutron in terms of quark flavour eigenstates,
   this is almost always glossed over, and these are more simply referred
   to as uud and udd.

   Similarly, the isopsin symmetry of the pions are given by:

          \vert \pi^+\rangle = \vert u\overline {d}\rangle
          \vert \pi^0\rangle = \frac{1}{\sqrt{2}} \left(\vert u\overline
          {u}\rangle - \vert d \overline{d} \rangle \right)
          \vert \pi^-\rangle = \vert d\overline {u}\rangle

   The overline denotes, as usual, the complex conjugate representation of
   SU(2), or, equivalently, the antiquark.

Weak isospin

   The quarks also feel the weak interaction; however, the mass
   eigenstates of the strong interaction are not exactly the same as the
   eigenstates of the weak interaction. Thus, while there are still a pair
   of quarks u and d that take part in the weak interaction, they are not
   quite the same as the strong u and d quarks. The difference is given by
   a rotation, whose magnitude is called the Cabibbo angle or more
   generally, the CKM matrix.
   Retrieved from " http://en.wikipedia.org/wiki/Isospin"
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