Information AboutIsospin |
| CATEGORIES ABOUT ISOSPIN | |
| particle physics | |
| nuclear physics | |
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SYMMETRY Isospin was introduced by Werner Heisenberg to explain several related symmetries:
In 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+, π0 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 Boson s 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 Meson s and Baryon s, made it clear that the concept of isospin symmetry could be broadened to an even larger symmetry group, now called Flavour Symmetry . Once the Kaon s 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 lead to Gell-Mann's proposal of the existence of Quark s. 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 Group s and Lie Algebra s modelled the physical reality of particles in the most exceptional and unexpected way. The discovery of the J/ψ meson and Charm lead to the expansion of flavour symmetry to SU(4) , and the discovery of the Upsilon Meson (and the corresponding top and bottom quarks) lead 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 Quark s. 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 flavor eigenstates, is described by : and the neutron by : where ''perms'' stands for Permutation s. Here, is the Up Quark flavour eigenstate, and 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: : : : The overline denotes, as usual, the Complex Conjugate representation of SU(2), or, equivalently, the Antiquark . WEAK ISOSPIN See Also: weak isospin The quarks also feel the Weak Interaction , however, the mass Eigenstate s 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 . REFERENCES
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