Weak isospin

Weak isospin is a quantum number associated with representations of the SU(2) gauge symmetry in the electroweak theory. It determines how a particle’s fields couple to the three weak gauge bosons.

SU(2): The group of two-dimensional unitary matrices with determinant one, whose representations describe spin-like symmetry multiplets. Weak isospin labels the SU(2) representations used by the weak gauge symmetry.
Weak isospin multiplet: A collection of particle fields transforming together as one irreducible representation of weak SU(2). The representation dimension specifies how many components share a weak-isospin assignment.
Standard Model: The quantum field theory describing known elementary particles and three of their fundamental interactions. Its electroweak sector assigns weak isospin to fermions and gauge fields.
Strong isospin: An approximate SU(2) flavor symmetry relating up and down quarks in strong-interaction physics. It is a distinct symmetry from the gauged weak SU(2), despite the shared name.
Gauge symmetry: A symmetry of a field description whose transformations can vary independently across spacetime. Weak isospin arises from a local SU(2) gauge symmetry.
Weak hypercharge: The U(1) gauge quantum number that combines with weak isospin to determine electric charge. It supplies the second charge needed alongside weak isospin in the electroweak theory.
Lepton doublet: A left-handed pair of lepton fields, such as the electron neutrino and electron, forming an SU(2) doublet. It is a direct example of particles grouped by weak isospin.
Spin: Intrinsic angular momentum carried by quantum particles, measured in units of Planck’s constant. Weak isospin is an internal symmetry label, not ordinary angular momentum.
Isospin: An approximate symmetry quantum number that groups protons and neutrons, or related hadrons, into multiplets. Its name echoes weak isospin, but ordinary isospin is an approximate strong-interaction symmetry.
Electric charge: The conserved quantum number that determines a particle’s electromagnetic coupling. Electric charge follows from the third weak-isospin component and weak hypercharge.

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