is isomorphic to that of SO(4). The unbroken subgroup, realized in the linear Wigner–Weyl mode, is which is locally isomorphic to SU(2) (V: isospin). To construct a non-linear realization of SO(4), the representation describing four-dimensional rotations of a vector for an infinitesimal rotation parametrized by six angles is given by where The four real quantities define the smallest nontrivial chiral multiplet and represent the field content of the linear sigma model. To switch from the above linear realization of SO(4) to the nonlinear one, we observe that, in fact, only three of the four components of are independent with respect to four-dimensional rotations. These three independent components correspond to coordinates on a hypersphere S3, where and are subjected to the constraint with F a (pion decay) constant of dimension mass. Utilizing this to eliminate yields the following transformation properties of under SO(4), The nonlinear terms (shifting ) on the right-hand side of the second equation underlie the nonlinear realization of SO(4). The chiral group is realized nonlinearly on the triplet of pions— which, however, still transform linearly under isospin rotations parametrized through the angles By contrast, the represent the nonlinear "shifts" (spontaneous breaking). Through the spinor map, these four-dimensional rotations of can also be conveniently written using 2×2 matrix notation by introducing the unitary matrix and requiring the transformation properties of U under chiral rotations to be where The transition to the nonlinear realization follows, where denotes the trace in the flavor space. This is a non-linear sigma model. Terms involving or are not independent and can be brought to this form through partial integration. The constant F2/4 is chosen in such a way that the Lagrangian matches the usual free term for massless scalar fields when written in terms of the pions, Alternate Parametrization An alternative, equivalent (Gürsey, 1960), parameterization yields a simpler expression for U, Note the reparameterized transform under so, then, manifestly identically to the above under isorotations, ; and similarly to the above, as under the broken symmetries, , the shifts. This simpler expression generalizes readily (Cronin, 1967) to light quarks, so References ; Georgi, H. (1984, 2009). Weak Interactions and Modern Particle Theory (Dover Books on Physics) online . Quantum field theory Quantum chromodynamics Nuclear physics