isospinxyz

◼ isospinxyz[c, i] returns the x, y, and z components of the I=1/2 isospin operator for the level described by the operator c. The phase is given by (((-1))^i).
◼ isospinx[c, i] returns the x component of the isospin operator.
◼ isospiny[c, i] returns the y component of the isospin operator.
◼ isospinz[c, i] returns the z component of the isospin operator.
◼ isospinplus[c, i] returns the + component of the isospin operator, \(TraditionalForm\`\(I\_+\) = I\_x\)+i (Subscript[I, y]).
◼ isospinminus[c, i] returns the - component of the isospin operator, \(TraditionalForm\`\(I\_-\) = I\_x\)-i (Subscript[I, y]).

◼ Operator c must be expressed in the stripped form, i.e. a form which excludes the first index (creation/annihilation) and the last index (spin).

◼ The default value of index i is 0.

◼ See also: spinxyz, nambu, manyisospin.

Further examples

Load sneg and declare c to be a fermionic operator

<< sneg`sneg`;
snegfermionoperators[c];

Check definition, phase index i=0

{(Subscript[I, x]), (Subscript[I, y]), (Subscript[I, z])} = isospinxyz[c[], 0]
komutator[(Subscript[I, x]), (Subscript[I, y])] == I (Subscript[I, z]) // 
      ExpandAll
komutator[(Subscript[I, y]), (Subscript[I, z])] == I (Subscript[I, x]) // 
      ExpandAll
komutator[(Subscript[I, z]), (Subscript[I, x])] == I (Subscript[I, y]) // ExpandAll

Check definition, phase index i=1

{(Subscript[I, x]), (Subscript[I, y]), (Subscript[I, z])} = isospinxyz[c[], 1]
komutator[(Subscript[I, x]), (Subscript[I, y])] == I (Subscript[I, z]) // 
      ExpandAll
komutator[(Subscript[I, y]), (Subscript[I, z])] == I (Subscript[I, x]) // 
      ExpandAll
komutator[(Subscript[I, z]), (Subscript[I, x])] == I (Subscript[I, y]) // ExpandAll

Total isospin operator is related to the total spin operator (see spinss).

II = 
      pow[(Subscript[I, x]), 2] + pow[(Subscript[I, y]), 2] + pow[(Subscript[I, z]), 2]
II == 3/4 - spinss[c[]] // ExpandAll

Verify some equalities

isospinplus[c[]] == isospinx[c[]] + I isospiny[c[]] // 
      ExpandAll
isospinminus[c[]] == isospinx[c[]] - I isospiny[c[]] // 
      ExpandAll

Isospin operators commute with the spin operators

{(Subscript[S, x]), (Subscript[S, y]), (Subscript[S, z])} = spinxyz[c[]]
Outer[komutator, {(Subscript[S, x]), (Subscript[S, y]), (Subscript[S, z])}, {(Subscript[I, x]), (Subscript[I, y]), (Subscript[I, z])}]