Why are Pauli Matrices Invariant under Rotation?

[shehry1]

Homework Statement

Can anyone tell me why Pauli Matrices remain invariant under a rotation.

Homework Equations

Probably the rotation operator in the form of the exponential of a pauli matrix having an arbitrary unit vector as its input. It may also be written as:
I*Cos(x/2) - i* (pauli matrix).(unit vector) * Sin(x/2) where x is the angle of rotation.

See Sakurai 3.2.44

The Attempt at a Solution

 

[olgranpappy]

Homework Statement

Can anyone tell me why Pauli Matrices remain invariant under a rotation.

Homework Equations

Probably the rotation operator in the form of the exponential of a pauli matrix having an arbitrary unit vector as its input. It may also be written as:
I*Cos(x/2) - i* (pauli matrix).(unit vector) * Sin(x/2) where x is the angle of rotation.

See Sakurai 3.2.44

The Attempt at a Solution

Pauli Matrices are just matrices... they are just arrays of numbers. They don't rotate.

 

[Avodyne]

It's because you need to rotate both the spinor indices AND the vector index; let
U = I*Cos(x/2) - i* (pauli matrix).(unit vector) * Sin(x/2) where x is the angle of rotation, and let R_ij be the corresponding matrix that would rotate a vector by the angle x about the unit vector. Then

sigma_i = R_ij (U sigma_j U^dagger)

where j is summed and the spinor indices are suppressed.