[Physics] Difference between vector and pseudo-scalar

Question

In physics, a pseudo-scalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as improper rotations while a true scalar does not.

Can someone show me the graphical picture or illustration that how a scalar changes under rotations?

Why we can't say it as vector, if just the sign of the quantity changes? What is the fundamental difference between a vector and pseudo-scalar?

Why is the Klein-Gordon equation an equation of motion for a pseudo-scalar field?

Answer 1

• A scalar is invariant under rotation
• A pseudo-scalar is also invariant under a proper rotation but changes sign with parity.
• A vector is not invariant under a general rotation (only invariant under rotation around a rotational axis parallel to the vector), but rather transforms according to multiplication with a rotation matrix.