Why do we care about complex inner products and what do they mean?

Question

If I have two vectors from $\mathbb{R}^q$ then their inner product gives the length of the projection of one on them onto the other multiplied by the other's length. I have searched but couldn't find an intuitive analoge interpretation for the complex inner product. And why do we need complex inner products?

(I'm learning from the book Linear Algebra Done Right of Axler, and they don't really give a motivation for it either, only for the real case..)

Answer 1

It's needed in quantum mechanics, for measures of overlap between physical states. For example,

• $P(a\to b)=(a,b)(b,a)/(a,a)(b,b)$ is the Born rule for the probability of transition between two states $a\to b$, and
• $\mathrm{e}^{2\mathrm{i}\phi(a\to b\to c \to a)}=\frac{(a,c)(c,b)(b,a)}{(a,b)(b,c)(c,a)}$ is a geometric phase acquired during a transition $a\to b\to c \to a$.