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I've been reading something about Quantum Mechanics where they introduce the maths slightly more rigorously. They talk about vector spaces and an inner product which yields a scalar. Moreover complex conjugation appears.

Of course I know about complex numbers, but is there a more general framework which defines more generally what a scalar means and also what conjugation (and inner product) means? Maybe some more general algebra which also satisfies some minimum axioms?

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1 Answer 1

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In the context of vector spaces, a scalar is a member of the underlying field.

An inner product is a special kind of bilinear form on a vector space over the reals or complexes. Inner products satisfy conjugate symmetry, which over the reals is just plain symmetry.

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I guess I have to look up more closely what field and vector space means. Can scalars be something else than reals or complex numbers? –  Gerenuk Feb 15 '12 at 12:37
I'll go through these, but I'm not an expert to see the general picture confidently enough and not make premature judgements. So can scalars in vector spaces be something different than real numbers or complex numbers? –  Gerenuk Feb 15 '12 at 14:58
@Gerenuk, sure. The reals are a vector space over the rationals and so are polynomials with rational coefficients and the set $\mathbb Q(\sqrt 2)=\{ a + b \sqrt 2 : a,b \in\mathbb Q\} \subset \mathbb R$. –  lhf Feb 15 '12 at 15:03
Thanks! Hmm, not sure if I understand it right. I basically wonder if the result of the inner product of two vectors can be something different than a real number or a complex number. You are saying they can? And if such algebra exists, how would you define the general conjugation neccessary for the definition or inner product? –  Gerenuk Feb 15 '12 at 15:25

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