A elementary study of Quantum Mechanics, following $[1]$, yields in the realization that the basic algebraic structure are the complex vector spaces $\mathbb{C}^{n}$. Then a contravariant vector (using the terminology of Differential Geometry/General Relativity) is simply defined as an element of $\mathbb{C}^{n}$ ;

$$\mid x \rangle \in \mathbb{C}^{n} \tag{1}$$

In $(1)$, these vectors are called Ket.

Futhermore, concerning the vector space $\mathcal{L}(V,U)$ there's a particular subspace called Dual vector space $\mathcal{L}(V,\mathbb{K}) \equiv V^{*}$ which is the Vector Space of the Covariant Vectors (Linear Functionals). So, for the quantum mechanics, you can define:

$$\begin{array}{rl} \langle y \mid:\mathbb{C}^{n} &\to \mathbb{C} \\ \mid x \rangle &\mapsto \langle y \mid (\mid x \rangle) \tag{2} \end{array}$$

In $(2)$, these vectors are called Bra.

And then, roughly speaking, by Riesz Representation Theorem $[2]$, we can say that the inner product $\langle y \mid x \rangle$ is well defined as:

$$\langle y \mid x \rangle =: \sum^{n}_{i=1} y^{*}_{i}x_{i} \tag{3}$$

Concerning the $(3)$, can we generalize to something like, $\langle y \mid x \rangle =: \sum^{n}_{i=1} g_{ij}y^{*i}x^{j}$ ? Where $g_{ij}$ is the metric tensor.

$$ * * * $$

$[1]$ NAKAHARA.M.; Quantum Computing: From Linear Algebra to Physical Realizations, CRC Press, 2008.

$[2]$ https://en.wikipedia.org/wiki/Riesz_representation_theorem

  • $\begingroup$ I haven't had mathematical training quite at this level, but was under under the impression one can define the inner product in a variety of ways. Perhaps you could consider the fundamental properties of inner products and whether they apply to the inner product you propose (commutativity, distributivity, associativity, and positive definiteness). $\endgroup$ – electronpusher Jun 17 at 13:17
  • $\begingroup$ You mean PT stuff? $\endgroup$ – Cosmas Zachos Jun 17 at 14:59

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