# A doubt about a naive generalization of inner product in elementary quantum mechanics

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

• 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). – electronpusher Jun 17 at 13:17
• You mean PT stuff? – Cosmas Zachos Jun 17 at 14:59