# Quantum observables in nonstandard Hilbert space

Consider a Hermitian $$(n \times n)$$-matrix $$A$$, and a Hilbert space $$\mathbb{C}^n$$, foreseen with a nonstandard inner product. (An inner product $$s(\cdot,\cdot)$$ is standard if for any two vectors $$x = (x_1,\ldots,x_n)$$ and $$y = (y_1,\ldots,y_n)$$ we have that $$s(x,y) = x_1y_1 + \cdots + x_ny_n$$.)

I understand that it is important in Quantum Theory that $$A$$ has an orthogonal base of eigenvectors, and relative to the standard inproduct, this is indeed the case.

But what about a nonstandard inproduct ? Is this property also true, or does one only work with standard inproducts in Quantum Theory ?

• Are you asking about equivalents to PT QM? Mar 4, 2020 at 19:50
• This review is popular. Mar 4, 2020 at 20:11