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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 ?

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

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