Physical interpretation of noncommutative algebraic varieties. (English) Zbl 1377.14004

I cite the abstract of the paper: “The theory of algebraic varieties gives an algebraic interpretation of differential geometry, thus of our physical world. To treat, among other physical properties, the theory of entanglement, we need to generalize the space parametrizing the objects of physics. We do this by introducing noncommutative varieties”.
The paper under review (3 pages long) contains many mistakes, I’m unsure if these are the mistakes of the author or the publisher. The author defines a “matrix polynomial algebra” \(R\), the non-commutative spectrum of a “matrix polynomial algebra” \(R\) and the non-commutative structure sheaf of \(R\). The points of \(\mathrm{Spec}(R)\) are the set of simple left \(R\)-modules of \(R\). The author gives some elementary examples and makes many claims without supplying any proofs. In the second chapter of the paper “The dynamics in non-commutative affine varieties” the author introduce the 1-radical of a matrix polynomial algebra \(R\) and the tangent space. He gives some examples and also calculates the tangent space of a free matrix polynomial algebra. The author ends the paper with a short section called “Entanglement: Blowing up a singularity” where the author studies a non-commutative version of the \(x^4+y^3\) singularity. The reviewer wants to encourage the author to write a more detailed paper on the subject where he includes all definitions and proofs – a self-contained paper. Send it to a journal where the refereeing is more thorough.


14A22 Noncommutative algebraic geometry
00A79 Physics
Full Text: Euclid Link