Addendum to “Quantum theory as a trace algebra”

Per‐Olov Löwdin

Abhijeet Alase, Salini Karuvade, Carlo Maria Scandolo

Abstract This document is our reply to the Comment (Miloslav Znojil 2023 J. Phys. A: Math. Theor. 56 038001) on our recent work titled ‘The operational foundations of PT-symmetric and quasi-Hermitian quantum theory’. The original Comment consists of three addenda to our work. The first addendum claims that our work is ill-motivated as the motivating question, namely whether PT-symmetric quantum theory extends the standard quantum theory, was already answered in the literature. The second addendum points to some missing references in our work, and the third addendum suggests what constraints could lead to an extension of standard quantum theory. In our reply, we explain that the claim in the first addendum is a result of a misinterpretation of our motivating question. When interpreted correctly, the third addendum in the Comment in itself elaborates on why our motivating question is interesting and relevant. We also briefly comment on the prospects of an extension of standard quantum theory along the lines suggested in the third addendum. As our response to the second addendum, we explain our rationale behind citing certain references while leaving out others.

Stephen L. Adler

We show that the recently proposed matrix model for M theory obeys the cyclic trace assumptions underlying generalized quantum or trace dynamics. This permits a verification of supersymmetry as an operator calculation, and a calculation of the supercharge density algebra by using the generalized Poisson bracket, in a basis-independent manner that makes no reference to individual matrix elements. Implications for quantization of the model are discussed. Our results are a special case of a general result presented elsewhere, that all rigid supersymmetry theories can be extended to give supersymmetric trace dynamics theories, in which the supersymmetry algebra is represented by the generalized Poisson bracket of trace supercharges, constructed from fields that form a noncommutative trace class graded operator algebra.

A. A. Albert

Ronald J. Adler, David Santiago

Luca Mezincescu, Rafael I. Nepomechie

In the above reference, we prove that certain quantum spin chains with quantumalgebra symmetry are integrable. Specifically, these are the quantum-algebra-invariant open chains associated with the affine Lie algebras A (1) 1, A(2) 2n, A(2) 2n−1, B(1) n, C (1) n, and D (1)