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arXiv:1606.05086

Process-theoretic characterisation of the Hermitian adjoint

John H. Selby, Bob Coecke

Published Jun 16, 2016• 2 citations
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Abstract

We show that the physical principle "the adjoint associates to each state a `test' for that state" fully characterises the Hermitian adjoint for pure quantum theory, therefore providing the adjoint with operational meaning beyond its standard mathematical definition. Also, we show that for general process theories, which all admit a diagrammatic representation, this physical principle induces a reflection operation.

DOI
10.48550/arxiv.1606.05086

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We show that the physical principle "the adjoint associates to each state a `test' for that state" fully characterises the Hermitian adjoint for pure quantum theory, therefore providing the adjoint with operational meaning beyond its standard mathematical definition. Also, we show that for general process theories, which all admit a diagrammatic representation, this physical principle induces a reflection operation.
Abstract The object of this paper is to reformulate the principles of the quantum theory as a sequence of propositions which shall be either summary statements of standard experimental procedure or hypotheses concerning the results of experiment and having an immediate physical interpretation. It is shown that the standard process in micro-physics is a generalised spectral analysis, whose properties are simply expressible in symbolic form by means of projective or “idempotent” operators (Einzeloperatoren). It appears that only two hypotheses need be made and that these relate to the existence and properties of transition probabilities. From these fundamental principles, which have a direct physical significance, it is possible to deduce the subsidiary principles which form the accepted basis of the mathematical analysis of the quantum theory and which deal with the representation of quantum states and physical quantities by vectors and linear operators respectively. In this paper the emphasis is laid on the experimental process determining a state of a system and on the associated operators rather than on the state itself or the vector representing it in the system space. Projective operators, which represent actual processes of measurement, and unitary operators, which represent actual transformations of systems of measurement, are given priority over the (statistical) operators which represent physical variables. This method of representation makes the physical meaning of the theory fundamental, instead of leaving it to be extracted from a purely mathematical system of non-commutative algebra or differential equations.
We are focused on the idea that observables in quantum physics are a bit more then just hermitian operators and that this is, in general, a “tricky business”. The origin of this idea comes from the fact that there is a subtle difference between symmetric, hermitian, and self-adjoint operators which are of immense importance in formulating Quantum Mechanics. The theory of self-adjoint extensions is presented through several physical examples and some emphasis is given on the physical implications and applications.
In arXiv:0709.0483 Gunther and Samsonov outline a ``generalization'' of quantum mechanics that involves simultaneous consideration of Hermitian and non-Hermitian operators and promises to be ``capable to produce effects beyond those of standard Hermitian quantum mechanics.'' We give a simple physical interpretation of Hermiticity and discuss in detail the shortcomings of the above-mentioned composite quantum theory. In particular, we show that the corresponding ``generalization of measurement theory'' suffers from a dynamical inconsistency and that it is by no means adequate to replace the standard measurement theory.