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Foundations of Physics

A Diagrammatic Derivation of the Hermitian Adjoint

John H. Selby, Bob Coecke

Published Jun 15, 2017• 4 citations
Publisher

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. Moreover, we demonstrate that for general process theories, which all admit a diagrammatic representation, this physical principle induces a diagrammatic reflection operation.

DOI
10.1007/s10701-017-0102-7

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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. Moreover, we demonstrate that for general process theories, which all admit a diagrammatic representation, this physical principle induces a diagrammatic 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.
The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of "physical experiment" and from five simple Postulates concerning "experimental accessibility and simplicity". For the infinite dimensional case, on the other hand, a C*-algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand-Naimark-Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in version 1. The main ingredient of the axiomatization is the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the "transposed" of a physical transformation. What is new in the present paper with respect to quant-ph/0603011 is the operational deduction of an involution corresponding to the "complex-conjugation" for effects, whose extension to transformations allows to define the "adjoint" of a transformation when the extension is composition-preserving.