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Operational axioms for C ∗ -algebra representation of transformations 1

A. Volta

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

It is shown how a C ∗ -algebra representation of the transformations of a physic al sys- tem can be derived from two operational postulates: 1) the existence of dynamically independent systems; 2) the existence of symmetric faithful states. Both notions are crucial for the possibility of performing experiments on the system, in preventing remote instantaneous influences and in al- lowing calibration of apparatuses. The case of Quantum Mechanics is thoroughly analyzed. The possibility that other no-signaling theories admit a C ∗ -algebra formulation is discussed.

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It is shown how a C ∗ -algebra representation of the transformations of a physic al sys- tem can be derived from two operational postulates: 1) the existence of dynamically independent systems; 2) the existence of symmetric faithful states. Both notions are crucial for the possibility of performing experiments on the system, in preventing remote instantaneous influences and in al- lowing calibration of apparatuses. The case of Quantum Mechanics is thoroughly analyzed. The possibility that other no-signaling theories admit a C ∗ -algebra formulation is discussed.
It is shown how a C*-algebra representation of the transformations of a physical system can be derived from two operational postulates: 1) the existence of dynamically independent systems}; 2) the existence of symmetric faithful states. Both notions are crucial for the possibility of performing experiments on the system, in preventing remote instantaneous influences and in allowing calibration of apparatuses. The case of Quantum Mechanics is thoroughly analyzed. The possibility that other no-signaling theories admit a C*-algebra formulation is discussed.
It is shown how a C*‐algebra representation of the transformations of a physical system can be derived from two operational postulates: 1) the existence of dynamically independent systems; 2) the existence of symmetric faithful states. Both notions are crucial for the possibility of performing experiments on the system, in preventing remote instantaneous influences and in allowing calibration of apparatuses. The case of Quantum Mechanics is thoroughly analyzed. The possibility that other no‐signaling theories admit a C*‐algebra formulation is discussed.
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 Ref. [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 Ref. [1], 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. The existence of such composition‐preserving extension among possible extensions is analyzed.
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.