A Reformulation of von Neumann’s Postulates on Quantum Measurement by Using Two Theorems in Clifford Algebra

摘要

According to a procedure previously introduced from Y. Ilamed and N. Salingaros, we start giving proof of two existing Clifford algebras, the Si that has isomorphism with that one of Pauli matrices and the N _i,±1 where N _i stands for the dihedral Clifford algebra. The salient feature is that we show that the N _i,±1 may be obtained from the Si algebra when we attribute a numerical value (+1 or −1) to one of the basic elements (e ₁,e ₂,e ₃) of the Si. We utilize such result to advance a criterium under which the Si algebra has as counterpart the description of quantum systems that in standard quantum mechanics are considered in absence of observation and quantum measurement while the N _i,±1 attend when a quantum measurement is performed on such system with advent of wave function collapse. The physical content of the criterium is that the quantum measurement with wave function collapse induces the passage in the considered quantum system from the Si to N _i,+1 or to the N _i,−1 algebras, where each algebra has of course its proper rules of commutation. After a proper discussion on the difference between decoherence and wave function collapse, we re-examine the von Neumann postulate on quantum measurement, and we give a proper justification of such postulate by using the Si algebra. Soon after we study some applications of the above mentioned criterium to some cases of interest in standard quantum mechanics, analyzing in particular a two state quantum system, the case of time dependent interaction of such system with a measuring apparatus and finally the case of a quantum system plus measuring apparatus developed at the order n=4 of the considered Clifford algebras and of the corresponding density matrix in standard quantum mechanics. In each of such cases examined, we find that the passage from the algebra Si to N _i,±1, considered during the quantum measurement of the system, actually describes the collapse of the wave function. Therefore we conclude that the actual quantum measurement has as counterpart in the Clifford algebraic description, the passage from the Si to the N _i,±1 Clifford algebras, reaching in this manner the objective to reformulate von Neumann postulate on quantum measurement and proposing a self-consistent formulation of quantum theory.