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Theory of large quantal systems and the quantum theory of measurement

T. F. Hill

Published Jan 1, 1974• 0 citations
OpenAlex

Abstract

The consistency problem of quantum mechanics may be resolved by regarding any measurement performed on an atomic system as essentially involving an interaction between the atomic system and a macroscopic measuring apparatus and the subsequent behaviour of the apparatus in producing a signal to indicate the result of the measurement. Of central importance in such an approach, therefore, is the study of large quantal systems. Along these lines, two approaches to the consistency problem have been provided; one, by the Italian group of Daneri, Loinger and Prosperi, based on a time-independent ergodic theory of large quantal systems; the other by the Belgian group of George, Prigogine and Rosenfeld, based on a time-dependent theory. In this thesis these two theories are compared and are shown to be equivalent.

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The consistency problem of quantum mechanics may be resolved by regarding any measurement performed on an atomic system as essentially involving an interaction between the atomic system and a macroscopic measuring apparatus and the subsequent behaviour of the apparatus in producing a signal to indicate the result of the measurement. Of central importance in such an approach, therefore, is the study of large quantal systems. Along these lines, two approaches to the consistency problem have been provided; one, by the Italian group of Daneri, Loinger and Prosperi, based on a time-independent ergodic theory of large quantal systems; the other by the Belgian group of George, Prigogine and Rosenfeld, based on a time-dependent theory. In this thesis these two theories are compared and are shown to be equivalent.
Two methods for the description of physical phenomena, microscopic and macroscopic, are necessary to understand the physical world. We now have two examples. The first one is mechanical, whether classical or quantal, vs thermodynamic treatments, which are characterized by reversibility and irreversibility, respectively. Another one is quantal vs classical theories, which have different forms of logic and lead to various paradoxes in measurement processes. To understand these contradicting aspects, it should be noted that in both cases, chaos and measurement are involved inevitably, and averages over small regions of space and time for measurement are required. In this way, chaos mediates between the two methods of description. We have already made an attempt to treat the former example in terms of ergodicity. In this paper the latter example will be treated with the theory of quantum measurement and the EPR paradox in mind.
I start from the observation that the usual treatment of the object and the apparatus as independent systems in discussions of the quantum measurement process is not obviously consistent with a field theoretic picture. In such connection I consider a non relativistic second quantization model in which the role of the object is played by a particle out of the sea of the other particles in a macroscopic equilibrium state of the global system and the role of the apparatus index by the number of ions produced in certain given regions. In the framework of a formalism developed in preceding papers and in connection with the problem of decoherence, I also consider a specific, even if rather artificial, prescription to introduce a formal continuous monitoring of the particle distribution functions which would give an objective character to the macroscopic variables.
The main theme of this book is the idea that quantum mechanics is valid not only for microscopic objects but also for the macroscopic apparatus used for quantum mechanical measurements. The author demonstrates the intimate relations that exist between quantum mechanics and its interpretation which are induced by the quantum mechanical measurement process. Consequently, the book is concerned both with the philosophical, metatheoretical problems of interpretations and with the more formal problems of quantum object theory. The consequences of this approach turn out to be partly very promising and partly rather disappointing. On the one hand, it is possible to give a rigorous justification of some important parts of interpretation, such as probability, by means of object theory. On the other hand, the problem of the objectification of measurement results leads to inconsistencies which cannot be resolved in an obvious way. This open problem has far-reaching consequences for the possibility of recognising an objective reality in physics.
The standard theory of measurements in quantum mechanics is reviewed with special emphasis on the conceptual and epistemological implications. It is concluded that the standard theory remains the only one which is compatible with present quantum mechanics. Hence, if one wants to avoid the conclusion that quantum mechanics only gives probability connections between subsequent observations, the quantum-mechanical equations would have to be modified. Particular attention is paid to the case that the measuring apparatus is macroscopic and its state vector not accurately known before the measurement.