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Fundamental theories of physics

Interpreting Quantum Mechanics

Gerard ’t Hooft

Published Jan 1, 2016• 0 citations
Publisher

Abstract

The world of atoms and molecules is described by exactly the same expressions as the determinisric world described in the previous chapter, yet here, a world consisting of ‘real things’, seems to be missing. As soon as one makes some very basic assumptions about locality and causality, it seems to be possible to prove that such a world cannot exist at all. Have other assumptions been made? Can deterministic theories exist that generate strong correlations in space-like directions? What does free will mean? These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-319-41285-6_3

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The world of atoms and molecules is described by exactly the same expressions as the determinisric world described in the previous chapter, yet here, a world consisting of ‘real things’, seems to be missing. As soon as one makes some very basic assumptions about locality and causality, it seems to be possible to prove that such a world cannot exist at all. Have other assumptions been made? Can deterministic theories exist that generate strong correlations in space-like directions? What does free will mean? These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
An outline is given of the interpretation of quantum mechanical expressions if one assumes that what is called Quantum Mechanics today, refers to an underlying reality as in Chap. 2 , even if many authors claim to have evidence to the contrary, as is usually brought forward. If there exists a sub-microscopic classical world of real things, is that the same classical world as the world of large things such as planets, rocks and people? How does the notion of probability enter into quantum mechanics? These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The conclusion of Part I, the non-technical (or somewhat less technical) part of this book, is that a completely convincing model of our universe is lacking. This should be a model that shows how quantum mechanical behaviour, with all its mysteries and paradoxes, can emerge from a local, deterministic theory. Yet we also find that one can assume its existence to arrive at a picture of our quantum mechanical universe that makes perfect sense. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The usual ‘proofs’ that deterministic hidden variable models for quantum mechanics cannot be completely local, are based on rather indirect arguments, using thought experiments, free will, and the like. In this chapter we show how difficulties emerge when we wish to construct mathematical models. We also show how, in principle, the problem can be addressed at the level of model building. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Can we guess the shape of theories yet to come? History dampens our expectations, with quantum mechanics itself throwing the coldest water. But indulging in some harmless speculations can be entertaining and may help clarify some issues. Before the free-associating begins, however, there are some needed preliminaries, which will make this chapter a bit of a hodge-podge. First, a tighter focus on material from previous chapters about determinism and locality will be useful. Then we require a few additional points about quantum mechanics which did not fit conveniently into the historical narrative. Finally we come to the shameless speculating. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The measurement problem of quantum mechanics was probably born in 1926: ‘Thus Schrödinger’s quantum mechanics gives a very definite answer to the question of the outcome of a collision; however, this does not involve any causal relationship. One obtains no answer to the question “what is the state after the collision,” but only to the question “how probable is a specific outcome of the collision” (in which the quantum-mechanical law of [conservation of] energy must of course be satisfied). This raises the entire problem of determinism. From the standpoint of our quantum mechanics, there is no quantity that could causally establish the outcome of a collision in each individual case; however, so far we are not aware of any experimental clue to the effect that there are internal properties of atoms that enforce some particular outcome. Should we hope to discover such properties that determine individual outcomes later (perhaps phases of the internal atomic motions)? Or should we believe that the agreement between theory and experiment concerning our inability to give conditions for a causal course of events is some pre-established harmony that is based on the non-existence of such conditions? I myself tend to relinquish determinism in the atomic world. But this is [also] a philosophical question, for which physical arguments alone are not decisive.’ (Born, 1926a, p. 866; translation by the author) These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In this chapter we introduce the Hilbert space which plays the central role in quantum mechanics. For a more detailed discussion of this subject we refer to the books of Balakrishnan [5], Richtmyer [137], Sewell [146], Stakgold [152], Steeb [163], Weidmann [181], Yosida [185]. Moreover the proofs of the theorems given in this chapter can be found in these books. We assume that the reader is familiar with the notation of a linear space. First we introduce the pre-Hilbert space. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.