Abstract
The great revolution of the nineteenth century started with the theory of special and general relativity and culminated in quantum theory. However, up to date, there are still some fundamental issues with quantum theory that are, yet, to be solved. Nonetheless, a great deal of effort in fundamental physics is spent on an elusive theory of quantum gravity, which is an attempt to combine the two above mentioned theories which seem, as they have been formulated, to be incompatible. In the last five decades various attempts to formulate such a theory of quantum gravity have been made, but none has fully succeeded in becoming the quantum theory of gravity. One possibility of the failure for reaching an agreement on a theory of quantum gravity might be the presence of unresolved fundamental issues already present in quantum theory, first pointed out by John Bell in (Speakable and Unspeakable in Quantum Mechanics, 2004; Toposes and Local Set Theories, 1988) (1987) and subsequently by others. Since then the scientific community has been split in two depending on whether or not the interpretation of quantum theory is regarded as problematic. In fact as a reaction to Bell’s ideas many new interpretations of quantum theory started emerging, each motivated from the belief that the conceptual framework of quantum theory was wrong or incomplete. The major examples of alternative formulations of quantum theory are: (i) many-worlds, (ii) many minds, (iii) consistent histories and (iv) hidden-variables.
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Notes
- 1.
This view is also supported by the fact that, effectively, isolated systems are idealisations and are not very frequent in practice, unless of course one is studying cosmology where the universe itself is an isolated system.
- 2.
By a ‘realist’ theory we mean one in which the following conditions are satisfied: (i) propositions form a Boolean algebra; and (ii) propositions can always be assessed to be either true or false. As it will be delineated in the following, in the topos approach to quantum theory both of these conditions are relaxed, leading to what Isham and Döring called a neo-realist theory.
- 3.
The reason is, roughly, because the internal logic of the topos used to describe quantum theory is multi-valued while the internal logic of the topos used to describe classical theory is bi-valued.
- 4.
Roughly the Kochen-Specker Theorem asserts that, if the dimension of \(\mathcal {H}\) is greater than 2, then it is not possible to assign values to all physical quantities, at the same time, in a consistent way. See Chap. 3.
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Flori, C. (2013). Introduction. In: A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol 868. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35713-8_1
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