A First Course in Topos Quantum Theory (Lecture Notes in Physics, 868) 2013th Edition by Cecilia Flori | (PDF) Free Download

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In the last five decades various attempts to formulate theories of quantum gravity have been made, but none has fully succeeded in becoming the quantum theory of gravity. One possible explanation for this failure might be the unresolved fundamental issues in quantum theory as it stands now. Indeed, most approaches to quantum gravity adopt standard quantum theory as their starting point, with the hope that the theory’s unresolved issues will get solved along the way. However, these fundamental issues may need to be solved before attempting to define a quantum theory of gravity. The present text adopts this point of view, addressing the following basic questions: What are the main conceptual issues in quantum theory? How can these issues be solved within a new theoretical framework of quantum theory? A possible way to overcome critical issues in present-day quantum physics – such as a priori assumptions about space and time that are not compatible with a theory of quantum gravity, and the impossibility of talking about systems without reference to an external observer – is through a reformulation of quantum theory in terms of a different mathematical framework called topos theory. This course-tested primer sets out to explain to graduate students and newcomers to the field alike, the reasons for choosing topos theory to resolve the above-mentioned issues and how it brings quantum physics back to looking more like a “neo-realist” classical physics theory again.

User’s Reviews

Editorial Reviews: Review From the reviews:“This text presents a reformulation of quantum theory in terms of modern mathematical framework called topos theory. … This is one of the first text to address all of these aspects of topos quantum theory at the graduate student level.” (S. Moskaliuk, zbMATH, Vol. 1280, 2014)“The book is self-contained and can be used as a textbook or self-study manual teaching the usage of category theory and topos theory in particular in theoretical physics or in investigating the foundations of quantum theory in mathematically rigorous terms. … the book is a very welcome contribution.” (Frank Antonsen, Mathematical Reviews, December, 2013) From the Back Cover In the last five decades various attempts to formulate theories of quantum gravity have been made, but none has fully succeeded in becoming the quantum theory of gravity. One possible explanation for this failure might be the unresolved fundamental issues in quantum theory as it stands now. Indeed, most approaches to quantum gravity adopt standard quantum theory as their starting point, with the hope that the theory’s unresolved issues will get solved along the way. However, these fundamental issues may need to be solved before attempting to define a quantum theory of gravity. The present text adopts this point of view, addressing the following basic questions: What are the main conceptual issues in quantum theory? How can these issues be solved within a new theoretical framework of quantum theory? A possible way to overcome critical issues in present-day quantum physics – such as a priori assumptions about space and time that are not compatible with a theory of quantum gravity, and the impossibility of talking about systems without reference to an external observer – is through a reformulation of quantum theory in terms of a different mathematical framework called topos theory. This course-tested primer sets out to explain to graduate students and newcomers to the field alike, the reasons for choosing topos theory to resolve the above-mentioned issues and how it brings quantum physics back to looking more like a “neo-realist” classical physics theory again.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐It is probably accurate to say that contemporary physicists are not shy about using complex or esoteric mathematics. As evidence one can cite not only this book but also the extreme mathematical adventures taken in quantum field theory, quantum information theory, and superstring theory. Anyone who has approached these subjects will appreciate the enormous mathematical background needed for their understanding but may also express skepticism as to the actual payoff in using this level of mathematics. Will its use be comparable to swatting a fly with a sledgehammer?Topos theory, which certainly qualifies as being esoteric, has found many applications in mathematics, especially in areas such as algebraic topology and algebraic geometry. This book attempts to apply it to quantum theory, but not just as another formalism for expressing it. Rather, the goal of the work is to express quantum theory in a topos-theoretic framework in order make it what the author calls a non-realist theory. A non-realist theory is to be contrasted with a realist theory, where in the latter propositions (in the quantum theory) form a Boolean algebra and are either true or false. A topos quantum theory would thus allow propositions to have multi-valued truth conditions.But one could perhaps impose such a multi-valued capability on the propositions of quantum theory arbitrarily, without using a complex mathematical formalism such as topos theory. Since it is arbitrary, such an imposition might be viewed as unnatural by some physicists. On the other hand, using topos theory may be viewed as a sledgehammer by others. Individuals who have ample time and the appropriate mathematical background may go ahead and commit to a study of this book, whereas others who do not may refrain from doing so.If one does make the decision to study this book it is worth the time and effort to do so, even though the text gets rather heavy going in places, and this is exacerbated sometimes by use of awkward notation.Some of the more helpful/interesting discussions in the book include:- The motivation given early on for why topos theory is necessary for viewing a quantum object as a collection of classical approximations, the connection with sheaf theory, and how to choose the correct topos formulation. Along these same lines the author also points out the topos formulation can contain distributive logic naturally. At this point in the book however it is still unclear as to whether it is really worth the effort to use topos theory (which is very abstract/esoteric) versus some other approach which might be a lot less complicated. The philosopher reader may object to the criteria used in the introduction to characterize a classical theory as a realist theory and choose not to read further. However, the next chapter (albeit short) is devoted entirely to philosophical musings on physical theories, which may affect the decision to continue with the study of the book.- The formal but understandable discussion on limits and colimits in category theory, which despite the short length of the chapter on category theory, is actually very insightful and the connection with universal properties and universal constructions is made, as well as the notion of cone and co-cone. The discussion of the important notion of pullback in a category theory is somewhat mirky because the author brings in an object D without explaining its explicit origin. The same problem occurs in the discussion of the notion of a pushout. For readers who are finding difficulty in accepting any significant difference between category theory and ordinary set theory, the author carefully differentiates between the categorical definition of a sub-object and the definition of a subset in set theory, this being done at the conceptual level. However, she makes the point that from an operational point of view, this distinction does not hold, and then gives an elementary example.- The author gives an illuminating discussion of sub-object classifiers in a general topos, in particular its ability to go beyond merely classical binary logic into one that is more intuitionistic in nature, i.e. the truth values can vary between zero and one. She gives a helpful example that illustrates the identification of the truth value of a proposition by the degree to which an original proposition must be generalized so that the new one is true. This example motivates well the need for defining a ‘distance from the truth’ in a general topos.- Yoneda’s Lemma is sometimes viewed as either excessively abstract by newcomers to category theory or its importance is not really appreciated. In this book the author relates its use to the ability to identify the elements of a particular topos as sieves. In other words, one can view Yoneda’s Lemma as being a resource for the enablement of multivalued logic, and not just imposing this type of logic arbitrarily.- The lack of presheaves to have global sections is related to giving a topos analogue of the Kochen-Specker theorem. This sheaf-theoretic framework may alleviate the skepticism of some readers as to the non-constructive nature of the standard proof of the Kochen-Specker theorem.- Interestingly, quantum theory in the context of topos theory in this book is viewed as ‘locally classical’, in the sense of being a collection of local classical approximations expressed in terms of abelian von Neumann sub-algebras which form what is called a ‘spectral presheaf.’ It is by the spectral presheaf that connection to the Kochen-Specker theorem is made Strategies from category theory are used to retain quantum information resources in this topos formulation (e.g. the categorical constructions will be aware of the operator non-commutativity that is typical in the quantum theory).- It is the spectral presheaf that gives the connection with propositions, which in the ordinary quantum formulation are given by projection operators. In the topos formulation propositions are represented as sub-objects of the spectral presheaf by a process that the author calls ‘daseinisation.’ This is basically an approximation or ‘coarse-graining’ strategy for ensuring that operators will be part of any given context. Due mostly to the notation used, the discussion of daseinization is difficult to follow, and the reviewer found a perusal of the references was necessary to gain better insight into this procedure. This occurred also for the discussion on the topos analogue of states.

⭐wow…the external observer resurfaces…this book will give you such a good headache. Topos is here to stay!

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