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This advanced course, a sequel to the first volume of this lecture
series on topos quantum theory, delves deeper into the theory,
addressing further technical aspects and recent advances. These
include, but are not limited to, the development of physical
quantities and self-adjoint operators; insights into the
quantization process; the description of an alternative, covariant
version of topos quantum theory; and last but not least, the
development of a new concept of spacetime. The book builds on the
concepts introduced in the first volume (published as Lect. Notes
Phys. 868), which presents the main building blocks of the theory
and how it could provide solutions to interpretational problems in
quantum theory, such as: What are the main conceptual issues in
quantum theory? And how can these issues be solved within a new
theoretical framework of quantum theory? These two volumes together
provide a complete, basic course on topos quantum theory, offering
a set of mathematical tools to readers interested in tackling
fundamental issues in quantum theory in general, and in quantum
gravity in particular. From the reviews of the first volume: 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
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.
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