A second course in topos quantum theory
type: Item
instance of: scholarly article
title: A second course in topos quantum theory
zbMATH Open document ID: 1398.81002
DOI: 10.1007/978-3-319-71108-9
author: Cecilia Flori
published in: Lecture Notes in Physics
publication date: 2018-01-09T00:00:00Z
review text: One of the main challenges in theoretical physics in the past 50 years has been to define a theory of quantum gravity, i.e., a theory which consistently combines general relativity and quantum theory in order to define a theory of space-time itself seen as a fluctuating field. The reason for such a difficulty is the seemingly incompatible roles of space-time put forward by general relativity and quantum theory. In fact, on the one hand, in general relativity, although the presence of both the 4-dimensional metric and the connection is assumed ab initio, they are both considered to be dynamical quantities and there is no preferred foliation of space-time. On the other hand, quantum theory assumes a fixed (with respect to its differentiable structure and metric) space-time, implied by the mathematical formalism of the theory. For example the Schroedinger's equation describing unitary evolution between measurements is \(i\hbar \frac{\partial {\psi}}{\partial t}=\hat{H}\psi\) where the notion of time is fixed, while the measurement process `reduces' the state vector via the second law of evolution: \({\psi} \rightarrow \frac{\partial {\hat{P} \psi}}{\partial \| {\hat{P} \psi} \|}\) where \({\hat{P}}\) is the projection on the outcome of the measurement. Therefore the state gets projected on a spacelike surface. Clearly, a fixed geometry of spacetime is needed to define both the time \(t\) in the Schroedinger equation and the spacelike surface on which the state vector is reduced. Therefore, although mathematically space-time is treated in an analogous way in both quantum theory and general relativity, its role in these two theories is very different. However, when defining a theory of quantum gravity, the very definition of space-time as a differentiable manifold is put into discussion. In fact, it is believed that at microscopic scales, space-time ceases to be continuous but acquires a discrete nature. Therefore, the continuum structure of space-time suggested by the t…
Mathematics Subject Classification ID: 81-02
Mathematics Subject Classification ID: 81S99
Mathematics Subject Classification ID: 81P05
Mathematics Subject Classification ID: 58A03
Mathematics Subject Classification ID: 16B50
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 83C45
Mathematics Subject Classification ID: 81P15
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 54B40
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 81R15
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 00A79
zbMATH DE Number: 6823466
zbMATH Keywords: quantum theory
zbMATH Keywords: category theory
zbMATH Keywords: topos quantum theory
zbMATH Keywords: presheaves and sheaves
zbMATH Keywords: problems of quantization
zbMATH Keywords: quantum gravity
reviewed by: Stepan Moskaliuk
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-71108-9
OpenAlex ID: W2792436936
description: scientific article; zbMATH DE number 6823466