Quantum Quandaries: A Category-Theoretic Perspective

Abstract

General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n − 1)-dimensional manifolds representing 'space' and whose morphisms are n-dimensional cobordisms representing 'spacetime'. Quantum theory makes heavy use of the category Hilb, whose objects are Hilbert spaces used to describe 'states', and whose morphisms are bounded linear operators used to describe 'processes'. Moreover, the categories nCob and Hilb resemble each other far more than either resembles Set, the category whose objects are sets and whose morphisms are functions. In particular, both Hilb and nCob but not Set are * -categories with a noncartesian monoidal structure. We show how this accounts for many of the famously puzzling features of quantum theory: the failure of local realism, the impossibility of duplicating quantum information, and so on. We argue that these features only seem puzzling when we try to treat Hilb as analogous to Set rather than nCob, so that quantum theory will make more sense when regarded as part of a theory of spacetime.

Axiomatic Method and Category Theory

“…6 -9). Unlike Curry Howard did not stress the philosophical motivation and the foundational significance of this result but formulated it in terms of structural 10 For more detailed historical accounts see [31], [224]. 11 In the above quote Curry uses the term "category" interchangeably with the the term "type" -and further in his book he uses the former term more often than the latter.…”

Scientific Explanation and Scientific Structuralism

“…needs to be explained causally is rather the loss of coherence, or the non-entangled nature of the macroscopic world. 13 Secondly, also in this case study there exists another, more abstract and general way to explain non-locality structurally or formally vis-à-vis classical separability, one that has been illustrated by John Baez (2006), and that hinges on the formal difference between the properties of the tensor product in the category "HILBERT SPACES" and the Cartesian product in the category "SET". In this explanatory approach, involving category theory, the classical intuition that a joint system can be accurately described by specifying the states of its parts corresponds to, or is denoted by, the mathematical properties of the Cartesian product: if the set of states of the first system is S and that of the second is T, the joint system has the Cartesian product S x T as its formal counterpart, where S x T is the set of all ordered pairs (s,…”

A categorial approach to relativistic locality

“…Topological quantum field theory is motivated by the difficulties of the path integral formalism, and it circumvents path integrals by specifying functors embodying crucial features of time evolutions. The work by Bartlett [26] gives an overview of the main ideas of topological quantum field theory, including some of its history; Baez [27] provides a compact introduction. The categorial approach generalizing the algebraic axiomatization [28], [29], [30] was initiated by Brunetti, Fredenhagen and Verch [1].…”