Quantum Quandaries: a Category-Theoretic Perspective

By John C. Baez. 2004

177 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ced55545d8465f90246ec9967cbfde34d28e4fed

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. Bob Coecke
2. É. O. Paquette
3. Dusko Pavlovic
4. A. Doering
5. C. Isham
6. Ross Duncan
7. Miriam Backens

In Mathematical Physics

1. Bob Coecke
2. Dusko Pavlovic
3. É. O. Paquette
4. A. Doering
5. C. Isham
6. Ross Duncan
7. Miriam Backens

In Statistical and Nonlinear Physics

1. Dusko Pavlovic
2. É. O. Paquette
3. A. Doering
4. Bob Coecke
5. C. Isham

Citations in the corpus

• “What is a Thing?”: Topos Theory in the Foundations of Physics (2008, background): “For example, Baez’s advocation of n-categories [5, 6]; ‘categorial quantum theory’ [2, 73]; Takeuti’s theory(19) of ‘quantum sets’ [71]; and Crane’s work on categorial models of space-time [17].”
• Quantum measurements without sums (2006, methodology): “Besides this reference, the interested reader may wish to consult [1, 18, 31] for methods and proofs, and [8, 9] and also Baez’s [4] for a more leisurely introduction into †-categories.”
• Categories for the practising physicist (2009, background): “For a more elaborated discussion and technical details we refer the reader to [9, 10, 39, 62].”
• Quantum picturalism (2009, background): “Taking closed n71-dimensional manifolds as objects, n-dimensional manifolds connecting these as morphisms (1⁄4 cobordisms), the disjoint union of these manifolds as the tensor, and reversal of the manifold as the dagger, results in a dagger compact category nCob [22].”