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Physics, Topology, Logic and Computation: A Rosetta Stone

Part of the Lecture Notes in Physics book series (LNP,volume 813)

Abstract

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology. Namely, a linear operator behaves very much like a “cobordism”: a manifold representing spacetime, going between two manifolds representing space. This led to a burst of work on topological quantum field theory and “quantum topology”. But this was just the beginning: similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of “closed symmetric monoidal category”. We assume no prior knowledge of category theory, proof theory or computer science.

Keywords

  • Inference Rule
  • Monoidal Category
  • Intuitionistic Logic
  • Lambda Calculus
  • Unit Object

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Fig. 2.1
Fig. 2.2
Fig. 2.3

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Acknowledgments

We owe a lot to participants of the seminar at UCR where some of this material was first presented: especially David Ellerman, Larry Harper, Tom Payne—and Derek Wise, who took notes [13]. This paper was also vastly improved by comments by Andrej Bauer, Tim Chevalier, Derek Elkins, Greg Friedman, Matt Hellige, Robin Houston, Theo Johnson–Freyd, Jürgen Koslowski, Todd Trimble, Dave Tweed, and other regulars at the n-Category Café. MS would like to thank Google for letting him devote 20& of his time to this research, and Ken Shirriff for helpful corrections. This work was supported by the National Science Foundation under Grant No. 0653646.

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Baez, J., Stay, M. (2010). Physics, Topology, Logic and Computation: A Rosetta Stone. In: Coecke, B. (eds) New Structures for Physics. Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12821-9_2

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