Abstract
The introduction of linear logic and its associated proof theory has revolutionized many semantical investigations, for example, the search for fully-abstract models of PCF and the analysis of optimal reduction strategies for lambda calculi. In the present paper we show how proof nets, a graph-theoretic syntax for linear logic proofs, can be interpreted as operators in a simple calculus.
This calculus was inspired by Feynman diagrams in quantum field theory and is accordingly called the φ-calculus. The ingredients are formal integrals, formal power series, a derivative-like construct and analogues of the Dirac delta function.
Many of the manipulations of proof nets can be understood as manipulations of formulas reminiscent of a beginning calculus course. In particular, the “box” construct behaves like an exponential and the nesting of boxes phenomenon is the analogue of an exponentiated derivative formula. We show that the equations for the multiplicative-exponential fragment of linear logic hold.
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Notes
- 1.
Actually a completely rigourous theory of path integration, due to Wiener, existed in the 1920s. It was, however, for statistical mechanics and worked with a gaussian measure rather than the kind of measure that Feynman needed.
- 2.
The grammatically correct way to name this is a “Green function”, but it is too late to change common practice.
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Acknowledgement
We would like to thank Samson Abramsky, Martin Hyland, Radha Jagadeesa and Mikhail Gromov for interesting discussions.
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Blute, R., Panangaden, P. (2010). Proof Nets as Formal Feynman Diagrams. 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_7
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