CONTRIBUTED TALK ABSTRACTS1 I M.A. NAIT ABDALLAH, On an application of Curry-Howard correspondence to quantum mechanics. Dep. Computer Science, UWO, London, Canada; INRIA, Rocquencourt, France. E-mail:
[email protected]. We present an application of Curry-Howard correspondence to the formalization of physical processes in quantum mechanics. Quantum mechanics' assignment of complex amplitudes to events and its use of superposition rules are puzzling from a logical point of view. We provide a Curry-Howard isomorphism based logical analysis of the photon interference problem [1], and develop an account in that framework. This analysis uses the logic of partial information [2] and requires, in addition to the resolution of systems of algebraic constraints over sets of λ-terms, the introduction of a new type of λ-terms, called phase λ-terms. The numerical interpretation of the λ-terms thus calculated matches the expected results from a quantum mechanics point of view, illustrating the adequacy of our account, and thus contributing to bridging the gap between constructive logic and quantum mechanics. The application of this approach to a photon traversing a Mach-Zehnder interferometer [1], which is formalized by context Γ = fx : s; hP ; πi : s ! a? ! a; Q : s ! b? ! b; hJ; πi : a ! a0; hJ 0; πi : b ! b0; hP 0; πi : b0 ! c? ! c; Q0 : b0 ! d? ! d; P 00 : a0 ! d0? ! d0;Q00 : a0 ! c0? ! c0;R : c _ c0 ! C; S : 0 ? ? 0 ? 0 ? 00 0? 00 0? d _ d ! D; ξ1 : a ; ξ2 : b ; ξ1 : c ; ξ2 : d ; ξ1 : d ; ξ2 : c g, yields inhabitation claims: 00 00 0 0 0 R(in1(Q (J(P xξ1))ξ2 )) + R(in2(P (J (Qxξ2))ξ1)) : C 00 00 0 0 0 S(in1(P (J(P xξ1))ξ1 )) + hS(in2(Q (J (Qxξ2))ξ2)); πi : D which are the symbolic counterpart of the probability amplitudes used in the standard quantum mechanics formalization of the interferometer, with constructive (resp.