A cameron-martin formula for feynman integrals (The origin of the Maslov indices)

Nikolaj Nikolaevich Shamarov

Sonia Mazzucchi

David Elworthy, Aubrey Truman

A Cameron-Martin formula is established for the Feynman map definition of the Feynman integral in non-relativistic quantum mechanics. This formula involves the Morse or Maslov indices and shows the connection with the Feynman-Hibbs definition of the path integral and the Fresnel integrals of Albeverio and Hoegh-Krohn. Applications are given to the Feynman-Kac-Ito formula for anharmonic oscillator potentials On etablit une formule de Cameron-Martin pour l'integrale de Feynman en mecanique quantique non relativiste. Cette formule fait intervenir les indices de Morse et de Maslov et montre la relation avec la definition de Feynman-Hibbs de l'integrale de chemins et les integrales de Fresnel considerees par Albeverio et Hoegh-Krohn. On donne des applications a la formule de Feynman-Kac-Ito pour des potentiels d'oscillateurs anharmoniques

Aubrey Truman

By introducing the family of Feynman maps ℱs, we show that our earlier definition of the Feynman path integral ℱ=ℱ1 can be obtained as the analytic continuation of the Wiener integral E =ℱ−i. This leads to some new results for the Wiener and Feynman integrals. We establish a translation and Cameron–Martin formula for the Feynman maps ℱs, having applications to nonrelativistic quantum mechanics. We also estalish a (weak) dominated convergence theorem for ℱ1=ℱ.

Gian‐Carlo Rota

Lawrence E. Thomas

Let H be the isotropic spin-s quantum ferromagnetic Heisenberg Hamiltonian associated with a finite volume in Zd. In an appropriate representation, the restriction of −H to the N-spin wave sector −HN can be regarded as the generator of a Markov semigroup exp(−tHN). For s= 1/2 the corresponding process XN(t) is identified as a simple N-particle exclusion process; for higher spin it is a more complicated exclusion process involving different species of particles. A Feynman–Kac, Cameron–Martin perturbation formula for exp(−tHN) is developed in the s= 1/2 case, and the formula is used to obtain some thermodynamic correlation inequalities.