On a Quantum System with Memory

We consider the integro-differential equation for the classical trajectory of an oscillator coupled to another one. On the quantum level the elimination of the coordinate A of the “unvisible” oscillator leads to an effective path integral (X, Ξ, μ) for the associated imaginary time stochastic process t ?, (-∞,∞) → x(t). We prove reflection positivity of the measure dμ ≈ F · dξ, where dξ governes the free oscillator x and F is the counterpart of Feynman's influence functional. Finally, realizing the Hamiltonian semigroup exp(-tH), t ≧ 0, in the physical Hilbert space ?? = L²(X, Γ, μ), where Γ ? Ξ₊, we try to understand what is memory.