It has been stated in the physics literature that the probability amplitude for a quantum system after a continuous position measurement can be expressed by the restricted or the weighted Feynman path integral containing information about the result and the resolution of its measurement. In the present paper it is proved that the restricted Feynman path integrals can be defined in a rigorous sense. More specifically, we consider a quantum system consisting of some particles and simultaneously perform a continuous measurement of the positions of all particles in the time interval
[0,T]. Then, it is shown rigorously that the restricted Feynman path integral after the measurement can be defined in the
L2 space and satisfies the corresponding non-self-adjoint Schroeodinger equation. In addition, our results are generalized to a quantum spin system where a continuous position measurement for all spin components of all particles is simultaneously performed, though the results obtained are inadequate for general use.
Feynman proposed a postulate or a method of quantization in his celebrated paper in 1948. Applying Feynman’s postulate to temporally continuous quantum measurements of the positions of particles, Mensky proposed the restricted Feynman path integrals (RFPIs) for continuous quantum measurements after phenomenological considerations. Our aim in this paper is to give a rigorous proof that Mensky’s RFPIs emerge out of Feynman’s postulate under a simple approximation. In addition, it is proved that the RFPIs emerge out of von Neumann’s postulate on instantaneous measurements as well as Feynman’s postulate. The quantum systems that we study include spin systems. These results are applied to formulations of the multi-split experiments, the quantum Zeno and the Aharonov–Bohm effects.