Rigorous Real-Time Feynman Path Integral for Vector Potentials

we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the $L^2$ transition probability amplitude via improper Riemann integrals. Our formulation will hold for vector potential Hamiltonian for which its potential and vector potential each carries at most a finite number of singularities and discontinuities

Drought and the Sacramento-San Joaquin Delta, 2012–2016: Environmental Review and Lessons

This paper reviews environmental management and the use of science in the Sacramento–San Joaquin Delta during California’s 2012–2016 drought. The review is based on available reports and data, and guided by discussions with 27 agency staff, stake-holders, and researchers. Key management actions for the drought are discussed relative to four major drought water management priorities stated by water managers: support public health and safety, control saltwater intrusion, preserve cold water in Shasta Reservoir, and maintain minimum protections for endangered species. Despite some success in streamlining communication through interagency task forces, conflicting management mandates sometimes led to confusion about priorities and actions during the drought (i.e., water delivery, the environment, etc.). This report highlights several lessons and offers suggestions to improve management for future droughts. Recommendations include use of pre-drought warnings, timely drought declarations, improved transparency and useful documentation, better scientific preparation, development of a Delta drought management plan (including preparing for salinity barriers), and improved water accounting. Finally, better environmental outcomes occur when resources are applied to improving habitat and bolstering populations of native species during inter-drought periods, well before stressful conditions occur.