Physical interpretation of non-normalizable harmonic oscillator states and relaxation to pilot-wave equilibrium.

Non-normalizable states are difficult to interpret in the orthodox quantum formalism but often occur as solutions to physical constraints in quantum gravity. We argue that pilot-wave theory gives a straightforward physical interpretation of non-normalizable quantum states, as the theory requires only a normalized density of configurations to generate statistical predictions. In order to better understand such states, we conduct the first study of non-normalizable solutions of the harmonic oscillator from a pilot-wave perspective. We show that, contrary to intuitions from orthodox quantum mechanics, the non-normalizable eigenstates and their superpositions are bound states in the sense that the velocity field [Formula: see text] at large [Formula: see text]. We argue that defining a physically meaningful equilibrium density for such states requires a new notion of equilibrium, named pilot-wave equilibrium, which is a generalisation of the notion of quantum equilibrium. We define a new H-function [Formula: see text], and prove that a density in pilot-wave equilibrium minimises [Formula: see text], is equivariant, and remains in equilibrium with time. We prove an H-theorem for the coarse-grained [Formula: see text], under assumptions similar to those for relaxation to quantum equilibrium. We give an explanation of the emergence of quantization in pilot-wave theory in terms of instability of non-normalizable states due to perturbations and environmental interactions. Lastly, we discuss applications in quantum field theory and quantum gravity, and implications for pilot-wave theory and quantum foundations in general.

Effect of Phosphorus Application on Arsenic Species Accumulation and Co-Deposition of Polyphenols in Rice Grain: Phyto and Food Safety Evaluation.

The present study was aimed at exploring the effect of soil application of different concentrations of orthophosphate (P) (0, 10, 20, 30, and 40 mg kg-1) on rice agronomic and yield parameters, arsenic (As) species accumulation, and polyphenol levels in the grain of rice grown under As spiked soil (10 mg kg-1). The contents of As species (As(V), As (III), MMA and DMA) and polyphenols in rice grain samples were estimated using LC-ICP-MS and LC-MS/MS, respectively. P treatments significantly reduced the toxic effects of As on agronomic parameters such as root weight and length, shoot and spike length, straw, and grain yield. Among the treatments studied, only the treatment of 30 mg kg-1 P helps to decrease the elevated levels of As (V), As (III), and DMA in rice grains due to As application. The study revealed that 30 mg kg-1 was the optimal P application amount to minimize AS accumulation in rice grains and As-linked toxicity on agronomic parameters and chlorophyll biosynthesis. Furthermore, the levels of trans-ferulic acid, chlorogenic acid, caffeic acid, and apigenin-7-glucoside increased in response to accumulation of As in the rice grain. In conclusion, the precise use of phosphorus may help to mitigate arsenic linked phytotoxicity and enhance the food safety aspect of rice grain.