Reference number : GC-2017-Q1-03363, amendment number : , agreement type : G, recipient legal name : Sigal, Israel Michael (University of Toronto), recipient country : CA, recipient province : nan, recipient city : Ontario, recipient postal code : nan, program name english : Discovery Grants Program - Individual, program name french : Programme de subventions à la découverte - individuelles, program purpose english : The subject of this proposal is the dynamics of systems of non-relativistic particles, interacting with quantized fields. Specifically, it deals with the dynamics of (i) electrons in vacuum, interacting with the nuclei and with the quantized electromagnetic field (photons); and (ii) electrons in solids interacting with the solid crystal atoms and with the quantized crystal lattice oscillations (phonons). 
 The quantum dynamics of such systems are described by the Schr&ouml;dinger-type equation, discovered by Dirac, Fermi and Pauli and denoted below as SE. The key problem we address here is to describe the long-time behaviour of the solutions of SE. 
 For n particles ( n could be as large as 10 24 ), SE is a linear evolution equation in 3 n variables if one neglects quantum fields, and in &infin; number of variables if one does not. It is surprising how much we can understand using this equation, given that even for n =2 and without quantum fields, it cannot be solved using computers, unless there is a hidden dimensional reduction. 
 To address specific questions we have to use a simpler, approximate dynamic. The latter gives a rougher description, zeroing on a desired class of properties; it is called the effective dynamic. Hence, the second problem we address is to derive the effective equations and describe their long- time behaviour of the solutions. 
 Constructing rigorous theory of quantum non-relativistic matter and radiation and developing the corresponding mathematical tools for this is critical for understanding the physical world around us. It is also a rich source of fundamental mathematical problems which could potentially give rise to new branches in Analysis. 
 The issues described in the proposal underlie the structure and properties of matter. Not surprisingly, they have garnered a disproportionate number of Nobel prizes in Physics and Chemistry. Nevertheless the mathematical apparatus dealing with them is only in the initial stages of construction, especially compared with the mathematical techniques developed in the last few centuries to deal with problems of classical physics. 
 Though the main thrust of this project is theoretical, its subject matter is closely related to the theory behind the current effort in nano devices, all optical devices and in quantum computing. On micro- and meso-scopic scales, not only the electromagnetic field, but also heat is quantized, and &lsquo;particles&rsquo; of heat are exactly the phonons discussed in our proposal. 
 To quote from a well-known Harvard physicist M. D. Lukin, &ldquo;the practical implementation of quantum information requires precise manipulation of quantum states of light and matter coming from many coupled quantum-mechanical systems, which is an extremely challenging task.&rdquo; A better mathematical understanding of quantum dynamics can help us to manage this task., program purpose french : The subject of this proposal is the dynamics of systems of non-relativistic particles, interacting with quantized fields. Specifically, it deals with the dynamics of (i) electrons in vacuum, interacting with the nuclei and with the quantized electromagnetic field (photons); and (ii) electrons in solids interacting with the solid crystal atoms and with the quantized crystal lattice oscillations (phonons). 
 The quantum dynamics of such systems are described by the Schr&ouml;dinger-type equation, discovered by Dirac, Fermi and Pauli and denoted below as SE. The key problem we address here is to describe the long-time behaviour of the solutions of SE. 
 For n particles ( n could be as large as 10 24 ), SE is a linear evolution equation in 3 n variables if one neglects quantum fields, and in &infin; number of variables if one does not. It is surprising how much we can understand using this equation, given that even for n =2 and without quantum fields, it cannot be solved using computers, unless there is a hidden dimensional reduction. 
 To address specific questions we have to use a simpler, approximate dynamic. The latter gives a rougher description, zeroing on a desired class of properties; it is called the effective dynamic. Hence, the second problem we address is to derive the effective equations and describe their long- time behaviour of the solutions. 
 Constructing rigorous theory of quantum non-relativistic matter and radiation and developing the corresponding mathematical tools for this is critical for understanding the physical world around us. It is also a rich source of fundamental mathematical problems which could potentially give rise to new branches in Analysis. 
 The issues described in the proposal underlie the structure and properties of matter. Not surprisingly, they have garnered a disproportionate number of Nobel prizes in Physics and Chemistry. Nevertheless the mathematical apparatus dealing with them is only in the initial stages of construction, especially compared with the mathematical techniques developed in the last few centuries to deal with problems of classical physics. 
 Though the main thrust of this project is theoretical, its subject matter is closely related to the theory behind the current effort in nano devices, all optical devices and in quantum computing. On micro- and meso-scopic scales, not only the electromagnetic field, but also heat is quantized, and &lsquo;particles&rsquo; of heat are exactly the phonons discussed in our proposal. 
 To quote from a well-known Harvard physicist M. D. Lukin, &ldquo;the practical implementation of quantum information requires precise manipulation of quantum states of light and matter coming from many coupled quantum-mechanical systems, which is an extremely challenging task.&rdquo; A better mathematical understanding of quantum dynamics can help us to manage this task., agreement title english : Mathematical Studies of Quantum Dynamics of Matter and Radiation, agreement title french : Mathematical Studies of Quantum Dynamics of Matter and Radiation, agreement number : RGPIN, agreement value : 185000.0, agreement start date : 2017-05-10, agreement end date : nan, description english : nan, description french : nan, expected results english : nan, expected results french : nan, additional information english : Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023), additional information french : Subvention ou bourse octroyée s'appliquant à plus d'un exercice financier. (2017-2018 à 2022-2023), owner organization : nserc-crsng, owner organization title : Natural Sciences and Engineering Research Council of Canada | Conseil de recherches en sciences naturelles et en génie du Canada