Mathematical Studies of Quantum Dynamics of Matter and Radiation

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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 Schrödinger-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 1024), SE is a linear evolution equation in 3n variables if one neglects quantum fields, and in 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 particles' of heat are exactly the phonons discussed in our proposal.*** To quote from a well-known Harvard physicist M. D. Lukin, “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.” A better mathematical understanding of quantum dynamics can help us to manage this task.