Physical and Mathematical Background

Pramode K. Verma, Mayssaa El Rifai, Kam Wai Clifford Chan

Ilya Kuprov

Valter Moretti

Valter Moretti

Anatoliy K. Prykarpatsky, Nikolai N. Bogolubov, Jolanta Golenia +1 more

Introductive backgrounds of a new mathematical physics discipline - Quantum Mathematics - are discussed and analyzed both from historical and analytical points of view. The magic properties of the second quantization method, invented by V. Fock in 1932, are demonstrated, and an impressive application to the nonlinear dynamical systems theory is considered.

Dariusz Chruściński, Andrzej Jamiołkowski

The mathematical background required for the study of geometric phases in classical and quantum mechanics is rather extensive. The aim of this introductory chapter is to provide a background of some basic notions of classical differential geometry and topology. Classical differential geometry is now a well established tool in modern theoretical physics. Many classical theories like mechanics, electrodynamics, Einstein’s General Relativity or Yang-Mills gauge theories are well known examples where the geometrical methods enter in the natural and very effective way. As we shall see throughout this book, also quantum physics shows its intricate beauty when one applies an appropriate geometric framework. All this proves Wigner’s celebrated statement about the “unreasonable effectiveness” of mathematics in natural sciences. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.