Advanced Quantum Mathematics Ready For Difficult Problems

A new quantization procedure, called affine quantization, using toy examples, introduces the half-harmonic oscillator, which involves only positive coordinates, and requires affine quantization, not canonical quantization. In addition, “The Particle in a Box” is properly formulated, and ready to be solved, while its affine analysis is valid, however, using canonical quantization, fails completely.  These toy examples point to improved quantum field theory results, as well as quantum gravity, exploiting either operators or path integrations for valid quantum results.