Emergence of the Quantum from the Classical

Mathematical Aspects of Quantum Processes

The emergence of quantum mechanics from classical world mechanics is now a well-established theme in mathematical physics. This book demonstrates that quantum mechanics can indeed be viewed as a refinement of Hamiltonian mechanics, and builds on the work of George Mackey in relation to their mathematical foundations. Additionally when looking at the differences with classical mechanics, quantum mechanics crucially depends on the value of Planck's constant h. Recent cosmological observations tend to indicate that not only the fine structure constant α but also h might have varied in both time and space since the Big Bang. We explore the mathematical and physical consequences of a variation of h; surprisingly we see that a decrease of h leads to transitions from the quantum to the classical.

Emergence of the Quantum from the Classical provides help to undergraduate and graduate students of mathematics, physics and quantum theory looking to advance into research in the field.

Contents:

• Hamiltonian Mechanics

• Hamilton–Jacobi Theory

• Matter Waves, Schrödinger's Equation, and Bohm's Theory

• The Metatron

• Uncertainties and Quantum Blobs

• Quantum States and the Density Matrix

• Varying Planck's Constant

• Appendices:

  • The Symplectic Group
  • The Metaplectic Representation
  • Born–Jordan Quantization
  • Twisted Product and Convolution

Readership: Undergraduate and graduate students of mathematics and physics, interested in analysis and differential equations, probability and statistics, geometry and topology and mathematical physics.