The Wigner Transform

This book provides an in-depth and rigorous study of the Wigner transform and its variants. They are presented first within a context of a general mathematical framework, and then through applications to quantum mechanics. The Wigner transform was introduced by Eugene Wigner in 1932 as a probability quasi-distribution which allows expression of quantum mechanical expectation values in the same form as the averages of classical statistical mechanics. It is also used in signal processing as a transform in time-frequency analysis, closely related to the windowed Gabor transform.

Written for advanced-level students and professors in mathematics and mathematical physics, it is designed as a complete textbook course providing analysis on the most important research on the subject to date. Due to the advanced nature of the content, it is also suitable for research mathematicians, engineers and chemists active in the field.

Contents:

• General Mathematical Framework:

  • Phase Space Translations and Reflections
  • The Cross-Wigner Transform
  • The Cross-Ambiguity Function
  • Weyl Operators
  • Symplectic Covariance
  • The Moyal Identity
  • The Feichtinger Algebra
  • The Cohen Class
  • Gaussians and Hermite Functions
  • Sub-Gaussian Estimates

• Applications to Quantum Mechanics:

  • Moyal Star Product and Twisted Convolution
  • Probabilistic Interpretation of the Wigner Transform
  • Mixed Quantum States and the Density Operator
  • The KLM Conditions and the Narcowich–Wigner Spectrum
  • Wigner Transform and Quantum Blobs

• Appendices:

  • A: Sp(n) and Mp(n)
  • B: The Symplectic Fourier Transform
  • C: Symplectic Diagonalization
  • D: Symplectic Capacities

Readership: Advanced-level students and professors in mathematics and mathematical physics; research mathematicians, engineers and chemists active in the field.