Lorentz Group in Optical Sciences

Squeezed States!

The picture is for Lorentz-boosted extended objects in particle physics. The supporting mathematics is called the Lorentz group. This group, by now, is the standard mathematical device for quantum and classical optics.
It is by now widely established that the Wigner function is the key scientific language for quantum optics, but not many people know that those Wigner functions in optics also serve as the representation of the Lorentz groups, such as O(2,1) and O(3,2) which are the fundamental languages for the one- and two-mode squeezed states. When you do squeezed states, you are doing the Lorentz groups.

On this subject, with Marilyn Noz, I have written a book entitled Phase Space Picture of Quantum Mechanics.

Recently, I have been publishing papers on applications of the Lorentz group to classical ray optics, mostly in Phys. Rev. E. Those papers were also archived in Los Alamos. Some of them are listed below.

My youngest co-authors!

Sibel Baskal

Elena Georgeiva

• Jones matrix formalism as a representation of the Lorentz group ArXiv. , published in J. Opt. Soc. Am. A /Vol.14, No.9 page 2290 (1997).

• Stokes parameters as a Minkowskian four-vector ArXiv.
  Physical Review E (1997).

• Wigner rotations and Iwasawa decompositions in polarization optics ArXiv. Physical Review E (1999).

• Optics computers for space-time symmetries ArXiv.
  presented at various conferences during the year 1999.

• Interferometers and decoherence matrices ArXiv.
  Physical Review E (2000).

• Iwasawa effect in multilayer optics ArXiv.
  Physical Review E (2001).

• Three-lenses at most in multi-lens system ArXiv.
  Physical Review E (2001)

• Wigner rotations in laser cavities ArXiv.
  Physical Review E (2002).

• Lens optics and group contractions ArXiv.
  Physical Review E (2003).

• Cyclic representations of multilayer optics ArXiv.
  Physical Review E (2003).

• Physics of two-by-two matrices ArXiv.
  presented at various conferences during the year 2003.

• Lorentz group in ray optics (Review Paper) AriXiv.
  Journal of Optics B: Quantum and Secmiclassical Optics, Vol. 6, S455-472 (2004).

• de Sitter group as a symmetry for optical decoherence ArXiv.
  Journal of Physics A (2006).

• ABCD matrices as similarity transformations of Wigner matrices and periodic systems in optics
  J. Opt. Soc. Am. A, 26 3049-2054 (2009).

• Optical activities as computing resources for space-time symmetries
  Journal of Modern Optics, 57, 17-22 (2010).
  

• One anlytic form for four branches of the ABCD matrix,
  J. Mod. Opt. [57], 1251-1259 (2010).
  ArXiv

• Possible Minkowskian Language in Two-level Systems
  Journal Optics and Spectroscopy [108], 297-300 (2010).
  ArXiv

• Internal Space-time Symmetries of Particles derivable from Periodic
  Arxiv to be published in the proceedings of the 10h Int'l Conference on Quantum Optics and Quantum Information.