Lorentz Group in Optical Sciences

Squeezed States!

Have you seen this picture? It came from my papers.

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 co-authors.

Marilyn Noz in 1975 and 2006

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).

• Illustratative Example of Feynman's Rest of the Universe
  Am. J. Phys. [67], 61 - 66 (1999).
  Preprint

• 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).

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

• 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

• Internal space-time symmetries of particles derivable from periodic systems in optics,
  Optics and Spectroscopy [111], 731-737 (2011).
  ArXiv.

• The Language of Two-by-two Matrices spoken by Optical Devices
  19th InternationalConference on Composites or Nano Engineering, Shanghai (China 2011).
  AxXiv

  with Roy Galuber, Robert Boyd, Masha Chekova, and Elizabeth Giacobino (Nuremberg 2103).

• Time separation as a hidden variable in the Copenhagen school of quantum mechanics, in Advances in Quantum Theory,
  AIP Conference Series, No.1327, pp 138-147 (2011).
  ArXiv.

• Lorentz Harmonics, Squeeze Harmonics and Their Physical Applications,
  Symmerty [3], 16-36 (2011).
  ArXiv.

• Poincaré Sphere and Decoherence Problems
  Arxiv Based on an invited talk presented at the Fedorov Memorial Symposium: International Conference "Spins and Photonic Beams at Interface," dedicated to the 100th anniversary of F.I.Fedorov (Minsk, Belarus, 2011).

• Lorentz Group in Ray and Polarization Optics
  Chapter 9 in "Mathematical Optics: Classical, Quantum and Computational Methods" edited by Vasudevan Lakshminarayanan, Maria L. Calvo, and Tatiana Alieva (CRC Taylor and Francis, New York 2013), pp 303-349.
  ArXiv.

• Symmetries shared by the Poincaré Group and Poincaré Sphere.
  Symmetry 5, 223-252 (2013).
  ArXiv.

  with two big boys from Vienna (Budapest 2013). Helmut Rausch (left) and Anton Zeilinger.

• Poincaré Sphere and a Unified Picture of Wigner's Little Groups, Proceedings of Wigner 111, Colorful & Deep Symposium, edited by S. Varro, P. Adam, T.S. Biro, G. G. Barnafoldi, and P. Levai,
  EPJ Web of Conferences, [78], 01005:1 - 8 (2014),
  doi10.151/ejpcon/20147801005.
  ArXiv.

• Lens optics and the continuity problems of the ABCD matrix
  J. Mod. Opt. [61], 161 - 166 (2014).
  ArXiv.

• Symmetries of Massive and Massless Neutrinos,
  Invited paper presented at the International Conference on Hadron Structure and QCD: from Low to High Energies (Gatchina, Russia, 2016).
  ArXiv.

  

• Photons in the Quantum World,
  with S. Baskal and M. E. Noz,
  Invited paper presented at the 15th International Conference on Squeezed States and Uncertainty Relations (Jeju, South Korea, August 12-17, 2017).
  ArXiv.

• Mathematical Devices for Optical Sciences.
  with S. Baskal and M.E. Noz
  IOP Book (2019)
  available from the amazon.com

• Role of Quantum Optics in Synthesizing Quantum Mechanics and Relativity,
  Invited paper presented at the 26th International Conference on Quantum Optics and Quantum Information (Minsk, Belarus, May 2019).
  ArXiv. For pdf with sharper images, click here.