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Random Walk on Quantum Blobs

Arkadiusz Jadczyk

Laboratoire de Physique Théorique, Université de Toulouse III & CNRS, 118 route de Narbonne, 31062 Toulouse, France

Ronin Institute, Montclair, NJ 0704, USA

https://doi.org/10.1142/S1230161222500135Cited by:0 (Source: Crossref)

Abstract

We describe the action of the symplectic group on the homogeneous space of squeezed states (quantum blobs) and extend this action to the semigroup. We then extend the metaplectic representation to the metaplectic (or oscillator) semigroup and study the properties of such an extension using Bargmann-Fock space. The shape geometry of squeezing is analyzed and noncommuting elements from the symplectic semigroup are proposed to be used in simultaneous monitoring of noncommuting quantum variables — which should lead to fractal patterns on the manifold of squeezed states.

Keywords:

References

1. M. de Gosson, Symplectic Geometry and Quantum Mechanics, Birkhäuser, 2006. Crossref, Google Scholar
2. M. de Gosson, Found. Phys. 43, 440 (2013). Crossref, Web of Science, Google Scholar
3. S. Berceanu, J. Geom. and Symm. in Physics 4, 1 (2005). Google Scholar
4. S. Berceanu, Rev. Math. Phys. 24, 1250024 (2012). Link, Web of Science, Google Scholar
5. S. Berceanu, Int. J. Geom. Meth. Phys. 11, 1450035 (2014). Link, Web of Science, Google Scholar
6. R. Ruskov, A. N. Korotkov, and M olmer, Phys. Rev. Lett. 105 100506 (2010). Crossref, Web of Science, Google Scholar
7. A. Cornelissen, J. Bausch, A. Gilyén, Scalable Benchmarks for Gate-Based Quantum Computers, 2021. Google Scholar
8. A. Jadczyk, Quantum Fractals, From Heisenberg’s Uncertainty to Barnsley’s Fractality, World Sci., 2014. Link, Google Scholar
9. L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979. Google Scholar
10. V. Bargmann, in: Analytic methods in mathematical physics, R. P. Gilbert and R. G. Newton, eds., Gordon and Breach Science Publ., 1970. Google Scholar
11. M. Brunet, Rep. Math. Phys. 22, 149 (1985). Crossref, Google Scholar
12. M. Brunet and P. Kramer, Rep. Math. Phys. 15, 287 (1979). Crossref, Google Scholar
13. M. Brunet and P. Kramer. Rep. Math. Phys. 17, 205 (1980). Crossref, Google Scholar
14. G. B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, 1989. Crossref, Google Scholar
15. B. C. Hall, Conteporary Mathematics 260, 1 (2000). Google Scholar
16. J Hilgert, Ann. Inst. Fourier, 39, 663 (1989). Crossref, Web of Science, Google Scholar
17. J. Hilgert, Applications of lie semigroups in analysis, in: The Analytical and Topological Theory of Semigroups: Trend and Developments, K. H. Hofmann, J. G. Lawson, and J. S. Pym, eds., Walter de Gruyter, 1990, pp. 27–50. Crossref, Google Scholar
18. J. Hilgert and K-H. Neeb, Lie Semigroups and their Applications, Springer, 1993. Crossref, Google Scholar
19. R. Honegger and A. Rieckers, Photons in Fock space and beyond, volume 1, World Scientific, 2015. Google Scholar
20. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, 1990. Google Scholar
21. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, 1991. Crossref, Google Scholar
22. R. Howe, The oscillator semigroup, in: The mathematical heritage of Hermann Weyl, Proc. Symposia in Pure Mathematics, vol. 48, R. O. Wells, ed., AMS, 1988, pp. 61–132. Crossref, Google Scholar
23. A. Jadczyk, Quantum spaceship, https://www.youtube.com/watch?v=IxHD7XxapWs. Google Scholar