Abstract
Using a left multiplication defined on a right quaternionic Hilbert space, we shall demonstrate that pure squeezed states, which are obtained by the sole action of the squeeze operator on the vacuum state, can be defined with all the desired properties on a right quaternionic Hilbert space. Further, we shall also demonstrate that squeezed states, which are obtained by the action of the squeeze operator on canonical coherent states, in other words they are obtained by the action of the displacement operator followed by the action of the squeeze operator on the vacuum state, can be defined on the same Hilbert space, but the non-commutativity of quaternions prevents us in getting the desired results. However, we will show that if one considers the quaternionic slice wise approach, then the desired properties can be obtained for quaternionic squeezed states.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995)
Adler, S.L., Millard, A.C.: Coherent states in quaternionic quantum mechanics. J. Math. Phys. 38, 2117–26 (1997)
Ali, S.T., Antoine, J.-P., Gazeau, J.-P.: Coherent States, Wavelets and Their Generalizations, 2nd edn. Springer, New York (2014)
Alpay, D., Colombo, F., Kimsey, D.P.: The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum. J. Math. Phys. 57, 023503 (2016)
Alpay, D., Colombo, F., Sabadini, I., Salomon, G: The Fock Space in the Slice Hyperholomorphic Setting, Hypercomplex Analysis: New Perspective and Applications, Trends in Mathematics, pp 43–59. Birkhüser, Basel (2014)
Alpay, D., Colombo, F., Sabadini, I.: Slice Hyperholomorphic Schur Analysis. Besel, Birkhauser (2016)
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)
Brian, C.: Hall, Quantum Theory for Mathematicians. Springer, New York (2013)
Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative Functional Calculus. Birkhäuser, Basel (2011)
Gazeau, J.-P.: Coherent States in Quantum Physics. Wiley, Berlin (2009)
Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternionic variable. Adv. Math. 216, 279–301 (2007)
Ghiloni, R., Moretti, W., Perotti, A.: Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys. 25, 1350006 (2013)
Gülebeck, K., Habetha, K., Spröbig, W.: Holomorphic Functions in the Plane and n-Dimensional Spaces. Birkhäuser, Basel (2008)
Loudon, R., Knight, P.L.: Squeezed light. J. Mod. Opt. 34, 709–759 (1987)
Loudon, R.: The Quantum Theory of Light, 3rd edn. Oxford University Press, New York (2000)
Muraleetharan, B., Thirulogasanthar, K.: Coherent states on quaternion slices and a measurable field of Hilbert spaces. J. Geom. Phys. 110, 233–247 (2016)
Muraleetharan, B., Thirulogasanthar, K.: Coherent state quantization of quaternions. J. Math. Phys. 56, 083510 (2015)
Muraleetharam, B., Thirulogasanthar, K., Sabadini, I.: A representation of Weyl-Heisenberg algebra in the quaternionic setting. Ann. Phys. 365, 180–213 (2017)
Thirulogasanthar, K., Twareque Ali, S.: Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states. J. Math. Phys. 54, 013506 (2013)
Thirulogasanthar, K., Honnouvo, G., Krzyzak, A.: Coherent states and Hermite polynomials on Quaternionic Hilbert spaces. J. Phys.A: Math. Theor. 43, 385205 (2010)
Viswanath, K.: Normal operators on quaternionic Hilbert spaces. Trans. Am. Math. Soc. 162, 337–350 (1971)
Youen, Y.P.: Two photon coherent states of the radiation field. Phys. Rev. A. 13, 2226–43 (1976)
Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)
Acknowledgments
K. Thirulogasanthar would like to thank the, FQRNT, Fonds de la Recherche Nature et Technologies (Quebec, Canada) for partial financial support Under the grant number 2017-CO-201915. Part of this work was done while he was visiting the Politecnico di Milano to which he expresses his thanks for the hospitality. He also thanks the program Professori Visitatori GNSAGA, INDAM for the support during the period in which this paper was partially written. The authors thank Prof. I Sabadini for discussions. The authors would also like to thank the reviewer for his/her valuable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Thirulogasanthar, K., Muraleetharan, B. Squeezed States in the Quaternionic Setting. Math Phys Anal Geom 23, 8 (2020). https://doi.org/10.1007/s11040-020-9332-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-020-9332-6