Skip to main content
Log in

Coherent states and Berezin quantization for non-scalar holomorphic representations

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

Let \(G\) be a quasi-Hermitian Lie group and let \(K\) be a maximal compactly embedded subgroup of \(G\). Let \(\pi \) be a unitary representation of \(G\) which is holomorphically induced from a unitary representation \(\rho \) of \(K\). Then \(\pi \) is usually realized in a Hilbert space of vector-valued holomorphic functions. Here we introduce a realization of \(\pi \) in a reproducing kernel Hilbert space of complex-valued functions and we compute the (scalar-valued) coherent states. We study the corresponding Berezin quantization map and, in particular, we show that this map is equivalent to that introduced by Cahen (Rend Istit Mat Univ Trieste 46:157–180, 2014).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ali, S.T., Antoine, J.-P., Gazeau, J.-P.: Coherent States, Wavelets and Their Generalizations, Graduate Texts in Contemporary Physics. Springer, Berlin (2000)

    Book  Google Scholar 

  2. Ali, S.T., Englis, M.: Quantization methods: a guide for physicists and analysts. Rev. Math. Phys. 17(4), 391–490 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ali, S.T., Englis, M.: Berezin–Toeplitz quantization over matrix domains. arXiv:math-ph/0602015v1

  4. Arazy, J., Upmeier, H.: Weyl Calculus for Complex and Real Symmetric Domains. Harmonic analysis on complex homogeneous domains and Lie groups (Rome, 2001). Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13(3–4), 165–181 (2002)

  5. Arazy, J., Upmeier, H.: Invariant symbolic calculi and eigenvalues of invariant operators on symmeric domains. Function spaces, interpolation theory and related topics (Lund, 2000) pp. 151–211. de Gruyter, Berlin (2002)

  6. Arnal, D., Cahen, M., Gutt, S.: Representations of compact Lie groups and quantization by deformation. Acad. R. Belg. Bull. Cl. Sc. 3e série LXXIV 45, 123–141 (1988)

    MathSciNet  Google Scholar 

  7. Benson, C., Jenkins, J., Lipsmann, R.L., Ratcliff, G.: A geometric criterion for Gelfand pairs associated with the Heisenberg group. Pac. J. Math. 178(1), 1–36 (1997)

    Article  Google Scholar 

  8. Benson, C., Jenkins, J., Ratcliff, G.: The orbit method and Gelfand pairs associated with nilpotent Lie groups. J. Geom. Anal. 9, 569–582 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Berezin, F.A.: Covariant and contravariant symbols of operators. Math. USSR Izv. 6(5), 1117–1151 (1972)

    Article  Google Scholar 

  10. Berezin, F.A.: Quantization. Math. USSR Izv. 8(5), 1109–1165 (1974)

    Article  Google Scholar 

  11. Brif, C., Mann, A.: Phase-space formulation of quantum mechanics and quantum-state reconstruction for physical systems with Lie-group symmetries. Phys. Rev. A 59(2), 971–987 (1999)

    Article  MathSciNet  Google Scholar 

  12. Cahen, B.: Weyl quantization for semidirect products. Differ. Geom. Appl. 25, 177–190 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cahen, B.: Berezin quantization on generalized flag manifolds. Math. Scand. 105, 66–84 (2009)

    MATH  MathSciNet  Google Scholar 

  14. Cahen, B.: Stratonovich–Weyl correspondence for compact semisimple Lie groups. Rend. Circ. Mat. Palermo 59, 331–354 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cahen, B.: Stratonovich–Weyl correspondence for discrete series representations. Arch. Math. (Brno) 47, 41–58 (2011)

    MathSciNet  Google Scholar 

  16. Cahen, B.: Berezin quantization for holomorphic discrete series representations: the non-scalar case. Beiträge Algebra Geom. 53, 461–471 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. Cahen, B.: Berezin transform for non-scalar holomorphic discrete series. Comment. Math. Univ. Carolin. 53(1), 1–17 (2012)

    MATH  MathSciNet  Google Scholar 

  18. Cahen, B.: Berezin quantization and holomorphic representations. Rend. Sem. Mat. Univ. Padova 129, 277–297 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Cahen, B.: Global parametrization of scalar holomorphic coadjoint orbits of a quasi-Hermitian Lie group. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 52, 35–48 (2013)

    MATH  MathSciNet  Google Scholar 

  20. Cahen, B.: Stratonovich–Weyl correspondence for the Jacobi group. Commun. Math. 22, 31–48 (2014)

    MATH  MathSciNet  Google Scholar 

  21. Cahen, B.: Stratonovich–Weyl correspondence via Berezin quantization. Rend. Istit. Mat. Univ. Trieste 46, 157–180 (2014)

    MathSciNet  Google Scholar 

  22. Cahen, B.: Berezin transform and Stratonovich–Weyl correspondence for the multi-dimensional Jacobi group. Rend. Sem. Mat. Univ. Padova (2015, to appear)

  23. Cahen, M., Gutt, S., Rawnsley, J.: Quantization on Kähler manifolds I, Geometric interpretation of Berezin quantization. J. Geom. Phys. 7, 45–62 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  24. Cariñena, J.F., Gracia-Bondìa, J.M., Vàrilly, J.C.: Relativistic quantum kinematics in the Moyal representation. J. Phys. A Math. Gen. 23, 901–933 (1990)

    Article  MATH  Google Scholar 

  25. Davidson, M., Òlafsson, G., Zhang, G.: Laplace and Segal–Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials. J. Funct. Anal. 204, 157–195 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. Figueroa, H., Gracia-Bondìa, J.M., Vàrilly, J.C.: Moyal quantization with compact symmetry groups and noncommutative analysis. J. Math. Phys. 31, 2664–2671 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  27. Folland, B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  28. Gazeau, J.-P.: Coherent States in Quantum Physics. Wiley-VCH Verlag, New York (2009)

    Book  Google Scholar 

  29. Gracia-Bondìa, J.M.: Generalized Moyal quantization on homogeneous symplectic spaces. Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990), pp. 93–114, Contemp. Math., vol. 134. Amer. Math. Soc., Providence (1992)

  30. Gracia-Bondìa, J.M., Vàrilly, J.C.: The Moyal representation for spin. Ann. Phys. 190, 107–148 (1989)

    Article  MATH  Google Scholar 

  31. Kilic, S.: The Berezin symbol and multipliers on functional Hilbert spaces. Proc. Am. Math. Soc. 123(12), 3687–3691 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  32. Kirillov, A.A.: Lectures on the Orbit Method, Graduate Studies in Mathematics, vol. 64. American Mathematical Society, Providence (2004)

    Book  Google Scholar 

  33. Kostant, B.: Quantization and unitary representations. In: Modern Analysis and Applications. Lecture Notes in Mathematics, vol. 170, pp. 87–207. Springer, Berlin (1970)

  34. Neeb, K.-H.: Holomorphy and Convexity in Lie Theory, de Gruyter Expositions in Mathematics, vol. 28. Walter de Gruyter, Berlin (2000)

    Book  Google Scholar 

  35. Nomura, T.: Berezin transforms and group representations. J. Lie Theory 8, 433–440 (1998)

    MATH  MathSciNet  Google Scholar 

  36. Ørsted, B., Zhang, G.: Weyl quantization and tensor products of Fock and Bergman spaces. Indiana Univ. Math. J. 43(2), 551–583 (1994)

    Article  MathSciNet  Google Scholar 

  37. Perelomov, A.M.: Generalized Coherent States and Their Applications. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  38. Satake, I.: Algebraic Structures of Symmetric Domains. Iwanami Sho-ten. Tokyo and Princeton University Press, Princeton (1971)

    Google Scholar 

  39. Stratonovich, R.L.: On distributions in representation space. Sov. Phys. JETP 4, 891–898 (1957)

    MathSciNet  Google Scholar 

  40. Unterberger, A., Upmeier, H.: Berezin transform and invariant differential operators. Commun. Math. Phys. 164(3), 563–597 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  41. Wallach, N.R.: Harmonic Analysis on Homogeneous Spaces. Marcel Dekker, New York (1973)

    MATH  Google Scholar 

  42. Wildberger, N.J.: On the Fourier transform of a compact semisimple Lie group. J. Aust. Math. Soc. A 56, 64–116 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  43. Zhang, G.: Berezin transform on compact Hermitian symmetric spaces. Manuscr. Math. 97, 371–388 (1998)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

I would like to thank the referee for some pertinent remarks and for pointing out the recent book [28] to me.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Benjamin Cahen.

Additional information

Communicated by Salvatore Rionero.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cahen, B. Coherent states and Berezin quantization for non-scalar holomorphic representations. Ricerche mat. 64, 115–135 (2015). https://doi.org/10.1007/s11587-015-0223-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-015-0223-2

Keywords

Mathematics Subject Classification

Navigation