ABSTRACT
Segal-Bargmann coherent state transforms can be viewed as unitary maps from L2 spaces of functions (or sections of an appropriate line bundle) on a manifold X to spaces of square integrable holomorphic functions (or sections) on Xℂ. It is natural to consider higher dimensional extensions of X based on Clifford algebras as they could be useful in studying quantum systems with internal, discrete, degrees of freedom corresponding to nonzero spins. Notice that the extensions of X based on the Grassmann algebra appear naturally in the study of supersymmetric quantum mechanics. In Clifford analysis, the zero mass Dirac equation provides a natural generalization of the Cauchy-Riemann conditions of complex analysis and leads to monogenic functions. For the simplest but already quite interesting case of X = ℝ, we introduce two extensions of the Segal-Bargmann coherent state transform from L2(ℝ, dx) ⊗ ℝm to Hilbert spaces of slice monogenic and axial monogenic functions and study their properties. These two transforms are related by the dual Radon transform. Representation theoretic and quantum mechanical aspects of the new representations are studied.
Acknowledgments
The authors would like to thank the referee for several suggestions and corrections. The authors were partially supported by Macau Government FDCT through the Project No. 099/2014/A2, Two related topics in Clifford analysis. The authors J.M. and J.P.N. were also partly supported by FCT/Portugal through the Project Nos. UID/MAT/04459/2013, EXCL/MAT-GEO/0222/2012, and PTDC/MAT-GEO/3319/2014. J.M. was also partially supported by the Emerging Field Project on Quantum Geometry from Erlangen–Nürnberg University.
- 1. Bargmann, V., “On a Hilbert space of analytic functions and an associated integral transform, Part I,” Commun. Pure Appl. Math. 14, 187–214 (1961). https://doi.org/10.1002/cpa.3160140303, Google ScholarCrossref
- 2. Brackx, F., Delanghe, R., and Sommen, F., “Clifford analysis,” in Research Notes in Mathematics (Pitman, Boston, 1982), Vol. 76. Google Scholar
- 3. Colombo, F., Lavicka, R., Sabadini, I., and Soucek, V., “The Radon transform between monogenic and generalized slice monogenic functions,” Math. Ann. 363, 733–752 (2015). https://doi.org/10.1007/s00208-015-1182-3, Google ScholarCrossref
- 4. Colombo, F., Sabadini, I., and Sommen, F., “The inverse Fueter mapping theorem,” Commun. Pure Appl. Anal. 10, 1165–1181 (2011). https://doi.org/10.3934/cpaa.2011.10.1165, Google ScholarCrossref
- 5. Colombo, F., Sabadini, I., and Struppa, D. C., “Slice monogenic functions,” Isr. J. Math. 171, 385–403 (2009). https://doi.org/10.1007/s11856-009-0055-4, Google ScholarCrossref
- 6. Colombo, F., Sabadini, I., and Struppa, D. C., “An extension theorem for slice monogenic functions and some of its consequences,” Isr. J. Math. 177, 369–389 (2010). https://doi.org/10.1007/s11856-010-0051-8, Google ScholarCrossref
- 7. Colombo, F., Sabadini, I., and Struppa, D. C., Noncommutative Functional Calculus (Birkhäuser, 2011). Google ScholarCrossref
- 8. De Schepper, N. and Sommen, F., “Cauchy–Kowalevski extensions and monogenic plane waves using spherical monogenics,” Bull. Braz. Math. Soc. 44, 321–350 (2013). https://doi.org/10.1007/s00574-013-0016-8, Google ScholarCrossref
- 9. Delanghe, R., Sommen, F., and Soucek, V., “Clifford algebra and spinor–valued functions,” in Mathematics and its Applications (Kluwer, 1992), Vol. 53. Google ScholarCrossref
- 10. Driver, B., “On the Kakutani-Itô-Segal-Gross and Segal-Bargmann-Hall isomorphisms,” J. Funct. Anal. 133, 69–128 (1995). https://doi.org/10.1006/jfan.1995.1120, Google ScholarCrossref
- 11. Fueter, R., “Die funktionentheorie der differetialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen variablen,” Comment. Math. Helv. 7, 307–330 (1935). https://doi.org/10.1007/BF01292723, Google ScholarCrossref
- 12. Hall, B. C., “The Segal-Bargmann coherent-state transform for Lie groups,” J. Funct. Anal. 122, 103–151 (1994). https://doi.org/10.1006/jfan.1994.1064, Google ScholarCrossref
- 13. Hall, B. C., “Holomorphic methods in analysis and mathematical physics,” in First Summer School in Analysis and Mathematical Physics (Cuernavaca, Morelos, 1998), pp. 1–59, Contemp. Math. 260, American Mathematical Society, Providence, RI, 2000. Google Scholar
- 14. Helgason, S., Integral Geometry and Radon Transforms (Springer-Verlag, New York, 2011). Google ScholarCrossref
- 15. Kou, K. I., Qian, T., and Sommen, F., “Generalizations of Fueter’s theorem,” Methods Appl. Anal. 9, 273–289 (2002). https://doi.org/10.4310/maa.2002.v9.n2.a5, Google ScholarCrossref
- 16. Peña Peña, D., Qian, T., and Sommen, F., “An alternative proof of Fueter’s theorem,” Complex Var. Elliptic Equations 51, 913–922 (2006). https://doi.org/10.1080/17476930600667650, Google ScholarCrossref
- 17. Qian, T., “Generalization of Fueter’s result to ℝn+1,” Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8, 111–117 (1997). Google Scholar
- 18. Sce, M., “Osservazioni sulle serie di potenze nei moduli quadratici (Italian),” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 23, 220–225 (1957). Google Scholar
- 19. Segal, I., “Mathematical characterization of the physical vacuum for a linear Bose-Einstein field,” Ill. J. Math. 6, 500–523 (1962). Google ScholarCrossref, CAS
- 20. Segal, I., “The complex wave representation of the free Boson field,” in Topics in Functional Analysis: Essays Dedicated to M.G. Krein on the Occasion of His 70th Birthday, edited byGohberg, I. and Kac, M., Advances in Mathematics Supplementary Studies Vol. 3 (Academic Press, New York, 1978), pp. 321–343. Google Scholar
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