No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

The full text of this article is not currently available.

f

Extending coherent state transforms to Clifford analysis

### Abstract

Segal-Bargmann coherent state
transforms can be viewed as unitary maps from L
^{2} spaces of functions (or sections of an appropriate line bundle) on a manifold X to spaces of square integrable holomorphic functions (or sections) on X
ℂ. 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 L
^{2}(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.

Published by AIP Publishing.

Received 21 April 2016
Accepted 26 September 2016
Published online 11 October 2016

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.

Article outline:

I. INTRODUCTION
II. PRELIMINARIES
A. Coherent statetransforms (CSTs)
B. Clifford analysis
III. MONOGENIC EXTENSIONS OF ANALYTIC FUNCTIONS
A. Slice monogenic extension
B. Axial monogenic extension and dual Radon transform
IV. CLIFFORD EXTENSIONS OF THE CST
A. Slice monogenic coherent statetransform (SCST)
B. Axial monogenic coherent statetransform (ACST)
V. REPRESENTATION THEORETIC AND QUANTUM MECHANICAL INTERPRETATION

/content/aip/journal/jmp/57/10/10.1063/1.4964448

2.

Brackx, F. , Delanghe, R. , and Sommen, F. , “Clifford analysis,” in Research Notes in Mathematics (Pitman, Boston, 1982), Vol. 76.

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).

http://dx.doi.org/10.1007/s00208-015-1182-3
7.

Colombo, F. , Sabadini, I. , and Struppa, D. C. , Noncommutative Functional Calculus (Birkhäuser, 2011).

9.

Delanghe, R. , Sommen, F. , and Soucek, V. , “Clifford algebra and spinor–valued functions,” in Mathematics and its Applications (Kluwer, 1992), Vol. 53.

11.

Fueter, R. , “Die funktionentheorie der differetialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen variablen,” Comment. Math. Helv. 7, 307–330 (1935).

http://dx.doi.org/10.1007/BF01292723
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.

14.

Helgason, S. , Integral Geometry and Radon Transforms (Springer-Verlag, New York, 2011).

17.

Qian, T. , “Generalization of Fueter’s result to ℝ^{n+1},” Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8, 111–117 (1997).

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).

19.

Segal, I. , “Mathematical characterization of the physical vacuum for a linear Bose-Einstein field,” Ill. J. Math. 6, 500–523 (1962).

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.

http://aip.metastore.ingenta.com/content/aip/journal/jmp/57/10/10.1063/1.4964448

Article metrics loading...

/content/aip/journal/jmp/57/10/10.1063/1.4964448

2016-10-11

2016-10-22

### Abstract

Segal-Bargmann coherent state
transforms can be viewed as unitary maps from L
^{2} spaces of functions (or sections of an appropriate line bundle) on a manifold X to spaces of square integrable holomorphic functions (or sections) on X
ℂ. 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 L
^{2}(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.

Full text loading...

/deliver/fulltext/aip/journal/jmp/57/10/1.4964448.html;jsessionid=uory2KDgHR3tiBSkCxWR9Idd.x-aip-live-06?itemId=/content/aip/journal/jmp/57/10/10.1063/1.4964448&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/jmp

###
Most read this month

Article

content/aip/journal/jmp

Journal

5

3

true

true

Commenting has been disabled for this content