×

The Riesz-Clifford functional calculus for non-commuting operators and quantum field theory. (English) Zbl 0853.47012

Summary: We present a Riesz-like hyperholomorphic functional calculus for a set of non-commuting operators based on Clifford analysis. Applications to the quantum field theory are described.

MSC:

47A60 Functional calculus for linear operators
47N50 Applications of operator theory in the physical sciences
81T10 Model quantum field theories
PDFBibTeX XMLCite
Full Text: arXiv

References:

[1] and , Theory of Linear Operators in Hilbert Space, Vol. 1, Pitman Advanced Publishing Program, London, 1981.
[2] Anderson, J. Funct. Anal. 4 pp 240– (1969)
[3] Berezin, Math. USSR Izvestija 8 pp 1109– (1974)
[4] and , Clifford Analysis, Research Notes in Mathematics, Vol. 76, Pitman Advanced Publishing Program, Boston, 1982.
[5] ’Berezin-Toeplitz quantization’, in: Algebraic Methods in Operator Theory, pp. 101-108, Birkhäuser, New York, 1994. · doi:10.1007/978-1-4612-0255-4_12
[6] Coburn, Comm. Math. Phys.
[7] Delanghe, Math. Ann. 196 pp 293– (1972)
[8] and , Clifford Algebra and Spinor-Valued Functions, Kluwer Academic Publishers, Dordrecht, 1992. · doi:10.1007/978-94-011-2922-0
[9] Lectures on Quantum Field Theory, Yeshiva University, New York, 1967.
[10] ed., C*-Algebras: 1943-1993, Contemporary Mathematics 167, AMS, Providence, RI, 1994.
[11] ’Some quantizations and reflections inspired by the Gelfand-Naimark theorem’, in: [10], pp. 99-113. · Zbl 0824.46072
[12] Feynman, Phys. Rev. 84 pp 108– (1951)
[13] Harmonic Analysis in Phase Space, Princeton University Press, Princeton, NJ, 1989. · Zbl 0682.43001 · doi:10.1515/9781400882427
[14] The Analysis of Linear Partial Differential Operators III: Pseudodifferential Operators, Springer, Berlin, 1985.
[15] Howe, Bull. AMS (New Series) 3 pp 821– (1980) · Zbl 0442.43002 · doi:10.1090/S0273-0979-1980-14825-9
[16] Howe, J. Funct. Anal. 38 pp 188– (1980)
[17] Geometric Quantization, Encyclopedia of Mathematical Sciences, Vol. 4, pp. 137-172, Springer, Berlin, 1990.
[18] ’Relative convolutions, I. Properties and applications’, Reporte Interno # 162, Departamento de Matemáticas, CINVESTAV del I.P.N., Mexico City, 1994;
[19] Adv Math.
[20] and , ’Polynomial solutions of the Fueter-Hurwitz equation’, in: The Madison Symposium on Complex Analysis ( and , eds.), Contemporary Mathematics 137, pp. 297-305, AMS, Providence, RI, 1992. · doi:10.1090/conm/137/1190991
[21] ’Sur un calcul symbolique de Feynmann’, Seminar d’ Analyse, Lecture Notes in Math. Vol. 1295, pp. 132-145, Springer, Berlin, 1987.
[22] McIntosh, Indiana Univ. Math. J. 36 pp 421– (1987)
[23] ’Hypercomplex differentiability and its applications’, in: Clifford Algebras and Applications in Mathematical Physics, and , eds., pp. 141-150, Kluwer Academic Publishers, Netherlands, 1993. · doi:10.1007/978-94-011-2006-7_17
[24] Operational Methods, Nauka, Moscow, 1973.
[25] ’Quantization and C*-algebras’, in: [10], pp. 68-97.
[26] and , Functional Analysis, Ungar, New York, 1955.
[27] ’C*-algebras and quantization’, in: [10], pp. 55-65. · Zbl 0853.46069
[28] Pseudodifferential Operators and Spectral Theory, Springer, Berlin, 1987. · Zbl 0616.47040 · doi:10.1007/978-3-642-96854-9
[29] Sudbery, Math. Proc. Camb. Phil. Soc. 85 pp 197– (1979)
[30] Taylor, Acta Math. 125 pp 1– (1970)
[31] Taylor, Adv. Math. 9 pp 183– (1972)
[32] Pseudodifferential Operators, Princeton Mathematical Series, Vol. 34, Princeton University Press, Princeton, NJ, 1981.
[33] Noncommutative Harmonic Analysis, Math. Surv. and Monographs, Vol. 22, American Mathematical Society, Providence, RI, 1986. · doi:10.1090/surv/022
[34] The Theory of Groups and Quantum Mechanics, Dover, New York, 1950.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.