Effects of absolute continuity in Feynman's operational calculus

Author:
Lance Nielsen

Journal:
Proc. Amer. Math. Soc. **131** (2003), 781-791

MSC (2000):
Primary 47A60, 46H30; Secondary 47A56, 47B38

DOI:
https://doi.org/10.1090/S0002-9939-02-06592-9

Published electronically:
July 17, 2002

MathSciNet review:
1937417

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Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the effects of having an absolute continuity relation between the time-ordering measures in Feynman's operational calculus. In particular, we obtain some theorems concerning the formation of functions of several noncommuting operators or operator-valued functions under specific absolute continuity assumptions on the time-ordering measures.

**1.**Defacio, B., Johnson, G.W., and Lapidus, M.L., Feynman's operational calculus and evolution equations.*Acta Applicandae Mathematicae***47**(1997), 155 - 211. MR**98f:58050****2.**Feynman, R.P., An operator calculus having applications in quantum electrodynamics,*Phys. Rev.***84**(1951), 108-128. MR**13:410e****3.**Jefferies, B., and Johnson, G.W., Feynman's operational calculi for noncommuting operators: definitions and elementary properties,*Russ. J. Math. Phys*.**8**(2001), to appear.**4.**-, Feynman's operational calcui for noncommuting operators: tensors, ordered support and disentangling an exponential factor,*Math. Notes***70**(2001), accepted for publication.**5.**Jefferies, B., Johnson, G.W., and Nielsen, L., Feynman's operational calculi for time-dependent noncommuting operators.*J. of Korean Math. Soc.***38**(2001), 193-226. MR**2001m:81154****6.**Johnson, G.W., and Lapidus, M.L., Generalized Dyson series, generalized Feynman diagrams, the Feynman integral and Feynman's operational calculus,*Mem. Amer. Math. Soc.***62**(1986), 1-78. MR**88f:81034****7.**-,*The Feynman Integral and Feynman's Operational Calculus*. Oxford Science Publications, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford and New York, 2000. MR**2001i:58015****8.**Johnson, G.W., Nielsen, L., A stability theorem for Feynman's operational calculus, Conference Proc. Canadian Math. Soc.: Conference in honor of Sergio Albeverio's 60th birthday, 2000. MR**2001k:47020****9.**Nielsen, L.,*Stability Properties for Feynman's Operational Calculus*, Ph.D. Dissertation, Mathematics, University of Nebraska, Lincoln, 1999.**10.**Nielsen, L., Stability properties of Feynman's operational calculus for exponential functions of noncommuting operators, submitted for publication.

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Additional Information

**Lance Nielsen**

Affiliation:
Department of Mathematics and Computer Science, Creighton University, Omaha, Nebraska 68178-2090

Email:
lnielsen@creighton.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06592-9

Keywords:
Functional calculus,
disentangling,
operational calculus

Received by editor(s):
July 10, 2001

Received by editor(s) in revised form:
October 10, 2001

Published electronically:
July 17, 2002

Communicated by:
N. Tomczak-Jaegermann

Article copyright:
© Copyright 2002
American Mathematical Society