Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus

The Time Independent Setting

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Abstract

Via a general construction, we are able to establish a quite general and comprehensive stability theory for Feynman’s operational calculus in the time independent setting. In particular, we are able to establish stability of the operational calculus with respect to general types of the time-ordering measures. While the domain of the operational calculus is somewhat restricted as compared to the “standard” version of the operational calculus (established by Jefferies and Johnson (Russ. J. Math. Phys. 8:153–171, 2001; Mat. Zametki 70:815–838, 2001; Adv. Appl. Clifford Algebras 11:239–264, 2001; Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5:171–199, 2002), the advantages of this relatively minor domain restriction are significant in that the stability theory (with respect to the time-ordering measures) as it stands to this time is contained, essentially in its entirety, in the principle result of this paper, Theorem 2. Moreover, Theorem 2 allows immediate, and rather far-reaching extensions of the stability theory that, until now, have not been possible.

Original languageEnglish
Pages (from-to)59-79
Number of pages21
JournalActa Applicandae Mathematicae
Volume138
Issue number1
DOIs
StatePublished - Jul 10 2014

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Operational Calculus
Stability Theory
Clifford Algebra
Theorem
Algebra
Minor
Restriction

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

Cite this

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title = "Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time Independent Setting",
abstract = "Via a general construction, we are able to establish a quite general and comprehensive stability theory for Feynman’s operational calculus in the time independent setting. In particular, we are able to establish stability of the operational calculus with respect to general types of the time-ordering measures. While the domain of the operational calculus is somewhat restricted as compared to the “standard” version of the operational calculus (established by Jefferies and Johnson (Russ. J. Math. Phys. 8:153–171, 2001; Mat. Zametki 70:815–838, 2001; Adv. Appl. Clifford Algebras 11:239–264, 2001; Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5:171–199, 2002), the advantages of this relatively minor domain restriction are significant in that the stability theory (with respect to the time-ordering measures) as it stands to this time is contained, essentially in its entirety, in the principle result of this paper, Theorem 2. Moreover, Theorem 2 allows immediate, and rather far-reaching extensions of the stability theory that, until now, have not been possible.",
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