### Abstract

In this paper we develop a method of forming functions of noncommuting operators (or disentangling) using functions that are not necessarily analytic at the origin in ℂ ^{n} . The method of disentangling follows Feynman's heuristic rules from in (Feynman in Phys. Rev. 84:18-128, 1951) a mathematically rigorous fashion, generalizing the work of Jefferies and Johnson and the present author in (Jefferies and Johnson in Russ. J. Math. 8:153-181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193-226, 2001). In fact, the work in (Jefferies and Johnson in Russ. J. Math. 8:153-181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193-226, 2001) allow only functions analytic in a polydisk centered at the origin in ℂ ^{n} while the method introduced in this paper enable functions that are not analytic at the origin to be used. It is shown that the disentangling formalism introduced here reduces to that of (Jefferies and Johnson in Russ. J. Math. 8:153-181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193-226, 2001) under the appropriate assumptions. A basic commutativity theorem is also established.

Original language | English |
---|---|

Pages (from-to) | 409-429 |

Number of pages | 21 |

Journal | Acta Applicandae Mathematicae |

Volume | 110 |

Issue number | 1 |

DOIs | |

State | Published - Apr 2010 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Applied Mathematics

### Cite this

**Feynman's operational calculi : Disentangling away from the origin.** / Nielsen, Lance.

Research output: Contribution to journal › Article

*Acta Applicandae Mathematicae*, vol. 110, no. 1, pp. 409-429. https://doi.org/10.1007/s10440-008-9428-7

}

TY - JOUR

T1 - Feynman's operational calculi

T2 - Disentangling away from the origin

AU - Nielsen, Lance

PY - 2010/4

Y1 - 2010/4

N2 - In this paper we develop a method of forming functions of noncommuting operators (or disentangling) using functions that are not necessarily analytic at the origin in ℂ n . The method of disentangling follows Feynman's heuristic rules from in (Feynman in Phys. Rev. 84:18-128, 1951) a mathematically rigorous fashion, generalizing the work of Jefferies and Johnson and the present author in (Jefferies and Johnson in Russ. J. Math. 8:153-181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193-226, 2001). In fact, the work in (Jefferies and Johnson in Russ. J. Math. 8:153-181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193-226, 2001) allow only functions analytic in a polydisk centered at the origin in ℂ n while the method introduced in this paper enable functions that are not analytic at the origin to be used. It is shown that the disentangling formalism introduced here reduces to that of (Jefferies and Johnson in Russ. J. Math. 8:153-181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193-226, 2001) under the appropriate assumptions. A basic commutativity theorem is also established.

AB - In this paper we develop a method of forming functions of noncommuting operators (or disentangling) using functions that are not necessarily analytic at the origin in ℂ n . The method of disentangling follows Feynman's heuristic rules from in (Feynman in Phys. Rev. 84:18-128, 1951) a mathematically rigorous fashion, generalizing the work of Jefferies and Johnson and the present author in (Jefferies and Johnson in Russ. J. Math. 8:153-181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193-226, 2001). In fact, the work in (Jefferies and Johnson in Russ. J. Math. 8:153-181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193-226, 2001) allow only functions analytic in a polydisk centered at the origin in ℂ n while the method introduced in this paper enable functions that are not analytic at the origin to be used. It is shown that the disentangling formalism introduced here reduces to that of (Jefferies and Johnson in Russ. J. Math. 8:153-181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193-226, 2001) under the appropriate assumptions. A basic commutativity theorem is also established.

UR - http://www.scopus.com/inward/record.url?scp=77952093989&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952093989&partnerID=8YFLogxK

U2 - 10.1007/s10440-008-9428-7

DO - 10.1007/s10440-008-9428-7

M3 - Article

VL - 110

SP - 409

EP - 429

JO - Acta Applicandae Mathematicae

JF - Acta Applicandae Mathematicae

SN - 0167-8019

IS - 1

ER -