### Abstract

The short-range behaviour of certain Feynman integrals reveals mathematical properties which are not those of either functions or distributions-they contain terms which are more singular than distributions and possess inherent ambiguities. Two classes of singularities exist: To the first one belong all those singularities which have a physical meaning in the sense that in a convergent (regularized) quantum field theory they contribute to observable quantities, frequently as renormalization constants. Most of the singularities of the second, the spurious type, violate the symmetries of the Lagrangian. We demonstrate that they are associated with certain mathematical difficulties of unregularized theories. Much of our analysis deals with the isolation of singularities of this type and with the study of the properties of the singular products of distribution. We argue that the four-dimensional integration leading to the S-matrix in the perturbation expansion must be carried out over an open domain which leaves out the contributions from singularities of the contact type, that is terms proportional to δ^{4}x-y.

Original language | English (US) |
---|---|

Pages (from-to) | 326-367 |

Number of pages | 42 |

Journal | Il Nuovo Cimento A |

Volume | 49 |

Issue number | 3 |

DOIs | |

State | Published - Feb 1979 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics
- Nuclear and High Energy Physics
- Astronomy and Astrophysics

### Cite this

*Il Nuovo Cimento A*,

*49*(3), 326-367. https://doi.org/10.1007/BF02773451

**S{cyrillic}i{cyrillic}n{cyrillic}g{cyrillic}u{cyrillic}l{cyrillic}ya{cyrillic}r{cyrillic}n{cyrillic}o{cyrillic}s{cyrillic}t{cyrillic}i{cyrillic} v{cyrillic} k{cyrillic}v{cyrillic}a{cyrillic}n{cyrillic}t{cyrillic}o{cyrillic}v{cyrillic}o{cyrillic}i{cyrillic, short} t{cyrillic}ye{cyrillic}o{cyrillic}r{cyrillic}i{cyrillic}i{cyrillic} p{cyrillic}o{cyrillic}l{cyrillic}ya{cyrillic}.** / Danos, M.; Rafelski, Johann.

Research output: Contribution to journal › Article

*Il Nuovo Cimento A*, vol. 49, no. 3, pp. 326-367. https://doi.org/10.1007/BF02773451

}

TY - JOUR

T1 - S{cyrillic}i{cyrillic}n{cyrillic}g{cyrillic}u{cyrillic}l{cyrillic}ya{cyrillic}r{cyrillic}n{cyrillic}o{cyrillic}s{cyrillic}t{cyrillic}i{cyrillic} v{cyrillic} k{cyrillic}v{cyrillic}a{cyrillic}n{cyrillic}t{cyrillic}o{cyrillic}v{cyrillic}o{cyrillic}i{cyrillic, short} t{cyrillic}ye{cyrillic}o{cyrillic}r{cyrillic}i{cyrillic}i{cyrillic} p{cyrillic}o{cyrillic}l{cyrillic}ya{cyrillic}

AU - Danos, M.

AU - Rafelski, Johann

PY - 1979/2

Y1 - 1979/2

N2 - The short-range behaviour of certain Feynman integrals reveals mathematical properties which are not those of either functions or distributions-they contain terms which are more singular than distributions and possess inherent ambiguities. Two classes of singularities exist: To the first one belong all those singularities which have a physical meaning in the sense that in a convergent (regularized) quantum field theory they contribute to observable quantities, frequently as renormalization constants. Most of the singularities of the second, the spurious type, violate the symmetries of the Lagrangian. We demonstrate that they are associated with certain mathematical difficulties of unregularized theories. Much of our analysis deals with the isolation of singularities of this type and with the study of the properties of the singular products of distribution. We argue that the four-dimensional integration leading to the S-matrix in the perturbation expansion must be carried out over an open domain which leaves out the contributions from singularities of the contact type, that is terms proportional to δ4x-y.

AB - The short-range behaviour of certain Feynman integrals reveals mathematical properties which are not those of either functions or distributions-they contain terms which are more singular than distributions and possess inherent ambiguities. Two classes of singularities exist: To the first one belong all those singularities which have a physical meaning in the sense that in a convergent (regularized) quantum field theory they contribute to observable quantities, frequently as renormalization constants. Most of the singularities of the second, the spurious type, violate the symmetries of the Lagrangian. We demonstrate that they are associated with certain mathematical difficulties of unregularized theories. Much of our analysis deals with the isolation of singularities of this type and with the study of the properties of the singular products of distribution. We argue that the four-dimensional integration leading to the S-matrix in the perturbation expansion must be carried out over an open domain which leaves out the contributions from singularities of the contact type, that is terms proportional to δ4x-y.

UR - http://www.scopus.com/inward/record.url?scp=0348182732&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0348182732&partnerID=8YFLogxK

U2 - 10.1007/BF02773451

DO - 10.1007/BF02773451

M3 - Article

AN - SCOPUS:0348182732

VL - 49

SP - 326

EP - 367

JO - Il Nuovo Cimento A Series 11

JF - Il Nuovo Cimento A Series 11

SN - 0369-3546

IS - 3

ER -