Quaternionic Analysis, Representation Theory and Physics II
Abstract
We develop further quaternionic analysis introducing left and right doubly regular functions. We derive CauchyFueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula for the second order pole, in addition to the one studied in the first paper with the same title. We also realize the doubly regular functions as a subspace of the quaternionicvalued functions satisfying a Euclidean version of Maxwell's equations for the electromagnetic field. Then we return to the study of the original quaternionic analogue of Cauchy's second order pole formula and its relation to the polarization of vacuum. We find the decomposition of the space of quaternionicvalued functions into irreducible components that include the spaces of doubly left and right regular functions. Using this decomposition, we show that a regularization of the vacuum polarization diagram is achieved by subtracting the component corresponding to the onedimensional subrepresentation of the conformal group. After the regularization, the vacuum polarization diagram is identified with a certain second order differential operator which yields a quaternionic version of Maxwell equations. Next, we introduce two types of quaternionic algebras consisting of spaces of scalarvalued and quaternionicvalued functions. We emphasize that these algebra structures are invariant under the action of the conformal Lie algebra. This uses techniques from our study of the vacuum polarization diagram. These algebras are not associative, but we can define an infinite family of nmultiplications, and we conjecture that they have structures of weak cyclic Ainfinity algebras. We also conjecture the relation between the multiplication operations of the scalar and nonscalar quaternionic algebras with the nphoton Feynman diagrams in the scalar and ordinary conformal QED.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1907.01594
 Bibcode:
 2019arXiv190701594F
 Keywords:

 Mathematics  Representation Theory;
 Mathematical Physics;
 Mathematics  Complex Variables
 EPrint:
 submitted, 87 pages, 8 figures