Mathematics – Representation Theory
Scientific paper
2007-11-16
Mathematics
Representation Theory
final version, published in Advances in Mathematics, 60 pages, 3 figures; Advances in Mathematics, 2008
Scientific paper
We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of quaternions. Representation theory also suggests a quaternionic version of the Cauchy formula for the second order pole. Remarkably, the derivative appearing in the complex case is replaced by the Maxwell equations in the quaternionic counterpart. We also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory, such as the spectrum of the hydrogen atom, polarization of vacuum, one-loop Feynman integrals. We also make some further conjectures. The main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis, representation theory and four-dimensional physics.
Frenkel Igor
Libine Matvei
No associations
LandOfFree
Quaternionic Analysis, Representation Theory and Physics does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Quaternionic Analysis, Representation Theory and Physics, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Quaternionic Analysis, Representation Theory and Physics will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-60882