\(\text{O}(3,\,1)\) harmonics and its application to Bethe-Salpeter equation. (English) Zbl 1098.81680

Seto, Kenji

Prog. Theor. Phys. 46, 1248-1260 (1971).

Summary: The \(O(3,1)\) harmonics are introduced as an extension of the \(O(2,1)\) ones given by Sertorio-Toller and are applied to solve the Bethe-Salpeter equation in the Wick-Cutkosky model without using the Wick-rotation. It is shown that the equation has physically unacceptable eigenvalues corresponding with complex-valued coupling constants which are connected to the eigenvalues known already through the \(O(4)\) symmetry by an analytical continuation in the four-dimensional angular momentum quantum number. On the other hand, it is shown that the solutions through the \(O(4)\) symmetry themselves cannot become solutions in our approach in a strict sense.
The lightlike limit of the harmonics are also discussed, where the \(E(2)\) harmonics are derived as a limit of the \(O(3,1)\) ones.

MSC:

81R05	Finite-dimensional groups and algebras motivated by physics and their representations
81Q40	Bethe-Salpeter and other integral equations arising in quantum theory