Metric walk in matrix formalism across $η$-pseudo-unitary groups in the S-matrix framework

Abstract

Pseudo-unitary circuits are silently recurrent in S-matrix theory. We propose a matrix and diagrammatic representation for the operation that maps S-matrices to T-matrices, and consequently a unitary group to a pseudo-unitary one. We call this operation partial inversion and show its diagrammatic representation in terms of permutations. We find the expressions for the deformed metrics and deformed dot products that preserve physical constraints after partial inversion. Subsequently, we define a special set that allows for the simplification of expressions containing infinities in matrix inversion. Finally, we proposed a renormalizedgrowth algorithm for the T-matrix as a possible application. Our studies furnish all the tools needed to build a family of pseudo-unitary and inter-pseudo-unitary circuits with full diagrammatic representation in three dimensions.