Generalized "bra-ket" formalism

TL;DR

This paper extends Dirac's bra-ket formalism to finite-dimensional vector spaces with indefinite metrics, providing a mathematical framework for basis transformations and group representations, especially for the $SL(2,\mathbb{C})$ group.

Contribution

It introduces a generalized bra-ket calculus for indefinite metrics, facilitating the study of basis transformations and finite-dimensional group representations.

Findings

Developed a mathematical framework for indefinite metric spaces.

Applied the formalism to finite-dimensional representations of $SL(2,\mathbb{C})$.

Provided calculation rules and notation for the generalized bra-ket calculus.

Abstract

The Dirac's bra-ket formalism is generalized to finite-dimensional vector spaces with indefinite metric in a simple mathematical context similar to thatof the theory of general tensors where, in addition, scalar products are introduced with the help of a metric operator. The specific calculation rules are given in a suitable intuitive notation. It is shown that the proposed bra-ket calculus is appropriate for the general theory of basis transformations and finite-dimensional representations of the symmetry groups of the metric operators. The presented application is the theory of finite-dimensional representations of the $SL(2,\Comp)$ group with invariant scalar products. Pacs: 02.10.Sp, 02.20.Qs