General Construction of Bra-Ket Formalism for Identical Particle Systems in Rigged Hilbert Space Approach

TL;DR

This paper develops a rigorous mathematical framework for Dirac's bra-ket formalism applied to identical particle systems within the rigged Hilbert space approach, ensuring consistency with particle symmetry and spectral expansion.

Contribution

It constructs bra-ket vectors and permutation operators in rigged Hilbert spaces for identical particles, extending the spectral theorem to this setting.

Findings

Bra-ket vectors are described in dual and anti-dual spaces for tensor products.

Permutation operators are constructed as operators on dual spaces.

Spectral expansion is consistent with identical particle symmetry when operators commute.

Abstract

This study discussed Dirac's bra-ket formalism for the identical particles system based on the rigged Hilbert space reformulated by R. Madrid [J. Phys A:Math. Gen. 37, 8129 (2004)]. The bra and ket vectors for a composite system that form the basis of an identical particle system are described in dual and anti-dual spaces for the tensor product of rigged Hilbert spaces. The permutation operator that characterizes the symmetry of identical particles is constructed as the operator on such dual spaces. We also show that the nuclear spectral theorem in the tensor product of rigged Hilbert spaces endows the spectral expansion of the self-adjoint operator in the dual and anti-dual spaces and the expansion is consistent with the identicle particle system when the permutation operator commutes the self-adjoint operator.