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

S. Ohmori, J. Takahashi

TL;DR

This work develops a rigorous bra-ket formalism for identical-particle systems within the rigged Hilbert space framework introduced by Madrid. It constructs bra and ket vectors in the dual spaces of a tensor-product RHS and extends the permutation symmetry to these duals, enabling generalized eigenvector decompositions of self-adjoint observables. The key contributions are a consistent spectral expansion in the dual spaces, a complete orthonormal system built from tensor-product eigenvectors, and a symmetry-preserving framework that links observable commutativity to symmetric/antisymmetric structures. The results provide a mathematically robust foundation for identical-particle quantum mechanics beyond $L^2$ spaces and hold potential for applications in open quantum systems, quantum statistics, and quantum field theory.

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.

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

TL;DR

This work develops a rigorous bra-ket formalism for identical-particle systems within the rigged Hilbert space framework introduced by Madrid. It constructs bra and ket vectors in the dual spaces of a tensor-product RHS and extends the permutation symmetry to these duals, enabling generalized eigenvector decompositions of self-adjoint observables. The key contributions are a consistent spectral expansion in the dual spaces, a complete orthonormal system built from tensor-product eigenvectors, and a symmetry-preserving framework that links observable commutativity to symmetric/antisymmetric structures. The results provide a mathematically robust foundation for identical-particle quantum mechanics beyond spaces and hold potential for applications in open quantum systems, quantum statistics, and quantum field theory.

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.
Paper Structure (11 sections, 1 theorem, 64 equations)

This paper contains 11 sections, 1 theorem, 64 equations.

Table of Contents

  1. Introduction
  2. Construction of the bra-ket vectors in the dual spaces
  3. General formulation
  4. Permutation operator on the dual spaces
  5. Observable
  6. Spectral expansion in the tensor product of RHS
  7. Eigenequations and Complete orthonormal system
  8. Extension to the dual spaces
  9. Symmetry of the bra-ket in the dual spaces
  10. Conclusion
  11. The relation (\ref{['eqn:extension_relation']})

Key Result

Lemma 4.1

Let $\Phi \subset \mathcal{H} \subset \Phi^\prime, \Phi^\times$ be an RHS and let $A:D(A)\to \mathcal{H}$ and $B:D(B)\to \mathcal{H}$ be self-adjoint operators in $\mathcal{H}$ such that they are continuous on $\Phi$ and the $A\Phi \subset \Phi$ and $B\Phi \subset \Phi$ are satisfied. If $A$ and $B$ are satisfied, where $\hat{A}$ and $\hat{B}$ are the extensions on $\Phi^\prime \cup \Phi^\times$.

Theorems & Definitions (2)

  • Lemma 4.1
  • proof