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Complex extension of quantum mechanics.

Carl M Bender1, Dorje C Brody, Hugh F Jones

  • 1Department of Physics, Washington University, St. Louis, MO 63130, USA.

Physical Review Letters
|January 7, 2003
PubMed
Summary

Quantum mechanics can be generalized using complex Hamiltonians with space-time reflection (PT) symmetry. This approach ensures unitary time evolution and positive definite norms, offering a broader framework for quantum theories.

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Area of Science:

  • Theoretical Physics
  • Quantum Mechanics
  • Mathematical Physics

Background:

  • The requirement of Hermitian Hamiltonians in quantum mechanics is a significant limitation.
  • Non-Hermitian Hamiltonians may lead to violations of unitarity, a fundamental principle.

Purpose of the Study:

  • To explore the possibility of a consistent quantum mechanics using non-Hermitian Hamiltonians.
  • To investigate the implications of space-time reflection (PT) symmetry in quantum systems.
  • To develop a generalized framework for quantum mechanics that accommodates non-Hermitian Hamiltonians.

Main Methods:

  • Consideration of complex Hamiltonians satisfying PT symmetry.
  • Construction of a novel symmetry operator C for PT-symmetric Hamiltonians.
  • Development of a positive definite inner product using the C operator.
  • Analysis of observable properties and time evolution.

Main Results:

  • A consistent quantum mechanical framework can be established with PT-symmetric, non-Hermitian Hamiltonians.
  • The presence of PT symmetry, when unbroken, allows for the construction of a positive definite norm.
  • A previously unnoticed symmetry C is identified and utilized.
  • Observables exhibit CPT symmetry, and time evolution remains unitary.

Conclusions:

  • The study presents a complex generalization of quantum mechanics, relaxing the Hermitian condition.
  • PT-symmetric Hamiltonians offer a viable alternative for constructing consistent quantum theories.
  • This work expands the theoretical foundations of quantum mechanics without contradicting conventional principles.

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