Geometric Aspects of Space-Time Reflection Symmetry in Quantum Mechanics

  • Carl M. Bender
  • Dorje C. Brody
  • Lane P. Hughston
  • Bernhard K. Meister
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 184)


For nearly two decades, much research has been carried out on properties of physical systems described by Hamiltonians that are not Hermitian in the conventional sense, but are symmetric under space-time reflection; that is, they exhibit \(\mathscr {PT}\) symmetry. Such Hamiltonians can be used to model the behavior of closed quantum systems, but they can also be replicated in open systems for which gain and loss are carefully balanced, and this has been implemented in laboratory experiments for a wide range of systems. Motivated by these ongoing research activities, we investigate here a particular theoretical aspect of the subject by unraveling the geometric structures of Hilbert spaces endowed with the parity and time-reversal operations, and analyze the characteristics of \(\mathscr {PT}\)-symmetric Hamiltonians. A canonical relation between a \(\mathscr {PT}\)-symmetric operator and a Hermitian operator is established in a geometric setting. The quadratic form corresponding to the parity operator, in particular, gives rise to a natural partition of the Hilbert space into two halves corresponding to states having positive and negative \(\mathscr {PT}\) norm. Positive definiteness of the norm can be restored by introducing a conjugation operator \(\mathscr {C}\); this leads to a positive-definite inner product in terms of \(\mathscr {CPT}\) conjugation.



We thank H. F. Jones and R. F. Streater for stimulating discussions. DCB was supported by The Royal Society and CMB was supported by the U.S. Department of Energy while the work presented here was carried out.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Carl M. Bender
    • 1
  • Dorje C. Brody
    • 2
    • 3
  • Lane P. Hughston
    • 2
    • 3
  • Bernhard K. Meister
    • 4
  1. 1.Department of PhysicsWashington UniversitySt. LouisUSA
  2. 2.Department of MathematicsBrunel University LondonUxbridgeUK
  3. 3.Department of Optical Physics and Modern Natural Science, ITMOSt PetersburgRussia
  4. 4.Department of PhysicsRenmin University of ChinaBeijingChina

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