Non-hermitian holography

Daniel Arean,Ignacio Salazar Landea,Karl Landsteiner

doi:10.21468/scipostphys.9.3.032

Daniel Arean, Ignacio Salazar Landea + Show 1 more

Open Access

https://doi.org/10.21468/scipostphys.9.3.032

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Abstract

Quantum theory can be formulated with certain non-Hermitian Hamiltonians. An anti-linear involution, denoted by PT, is a symmetry of such Hamiltonians. In the PT-symmetric regime the non-Hermitian Hamiltonian is related to a Hermitian one by a Hermitian similarity transformation. We extend the concept of non-Hermitian quantum theory to gauge-gravity duality. Non-Hermiticity is introduced via boundary conditions in asymptotically AdS spacetimes. At zero temperature the PT phase transition is identified as the point at which the solutions cease to be real. Surprisingly at finite temperature real black hole solutions can be found well outside the quasi-Hermitian regime. These backgrounds are however unstable to fluctuations which establishes the persistence of the holographic dual of the PT phase transition at finite temperature.

Highlights

One of the basic axioms of quantum mechanics is that the dynamics of a quantum system is generated by a Hermitian Hamiltonian
It comes as a surprise that meaningful quantum mechanics can be formulated for certain non-Hermitian Hamiltonians, the so-called PT-symmetric quantum mechanics [1, 2]
In order to assess the stability of our finite temperature solutions we study the quasinormal modes (QNM) of the system

Summary

Introduction

The gain/loss terms are exchanged by time-reversal T , which in quantum mechanics is just complex conjugation They are exchanged by the permutation of the subsystems A and B represented by the matrix P =. We can generate the non-Hermitian Hamiltonian from the Hermitian one by transforming the couplings g = (g , 0, 0) with cosh(α) 0 i sinh(α). The Hamiltonian is no longer quasi-Hermitian but it can be reached by taking the limit α → ∞, g → 0 while keeping the product fixed These special values of the couplings are generically known as “exceptional points”. Starting from the usual mass term and doing a complexified axial transformation one generates the non-Hermitian operator Ψγ5Ψ [6,7,8,9]1 Once the quasi-Hermitian theory is obtained it can be extended to the exceptional point and beyond

Holography

L2 such that and set

Conclusion and Outlook