January 12, 2006 Physics Institute, Prague 1

PT-symetrické modely v kvantové fyzice:

novinky z konference (Stellenbosch, listopad 2005)

January 12, 2006 Physics Institute, Prague 2

Otázka první: No jo, novinky,

ale v jaké oblasti fyziky?

Odpověď (v uspořádání podle řečníků):

Scholz, Geyer, Heiss: jaderná fyzika Mostafazadeh: relativistická QM Bender: teorie pole a kosmologie Berry: optika Günther: magnetohydrodynamika a další: statistika,nelin.f.,SUSY,atom.f.

January 12, 2006 Physics Institute, Prague 3

Otázka druhá: jestliže novinky, pak napřed to nejzajímavější

Uspořádání vyjde skoro stejně: Scholz, Geyer: o jednoznačnosti Θ Mostafazadeh: klasická limita PTSQM Bender: usnadnění regularizace Feyn.d. Berry, Ahmed: singularity, wedge-edge Günther, Kirillov: okolí EP, klasifikace Jones: Thirring, sine-Gordon (+rand.m.)

January 12, 2006 Physics Institute, Prague 4

Napřed ale krátká rekapitulace PTT: nehermitovské H – levé a pravé stavy matem.: nutnost dokázat realitu spektra modif. Dir. značení: << L | a | R > biorthogonalita, bi-úplnost, spektr.repr. cíl: fyzika. Potřeba vybrat metriku Ө a inovovat principy korespondence; v tomto smyslu je PT pouhá technikalita

January 12, 2006 Physics Institute, Prague 5

No a teď k prvnímu tématu:

připomínám: jedná se o starý problém nejednoznačnosti Θ s jadernou motivací v Dysonově IBM trfci herm. -> neherm.

hlavní výhoda: zjednodušení výpočtů hlavní pokrok: Scholz (anglicky: PTO)

January 12, 2006 Physics Institute, Prague 6

1. Important results on the ambiguity of physical metrics

Scholz (and Kriel): a transfer of ambiguity to PDE boundary conditions (inspired by Moyal, using Weber’s flow equations)

added by A. M.: the disappearance of the ambiguity in the classical limit

added by H.G. and H. J.: illustration in the Swanson's creation/annihilation model

January 12, 2006 Physics Institute, Prague 7

a. A transfer of ambiguity to PDE boundary conditions

Frederik ScholtzMetrics and more from Moyal products [idea and ix^3 illustration]

Johannes N KrielSolving non-perturbative flow

equations [Poster, some details of the method]

January 12, 2006 Physics Institute, Prague 8

b. Ambiguities and their disappearance in classical limit

Ali Mostafazadeh (Koc/Istanbul, Turkey)

Dynamically Equivalent but Kinematically Distinct Pseudo-Hermitian Quantum Systems

[example: ix^3]

January 12, 2006 Physics Institute, Prague 9

c. Ambiguities in the Swanson's creation/annihilation model

Hendrik GeyerChoice of physical operators and

uniqueness of the metric

[three different closed formulae for the metric in Hilbert space]

January 12, 2006 Physics Institute, Prague 10

Přišel čas na druhé téma:

připomínám: očekává se vybudování klasické mechaniky s komplexním x či p

hlavní nevýhoda: třista let to nikoho nenapadlo

hlavní pokrok: Ali M. (anglicky: PTO)

January 12, 2006 Physics Institute, Prague 11

2. PT-symmetric-like systems in classical kinematical regime

Bender: HO phase-space trajectories

Berry: numerical experiments in optics

Uwe G.: α-dynamos

January 12, 2006 Physics Institute, Prague 12

a. HO phase-space trajectories from complex initial bc's

Carl M Bender (St Louis, USA) Some new results on PT-symmetric classical and quantum mechanics

[N-tuple winding phenomenon – chaos?]

January 12, 2006 Physics Institute, Prague 13

b. Numerical experiments in optics

Michael V Berry (Bristol, UK) The optical singularities of

bianisotropic crystals

(-> D. Heiss: two-by-two model)

January 12, 2006 Physics Institute, Prague 14

c. Fourth-order spectral branch points in MHD dynamos

Uwe Günther Bundle stratification of PT-

symmetric 4x4 matrix systems

[exceptional points - cf. also K. and M.]

January 12, 2006 Physics Institute, Prague 15

Třetí okruh otázek a odpovědí

věnován bohapustějším spekulacím

a proto nejzajímavější pro lidi jako já;

ale přejděme raději zase do angličtiny:

January 12, 2006 Physics Institute, Prague 16

3. Key innovations within quasihermitian QM itself

reflectionless models for cosmology [edge-of-the-wedge bc’s, ABB] Kleefeld: De-Broglie-Bohm

quantization M. Z.: coupled-channel models and

the concept of RT symmetry (details later)

January 12, 2006 Physics Institute, Prague 17

a. Edge-of-the-wedge bc's

Zafar Ahmed (Bhabha Atomic Research Centre, Mumbai, India)

[with Bender and Berry – both see elsewhere]

January 12, 2006 Physics Institute, Prague 18

b. De-Broglie-Bohm-like approach to PT quantization

Frieder Kleefeld Non-Hermitian Quantum Theory and the Correspondence

Principle

[remember Prague]

January 12, 2006 Physics Institute, Prague 19

c. The concept of RT symmetry

Miloslav Znojil (CAS, Prague) Coupling of channels in PT-symmetric models [Rotation R need not be Hermitian] Lecture on the web: [.pdf] - 131kb

January 12, 2006 Physics Institute, Prague 20

Dříve než dodám podrobnosti,

o svém díle, vraťme se ještě v rychlosti

k nelineárním modelům, k relativistickým modelům, k některým vzrušujícím leč čistě

matematickým studiím, a konečně i k seznamu témat o nichž se nezmíním vrátane „satelitního” tématu „RMT”.

January 12, 2006 Physics Institute, Prague 21

4. Nonlinear physical models

Hugh Jones: Sine-Gordon and massive Thirring

E-M. Graefe: Finite-dimensional matrix

nonlinear Schroedinger equation

January 12, 2006 Physics Institute, Prague 22

a. Sine-Gordon and massive Thirring models

Hugh Jones

Equivalent Hamiltonians for PT-symmetric versions of dual 2-dimensional field theories

[form invariance]

January 12, 2006 Physics Institute, Prague 23

5. Relativistic physical models

Ali Mostafazadeh: First-quantized Klein Gordon

systems

C. Berkdemir: Dirac equation – solvable cases

January 12, 2006 Physics Institute, Prague 24

a. First-quantized Klein Gordon system

Ali Mostafazadeh (Koc/Istanbul, Turkey)

Dynamically Equivalent but Kinematically Distinct Pseudo-Hermitian Systems

[freedom and limitations]

January 12, 2006 Physics Institute, Prague 25

6. New mathematical methods

Rigorous proofs of the reality of energies

Perturbation theory

Zeros of polynomials

January 12, 2006 Physics Institute, Prague 26

a. Rigorous proofs using the method of TQ equation Patrick Dorey On the reality (and unreality) of PT-symmetric quantum

mechanics

[see Shizuoka]

January 12, 2006 Physics Institute, Prague 27

b. Perturbation theory

Oleg Kirillov

Perturbation theory for non-self-adjoint operator matrices and its application to the MHD a^2 -dynamo

January 12, 2006 Physics Institute, Prague 28

c. Zeros of polynomials

Stefan Weigert

Detecting “Broken” PT-Symmetry

[19th century mathematics]

January 12, 2006 Physics Institute, Prague 29

7. and miscellanea

delta pot.: Uncu, num.: Weideman stability of perturbations: Caliceti collective Field Theory (Feinberg) Lipkin model (Heiss) pseudo-norms (Lévai) exceptional points (Mailybaev) unbound states (Mondragon)

January 12, 2006 Physics Institute, Prague 30

Satellite section: Non-Hermitian random-matrix models

Hatano-Nelson, QCD with chem. pot. Jacobus Verbaarschot (Stony Brook, USA)

Statistical physics structures Joshua Feinberg (Haifa, Israel)

Unexpected links Maciej Nowak (Krakow, Poland)

January 12, 2006 Physics Institute, Prague 31

No a dočkali jsme se:

teď moje vlastní nejnovější výsledky