Calmest Quixotic Henchman: Pandemonium! Car Blender

Transcription

1 Calmest Quixotic Henchman: Pandemonium! Car Blender

2 Quantum Mechanics in the Complex Domain Carl Bender

3 Quantum Mechanics is complicated! Anyone who thinks he can contemplate quantum mechanics without getting dizzy hasn t properly understood it. Niels Bohr Anyone who thinks they know quantum mechanics doesn t. Richard Feynman I don t like it, and I m sorry I ever had anything to do with it. Erwin Schrödinger

4 Quantum mechanics is not easy!

5 So, for my thesis I decided to work on something easy, like perturbation theory for the anharmonic oscillator... I walked into T. T. Wu s office one day with a third-order calculation...

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8 This led to studies of the large-order behavior of perturbation theory...

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10 Later complex analysis... (1)The delta expansion with F. Cooper, A. Duncan, H. Jones, K. Milton, M. Moshe, S. Pinsky, A. Rebhan, L. M. Simmons,... (2)Stokes wedges and analytic eigenvalue problems with A. Turbiner This meant that I was intellectually prepared to deal with the following problem...

11 (Bessis and Zinn-Justin looked at this Hamiltonian as a model for the Lee-Yang edge singularity...) 2 3 H = p + ix

12 5 or 6 years later H = p + ix Wait a minute This model has PT symmetry

13 CMB and Boettcher

14 Some references CMB and S. Boettcher, PRL 80, 5243 (1998) CMB, D. Brody, H. Jones, PRL 89, (2002) CMB, D. Brody, and H. Jones, PRL 93, (2004) CMB, D. Brody, H. Jones, B. Meister, PRL 98, (2007) CMB and P. Mannheim, PRL 100, (2008) CMB, Reports on Progress in Physics 70, 947 (2007) P. Dorey, C. Dunning, and R. Tateo, JPA 34, 5679 (2001) P. Dorey, C. Dunning, and R. Tateo, JPA 40, R205 (2007)

15 Axioms of Quantum Mechanics Causality Locality Stability of the vacuum Relativity Probabilistic interpretation

16 Dirac Hermiticity guarantees real energy and conserved probability but is mathematical and not physical (Assuming Dirac Hermiticity is like postulating g μ in general relativity!)

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19 Picture from Prostejov (M. Znojil)

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21 The original discoverers of PT symmetry:

22 PT phase transition

23 OK, so the eigenvalues are real But is this quantum mechanics?? Probabilistic interpretation?? Hilbert space with a positive metric?? Unitarity??

24 The Hamiltonian determines its own adjoint

25 EXAMPLE: A non-hermitian PT-symmetric Hamiltonian where

26 Unitarity With respect to the CPT adjoint the theory has UNITARY time evolution. Norms are strictly positive! Probability is conserved!

27 OK, we have unitarity But is PT quantum mechanics useful?? It revives quantum theories that were thought to be dead It is beginning to be observed experimentally

28 The Lee Model

29 The problem:

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31 A non-hermitian Hamiltonian is unacceptable partly because it may lead to complex energy eigenvalues, but chiefly because it implies a nonunitary S matrix, which fails to conserve probability and makes a hash of the physical interpretation.

32 GHOSTBUSTING!

33 Pais-Uhlenbeck action Gives a fourth-order field equation: CMB and P. Mannheim, Phys. Rev. Lett. 100, (2008) CMB and P. Mannheim, Phys. Rev. D 78, (2008)

34 The problem: A fourth-order field equation gives a propagator like GHOST!

35 TOTALITARIAN PRINCIPLE Everything which is not forbidden is compulsory. ---M. Gell-Mann

36 Date: Thu, 13 Mar :04: From: Demetrios Christodoulides To: Carl M. Bender Subject: Re: Benasque workshop on non-hermitian Hamiltonians Dear Carl, I have some good news from Greg Salamo (U. of Arkansas). His students (who are now visiting us here in Florida) have just observed a PT phase transition in a passive AlGaAs waveguide system. We will be submitting soon these results as a post-deadline paper to CLEO/QELS and subsequently to a regular journal. We are still fighting against the Kramers-Kronig relations, but the phase transition effect is definitely there. We expect even better results under TE polarization conditions. I will bring them over to Israel. In close collaboration with us, more teams (also best friends!) are moving ahead in this direction. Moti Segev (from Technion) is planning an experiment in an active-passive dual core optical fiber -- fabricated in Southampton, England. More experiments will be carried later in Germany by Detlef Kip. Christian (his post doc) just left from here with a possible design. If everything goes well, with a bit of luck we may have an experimental explosion in the PT area. I wish the funding situation was a bit better. So far everything is done on a shoe-string budget (it is subsidized by other projects). Let us see... All the best Demetri

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42 Quantum experiment now being designed in Heidelberg by Maarten F. M. DeKieviet Stern- Gerlach Magnetic Field Magnetic Field + + Interference pattern! (spin echo)

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45 Complex classical mechanics Provides an intuitive explanation of what is going on

46 Motion on the Real Axis Motion of particles is governed by Newton s Law: F=ma In freshman physics this motion is restricted to the REAL AXIS.

47 Harmonic Oscillator: Particle on a Spring Back and forth motion on the real axis: Turning point Turning point ( = 0)

48 Hamilton s equations

49 Harmonic Oscillator: Motion in the complex plane: Turning point Turning point ( = 0)

50 2 ix 3 p H + = ( = 1)

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52 ( < 0) Broken PT symmetry

53 = 2 11 sheets

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55 The Beast = 16/15

56 Lotka-Volterra equations

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60 Conduction bands for a classical periodic potential V(x)=sin x T. Arpornthip

61 Other PT-symmetric classical systems KdV equation Camassa-Holm equation Sine-Gordon equation Boussinesq equation

62 The shortest path between two truths in the real domain passes through the complex domain. -- Jacques Hadamard [The Mathematical Intelligencer 13 (1991)] PT Meep! Meep! ptarmigan