Making sense of non Hermitian Hamiltonians. Crab Lender Practised Nymphets Washing Nervy Tuitions

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1 Making sense of non Hermitian Hamiltonians Crab Lender Practised Nymphets Washing Nervy Tuitions

2 Making sense of non Hermitian Hamiltonians Carl Bender Physics Department Washington University

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

5 Dirac Hermiticity guarantees real energy and conserved probability but is mathematical and not physical

6 2 3 H = p + ix

7 2 3 H = p + ix Wait a minute This model has PT symmetry

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9 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)

10 The original discoverers of PT symmetry:

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19 MY TALK

20 Outline of Talk Beginning Middle End (applause)

21 How to prove that the eigenvalues are real

22 PT Boundary

23 How to prove that the eigenvalues are real The proof is really hard! You need to use (3)Bethe ansatz (4)Monodromy group (5)Baxter T Q relation (6)Functional Determinants

24 PT Boundary Greatest murder mystery of all time Extinction of over 90% of species! Permian era Triassic era PT Boundary

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

26 Dirac, Bakerian Lecture 1941, Proceedings of the Royal Society A

27 The Hamiltonian determines its own adjoint

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

29 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

30 The Lee Model

31 The problem:

32

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

34 PT quantum mechanics to the rescue PT Meep! Meep!

35 GHOSTBUSTING: Reviving quantum theories that were thought to be dead

36 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)

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

38 There are now two possible realizations

39 There can be many realizations! Equivalent Dirac Hermitian Hamiltonian:

40 No ghost theorem for the fourth order derivative Pais Uhlenbeck model, CMB and P. Mannheim, PRL 100, (2008)

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

42 And there are now observations in table top optics experiments! Observing PT symmetry using wave guides: Z. Musslimani, K. Makris, R. El Ganainy, and D. Christodoulides, PRL 100, (2008) K. Makris, R. El Ganainy, D. Christodoulides, and Z. Musslimani, PRL 100, (2008)

43 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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49 OK, but how do we interpret a non Hermitian Hamiltonian?? Solve the quantum brachistochrone problem

50 Classical Brachistochrone Newton Bernoulli Leibniz L'Hôpital

51 Classical Brachistochrone is a cycloid Gravitational field

52 Quantum Brachistochrone Constraint:

53 Hermitian case

54 becomes

55 Minimize t over all positive r while maintaining constraint Minimum evolution time: Looks like uncertainty principle but is merely rate times time = distance

56 Non Hermitian PT symmetric Hamiltonian where

57 Exponentiate H

58 Interpretation Finding the optimal PT symmetric Hamiltonian amounts to constructing a wormhole in Hilbert space!

59 The shortest path between two truths in the real domain passes through the complex domain. Jacques Hadamard [The Mathematical Intelligencer 13 (1991)]

60 Overview of talk:

61 Classical PT symmetry Provides an intuitive explanation of what is going on

62 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.

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

64 Hamilton s equations

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

66 2 3 H = p + ix ( = 1)

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68 Broken PT symmetry (ε < 0)

69 = 2 11 sheets

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76 The Beast = 16/15

77 Lotka Volterra equations

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

82 Making sense of non Hermitian Hamiltonians Crab Lender Practised Nymphets Washing Nervy Tuitions 1

83 Making sense of non Hermitian Hamiltonians Carl Bender Physics Department Washington University 2

84 Quantum Mechanics 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 3

85 Axioms of Quantum Mechanics Causality Locality Stability of the vacuum Relativity Probabilistic interpretation 4

86 Dirac Hermiticity guarantees real energy and conserved probability but is mathematical and not physical 5

87 H = p2 + ix3 6

88 H = p2 + ix3 Wait a minute This model has PT symmetry 7

89 8

90 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) 9

91 The original discoverers of PT symmetry: 10

92 11

93 12

94 13

95 14

96 15

97 16

98 17

99 18

100 MY TALK 19

101 Outline of Talk Beginning Middle End (applause) 20

102 How to prove that the eigenvalues are real 21

103 PT Boundary 22

104 How to prove that the eigenvalues are real The proof is really hard! You need to use (3)Bethe ansatz (4)Monodromy group (5)Baxter T Q relation (6)Functional Determinants 23

105 PT Boundary Greatest murder mystery of all time Extinction of over 90% of species! Permian era Triassic era PT Boundary 24

106 OK, so the eigenvalues are real But is this quantum mechanics?? Probabilistic interpretation?? Hilbert space with a positive metric?? Unitarity?? 25

107 Dirac, Bakerian Lecture 1941, Proceedings of the Royal Society A 26

108 The Hamiltonian determines its own adjoint 27

109 Unitarity With respect to the CPT adjoint the theory has UNITARY time evolution. Norms are strictly positive! Probability is conserved! 28

110 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 29

111 The Lee Model 30

112 The problem: 31

113 32

114 A non Hermitian Hamiltonian is unacceptable partly because it may lead to complex energy eigenvalues, but chiefly because it implies a non unitary S matrix, which fails to conserve probability and makes a hash of the physical interpretation. 33

115 PT quantum mechanics to the rescue PT Meep! Meep! 34

116 GHOSTBUSTING: Reviving quantum theories that were thought to be dead 35

117 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) 36

118 The problem: A fourth order field equation gives a propagator like GHOST! 37

119 There are now two possible realizations 38

120 There can be many realizations! Equivalent Dirac Hermitian Hamiltonian: 39

121 No ghost theorem for the fourth order derivative Pais Uhlenbeck model, CMB and P. Mannheim, PRL 100, (2008) 40

122 TOTALITARIAN PRINCIPLE Everything which is not forbidden is compulsory. M. Gell Mann 41

123 And there are now observations in table top optics experiments! Observing PT symmetry using wave guides: Z. Musslimani, K. Makris, R. El Ganainy, and D. Christodoulides, PRL 100, (2008) K. Makris, R. El Ganainy, D. Christodoulides, and Z. Musslimani, PRL 100, (2008) 42

124 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 43

125 44

126 45

127 46

128 47

129 48

130 OK, but how do we interpret a non Hermitian Hamiltonian?? Solve the quantum brachistochrone problem 49

131 Classical Brachistochrone Newton Bernoulli Leibniz L'Hôpital 50

132 Classical Brachistochrone is a cycloid Gravitational 51 field

133 Quantum Brachistochrone Constraint: 52

134 Hermitian case 53

135 becomes 54

136 Minimize t over all positive r while maintaining constraint Minimum evolution time: Looks like uncertainty principle but is merely rate times time = distance 55

137 Non Hermitian PT symmetric Hamiltonian where 56

138 Exponentiate H 57

139 Interpretation Finding the optimal PT symmetric Hamiltonian amounts to constructing a wormhole in Hilbert space! 58

140 The shortest path between two truths in the real domain passes through the complex domain. Jacques Hadamard [The Mathematical Intelligencer 13 (1991)] 59

141 Overview of talk: 60

142 Classical PT symmetry Provides an intuitive explanation of what is going on 61

143 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. 62

144 Harmonic Oscillator: Particle on a Spring Back and forth motion on the real axis: Turning point Turning point ( = 0) 63

145 Hamilton s equations 64

146 Harmonic Oscillator: Motion in the complex plane: Turning point Turning point ( = 0) 65

147 H = p2 + ix3 ( = 1) 66

148 67

149 Broken PT symmetry (ε < 0) 68

150 = 2 11 sheets 69

151 70

152 71

153 72

154 73

155 74

156 75

157 The Beast = 16/15 76

158 Lotka Volterra equations 77

159 78

160 79

161 80

162 Other PT symmetric classical systems KdV equation Camassa Holm equation Sine Gordon equation Boussinesq equation 81