View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by City Research Online Assis, Paulo (2009). Non-Hermitian Hamiltonians in Field Theory. (Doctoral thesis, City University London) City Research Online Original citation: Assis, Paulo (2009). Non-Hermitian Hamiltonians in Field Theory. (Doctoral thesis, City University London) Permanent City Research Online URL: http://openaccess.city.ac.uk/2118/ Copyright & reuse City University London has developed City Research Online so that its users may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders. All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. 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Contents Acknowledgements VI Declaration VII Abstract VIII 1 Introduction 1 1.1 Outline . 7 2 Aspects of non-Hermitian Hamiltonians 10 2.1 Dissipative systems . 10 2.2 The role of PT -symmetry in Physics . 13 2.2.1 A family of non-Hermitian Hamiltonians with real spectra . 17 2.3 Non-Hermitian Hamiltonians as fundamental theories . 21 2.3.1 Anti-unitary symmetries in Quantum Mechanics . 21 2.3.2 Darboux transformations and superpartner potentials . 27 2.3.3 Dyson map and change of metric . 29 2.3.4 Quasi-Pseudo-Hermiticity . 31 2.3.5 Perturbative approach . 36 2.3.6 Uniqueness of the metric . 37 2.3.7 Non-Hermitian time evolution . 39 3 The quantum brachistochrone for non-Hermitian Hamiltonians 40 3.1 The classical brachistochrone problem . 40 3.2 Hermitian evolution in Quantum Physics . 41 3.3 Fast transitions with PT -symmetric non-Hermitian Hamiltonians . 44 3.4 Dissipative evolution . 47 III 3.5 Geometry of the state space for the quantum brachistochrone . 53 3.6 Interpretation of the physical setup . 54 4 Symmetries, Integrability and Solvability 56 4.1 The importance of symmetries in Physics . 56 4.1.1 Lie groups and Lie algebras . 58 4.1.2 Representation theory of semi-simple Lie algebras . 60 4.2 Integrability . 65 4.2.1 Solitons and Compactons . 72 4.3 Solvability . 75 5 Mapping non-Hermitian into Hermitian Hamiltonians 77 5.1 Lie algebras and similarity transformations with operators . 77 5.2 Hamiltonians of sl(2; R)-Lie algebraic type . 78 5.3 sl(2; R) Metrics and Hermitian counterparts . 80 5.4 Hamiltonians of su(1; 1)-Lie algebraic type and generalized Swanson models 81 5.5 su(1; 1) Metrics and Hermitian counterparts . 83 5.5.1 Non-Hermitian linear term and Hermitian bilinear combinations . 86 5.5.2 Hermitian linear term and non-Hermitian bilinear combinations . 86 5.5.3 Generic non-Hermitian reducible Hamiltonian . 87 5.5.4 Generic non-Hermitian non-reducible Hamiltonian . 90 5.5.5 Generalised Bogoliubov transformation in the construction of metrics 93 5.5.6 Some concrete realisations of the generalised Swanson Hamiltonian . 100 5.6 Hamiltonians of su(2)-Lie algebraic type and bosonic spin chains . 103 6 Moyal products and isospectral transformations 106 6.1 Moyal Products to construct metric operators . 106 6.2 Generic cubic PT -symmetric non-Hermitian Hamiltonians . 110 6.2.1 Non-vanishingp ^ x^2-term . 113 6.2.2 Non-vanishingp ^ x^2-term and vanishingx ^3-term . 115 6.2.3 Vanishingp ^ x^2-term and non-vanishingx ^2-term . 116 6.2.4 Vanishingp ^ x^2-term and non-vanishingp ^ -term or non-vanishing p^2-term . 117 6.3 The single site lattice Reggeon model . 118 6.3.1 Perturbative solution . 120 IV 6.4 Solvable examples . 122 6.5 Limitations of the method . 124 7 Integrable PT -symmetric deformations of classical models 126 7.1 PT -symmetry in classical theories . 126 7.2 Integrability and the Painlev¶etest . 128 7.3 PT -symmetrically deformed Burgers' equation . 131 7.4 PT -symmetrically deformed KdV equation . 138 7.5 PT -symmetrically generalized KdV Hamiltonian: Hl;m;p ........... 142 7.5.1 Generalized KdV-equation: m=2 . 145 7.5.2 PT -symmetric generalized KdV-equation . 146 7.5.3 Deformations of Burgers equation: m = 1; p = 1; l = 3 . 148 7.5.4 Coexistence of solitons and compactons . 148 8 PT -symmetric ¯elds and particles 150 8.1 PT -symmetric constraints on real ¯elds . 150 8.2 Calogero models . 151 8.3 Poles of nonlinear waves as interacting particles . 154 8.4 The motion of Boussinesq singularities . 157 8.5 Di®erent constraints in nonlinear wave equations . 162 8.6 Complex Calogero systems and PT -symmetric deformations . 164 9 Concluding remarks 170 Bibliography 176 V Acknowledgements Unfortunately - or should I say fortunately? - gratitude is not a measurable object. This makes this recognition demonstration imprecise. Nonetheless some people's contri- butions are colossal and cannot be left out. I do not intend to be long and must clarify that this is by no means an indication of lack of appreciation. First of all, I am enormously indebted with the person most directly involved in this work, Prof. Andreas Fring, supervisor and friend. Without him certainly none of this would have been accomplished. He has always demonstrated faith in me, even before we actually met. This support was crucial for my settlement in the United Kingdom and for the progress of this work. I owe so much to your help, Andreas. It has been a real pleasure working with you and I hope to collaborate with you in the future. Secondly, I give many thanks to the members of the panel who kindly accepted the arduous task of assessing this thesis, Prof. Carl Bender and Prof. Joe Chuang. I feel extremely honourned by their assistance. I am also very grateful for ¯nancial support provided by City University London. My stay in London is memorable, de¯nitely because of the amazing people I have met here. New friends who came into my life and will forever be remembered. Exactly like the friends I left in my home country and who have also contributed immensely in a very supportive way. I miss you, friends. I miss you, family. Wonderful family. I have no words to express my admiration and love for you. Thanks for everything. Thank you, life. You have been extraordinary, blessed and cursed and won. The writing of these words ¯lled my heart with joy. Few names but many faces. To everyone here briefly mentioned, one common sentiment. Simple but strong and sincere. Muito obrigado. VI Declaration The work presented in this thesis is based on investigations believed to be original and carried out at the Centre for Mathematical Science, City University London, in collabora- tion with Prof. Andreas Fring. It has not been presented elsewhere for a degree, diploma, or similar quali¯cation at any university or similar institution. I have clearly stated my contributions, in jointly-authored works, as well as referenced the contributions of other people working in the area. Powers of discretion are granted to the University Librarian to allow the thesis to be copied in whole or in part without further reference to the author. This permission covers only single copies made for study purposes, subject to normal conditions of acknowledge- ment. VII Abstract This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachis- tochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calcu- lated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of in¯nite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The ¯nding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commu- tation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of di®erential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our e®orts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear di®erential equations for which the presence of hidden symmetries allows one to assemble exact solu- tions.