\r\n\tThe discovery of Nylon by Wallace Hume Carothers, a Harvard-educated world-renowned organic chemist born in Burlington, IA in 1896, successfully crowned the attempts developed by E.I du Pont de Nemours & Company to investigate the structure of high molecular weight polymers and to synthesize the first synthetic polymeric fibre.
\r\n\tWhen it hit the market, it was in the form of stockings and all the women in the US wanted to get their hands on a pair. Despite the successful launch of Nylon on the synthetic fibre market and the high expectations created by its extraordinary features, the unexpected war events in 1941 diverted the production of the new synthetic fibre almost exclusively on military applications. Parachutes, ropes, bootlaces, fuel tanks, mosquito nets and hammocks absorbed the production of Nylon, which helped to determine the WWII events. When the war ended and production returned to pre-war levels, consumers rushed to the department stores in search of stockings, accessories and high-fashion garments.
\r\n\tEven if the world of high fashion now seems to more appreciate the use of natural fibres, Nylon is one of the most widely used polymers for the production of technical fibres and fabrics, automotive and micromechanical components. The global nylon 6 & 66 market is expected to reach USD 41.13 billion by 2025, by the following growth at 6.1% CAGR owing to the Increasing focus on fuel-efficient and less polluting vehicles.
\r\n\r\n\t
\r\n\tThe amazing success story of Nylon still continues. While its wide availability inspired the development of innovative applications, such as the additive manufacturing, on the other hand, proper disposal after use of high amounts of Nylon resin energised the development of efficient recycling methodology, including chemical recycling. Moreover, the production of Nylon precursors from biomass has become desirable due to the depletion of fossil hydrocarbons and to reduce greenhouse gas (GHG) emissions. This unique combination of technical and socio-economic driving forces is one that aims to further promote the development of Nylon as one of the most suitable ""best polymers"" with a low ecological footprint.
\r\n\r\n\t
\r\n\tThe aim of this publication is to unveil the relationships between the chemical structure and the outstanding properties of the broad family of polyamides and to describe the most recent use of Nylon in fostering new applications and promoting a culture aware of environmental sustainability.
",isbn:"978-1-78985-186-1",printIsbn:"978-1-78984-877-9",pdfIsbn:"978-1-78985-331-5",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"7e4a44212c61af8f5ce0b572e32171b6",bookSignature:"Dr. Eugenio Amendola",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/8182.jpg",keywords:"Linear Polymer, Synthetic Fiber, Step Growth Polymerization, Solid-State Condensation, Hydrogen Bonding, Nylon Innovative Applications, Polyamide Recycling, Chemical Decomposition, Thermoplastic Reworkability, Biopolymers, Renewable Resources, Ring Opening Polymerization",numberOfDownloads:48,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"May 11th 2020",dateEndSecondStepPublish:"June 1st 2020",dateEndThirdStepPublish:"July 31st 2020",dateEndFourthStepPublish:"October 19th 2020",dateEndFifthStepPublish:"December 18th 2020",remainingDaysToSecondStep:"8 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Dr. Eugenio Amnedola has more than 263 publications among which over 97 journal articles and 15 book chapters, he was appointed Director (2012-2014) of the same Institute within National Research Council (CNR), where he is currently affiliated as a researcher.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"235225",title:"Dr.",name:"Eugenio",middleName:null,surname:"Amendola",slug:"eugenio-amendola",fullName:"Eugenio Amendola",profilePictureURL:"https://mts.intechopen.com/storage/users/235225/images/system/235225.jpg",biography:"Eugenio Amendola, an industrial chemist, graduated from the University of Naples “Federico II” in 1986. He is a senior researcher at the Institute of Polymer, Composites and Biomaterials of the National Research Council. He was appointed Director of the same Institute from 2012 to 2014, and President of the Campania section of Italian Chemical Society from 2013 to 2015. He was appointed Professor in Polymer Technology, Chemistry and Applied Chemistry at the University of Naples “Federico II” and the University of Salerno from 1988 to 2012. He was an associate researcher at the University of Connecticut and Gotemburg Polytechnic.\nHis scientific activity was focused on the design and synthesis of Liquid Crystalline polymers and thermosets for structural and functional applications, development of smart nanocomposites based on hybrid structure, advanced polymeric materials for cultural heritage restoration and preservation. Recently he is developing self-healing thermosets and composites based on thermo-reversible chemical reactions for prolonged service use and for unprecedented recyclability and reworkability of composite materials based on thermosetting resins.",institutionString:"Institute of Polymers, Composites and Biomaterials",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"Institute of Polymers, Composites and Biomaterials",institutionURL:null,country:{name:"Italy"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"14",title:"Materials Science",slug:"materials-science"}],chapters:[{id:"74245",title:"Flame Retardant Treatments of Nylon Textiles: A Shift towards Eco-Friendly Approaches",slug:"flame-retardant-treatments-of-nylon-textiles-a-shift-towards-eco-friendly-approaches",totalDownloads:48,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"205697",firstName:"Kristina",lastName:"Kardum Cvitan",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/205697/images/5186_n.jpg",email:"kristina.k@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"6188",title:"Solidification",subtitle:null,isOpenForSubmission:!1,hash:"0405c42586170a1def7a4b011c5f2b60",slug:"solidification",bookSignature:"Alicia Esther Ares",coverURL:"https://cdn.intechopen.com/books/images_new/6188.jpg",editedByType:"Edited by",editors:[{id:"91095",title:"Dr.",name:"Alicia Esther",surname:"Ares",slug:"alicia-esther-ares",fullName:"Alicia Esther Ares"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"6802",title:"Graphene Oxide",subtitle:"Applications and Opportunities",isOpenForSubmission:!1,hash:"075b313e11be74c55a1f66be5dd56b40",slug:"graphene-oxide-applications-and-opportunities",bookSignature:"Ganesh Kamble",coverURL:"https://cdn.intechopen.com/books/images_new/6802.jpg",editedByType:"Edited by",editors:[{id:"236420",title:"Dr.",name:"Ganesh Shamrao",surname:"Kamble",slug:"ganesh-shamrao-kamble",fullName:"Ganesh Shamrao Kamble"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"6517",title:"Emerging Solar Energy Materials",subtitle:null,isOpenForSubmission:!1,hash:"186936bb201bb186fb04b095aa39d9b8",slug:"emerging-solar-energy-materials",bookSignature:"Sadia Ameen, M. Shaheer Akhtar and Hyung-Shik Shin",coverURL:"https://cdn.intechopen.com/books/images_new/6517.jpg",editedByType:"Edited by",editors:[{id:"52613",title:"Dr.",name:"Sadia",surname:"Ameen",slug:"sadia-ameen",fullName:"Sadia Ameen"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"6320",title:"Advances in Glass Science and Technology",subtitle:null,isOpenForSubmission:!1,hash:"6d0a32a0cf9806bccd04101a8b6e1b95",slug:"advances-in-glass-science-and-technology",bookSignature:"Vincenzo M. Sglavo",coverURL:"https://cdn.intechopen.com/books/images_new/6320.jpg",editedByType:"Edited by",editors:[{id:"17426",title:"Prof.",name:"Vincenzo Maria",surname:"Sglavo",slug:"vincenzo-maria-sglavo",fullName:"Vincenzo Maria Sglavo"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"10049",title:"Advanced Functional Materials",subtitle:null,isOpenForSubmission:!1,hash:"58745a56d54c143e4de8433f3d6eb62e",slug:"advanced-functional-materials",bookSignature:"Nevin Tasaltin, Paul Sunday Nnamchi and Safaa Saud",coverURL:"https://cdn.intechopen.com/books/images_new/10049.jpg",editedByType:"Edited by",editors:[{id:"94825",title:"Associate Prof.",name:"Nevin",surname:"Tasaltin",slug:"nevin-tasaltin",fullName:"Nevin Tasaltin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7666",title:"Synthesis Methods and Crystallization",subtitle:null,isOpenForSubmission:!1,hash:"cd26687924373b72a27a0f69e7849486",slug:"synthesis-methods-and-crystallization",bookSignature:"Riadh Marzouki",coverURL:"https://cdn.intechopen.com/books/images_new/7666.jpg",editedByType:"Edited by",editors:[{id:"300527",title:"Dr.",name:"Riadh",surname:"Marzouki",slug:"riadh-marzouki",fullName:"Riadh Marzouki"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"8812",title:"Contemporary Topics about Phosphorus in Biology and Materials",subtitle:null,isOpenForSubmission:!1,hash:"86c427901f631db034a54b22dd765d6a",slug:"contemporary-topics-about-phosphorus-in-biology-and-materials",bookSignature:"David G. Churchill, Maja Dutour Sikirić, Božana Čolović and Helga Füredi Milhofer",coverURL:"https://cdn.intechopen.com/books/images_new/8812.jpg",editedByType:"Edited by",editors:[{id:"219335",title:"Dr.",name:"David",surname:"Churchill",slug:"david-churchill",fullName:"David Churchill"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7960",title:"Assorted Dimensional Reconfigurable Materials",subtitle:null,isOpenForSubmission:!1,hash:"bc49969c3a4e2fc8f65d4722cc4d95a5",slug:"assorted-dimensional-reconfigurable-materials",bookSignature:"Rajendra Sukhjadeorao Dongre and Dilip Rankrishna Peshwe",coverURL:"https://cdn.intechopen.com/books/images_new/7960.jpg",editedByType:"Edited by",editors:[{id:"188286",title:"Associate Prof.",name:"Rajendra",surname:"Dongre",slug:"rajendra-dongre",fullName:"Rajendra Dongre"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7676",title:"Zeolites",subtitle:"New Challenges",isOpenForSubmission:!1,hash:"4dc664fa55f94b38c13af542041fc3cc",slug:"zeolites-new-challenges",bookSignature:"Karmen Margeta and Anamarija Farkaš",coverURL:"https://cdn.intechopen.com/books/images_new/7676.jpg",editedByType:"Edited by",editors:[{id:"216140",title:"Dr.",name:"Karmen",surname:"Margeta",slug:"karmen-margeta",fullName:"Karmen Margeta"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9321",title:"Advances in Microporous and Mesoporous Materials",subtitle:null,isOpenForSubmission:!1,hash:"d5b349cbde0b129c20f31dc02b94d33b",slug:"advances-in-microporous-and-mesoporous-materials",bookSignature:"Rafael Huirache Acuña",coverURL:"https://cdn.intechopen.com/books/images_new/9321.jpg",editedByType:"Edited by",editors:[{id:"181660",title:"Dr.",name:"Rafael",surname:"Huirache Acuña",slug:"rafael-huirache-acuna",fullName:"Rafael Huirache Acuña"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"413",title:"Applications of Petri Nets to Human-in-the-Loop Control for Discrete Automation Systems",doi:"10.5772/5049",slug:"applications_of_petri_nets_to_human-in-the-loop_control_for_discrete_automation_systems",body:null,keywords:null,chapterPDFUrl:"https://cdn.intechopen.com/pdfs/413.pdf",chapterXML:null,downloadPdfUrl:"/chapter/pdf-download/413",previewPdfUrl:"/chapter/pdf-preview/413",totalDownloads:2741,totalViews:71,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:null,dateReviewed:null,datePrePublished:null,datePublished:"July 1st 2006",dateFinished:null,readingETA:"0",abstract:null,reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/413",risUrl:"/chapter/ris/413",book:{slug:"manufacturing_the_future"},signatures:"Jin-Shyan Lee and Pau-Lo Hsu",authors:null,sections:null,chapterReferences:null,footnotes:null,contributors:null,corrections:null},book:{id:"6108",title:"Manufacturing the Future",subtitle:null,fullTitle:"Manufacturing the Future",slug:"manufacturing_the_future",publishedDate:"July 1st 2006",bookSignature:"V. Kordic, A. Lazinica and M. Merdan",coverURL:"https://cdn.intechopen.com/books/images_new/6108.jpg",licenceType:"CC BY-NC-SA 3.0",editedByType:"Edited by",editors:[{id:"396",title:"Dr.",name:"Vedran",middleName:null,surname:"Kordic",slug:"vedran-kordic",fullName:"Vedran Kordic"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},chapters:[{id:"407",title:"Multi-Agent Based Distributed Manufacturing",slug:"multi-agent_based_distributed_manufacturing",totalDownloads:2917,totalCrossrefCites:0,signatures:"J. Li, J.Y. H. Fuh, Y.F. Zhang and A.Y.C. Nee",authors:[null]},{id:"408",title:"The Cobasa Architecture as an Answer to Shop Floor Agility",slug:"the_cobasa_architecture_as_an_answer_to_shop_floor_agility",totalDownloads:2153,totalCrossrefCites:9,signatures:"Jose Barata",authors:[null]},{id:"409",title:"Development of Holonic Manufacturing Execution Systems",slug:"development_of_holonic_manufacturing_execution_systems",totalDownloads:2929,totalCrossrefCites:2,signatures:"Fan-Tien Cheng, Chih-Feng Chang and Shang-Lun Wu",authors:[null]},{id:"410",title:"Bio-Inspired Approach for Autonomous Routing in FMS",slug:"bio-inspired_approach_for_autonomous_routing_in_fms",totalDownloads:1897,totalCrossrefCites:0,signatures:"T. Berger, Y. Sallez and C. Tahon",authors:[null]},{id:"411",title:"Modular Machining Line Design and Reconfiguration: Some Optimization Methods",slug:"modular_machining_line_design_and_reconfiguration__some_optimization_methods",totalDownloads:2111,totalCrossrefCites:2,signatures:"S. Belmokhtar, A.I. Bratcu and A. Dolgui",authors:[null]},{id:"412",title:"Flexible Manufacturing System Simulation Using Petri Nets",slug:"flexible_manufacturing_system_simulation_using_petri_nets",totalDownloads:2739,totalCrossrefCites:0,signatures:"Carlos Mireles, Alfonso Noriega and Gerardo Leyva",authors:[null]},{id:"413",title:"Applications of Petri Nets to Human-in-the-Loop Control for Discrete Automation Systems",slug:"applications_of_petri_nets_to_human-in-the-loop_control_for_discrete_automation_systems",totalDownloads:2741,totalCrossrefCites:0,signatures:"Jin-Shyan Lee and Pau-Lo Hsu",authors:[null]},{id:"414",title:"Application Similarity Coefficient Method to Cellular Manufacturing",slug:"application_similarity_coefficient_method_to_cellular_manufacturing",totalDownloads:5141,totalCrossrefCites:2,signatures:"Yong Yin",authors:[null]},{id:"415",title:"Maintenance Management and Modeling in Modern Manufacturing Systems",slug:"maintenance_management_and_modeling_in_modern_manufacturing_systems",totalDownloads:5567,totalCrossrefCites:0,signatures:"Mehmet Savsar",authors:[null]},{id:"416",title:"Zadehian Paradigms for Knowledge Extraction in Intelligent Manufacturing",slug:"zadehian_paradigms_for_knowledge_extraction_in_intelligent_manufacturing",totalDownloads:2061,totalCrossrefCites:2,signatures:"A.M.M. Sharif Ullah and Khalifa H. Harib",authors:[null]},{id:"417",title:"PURE: A Fuzzy Model for Product Upgradability and Reusability Evaluation for Remanufacture",slug:"pure__a_fuzzy_model_for_product_upgradability_and_reusability_evaluation_for_remanufacture",totalDownloads:2230,totalCrossrefCites:1,signatures:"Ke Xing, Kazem Abhary and Lee Luong",authors:[null]},{id:"418",title:"Distributed Architecture for Intelligent Robotic Assembly Part I: Design and Multimodal Learning",slug:"distributed_architecture_for_intelligent_robotic_assembly_part_i__design_and_multimodal_learning",totalDownloads:2145,totalCrossrefCites:0,signatures:"Ismael Lopez Juarez and Reyes Rios Cabrera",authors:[null]},{id:"419",title:"Distributed Architecture for Intelligent Robotic Assembly Part II: Design of the Task Planner",slug:"distributed_architecture_for_intelligent_robotic_assembly_part_ii__design_of_the_task_planner",totalDownloads:1625,totalCrossrefCites:0,signatures:"Jorge Corona Castuera and Ismael Lopez Juarez",authors:[null]},{id:"420",title:"Distributed Architecture for Intelligent Robotic Assembly Part III: Design of the Invariant Object Recognition System",slug:"distributed_architecture_for_intelligent_robotic_assembly_part_iii__design_of_the_invariant_object_r",totalDownloads:2138,totalCrossrefCites:0,signatures:"Mario Pena Cabrera and Ismael Lopez Juarez",authors:[null]},{id:"421",title:"Assembly Sequence Planning Using Neural Network Approach",slug:"assembly_sequence_planning_using_neural_network_approach",totalDownloads:1934,totalCrossrefCites:0,signatures:"Cem Sinanoglu and Huseyin Riza Borklu",authors:[null]},{id:"422",title:"Evolutionary Optimisation of Mechanical Structures or Systems",slug:"evolutionary_optimisation_of_mechanical_structures_or_systems",totalDownloads:2320,totalCrossrefCites:0,signatures:"Marcelin Jean-Luc",authors:[null]},{id:"423",title:"Improving Machining Accuracy Using Smart Materials",slug:"improving_machining_accuracy_using_smart_materials",totalDownloads:2639,totalCrossrefCites:0,signatures:"Maki K. Rashid",authors:[null]},{id:"424",title:"Concurrent Process Tolerancing Based on Manufacturing Cost And Quality Loss",slug:"concurrent_process_tolerancing_based_on_manufacturing_cost_and_quality_loss",totalDownloads:3760,totalCrossrefCites:0,signatures:"M. F. Huang and Y. R. Zhong",authors:[null]},{id:"425",title:"Optimize Variant Product Design Based on Component Interaction Graph",slug:"optimize_variant_product_design_based_on_component_interaction_graph",totalDownloads:2278,totalCrossrefCites:0,signatures:"Elim Liu and Shih-Wen Hsiao",authors:[null]},{id:"426",title:"Applying a Hybrid Data Mining Approach in Machining Operation for Surface Quality Assurance",slug:"applying_a_hybrid_data_mining_approach_in_machining_operation_for_surface_quality_assurance",totalDownloads:2335,totalCrossrefCites:0,signatures:"Tzu-Liang (Bill) Tseng, Yongjin Kwon and Ryan B. Wicker",authors:[null]},{id:"427",title:"Sequential Design of Optimum Sized and Geometric Tolerances",slug:"sequential_design_of_optimum_sized_and_geometric_tolerances",totalDownloads:2984,totalCrossrefCites:0,signatures:"M. F. Huang and Y. R. Zhong",authors:[null]},{id:"428",title:"A New Rapid Tooling Process",slug:"a_new_rapid_tooling_process",totalDownloads:2424,totalCrossrefCites:0,signatures:"Xiaoping Jiang and Chao Zhang",authors:[null]},{id:"429",title:"SCM Innovation for Business Globalization Based on Coupling Point Inventory Planning",slug:"scm_innovation_for_business_globalization_based_on_coupling_point_inventory_planning",totalDownloads:2392,totalCrossrefCites:0,signatures:"Koshichiro Mitsukuni, Yuichi Nakamura and Tomoyuki Aoki",authors:[null]},{id:"430",title:"Relative Control and Management Philosophy",slug:"relative_control_and_management_philosophy",totalDownloads:2240,totalCrossrefCites:0,signatures:"Che-Wei Chang",authors:[null]},{id:"431",title:"Multidimensional of Manufacturing Technology, Organizational Characteristics, and Performance",slug:"multidimensional_of_manufacturing_technology__organizational_characteristics__and_performance",totalDownloads:2471,totalCrossrefCites:0,signatures:"Tritos Laosirihongthong",authors:[null]},{id:"432",title:"Engineering Change Management in Distruted Environment with PDM/PLM Support",slug:"engineering_change_management_in_distruted_environment_with_pdm_plm_support",totalDownloads:3276,totalCrossrefCites:1,signatures:"Joze Tavcar and Joze Duhovnik",authors:[null]},{id:"433",title:"Study of Flexibility and Adaptability in Distributed Supply Chains",slug:"study_of_flexibility_and_adaptability_in_distributed_supply_chains",totalDownloads:2256,totalCrossrefCites:0,signatures:"Felix T. S. Chan and H. K. Chan",authors:[null]},{id:"434",title:"An Autonomous Decentralized Supply Chain Planning and Scheduling System",slug:"an_autonomous_decentralized_supply_chain_planning_and_scheduling_system",totalDownloads:2547,totalCrossrefCites:1,signatures:"Tatsushi Nishi",authors:[null]},{id:"435",title:"Simulation Modeling and Analysis of the Impacts of Component Commonality and Process Flexibility on Integrated Supply Chain Network Performance",slug:"simulation_modeling_and_analysis_of_the_impacts_of_component_commonality_and_process_flexibility_on_",totalDownloads:2390,totalCrossrefCites:0,signatures:"Ming Dong and F. Frank Chen",authors:[null]},{id:"436",title:"On Direct Adaptive Control for Uncertain Dynamical Systems - Synthesis and Applications",slug:"on_direct_adaptive_control_for_uncertain_dynamical_systems_-_synthesis_and_applications",totalDownloads:1980,totalCrossrefCites:0,signatures:"Simon Hsu-Sheng Fu and Chi-Cheng Cheng",authors:[null]}]},relatedBooks:[{type:"book",id:"5787",title:"Cutting Edge Robotics",subtitle:null,isOpenForSubmission:!1,hash:"f5caeb19605b2ebe7260f03131c26a24",slug:"cutting_edge_robotics",bookSignature:"Vedran Kordic, Aleksandar Lazinica and Munir Merdan",coverURL:"https://cdn.intechopen.com/books/images_new/5787.jpg",editedByType:"Edited by",editors:[{id:"396",title:"Dr.",name:"Vedran",surname:"Kordic",slug:"vedran-kordic",fullName:"Vedran Kordic"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"},chapters:[{id:"1",title:"Dynamic Modelling and Adaptive Traction Control for Mobile Robots",slug:"dynamic_modelling_and_adaptive_traction_control_for_mobile_robots",signatures:"Abdulgani Albagul, Wahyudi Martono and Riza Muhida",authors:[null]},{id:"2",title:"Rapid Prototyping for Robotics",slug:"rapid_prototyping_for_robotics",signatures:"Imme Ebert-Uphoff, Clement M. Gosselin, David W. Rosen and Thierry Laliberte",authors:[null]},{id:"3",title:"The Role of 3D Simulation in the Advanced Robotic Design, Test and Control",slug:"the_role_of_3d_simulation_in_the_advanced_robotic_design__test_and_control",signatures:"Laszlo Vajta and Tamas Juhasz",authors:[null]},{id:"4",title:"Mechatronics Design of a Mecanum Wheeled Mobile Robot",slug:"mechatronics_design_of_a_mecanum_wheeled_mobile_robot",signatures:"Peter Xu",authors:[null]},{id:"5",title:"Tracking Skin-Colored Objects in Real-Time",slug:"tracking_skin-colored_objects_in_real-time",signatures:"Antonis A. Argyros and Manolis I.A. Lourakis",authors:[null]},{id:"6",title:"Feature Extraction and Grouping for Robot Vision Tasks",slug:"feature_extraction_and_grouping_for_robot_vision_tasks",signatures:"Miguel Cazorla and Francisco Escolano",authors:[null]},{id:"7",title:"Comparison of Demosaicking Methods for Color Information Extraction",slug:"comparison_of_demosaicking_methods_for_color_information_extraction",signatures:"Flore Faille",authors:[null]},{id:"8",title:"Robot Motion Trajectory-Measurement with Linear Inertial Sensors",slug:"robot_motion_trajectory-measurement_with_linear_inertial_sensors",signatures:"Bernard Favre Bulle",authors:[null]},{id:"9",title:"Supervisory Controller for Task Assignment and Resource Dispatching in Mobile Wireless Sensor Networks",slug:"supervisory_controller_for_task_assignment_and_resource_dispatching_in_mobile_wireless_sensor_networ",signatures:"Vincenzo Giordano, Frank Lewis, Prasanna Ballal and Biagio Turchiano",authors:[null]},{id:"10",title:"Design of a Generic, Vectorised, Machine-Vision library",slug:"design_of_a_generic__vectorised__machine-vision_library",signatures:"Bing-Chang Lai and Phillip John McKerrow",authors:[null]},{id:"11",title:"An Active Stereo Vision-Based Learning Approach for Robotic Tracking, Fixating and Grasping Control",slug:"an_active_stereo_vision-based_learning_approach_for_robotic_tracking__fixating_and_grasping_control",signatures:"Nan-Feng Xiao and Saeid Nahavandi",authors:[null]},{id:"12",title:"Managing Limited Sensing Resources for Mobile Robots Obstacle Avoidance",slug:"managing_limited_sensing_resources_for_mobile_robots_obstacle_avoidance",signatures:"Juan Carlos Alvarez, Rafael C. Gonzalez, Diego Alvarez and Antonio M. Lopez",authors:[null]},{id:"13",title:"Behaviour Based Mobile Robot Navigation with Dynamic Weighted Voting Technique",slug:"behaviour_based_mobile_robot_navigation_with_dynamic_weighted_voting_technique",signatures:"Shamsudin H.M. Amin, Rosbi Mamat and Tan Chee Kwong",authors:[null]},{id:"14",title:"Stochastic State Estimation for Simultaneous Localization and Map Building in Mobile Robotics",slug:"stochastic_state_estimation_for_simultaneous_localization_and_map_building_in_mobile_robotics",signatures:"Juan Andrade Cetto, Teresa A. Vidal Calleja and Alberto Sanfeliu",authors:[null]},{id:"15",title:"Neural Networks in Mobile Robot Motion",slug:"neural-network-in-mobile-robot-motion",signatures:"Danica Janglova",authors:[null]},{id:"16",title:"Generating Timed Trajectories for Autonomous Robotic Platforms: A Non-Linear Dynamical Systems Approach",slug:"generating_timed_trajectories_for_autonomous_robotic_platforms__a_non-linear_dynamical_systems_appro",signatures:"Cristina Manuela Peixoto dos Santos",authors:[null]},{id:"17",title:"Coevolution Based Adaptive Monte Carlo Localization",slug:"coevolution_based_adaptive_monte_carlo_localization",signatures:"Luo Ronghua, Hong Bingrong and Li Maohai",authors:[null]},{id:"18",title:"Autonomous Navigation of Unmanned Vehicles: A Fuzzy Logic Perspective",slug:"autonomous_navigation_of_unmanned_vehicles__a_fuzzy_logic_perspective",signatures:"Nikos C. Tsourveloudis, Lefteris Doitsidis and Kimon P. Valavanis",authors:[null]},{id:"19",title:"Integrating Behaviors for Mobile Robots: an Ethological Approach",slug:"integrating_behaviors_for_mobile_robots__an_ethological_approach",signatures:"Jose Maria Canas Plaza and Vicente Matellan Olivera",authors:[null]},{id:"20",title:"Stabilization of Fuzzy Takagi - Sugeno Descriptor Models; Application to a Double Inverted Pendulum",slug:"stabilization_of_fuzzy_takagi_-_sugeno_descriptor_models__application_to_a_double_inverted_pendulum",signatures:"Thierry Marie Guerra, Sebastien Delprat and Salim Labiod",authors:[null]},{id:"21",title:"Adaptive Control of Nonlinear Dynamics Systems Based on RBF Network",slug:"adaptive_control_of_nonlinear_dynamics_systems_based_on_rbf_network",signatures:"Ho Dac Loc, Nguyen Thi Phuong Ha and Luong Van Lang",authors:[null]},{id:"22",title:"Multi-Layered Learning System for Real Robot Behavior Acquisition",slug:"multi-layered_learning_system_for_real_robot_behavior_acquisition",signatures:"Yasutake Takahashi and Minoru Asada",authors:[null]},{id:"23",title:"The Design of a Pair of Identical Mobile Robots to Investigate Cooperative Behaviours",slug:"the_design_of_a_pair_of_identical_mobile_robots_to_investigate_co_-_operative_behaviours",signatures:"Dale Carnegie, Andrew Payne and Praneel Chand",authors:[null]},{id:"24",title:"Cooperative Hunting by Multiple Mobile Robots Based on Local Interaction",slug:"cooperative_hunting_by_multiple_mobile_robots_based_on_local_interaction",signatures:"Zhi-Qiang Cao, Min Tan, Saeid Nahavandi and Nong Gu",authors:[null]},{id:"25",title:"Market-Driven Multi-Agent Collaboration in Robot Soccer Domain",slug:"market-driven_multi-agent_collaboration_in_robot_soccer_domain",signatures:"Hatice Kose, Kemal Kaplan, Cetin Mericli, Utku Tatlidede and Levent Akin",authors:[null]},{id:"26",title:"The SocRob Project: Soccer Robots or Society of Robots",slug:"the_socrob_project__soccer_robots_or_society_of_robots",signatures:"Pedro U. Lima and Luis M. M. Custodio",authors:[null]},{id:"27",title:"RoboCup is a Stage which Impulse the Research of Basic Technology in Robot",slug:"robocup_is_a_stage_which_impulse_the_research_of_basic_technology_in_robot",signatures:"Cheng Xian-yi and Xia De-shen",authors:[null]},{id:"28",title:"A Multi-Robot System Architecture for Trajectory Control of Groups of People",slug:"a_multi-robot_system_architecture_for_trajectory_control_of_groups_of_people",signatures:"Edgar A. Martinez-Garcia, Akihisa Ohya and Shinichi Yuta",authors:[null]},{id:"29",title:"Sharing and Trading in a Human-Robot System",slug:"sharing_and_trading_in_a_human-robot_system",signatures:"Kai Wei Ong, Gerald Seet and Siang Kok Sim",authors:[null]},{id:"30",title:"A Robotic System for Volcano Exploration",slug:"a_robotic_system_for_volcano_exploration",signatures:"Daniele Caltabiano and Giovanni Muscato",authors:[null]},{id:"31",title:"A Simulator for Helping in Design of a New Active Catheter Dedicated to Coloscopy",slug:"a_simulator_for_helping_in_design_of_a_new_active_catheter_dedicated_to_coloscopy",signatures:"Georges Dumont and Christofer Kuehl",authors:[null]},{id:"32",title:"Development of a Range of Robot and Automation Prototypes for Service Applications",slug:"development_of_a_range_of_robot_and_automation_prototypes_for_service_applications",signatures:"Bing Lam Luk, Alexandar Djordjevic, Shiu Kit Tso and King Pui Liu",authors:[null]},{id:"33",title:"Legged Robotic Systems",slug:"legged_robotic_systems",signatures:"Giuseppe Carbone and Marco Ceccarelli",authors:[null]},{id:"34",title:"Humanoid Robot Motion in Unstructured Environment - Generation of Various Gait Patterns from a Single Nominal",slug:"humanoid_robot_motion_in_unstructured_environment_-_generation_of_various_gait_patterns_from_a_singl",signatures:"Miomir Vukobratovic, Dejan Andric and Branislav Borovac",authors:[null]},{id:"35",title:"Trajectory Planning of a Constrained Flexible Manipulator",slug:"trajectory_planning_of_a_constrained_flexible_manipulator",signatures:"Atef A. Ata and Habib Johar",authors:[null]},{id:"36",title:"Position / Force Hybrid Control of a Manipulator with a Flexible Tool Using Visual and Force Information",slug:"position___force_hybrid_control_of_a_manipulator_with_a_flexible_tool_using_visual_and_force_informa",signatures:"Jian Huang, Isao Todo and Tetsuro Yabuta",authors:[null]},{id:"37",title:"A Novel Parallel Engraving Machine Based on 6-PUS Mechanism and Related Technologies",slug:"a_novel_parallel_engraving_machine_based_on_6-pus_mechanism_and_related_technologies",signatures:"Kong Ling-fu and Zhang Shi-hui",authors:[null]},{id:"38",title:"Pose Estimating the Human Arm Using Kinematics and the Sequential Monte Carlo Framework",slug:"pose_estimating_the_human_arm_using_kinematics_and_the_sequential_monte_carlo_framework",signatures:"Thomas Moeslund",authors:[null]},{id:"39",title:"Cartesian Impedance Control of Flexible Joint Robots: A Decoupling Approach",slug:"cartesian_impedance_control_of_flexible_joint_robots__a_decoupling_approach",signatures:"Christian Ott, Alin Albu Schaffer, Andreas Kugi and Gerd Hirzinger",authors:[null]},{id:"40",title:"Collision-Free Path Planning in Robot Cells Using Virtual 3D Collision Sensors",slug:"collision-free_path_planning_in_robot_cells_using_virtual_3d_collision_sensors",signatures:"Tomislav Reichenbach and Zdenko Kovacic",authors:[null]},{id:"41",title:"Exploring Open-Ended Design Space of Mechatronic Systems",slug:"exploring_open-ended_design_space_of_mechatronic_systems",signatures:"Zhun Fan, Jiachuan Wang and Erik Goodman",authors:[null]},{id:"42",title:"Online Identification for the Automated Threaded Fastening Using GUI Format",slug:"online_identification_for_the_automated_threaded_fastening_using_gui_format",signatures:"Nicola Ivan Giannoccaro and Mongkorn Klingajay",authors:[null]},{id:"43",title:"Multilevel Intelligent Control of Mechatronical Technological Systems",slug:"multilevel_intelligent_control_of_mechatronical_technological_systems",signatures:"Tugengold Andrei Kirillovich, Ryzhkin Anatoliy Andreevich, Lukianov Evjeny Anatolievich and Wojciechowicz Boleslav",authors:[null]},{id:"44",title:"A Robot System for High Quality Belt Grinding and Polishing Processes",slug:"a_robot_system_for_high_quality_belt_grinding_and_polishing_processes",signatures:"Bernd Kuhlenkoetter and Xiang Zhang",authors:[null]},{id:"45",title:"Reconfigurable Mechatronic Robotic Plug-and-Play Controller",slug:"reconfigurable_mechatronic_robotic_plug-and-play_controller",signatures:"Johan Potgieter, Jonathan Zyzalo and Olaf Diegel",authors:[null]}]}]},onlineFirst:{chapter:{type:"chapter",id:"74091",title:"Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos",doi:"10.5772/intechopen.94491",slug:"classical-and-quantum-integrability-a-formulation-that-admits-quantum-chaos",body:'1. Introduction
In classical mechanics, a Hamiltonian system with N degrees of freedom is defined to be integrable if a set of N constants of the motion Ui which are in involution exist, so their Poisson bracket satisfies UiUj=0, i,j=1,…,N. For an integrable system, the motion is confined to an invariant two-dimensional torus in 2N-dimensional phase space. If the system is perturbed by a small nonintegrable term, the KAM theorem states that its motion may still be confined to the N-torus but deformed in some way [1, 2, 3]. The first computer simulation of nonequilibrium dynamics for a finite classical system was carried out by Fermi and his group. They considered a one-dimensional classical chain of anharmonic oscillators and found it did not equilibrate.
Classically, chaotic motion is longtime local exponential divergence with global confinement, a form of instability. Confinement with any kind of divergence is produced by repeatedly folding, a type of mixing that can only be analyzed by using probability theory. The motion of a Hamiltonian system is usually neither completely regular nor properly described by statistical mechanics, but shows both regular and chaotic motion for different sets of initial conditions. There exists generally a transition between the two types of motion as initial conditions are changed which may exhibit complicated behavior. As entropy or the phase space area quantifies the amount of decoherence, the rate of change of the phase space area quantifies the decoherence rate. In other words, the decoherence rate is the rate at which the phase space area changes.
It is important to extend the study of chaos into the quantum domain to better understand concepts such as equilibration and decoherence. Both integrable as well as nonintegrable finite quantum systems can equilibrate [4, 5]. Integrability does not seem to play a crucial role in the structure of the quasi-stationary state. This is in spite of the fact that integrable and nonintegrable quantum systems display different level-spacing statistics and react differently to external perturbation. Although integrable systems can equilibrate, the main difference from nonintegrable systems may be longer equilibration times. This kind of behavior is contrary to integrable classical finite systems that do not equilibrate at all. Nonintegrable classical systems can equilibrate provided they are chaotic.
The properties of a quantum system are governed by its Hamiltonian spectrum. Its form should be important for equilibration of a quantum system. The equilibration of a classical system depends on whether the system is integrable or not. Integrable classical systems do not tend to equilibrate, they have to be nonintegrable. Quantum integrability in n dimensions may be defined in an analogous way requiring the existence of n mutually commuting operators, but there is no corresponding theorem like the Liouville theorem. An integrable system in quantum mechanics is one in which the spectral problem can be solved exactly, and such systems are few in number [6, 7].
In closed classical systems, equilibration is usually accompanied by the appearance of chaos. Defining quantum chaos is somewhat of an active area of study now. The correspondence principle might suggest we conjecture quantum chaos exists provided the corresponding classical system is chaotic and the latter requires the system to be nonintegrable. Classical chaos does not necessarily imply quantum chaos, which seems to be more related to the properties of the energy spectrum.
It was proposed that the spectrum of integrable and nonintegrable quantum systems ought to be qualitatively different. This would be seen in the qualitative difference of the density of states. At a deeper level, one may suspect that changes in the energy spectrum as a whole may be connected to the breaking of some symmetry or dynamical symmetry. This is the direction taken here [8, 9, 10].
It is the objective to see how algebraic and geometric approaches to quantization can be used to give a precise definition of quantum degrees of freedom and quantum phase space. Thus a criterion can be formulated that permits the integrability of a given system to be defined in a mathematical way. It will appear that if the quantum system possesses dynamical symmetry, it is integrable. This suggests that dynamical symmetry breaking should be linked to nonintegrability and chaotic dynamics at the quantum level [11, 12, 13].
Algebraic methods first appeared in the context of the new matrix mechanics in 1925. The importance of the concept of angular momentum in quantum mechanics was soon appreciated and worked out by Wigner, Weyl and Racah [14, 15, 16]. The close relationship of the angular momentum and the SO3 algebra goes back to the prequantum era. The realization that SO4 is the symmetry group of the Kepler problem was first demonstrated by Fock. A summary of the investigation is as follows. To familiarize those who are not familiar with algebraic methods in solving quantum problems, an introduction to the algebraic solution of the hydrogen atom is presented as opposed to the Schrödinger picture. This approach provides a platform for which a definition of quantum integrability of quantum systems can be established. Thus, at least one approach is possible in which a definition of concepts such as quantum phase space, degrees of freedom as well as how an idea of quantum integrability and so forth can be formulated [17, 18, 19, 20, 21]. After these issues are addressed, a number of quantum models will be discussed in detail to show how the formalism is to be used [22, 23, 24].
1.1 The hydrogen atom
The hydrogen atom is a unique system. In this system, almost every quantity of physical interest can be computed analytically as it is a completely degenerate system. The classical trajectories are closed and the quantum energy levels only depend on the principle quantum number. This is a direct consequence of the symmetry properties of the Coulomb interaction. Moreover, the properties of the hydrogen atom in an external field can be understood using these symmetry properties. They allow a parallel treatment in the classical and quantum formalisms.
The Hamiltonian of the hydrogen atom in atomic units is
H0=p22−1r.E1
The corresponding quantum operator is found by replacement of p by −i∇. Due to the spherical symmetry of the system, the angular momentum components are constants of the motion,
L=r×p,H0L=0.E2
So H0L2Lz is a complete set of commuting operators classically, so three quantities in mutual involution, which implies integrability of the system.
The Coulomb interaction has another constant of the motion associated with the Runge-Lenz vector R. This has the symmetrized quantum definition
R=12p×L−L×p−rr,H0R=0.E3
If the R direction is chosen as the reference axis of a polar coordinate system in the plane perpendicular to L, one deduces the equation of the trajectory as
r=L21+‖R‖cosϑ.E4
The modulus determines whether the trajectory is an ellipse, a parabola or a hyperbola.
There are then 7 constants of the motion LRH0 are not independent and satisfy
R⋅L=0,L2−R22H0=−12H0−1.E5
The minus one on the right in (5) is not present in classical mechanics. The mutual commutation relations are given in terms of εijk, the fully antisymmetric tensor as follows,
LiLj=iεijkLk,LiRj=iεijkRk,RiRj=iεijk−2H0Lk,E6
Let us look at the symmetry group of the hydrogen atom. The symmetry group is the set of phase space transformations which preserve the Hamiltonian and the equations of motion. It can be identified from the commutation relations between constants of motion. For hydrogen, for negative energies, the group of rotations in 4-dimensional space is called SO4.
The generators of the rotation group in an n-dimensional space are the n−1n/2 components of the n-dimensional angular momentum
Lij=xipj−xjpi,1≤i,j≤n.E7
In (7), Lij is the generator of the rotations in the ij-plane and has the following commutation relations
LijLkl=0,LijLik=iLjk.E8
The first bracket in (8) holds if all four indices are different.
Define the reduced Runge-Lenz vector to be
R′=R−2H0.E9
The commutation relations (6) are those of a four-dimensional angular momentum with the identification
L12=LzL23=LzL31=LyL14=Rz′L24=Ry′L34=Rz′,E10
and Casimir operator
L2=∑i<jLij2=L2+R2=−12H0−1.E11
The classical trajectory is thus uniquely defined with the 6 components of L and L⋅R′=0. Any trajectory can be transformed into any other one having the same energy by a 4-dimensional rotation. An explicit realization of this four-dimensional invariance is to use a stereographic projection from the momentum space onto the 4-dimensional sphere with radius p0=−2H0. On this sphere, the solutions to Schrödinger’s equation as well as the classical equations of motion are those of the free motion. Schrödinger’s equation on the four-dimensional sphere can be separated into six different types of coordinates each associated with a set of commuting operators.
Spherical coordinates correspond to the most natural set, and choosing the quantization axis in the 4 direction and inside the (1,2,3) subspace, the z-axis or usual 3-axis as reference axis, the three operators can be simultaneously diagonalized,
L2=−12H0−1,L2=L122+L312+L232=Lx2+Ly2+Lz2,Lz=L12,E12
The respective eigenvalues of these operators are n2−1, ll+1 and M such that ∣M∣≤l≤n−1, so the total degeneracy is n2. It corresponds to a particular subgroup chain given by
SO4n⊃SO3l⊃SO2M.E13
Other choices are possible, such as other spherical coordinates obtained from the previous by interchanging the role of the 3 and 4 axes. This simultaneously diagonalizes the three operators
L2=−12H0−1,
λ2=L122+L142+L242=Rx\'2+R`′y2+L′z2,Lz=L12.E14
The respective eigenvalues of these operators are n2−1, λλ+1 and M such that ∣M∣≤λ≤n−1. The subgroup chain for this situation is
SO4n⊃SO3λ⊃SO2M.E15
Another relevant case is the adoption of cylindrical coordinates on the 4-dimensional sphere associated with the following set of commuting operators
L2=−12H0−1,L12=Lz,L34=Rz′.E16
This set has the following associated subgroup chain,
SO4⊃SO2⊗SO2.E17
In configuration space, this is associated with separability in parabolic coordinates. This is a specific system but it exhibits many of the mathematical and physical properties that will appear here.
2. Quantum degrees of freedom
The time evolution of a system in classical mechanics in time is usually represented by a trajectory in phase space and the dynamical variables are functions defined on this space. The dimension of phase space is twice the number of degrees of freedom, and a point represents a physical state. The space is even-dimensional and it is endowed with a symplectic Poisson bracket structure. Dynamical properties of the system are described completely by Hamilton’s equations within this space.
For a quantum system, on the other hand, the dynamical properties are discussed in the setting of a Hilbert space. Dynamical observables are self-adjoint operators acting on elements of this space. A physical state is represented by a ray of the space, so the Hilbert space plays a role similar to phase space for a classical system. The Hilbert space cannot play the role of a quantum phase space since its dimension does not in general relate directly to degrees of freedom. Nor can it be directly reduced to classical phase space in the classical limit. Let us define first the quantum degrees of freedom as well as giving a suitable meaning to quantum phase space.
Suppose H is a Hilbert space of a system characterized completely by a complete set of observables denoted C. Set C is composed of the basic physical observables, such as coordinates, momenta, spin and so forth, but excludes the Hamiltonian. The basis vectors of the space can be completely specified by a set of quantum numbers which are related to the eigenvalues of what are usually referred to as the fully non-degenerate commuting observables C′ of C. A fully degenerate operator or observable O∈C has for some constant λ the action
O∣ψi〉=λ∣ψi〉,∣ψi〉∈H.E18
Definition 1: (Quantum Dynamical Degrees of Freedom) Let C:OjOiOj=0ij=1…N be a complete set of commuting observables of a quantum system. A basis set of its Hilbert space H can be labeled completely by M numbers αi:i=1…M called quantum numbers which are related to the eigenvalues of the non-fully degenerate observables Oi:i=1…MM≤N, a subset of C. Then the number M is defined to be the number of quantum dynamical degrees of freedom. □
Since the members of C are provided by the system, not including the Hamiltonian, it depends only on the structure of the system’s dynamical group G. Thus the number of quantum dynamical degrees of freedom based on this definition is unique for a given system with a specific Hilbert space H.
The physical and mathematical considerations for defining the dimension of the nonfully degenerate operator subset C′ of C, not the dimension of C itself, as the number of quantum dynamical degrees of freedom is as follows. In a given H, all fully degenerate operators in C are equivalent to a constant multiple of the identity operator guaranteeing the irreducibility of H. The expectation values of any fully degenerate operator is a constant and contains no dynamical information.
A given quantum system generally has associated with it a well-defined dynamical group structure due to the fact that the mathematical image of a quantum system is an operator algebra g in a linear Hilbert space. This was seen in the case of hydrogen. It comes about from the mathematical structure of quantum mechanics. The dynamical group G with algebra g is generated out of the basic physical variables, with the corresponding algebraic structure defined by the commutation relations.
The Hamiltonian H and all transition operators O can be expressed as functions of a closed set of operators
H=HTi,O=OTi,TiTj=∑kCijkTk.E19
The Cijk in (19) are called the structure constants of algebra g. The Hilbert space is decomposed into a direct sum of the carrier spaces of unitary irreducible (irrep) representations of the group. Consequently, the dynamical symmetry properties of the system can be restricted to an irreducible Hilbert space which acts as one of the irrep carrier spaces of G.
From group representation theory, it will be given that a total of σ subgroup chains exist for a given group
Gα=Gsαα⊃Gsα−1α⊃⋯⊃G1α,α=1,…,σ.E20
For each subgroup chain Gα of G, there is a complete set of commuting operators C which specifies a basis set of its irreducible basis carrier space H, so the dimension of C for all subgroup chains of G is the same. A subgroup chain of dynamical group G serves to determine the M quantum dynamical degrees of freedom for a given quantum system with Hilbert space H an irrep carrier space of G.
Definition 2: For a quantum system with M independent quantum dynamical degrees of freedom the quantum phase space is defined to be a 2M-dimensional topological space. The space is isomorphic to the coset space G/R with explicit symplectic structure. Here G is the dynamical group of the system and R⊂G is the maximal stability subgroup of the Hilbert space. □
3. Quantum integrability and dynamical symmetry
Quantum phase space defined here can be compact or noncompact depending on the finite or infinite nature of the Hilbert space. A consequence of this development is that the classical definition of integrability can in general be directly transferred to the quantum case.
Definition 3: (Quantum Integrability) A quantum system with M independent dynamical degrees of freedom, hence a 2M-dimensional quantum phase space, is integrable if and only if there are M quantum constraints of motion, or good quantum numbers, which are related to the eigenvalues of M non-fully degenerate observables: O1,O2,…,On.□
Any set of variables that commute may be put in the form of a complete set of commuting observables C by including certain additional observables with it. The definition then says that if the system is integrable, a complete set of commuting variables C can be found so that the Hamiltonian is always diagonal in the basis referred to by C. In the reverse sense, the definition implies that if the system is integrable, simultaneous accurate measurements of M non-fully degenerate observables in the energy eigenvalues can be carried out.
The link with the dynamical group structure can be developed. This specifies exactly the integrability of a quantum system. To this end the definition of dynamical symmetry is needed.
Definition 4: (Dynamical Symmetry) A quantum system with dynamical group G possesses a dynamical symmetry if and only if the Hamiltonian operator of the system can be written and presented in terms of the Casimir operators of any specific chain with α fixed
H=FCkjαE21
The index of a particular subgroup chain Ckjα the i-th Casimir operator of subgroup Gkα, k=sα,…,1, i=1,…,lkα and lkα denotes that the rank of subgroup Gkα is l. It is now possible to state a theorem which gives a condition for integrability to apply.
Proposition 1: (Quantum Integrability) A quantum system with dynamical group G is said to be integrable if it possesses a dynamical symmetry of G.
To prove this, note that it can be broken down into two cases or subgroup classes for a given dynamical group G and are referred to as canonical and noncanonical.
First consider the case in which Gα is a canonical subgroup chain of G. The Casimir operators of G, CGi and all Casimir operators Ckiα corresponding to the subgroups in chain Gα form a complete set of commuting operators Cα of any carrier irrep space H of Gα so for fixed α,
Cα:CGi∪Ckiα≡Qjj=1⋯N.E22
When Gα is the dynamical symmetry of the system, all operators in Cα are constants of motion
HQj=0.E23
There are always M nonfully dynamical operators in Cα. By the third definition, the system is integrable.
For a non-canonical subgroup chain Gα the number of Casimir operators CGi of G and all Casimir operators of Ckiα of Gα is less than the number of the complete set of commuting operators C of any irrep carrier space of G. By definition, of any complete set of commuting operators, there must exist other commuting operators Oj that commute with CGi and Ckiα. These have to be included in the union as well when putting together Cα
Cα:CGi∪Ckiα∪Oj≡Qj:j=1…N.E24
When the system is characterized by the dynamical symmetry of Gα, the operators in (24) satisfy relation (23) as well. In this case as well, there must exist M non-fully degenerate operators of constants of motion as in the previous case.
Based on this proposition, it can be stated that nonintegrability of a quantum system involves the breaking of the dynamical symmetry of the system. It may be concluded that dynamical symmetry breaking can be said to be a property which characterizes quantum nonintegrability. □
Let us summarize what has been found as to what quantum mechanics tells us. In a given quantum system with dynamical Lie group G which is of rank l and dimension n, the dimension of a complete set of commuting operators C of G with any particular subgroup chain is d=l+n−l/2 in which the l operators are Casimirs of G and are fully degenerate for any given irrep of G. The number M of the non-fully degenerate operators in C for a given irrep of G cannot exceed M≤n−l/2. When dynamical symmetry is broken such that any of the M constants of the motion for the system is destroyed the system becomes nonintegrable.
4. Quantum phase space
It is of interest then to develop a model for phase space for quantum mechanics which may be regarded as an analogue to classical physics. By what has been said so far, the Hilbert space H of the system can be broken up into a direct sum of the unitary irreps carrier spaces of G,
H=∑Λ⊕YΛHΛ.E25
In (25), the subscript Λ labels a particular irrep of Lie group G, Λ is the largest weight of the irrep and YΛ the degeneracy of Λ in H with no correlations existing between various HΛ. The study of the dynamical properties of the system can be located on one particular irreducible subspace HΛ of H. For a quantum system with MΛ independent quantum dynamical degrees of freedom, the corresponding quantum phase space should be a 2MΛ-dimensional, topological phase space without additional constraints.
To construct the quantum phase space from the quantum dynamical degrees of freedom for an arbitrary quantum system, the elementary excitation operators can be obtained from the structures of G and HΛ. Let ai† be a subset of generators of G such that any states ∣Ψ〉 of the system are generated for all ∣Ψ〉∈HΛ by means of
∣Ψ〉=Fai†∣0〉.E26
Moreover F(ai†) is a polynomial in the operators ai† and ∣0〉∈HΛ is the reference state. The requirement placed on state ∣0〉 is that one can use a minimum subset of g to generate the entire subspace HΛ from ∣0〉. In this event, the collection ai† is called the set of elementary excitation operators of the quantum dynamical degrees of freedom. If G is compact, ∣0〉 is the lowest ∣Λ,−Λ〉 or highest weight ∣Λ,Λ〉 state of HΛ. If G is noncompact, it is merely the lowest state. The number of ai† is the same as the number of quantum dynamical degrees of freedom. Physically this has to be the case since that is how the operators are defined. Thus the set ai† and Hermitian conjugate ai in gΛ form a dynamical variable subspace μ of g so we can write
μ:ai†aii=1…MΛ.E27
With respect to μ there exists a manifold whose dimension is twice that of the quantum dynamical degrees. It can be realized by means of a unitary exponential mapping of the dynamical variable operator subspace μ
Ω=exp∑i=1MΛηiai†−ηi∗ai∈∐.E28
The ηi are complex parameters and i=1,…,MΛ. In fact, Ω is a unitary coset representation of G/R, where R⊂G is generated by the subalgebra κ=g−μ. Thus (28) shows that q is isomorphic to the 2MΛ-dimensional coset space G/R, and will be denoted this way from now on. The discussion will apply just to semi-simple Lie groups whose g satisfies the usual Cartan decomposition g=κ+μ and κκ⊂κ, κμ⊂μ and μμ⊂κ. Thus G/R will be a complex homogeneous space with topology and a group transformation acting on G/R is a homomorphic mapping of G/R into itself.
The homogeneous space G/R has a Riemannian structure with metric
gij=∂2logKzz¯∂zi∂z¯jE29
The function Kzz¯ is called the Bergmann kernel of G/R and can be represented as
Kzz¯=∑λfλzfλ∗z¯.E30
The functions fλz in (30) constitute an orthogonal basis for a closed linear subspace L2G/R of L2G/R such that
∫G/Rfλzfλ′∗z¯K−1zz¯dνzz¯=δλλ′,E31
and dνzz¯ is the group invariant measure on the space G/R. It will be written
dνzz¯=ζdetgij∏i=1Mλdzidz¯iπ.E32
In (32)ζ is a normalization factor given by the condition that (32) integrated over the space G/R is equal to one. There is also a closed, nondegenerate two-form on G/R which is expressed as,
ω=iℏ∑i,jgijdzi∧dz¯j.E33
Corresponding to this two form there is a Poisson bracket which is given by
fh=1iℏ∑i,jgij∂f∂zi∂h∂z¯j−∂f∂z¯j∂h∂zi.E34
In (34)f and h are functions defined on G/R. By introducing canonical coordinates qp these quantities can be rewritten in terms of these coordinates.
4.1 Phase space quantum dynamics
Based on what has been stated about G/R, it would be useful to describe the quantum phase space. This means for a given quantum system a phase space representation must exist. Such a representation can be found if there exists an explicit mapping such that
OTi→Uqp,∣Ψ〉→ρq+ip.E35
Here O is given by (19), and ρqp∈L2. For a quantum system with a quantum phase space G/R, this mapping can be realized by coherent states. To construct coherent states of G and HΛ defined on G/R, the fixed state ∣0〉 is chosen as the initial state
g∣0〉=Ωr∣0〉=∣Λ,Ω〉eiφr,g∈G,r∈R,Ω∈G/R.E36
Then R is the maximal stability subgroup of ∣0〉 so any r∈R acting on ∣0〉 will leave ∣0〉 invariant up to a phase factor
r∣0〉=eiφr∣0〉.E37
The ∣Λ,Ω〉 are the coherent states which are isomorphic to G/R. Therefore,
∣Λ,Ω〉≡Ω∣0〉=exp∑i=1MΛηiai†−ηi∗ai∣0〉=K1/2zz¯exp∑i=1MΛziai†∣0〉=K−1/2zz¯‖Λ,z〉.E38
Kzz¯=0exp∑i=1MΛz¯iaiexp∑i=1MΛziai†0=Λz‖Λz=∣〉0Λ0〉2=∑λfΛλzfΛλ∗z.
The Bargmann kernel was introduced in (30), and for a semisimple Lie group, the parameters zi are given by
z=ηtanη†η1/2η†η1/2,Gcompact,ηtanhη†η1/2η†η1/2,Gnoncompact.E39
Here η represents the nonzero k×p block matrix of the operator ∑i=1MΛηiai−ηi∗ai†. The state ‖Λ,z〉 in (38) is an unnormalized form of ∣Λ,Ω〉 and fΛ,λz is the orthogonal basis of L2G/R the function space
fΛ,λz=Λλ‖Λλ,E40
where ∣Λ,λ〉 is a basis for HΛ, a particular irreducible subspace of the Hilbert space. The coherent states of (38) are over-complete
∫G/R∣Λ,Ω〉〈Λ,Ω∣dνz=I.E41
A classical-like framework or analogy has been established in the form of a quantum phase space specified by G and HΛ. Variables which reside in this classical analogy are denoted thus c˜. The 2MΛ-dimensional quantum phase space G/R has all the required structures of a classical mechanical system. It is always possible a classical dynamical theory can be established in G/R whose motion is confined to G/R and is determined by the following equations of motion
dU˜dt=U˜qpH˜qp,q,p∈G/R.E42
This equation can be replaced by Hamilton’s equations
dqidt=∂H˜qp∂pi,dpidt=−∂H˜qp∂qi.E43
In (42) and (43), H˜qp is the Hamiltonian of the system, and U˜qp is a physical observable. A correspondence principle is implied here and requires that suitable conditions can be found such that the quantum dynamical Heisenberg equations can be written this way.
Clearly, if suitable conditions hold the phase space representation of the commutator of any two operators is equal to the Poisson bracket of the phase space representation of these two operators so that
1iℏΛΩAHBHΛΩ=A˜B˜.E44
Then the phase space representation of the Heisenberg equation
dAHdt=1iℏAHHH,E45
given by (42) is therefore equivalent to (43). In (45), AH is the Heisenberg operator
AH=UAU−1,U=eiHt/ℏ,E46
and A is time-independent in the Schrödinger picture. The coherent state on the left of (44) is time-independent. Observables on the right side are the expectation values of the Schrödinger operators in the time-dependent coherent state. The quantum phase space maintains many of the quantum properties which are important, such as internal degrees of freedom, the Pauli principle, statistical properties and dynamical symmetry. Formally the equation of motion is classical. The phase space representation is based on the whole quantum structure of the coset space G/R.
Let us discuss integrability and dynamical symmetry. A quantum system with MΛ independent degrees of freedom is integrable if and only if the MΛ non-fully degenerate observables can simultaneously be measured in the energy representation. There exist non-fully degenerate observables Ci:i=1…MΛ−1 which commute with each other and H
CiCj=0,CiH=0.E47
It follows that in the classical limit which has been formulated,
C˜iC˜j=0,C˜iH˜=0.E48
Together with the Hamilton equations, (47) also formally defines classical integrability, so quantum integrability is completely consistent with the classical theory. In the classical analogy, the group structure of the system is defined by Poisson brackets. The concept of dynamical symmetry is naturally preserved in the classical analogy, so the theorem on dynamical symmetry and integrability is also meaningful for the classical analogy. If the Hamiltonian has the symmetry S, then its phase space picture representation has the same symmetry. To see this, if
SHS−1=H,E49
in the phase space representation, it holds that
ΛΩHΛΩ=ΛΩSHS−1ΛΩ=ΛΩ′HΛΩ′.E50
To put this concisely, we write
H˜qp=H˜q′p′,E51
where S−1∣Λ,Ω〉=S−1Ω∣0〉=∣Λ,Ω′〉eiφh.
5. Applications to physical systems
5.1 Harmonic oscillator
The harmonic oscillator has dynamical group H4 and is a single-degree of freedom system [13, 14, 15, 23]. To the dynamical group corresponds the algebra h4 defined by the set a†aa†aI with Hilbert space the Fock space VF:n〉n=12…, so the fixed state is the ground state ∣0〉, and elementary excitation operator a†. The quantum phase space is constructed from the unitary exponential mapping of the subspace μ:a†a of h4,
Ωz=expza†−z¯a∈H4/U1⊗U1.E52
With generators a†a and I, U1⊗U1 in (52) is the maximal stability subgroup of ∣0〉. As H4/U1⊗U1 is isomorphic to the one-dimensional complex plane, the quantum phase space has metric gij=δij and dνz=dzdz¯/π. It is noncompact due to the infiniteness of the Fock space. There is a well-known symplectic structure on the complex plane with Poisson bracket of two functions F˜1,F˜2 defined by
F˜1F˜2=1iℏ∂F˜1∂z∂F˜2∂z¯−∂F˜1∂z¯∂F˜2∂z.E53
It is useful to introduce the standard canonical position and momentum coordinates
z=12ℏq+ip,z¯=12ℏq−ip.E54
The Glauber coherent states can be realized by the states ∣z〉 with the set of these states isomorphic to H4/U1⊗U1 and given as
∣z〉≡Ωz∣0〉=expza†−z¯a∣0〉=e−z2/2expza†∣0〉.E55
The normalization constant in (55) is the Bargmann kernel
Kzz¯=ez2.E56
The phase space representation of the wavefunction ∣Ψ〉∈VF is
fz=Ψ‖z=∑n=0∞fnznn!.E57
By Wick’s Theorem, it is always possible to write an operator A in normal product form
A=Aa†a=∑k,lAk,lna†kal.E58
The phase space representation of A is just
U˜zz¯=zAz=∑k,lAk,lnz¯kzl.E59
In the case A is simply a generator of H4, we can write (59) as
a˜†=za†z,a˜=zaz,a˜†a˜=za†az=z2,I˜=zIz=I.E60
The corresponding algebraic structure of H4 in the phase-space representation is
iℏca˜a˜†=I˜,iℏca˜†a˜=−a˜,iℏca˜†a˜a˜†=a˜†.E61
Here ℏc is used in the classical analogy. The algebraic structure of the H4 generators is preserved when commutators are replaced by Poisson brackets in phase space. Using (54) the Dirac quantization condition and H˜ are given by
qp=iℏcqp,H˜qp=zHz.E62
For the forced harmonic oscillator, the classical analogy of the Hamiltonian is given by
H˜qp=ω2p2+q2+i2ℜλtq−2ℑλtp=ω2p2+q2+12λt+λ¯tq)+12iλt−λ¯tp.E63
Hamilton’s equations in (44) can be used to evaluate the t derivatives of q and p:
dqdt=ωp+12iλt−λ¯t,dpdt=−ωq−12λt+λ¯t.E64
Hence combining these two derivatives, we obtain
ddtq+ip=−iωq+ip−2iλt.E65
Multiplying both sides by the integrating factor eiωt and then integrating with respect to t, the solution is
qt+ipt=e−iωtq0+ip0−i2e−iωt∫0tλτeiωτdτ=zt2ℏc.E66
If the initial state is ∣0〉 or a coherent state ∣z0〉, then the exact quantum solution is
∣ψt〉=∣zt〉eiφtE67
and zt is given by (66). The phase φ is a quantum effect obtained from zt
φt=−12ωt−∫0tℜλτzτdτ.E68
This seems to imply the classical analogy provides an exact quantum solution if the Hamiltonian is a linear function of the generators of G.
5.2 SU2 spin system
The phase space structure of a spin system will be constructed and as well the phase-space distribution and classical analogy.
Since the dynamical group of the spin system is SU2 and the Hilbert space is described by the states V2j+1=jm〉 where m=−j,−j+1,…,j and j is an integer or half-integer, the fixed state is ∣j,−j〉. This is the lowest weight state of V2j+1. Thus the elementary excitation operator of the spin system is J+ and the explicit form of ∣j,m〉 is
∣j,m〉=1j+m!2jj+m−1/2J+j+m∣j,−j〉.E69
Any state ∣Ψ〉=∑m=−jjfm∣j,m〉∈V2j+1 can be generated by a polynomial of J+ acting on ∣j,−j〉. This means the number of quantum dynamical degrees of freedom is equal to the number of elementary excitation operators. The quantum phase space can be found by mapping μ:J+J− to the coset space SU2/U1 by means of ηJ+−η¯J−→expηJ+−η¯J− where η=ϑ/2e−iφ, 0≤ϑ≤π, 0≤φ≤2π. The coset space SU2/U1 is isomorphic to a two-dimensional sphere. The coherent states of SU2/U1 are well known
∣jΩ〉=expηJ+−η¯J−∣j,−j〉=1+z2−jexpzJ+∣j,−j〉=1+z2−j‖jz〉,z=tanϑ2e−iφ.E70
The generalized Bargmann kernel on S2 is Kzz¯=1+z22j. Then the metric gij and measure are given by
gij=δij2j1+z22,dν=1π2j+1dzdz¯1+z2)2.E71
Given the canonical coordinates
14jℏq+ip=z1+z2=sinϑ2e−iφ,E72
there obtains the bracket
F˜1F˜2=∂F˜1∂q∂F˜2∂p−∂F˜1∂p∂F˜2∂q,E73
where q2+p2≤4jℏ, which implies the phase space of a spin system is compact.
The phase space representation of the state ∣Ψ〉∈V2j+1 is for f∈L2S2,
fz=Ψ‖jΩ=∑n=0∞fn2jj+m1/2zj+m,E74
The phase space representation of an operator B=BJi is
B˜zz¯=jΩBjΩ.E75
When the operator B in (75) is chosen to be one of the three operators J+, J− or J0, the results are
J˜+=jΩJ+jΩ=2jz¯1+z2,J˜−=jΩJ−jΩ=2jz1+z2,J˜0=jΩJ0jΩ=jz2−11+z2.E76
These can also be given in terms of q,p by using (72). The algebraic structure of SU2 in the phase space representation is given by the Poisson bracket
iℏcJ˜−J˜+=−2J˜0,iℏcJ˜0J˜±=±J˜±.E77
The classical analogy of an observable BJi is given by the following expression
B˜qp=jΩBJijΩ.E78
The classical limit is found by taking j→∞ and the classical Hamiltonian function is
H˜Cqp=HjΩJijΩ=HJ˜+J˜−J˜0.E79
5.3 SU11 quantum systems and a two-level atom
A two-level atom is considered which interacts with two coupled quantum systems that can be represented in terms of a su11 Lie algebra. When for example mixed four-waves are injected into a cavity containing a single two level-atom an interaction occurs between the four waves and the atom that is electromagnetic radiation and matter. The Hamiltonian has the form
1ℏH=∑i=12ωiai†ai+12+12ω0σz+λa12a22σ++a1†2a2†2σ−.E80
It is similar to 5.1, so we sketch the physical situation. The σ±, σz are raising lowering and inversion operators which satisfy the commutation relations σzσ±=2σ±, σ+σ−=σz, whereas the ai†, ai are basic creation and annihilation operators with ajaj†=δij. The interaction term in (80) can be thought of as the interaction between two different second harmonic modes. This can be cast in terms of three su11 Lie algebra generators K+, K− and Kz which satisfy the commutation relations,
KzK±=±K±,K−K+=2Kz.E81
The corresponding Casimir K which has eigenvalue kk−1 given by
K2=Kz2−12K+K−+K−K+.E82
Given that this is the Lie algebra, it can be said that the Fock space is spanned by the set of vectors VF:mk〉 and the operators in (81) act on these states as follows,
Kz∣m;k〉=m+k∣m;k〉,K2∣m;k〉=kk−1∣m;k〉,K+∣m;k〉=m+1m+2k∣m+1;k〉,K−∣m;k〉=mm+2k−1∣m−1;k〉.E83
It is the case that K−∣0;m〉=0 so this is the lowest level state. The su11 Lie algebra can be realized in terms of boson annihilation and creation operators and it is isomorphic to the Lie algebra of the non-compact SU11 group. For the Hamiltonian (80) define operators K±i and Kzi as
K+i=12ai†2,K−i=12ai2,Kzi=12ai†ai+12,i=1,2,E84
where the Bargmann index k is either 1/4 for the even parity states while 3/4 applies to the odd-parity states.
Using these operators, (80) is written in terms of su2 and su11 operators such that it has the form,
1ℏH=∑i=12ηiKzi+ω2σz+λK+1K+2σ−+K−1K−2σ+.E85
The Heisenberg equations of motion obtained from (85) gives
iddtKz1=λK+1K+2σ−−K−1K−2σ+,iddtKz2=λK+1K+1σ−−K−1K−2σ+,iddtσz=λK−1K−2σ+−K+1K+1σ−.E86
The following two operators N1 and N2 are constants of the motion
N1=Kz1+σz,N2=Kz2+σz.E87
Hamiltonian (80) can now be put in the equivalent form
1ℏH=N+C+I,E88
where I is the identity operator and N and C are the operators
N=∑i=12ηiNi,C=Δσz+λK+1K+2σ−+K−1K−2σ+.E89
The constant Δ is the detuning parameter defined as
Δ=ω2−η1−η2.E90
As N and C commute, each commutes with the Hamiltonian H so N and C are constants of the motion. The time evolution operator Ut is given by
Ut=exp−iHℏt⋅exp−iNt⋅expiCt.E91
In the space of the two-level eigenstates
e−iNt=e−iW1t00e−iW2t.E92
The operators Wi, i=1,2 are defined by W1=η1Kz1+η2Kz2+1 and W2=η1Kz1+η2Kz2−1. The second exponential on the right of (91) takes the form,
exp−iCt=cosτit−iΔτsinτ1t−iλsinτ1tτ1K−1K−2−iλK+1K+2sinτ1tτ1cosτ2t−iΔτ2sinτ2tE93
where τj2=Δ2+νj, j=1,2 and
τ1=λ2K−1K+1K−2K+2,τ2=λ2K+1K−1K+2K−2E94
The coherent atomic state ∣ϑ,φ〉 is considered to be the initial state that contains both excited and ground states and has the structure,
∣ϑ,φ〉=cosϑ2∣e〉+sinϑ2e−iφ∣g〉.E95
where ϑ is the coherence angle, φ the relative phase of the two atomic states. The excited state is attained by taking ϑ→0, while the ground state of the atom is derived from the limit ϑ→π. The initial state of the system that describes the two su11 Lie algebras is assumed to be prepared in the pair correlated state ∣ξ,q〉 defined by
K−1K−2∣ξ,q〉=ξ∣ξ,q〉,Kz1−Kz1∣ξ,q〉=q∣ξ,q〉.E96
Since the operators K−1K−2 and Kz1−Kz2 commute, ∣ξ,q〉 can be introduced which is simultaneously an eigenstate of both operators,
∣ξ,q〉=∑n=0∞Cn∣q+n+k2−k1;k1;n,k2〉.E97
Then applying K−2 and then K−1 we obtain,
K−1K−2∣ξ,q〉=∑n=0∞Cnnn+2k−1q+n+k2−k1(q+n+k2−k1+2k1−1∣q+n+k2−k1−1,k1;n−1,k2〉,=∑n=0∞Cn+1n+1n+2kq+n+k2−k1+1(q+n+k2−k1+2k1∣q+n+k2−k1,k1;n,k2〉.E98
This calculation implies that the normalization constant Cn can be obtained by solving
n+1n+2kq+n+k2−k1+1n+q+k1+k2Cn+1=ξC0.E99
The new state is of the form,
∣ξ,q〉=Nq∑n=0∞Cn∣q+n+k2−k1,k;n,k2〉,Nq−2=∑n=0∞Cn2.E100
If it is assumed that at t=0 the wave function of the system is ∣ψ0〉=∣ϑ,φ〉⊗∣ξ,q〉, using (91) on ∣ψ0〉, the state can be calculated for t>0 can be determined
∣ψt〉=e−iW1tcosτ1t−iΔτ1sinτ1t)cosϑ2−iλτ1sinτ1tK−1K−2e−iφsinϑ2∣e〉⊗∣ξ,q〉+eiW2tcosτ2t+iΔτ2sinτ2te−iφsinϑ2−iλτ2sinτ2tK+1K+2cosϑ2∣g〉⊗∣ξ,q〉.E101
The reduced density matrix is constructed from this
ρft=Tratom∣ψt〉〈ψt∣.E102
6. Summary and conclusions
Explicit structures for quantum phase space have been examined. Quantum phase space provides an inherent geometric structure for an arbitrary quantum system. It is naturally endowed with sympletic and quantum structures. The number of quantum dynamical degrees of freedom has a great effect on determining the quantum phase space. Inherent properties of quantum theory, the Pauli principle, quantum internal degrees of freedom and quantum statistical properties are included. A procedure can be stated for constructing this quantum phase space and canonical coordinates should be derivable for all semi-simple dynamical Lie groups with Cartan decomposition. The coset space G/R provides a way to define coherent states which link physical Hilbert space and quantum phase space. This motivates the study of the algebraic structure of the phase space representation of observables. The algebraic structure of operators is preserved in phase space if the operators are those of the dynamical group G. Through this approach, this property results in an explicit realization of the classical limit of quantum systems. A classical analogy was developed and seen in the examples as well for an arbitrary quantum system independently of the existence of the classical counterpart, so the classical limit of the quantum system can be obtained explicitly if it exists. The classical analogy will contain the first-order quantum correlation. A theorem which pertains to the relationship between dynamical symmetry and integrability has been proved, and is also valid in classical mechanics. It is then possible to construct a way to look for the quantum manifestation of chaos. Finally, it is then consistent with Berry’s definition, the study of semi-classical but nonclassical, behavior characteristic of systems whose classical motion exhibits chaos.
\n',keywords:"classical, quantum, chaos, integrability, conservation law, algebra",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/74091.pdf",chapterXML:"https://mts.intechopen.com/source/xml/74091.xml",downloadPdfUrl:"/chapter/pdf-download/74091",previewPdfUrl:"/chapter/pdf-preview/74091",totalDownloads:44,totalViews:0,totalCrossrefCites:0,dateSubmitted:"July 27th 2020",dateReviewed:"October 14th 2020",datePrePublished:"November 17th 2020",datePublished:null,dateFinished:null,readingETA:"0",abstract:"The concept of integrability of a quantum system is developed and studied. By formulating the concepts of quantum degree of freedom and quantum phase space, a realization of the dynamics is achieved. For a quantum system with a dynamical group G in one of its unitary irreducible representative carrier spaces, the quantum phase space is a finite topological space. It is isomorphic to a coset space G/R by means of the unitary exponential mapping, where R is the maximal stability subgroup of a fixed state in the carrier space. This approach has the distinct advantage of exhibiting consistency between classical and quantum integrability. The formalism will be illustrated by studying several quantum systems in detail after this development.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/74091",risUrl:"/chapter/ris/74091",signatures:"Paul Bracken",book:{id:"10413",title:"Chaotic Systems",subtitle:null,fullTitle:"Chaotic Systems",slug:null,publishedDate:null,bookSignature:"Prof. Paul Bracken and Dr. Dimo I. Uzunov",coverURL:"https://cdn.intechopen.com/books/images_new/10413.jpg",licenceType:"CC BY 3.0",editedByType:null,editors:[{id:"92883",title:"Prof.",name:"Paul",middleName:null,surname:"Bracken",slug:"paul-bracken",fullName:"Paul Bracken"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"92883",title:"Prof.",name:"Paul",middleName:null,surname:"Bracken",fullName:"Paul Bracken",slug:"paul-bracken",email:"paul.bracken@utrgv.edu",position:null,institution:{name:"The University of Texas Rio Grande Valley",institutionURL:null,country:{name:"United States of America"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_1_2",title:"1.1 The hydrogen atom",level:"2"},{id:"sec_3",title:"2. Quantum degrees of freedom",level:"1"},{id:"sec_4",title:"3. Quantum integrability and dynamical symmetry",level:"1"},{id:"sec_5",title:"4. Quantum phase space",level:"1"},{id:"sec_5_2",title:"4.1 Phase space quantum dynamics",level:"2"},{id:"sec_7",title:"5. Applications to physical systems",level:"1"},{id:"sec_7_2",title:"5.1 Harmonic oscillator",level:"2"},{id:"sec_8_2",title:"5.2 SU2 spin system",level:"2"},{id:"sec_9_2",title:"5.3 SU11 quantum systems and a two-level atom",level:"2"},{id:"sec_11",title:"6. Summary and conclusions",level:"1"}],chapterReferences:[{id:"B1",body:'[Baker GL Gollub JP. Chaotic Dynamics. Cambridge: Cambridge University Press 1990.]'},{id:"B2",body:'[Ott E. Chaos in Dynamical Systems. Cambridge: Cambridge University Prees 1993.]'},{id:"B3",body:'[Eckhart B. Quantum Mechanics of Classically Non-Integrable Systems. Phys Rep 1988; 163: 205–297.]'},{id:"B4",body:'[Arnold VI. Mathematical Methods of Classical Mechanics. Springer, New York. 1978.]'},{id:"B5",body:'[Giannoni MJ, Voros A, Zinn-Justin J. eds. Chaos and Quantum Physics. Les Houches, Session LII. Amsterdam, North Holland. 1991.]'},{id:"B6",body:'[Berry MV. Semiclassical mechanics of regular and irregular motion, in: Chaotic Behavior in Deterministic Systems. eds Iooss G Helleman RHG Stora R. Amsterdam North Holland 171–271.]'},{id:"B7",body:'[Berry MV. Semiclassical theory of spectral rigidity. Proc R Soc London. 1995; A 400: 229–251.]'},{id:"B8",body:'[Berry MV Mount KE. Semiclassical wave mechanics. Rep Prog Physics. 1972; 35: 315–397.]'},{id:"B9",body:'[Simon B. Nonclassical eigenvalue asymptotics. J. Fuct Anal. 1983; 53: 84–98.]'},{id:"B10",body:'[Simon B. Holonomy, the quantum adiabatic theorem and Berry’s phase. Phys. Rev. Lett. 1983; 51: 2167–2170.]'},{id:"B11",body:'[Nelson P Alvarez-Gaumé L. Hamiltonian interpretation of anomalies. Commun Math Phys. 1985; 99: 103–114,]'},{id:"B12",body:'[Teller E. Crossing of potential surfaces. J Chem Phys 1937; 41: 109–116.]'},{id:"B13",body:'[Zhang W-M Feng PH Yuan J-M Wang S-J. Integrability and nonintegrability of quantum systems: Quantum integrability and dynamical symmetry. Phys Rev 1989; A 40: 438–447.]'},{id:"B14",body:'[Zhang W-M Feng DH Yuan J-M. Integrability and nonintegrability of quantum systems II. Dynamics in quantum phase space. Phys Rev 1990; A 42: 7125–7150.]'},{id:"B15",body:'[Zhang W-M Feng DH Pan Q Tjon J. Quantum fluctuations in quantum chaos. Phys Rev 1990; 42: 3646–3649.]'},{id:"B16",body:'[Adams BG Cizek J Paldus J. Lie Algebraic Methods and Their Application to Single Quantum Systems. Adv Quantum Chem. 1988; 1988: 1–85.]'},{id:"B17",body:'[Stone AD. Einstein’s Unknown Insight and the Problem of Quantizing Chaos. Physics Tod. 2005; 8: 37–43.]'},{id:"B18",body:'[Burić N. Hamiltonian quantum dynamics with separability constraints. Ann Phys. 2008; 323: 17–33.]'},{id:"B19",body:'[Yazbashyan EA. Generalized microcanonical and Gibbs ensembles in classical and quantum integrable dynamics. Ann Phys. 2016; 367: 288–296.]'},{id:"B20",body:'[Balain R Block C. Distribution of eigenfrequencies for the wave equation in a finite domain III. Ann Phys. 1972; 69: 76–160.]'},{id:"B21",body:'[Dirac PAM. The adiabatic invariance of the quantum integrals. Proc Roy Soc. 1925; 107: 725–734.]'},{id:"B22",body:'[Clemente-Gallardo J Marmo G. Towards a Definition of Quantum Integrability, Int. J. of Geometric Methods in Modern Physics. 2009; 6: 120–172.]'},{id:"B23",body:'[Bohm A. Quantum Mechanics. Springer, New York, 1979.]'},{id:"B24",body:'[Abdalla MS Khalil EM Obada AS-F. Interaction between two SU11 quantum systems and a two-level atom. Physics A. 2016; 454: 99–109.]'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Paul Bracken",address:"paul.bracken@utrgv.edu",affiliation:'- Department of Mathematics, University of Texas, Edinburg, TX 78540, USA
'}],corrections:null},book:{id:"10413",title:"Chaotic Systems",subtitle:null,fullTitle:"Chaotic Systems",slug:null,publishedDate:null,bookSignature:"Prof. Paul Bracken and Dr. Dimo I. Uzunov",coverURL:"https://cdn.intechopen.com/books/images_new/10413.jpg",licenceType:"CC BY 3.0",editedByType:null,editors:[{id:"92883",title:"Prof.",name:"Paul",middleName:null,surname:"Bracken",slug:"paul-bracken",fullName:"Paul Bracken"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}}},profile:{item:{id:"138062",title:"Dr.",name:"Fabian",middleName:null,surname:"Leinisch",email:"leinischf@niehs.nih.gov",fullName:"Fabian Leinisch",slug:"fabian-leinisch",position:null,biography:null,institutionString:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",totalCites:0,totalChapterViews:"0",outsideEditionCount:0,totalAuthoredChapters:"1",totalEditedBooks:"0",personalWebsiteURL:null,twitterURL:null,linkedinURL:null,institution:null},booksEdited:[],chaptersAuthored:[{title:"Fluorescent Nitrones for the Study of ROS Formation with Subcellular Resolution",slug:"fluorescent-nitrones-for-the-study-of-ros-formation-with-subcellular-resolution",abstract:null,signatures:"Stefan Hauck, Yvonne Lorat, Fabian Leinisch, Christian Kopp, Jessica Abrossinow and Wolfgang E. Trommer",authors:[{id:"137478",title:"Prof.",name:"Wolfgang",surname:"Trommer",fullName:"Wolfgang Trommer",slug:"wolfgang-trommer",email:"trommer@chemie.uni-kl.de"},{id:"138021",title:"Mr.",name:"Christian",surname:"Kopp",fullName:"Christian Kopp",slug:"christian-kopp",email:"kopp@chemie.uni-kl.de"},{id:"138022",title:"Ms.",name:"Jessica",surname:"Abrossinow",fullName:"Jessica Abrossinow",slug:"jessica-abrossinow",email:"abrossinow@chemie.uni-kl.de"},{id:"138061",title:"Dr.",name:"Stefan",surname:"Hauck",fullName:"Stefan Hauck",slug:"stefan-hauck",email:"stefanhauck1977@yahoo.de"},{id:"138062",title:"Dr.",name:"Fabian",surname:"Leinisch",fullName:"Fabian Leinisch",slug:"fabian-leinisch",email:"leinischf@niehs.nih.gov"},{id:"138063",title:"Mrs.",name:"Yvonne",surname:"Lorat",fullName:"Yvonne Lorat",slug:"yvonne-lorat",email:"yvonne.lorat@uniklinikum-saarland.de"}],book:{title:"Nitroxides",slug:"nitroxides-theory-experiment-and-applications",productType:{id:"1",title:"Edited Volume"}}}],collaborators:[{id:"136943",title:"MSc.",name:"Toshihide",surname:"Yamasaki",slug:"toshihide-yamasaki",fullName:"Toshihide Yamasaki",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"136944",title:"MSc.",name:"Fumiya",surname:"Mito",slug:"fumiya-mito",fullName:"Fumiya Mito",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"137323",title:"Dr.",name:"Vasily",surname:"Sen'",slug:"vasily-sen'",fullName:"Vasily Sen'",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"137352",title:"Dr.",name:"Eugene",surname:"Pliss",slug:"eugene-pliss",fullName:"Eugene Pliss",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"137465",title:"Prof.",name:"Rui",surname:"Tamura",slug:"rui-tamura",fullName:"Rui Tamura",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"138494",title:"PhD.",name:"Ken-Ichi",surname:"Yamada",slug:"ken-ichi-yamada",fullName:"Ken-Ichi Yamada",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"140109",title:"Mr.",name:"Yuta",surname:"Matsuoka",slug:"yuta-matsuoka",fullName:"Yuta Matsuoka",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"150722",title:"Dr.",name:"Mayumi",surname:"Yamato",slug:"mayumi-yamato",fullName:"Mayumi Yamato",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"151038",title:"Dr.",name:"Yoshiaki",surname:"Uchida",slug:"yoshiaki-uchida",fullName:"Yoshiaki Uchida",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"151039",title:"Mr.",name:"Katsuaki",surname:"Suzuki",slug:"katsuaki-suzuki",fullName:"Katsuaki Suzuki",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null}]},generic:{page:{slug:"order-print-copies",title:"Order Print Copies",intro:'Our books are published online and are accessible for free. However, if you are interested in ordering your hardcover copy, you can do so by contacting our Print Sales Department at orders@intechopen.com. All IntechOpen books are printed on demand in full-colour and delivered in signature packaging through free DHL Express delivery. A selection of our books in soft cover is also available through Amazon.
',metaTitle:"Order Print Copies",metaDescription:"Our books are published online and are accessible for free. However, if you are interested in ordering your printed copy, you can do so by contacting our Print Sales Department at orders@intechopen.com.\n\nOur hardcover books are carefully designed and printed on wood-free premium quality paper.\n\nThe paper size is 155 mm x 225 mm (6.1 X 8.8 inches).",metaKeywords:null,canonicalURL:"/page/order-print-copies",contentRaw:'[{"type":"htmlEditorComponent","content":"InTechOpen contributors can order print books at a special price ranging from:
\\n\\n55-90 GBP for Hardcover full-colour books
\\n\\n\\n\\t- Discounts are granted for orders of 10 copies and more.
\\n\\t- All library orders receive a 20% discount.
\\n\\t- There is no minimum or maximum threshold on the quantity of book orders.
\\n
\\n\\nFor a quote please contact us directly at orders@intechopen.com The quote will be sent to you within 1-2 business days.
\\n\\nAll of the books and chapters can be browsed online. To obtain InTechOpen's full book catalogue in PDF, please contact us.
\\n\\nTERMS (find out more)
\\n\\nPrint on Demand (POD)
\\n\\n\\n\\t- Books and offprints printed specifically for your order
\\n\\t- Ordered, printed, and shipped in 12-20 business days
\\n\\t- Available for purchase at any time
\\n\\t- No minimum or maximum threshold on book order quantity
\\n
\\n\\nIntechOpen works with award winning print-houses and we hold to the fact that all of our printed products are of the highest quality.
\\n\\nRepresentatives
\\n\\nPrint copies of our publications are most often purchased as individual purchases by universities, libraries, institutions and academia personnel, hence increasing the visibility and outreach of our authors' published work among science communities and institutions. Our books are available at our direct Print Sales Department and through selected representatives throughout the world.
\\n\\nIndia - CBS Publishers & Distributors Pvt. Ltd.
\\n\\n\\n\\t- Representative for: India, Bangladesh, Pakistan, Sri Lanka, Bhutan, Nepal, Maldives, Iran, Algeria, Bahrain, Egypt, Iraq, Israel, Jordan, Kuwait, Lebanon, Libya, Malta, Morocco, Oman, Qatar, Saudi Arabia, Syria, Tunis, United Arab Emirates and Yemen
\\n
\\n\\nASEAN - Books International
\\n\\n\\n\\t- Representative for: Brunei, Cambodia, Indonesia, Indonesia, Laos, Malaysia, Myanmar, Philippines, Singapore, Thailand, Vietnam
\\n
\\n\\nChina Publishers Services Ltd - CPS
\\n\\n\\n\\t- Representative for: China, Taiwan, Hong Kong
\\n
\\n\\nMallory International Ltd
\\n\\n\\n\\t- Representative for: Africa
\\n
\\n\\nFor partnership opportunities, please contact orders@intechopen.com.
\\n\\n*The price does not include Value-Added Tax (VAT). Residents of European Union countries need to add a Book Value-Added Tax of 5%. Institutions and companies registered as VAT taxable entities in their own EU member state, will not pay VAT by providing us with their VAT registration number. This is made possible by the EU reverse charge method.
\\n"}]'},components:[{type:"htmlEditorComponent",content:'InTechOpen contributors can order print books at a special price ranging from:
\n\n55-90 GBP for Hardcover full-colour books
\n\n\n\t- Discounts are granted for orders of 10 copies and more.
\n\t- All library orders receive a 20% discount.
\n\t- There is no minimum or maximum threshold on the quantity of book orders.
\n
\n\nFor a quote please contact us directly at orders@intechopen.com The quote will be sent to you within 1-2 business days.
\n\nAll of the books and chapters can be browsed online. To obtain InTechOpen's full book catalogue in PDF, please contact us.
\n\nTERMS (find out more)
\n\nPrint on Demand (POD)
\n\n\n\t- Books and offprints printed specifically for your order
\n\t- Ordered, printed, and shipped in 12-20 business days
\n\t- Available for purchase at any time
\n\t- No minimum or maximum threshold on book order quantity
\n
\n\nIntechOpen works with award winning print-houses and we hold to the fact that all of our printed products are of the highest quality.
\n\nRepresentatives
\n\nPrint copies of our publications are most often purchased as individual purchases by universities, libraries, institutions and academia personnel, hence increasing the visibility and outreach of our authors' published work among science communities and institutions. Our books are available at our direct Print Sales Department and through selected representatives throughout the world.
\n\nIndia - CBS Publishers & Distributors Pvt. Ltd.
\n\n\n\t- Representative for: India, Bangladesh, Pakistan, Sri Lanka, Bhutan, Nepal, Maldives, Iran, Algeria, Bahrain, Egypt, Iraq, Israel, Jordan, Kuwait, Lebanon, Libya, Malta, Morocco, Oman, Qatar, Saudi Arabia, Syria, Tunis, United Arab Emirates and Yemen
\n
\n\nASEAN - Books International
\n\n\n\t- Representative for: Brunei, Cambodia, Indonesia, Indonesia, Laos, Malaysia, Myanmar, Philippines, Singapore, Thailand, Vietnam
\n
\n\nChina Publishers Services Ltd - CPS
\n\n\n\t- Representative for: China, Taiwan, Hong Kong
\n
\n\nMallory International Ltd
\n\n\n\t- Representative for: Africa
\n
\n\nFor partnership opportunities, please contact orders@intechopen.com.
\n\n*The price does not include Value-Added Tax (VAT). Residents of European Union countries need to add a Book Value-Added Tax of 5%. Institutions and companies registered as VAT taxable entities in their own EU member state, will not pay VAT by providing us with their VAT registration number. This is made possible by the EU reverse charge method.
\n'}]},successStories:{items:[]},authorsAndEditors:{filterParams:{sort:"featured,name"},profiles:[{id:"160349",title:null,name:null,middleName:null,surname:null,slug:"",fullName:null,position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"25887",title:"Dr.",name:null,middleName:null,surname:"Abbasi",slug:"abbasi",fullName:"Abbasi",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"161332",title:"Dr",name:null,middleName:null,surname:"Abu-El Hassan",slug:"abu-el-hassan",fullName:"Abu-El Hassan",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"212347",title:"Dr.",name:null,middleName:null,surname:"Abubakar",slug:"abubakar",fullName:"Abubakar",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"93806",title:"Dr",name:null,middleName:null,surname:"Adani",slug:"adani",fullName:"Adani",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"158756",title:"Dr",name:null,middleName:null,surname:"Adler",slug:"adler",fullName:"Adler",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"63002",title:"Dr.",name:null,middleName:null,surname:"Agius",slug:"agius",fullName:"Agius",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"34637",title:"Dr.",name:null,middleName:null,surname:"Ahmed",slug:"ahmed",fullName:"Ahmed",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"118228",title:"Dr",name:null,middleName:null,surname:"Ahmed",slug:"ahmed",fullName:"Ahmed",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"89784",title:"Dr",name:null,middleName:null,surname:"Ai",slug:"ai",fullName:"Ai",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"158339",title:"Dr",name:null,middleName:null,surname:"Akahoshi",slug:"akahoshi",fullName:"Akahoshi",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"158540",title:"Dr",name:null,middleName:null,surname:"Al-Jumaily",slug:"al-jumaily",fullName:"Al-Jumaily",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null}],filtersByRegion:[{group:"region",caption:"North America",value:1,count:5682},{group:"region",caption:"Middle and South America",value:2,count:5165},{group:"region",caption:"Africa",value:3,count:1682},{group:"region",caption:"Asia",value:4,count:10206},{group:"region",caption:"Australia and Oceania",value:5,count:886},{group:"region",caption:"Europe",value:6,count:15610}],offset:12,limit:12,total:117097},chapterEmbeded:{data:{}},editorApplication:{success:null,errors:{}},ofsBooks:{filterParams:{topicId:"14"},books:[{type:"book",id:"10216",title:"Paraffin - Thermal Energy Storage Applications",subtitle:null,isOpenForSubmission:!0,hash:"456090b63f5ba2290e24e655abd119bf",slug:null,bookSignature:"Dr. Elsayed Zaki and Dr. Abdelghaffar S. Dhmees",coverURL:"https://cdn.intechopen.com/books/images_new/10216.jpg",editedByType:null,editors:[{id:"220156",title:"Dr.",name:"Elsayed",surname:"Zaki",slug:"elsayed-zaki",fullName:"Elsayed Zaki"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"10411",title:"Nanocrystals",subtitle:null,isOpenForSubmission:!0,hash:"be29908600b7067c583ac21da1544a2d",slug:null,bookSignature:"Dr. Awadesh Mallik",coverURL:"https://cdn.intechopen.com/books/images_new/10411.jpg",editedByType:null,editors:[{id:"178218",title:"Dr.",name:"Awadesh",surname:"Mallik",slug:"awadesh-mallik",fullName:"Awadesh Mallik"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"10412",title:"Transition Metals",subtitle:null,isOpenForSubmission:!0,hash:"bd7287b801dc0ac77e01f66842dc1d99",slug:null,bookSignature:"Dr. Sajjad Haider and Dr. Adnan Haider",coverURL:"https://cdn.intechopen.com/books/images_new/10412.jpg",editedByType:null,editors:[{id:"110708",title:"Dr.",name:"Sajjad",surname:"Haider",slug:"sajjad-haider",fullName:"Sajjad Haider"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"10609",title:"Zeolites",subtitle:null,isOpenForSubmission:!0,hash:"90681a8fef45a03f68f4b9276acba2d3",slug:null,bookSignature:"Dr. Pavel Krivenko",coverURL:"https://cdn.intechopen.com/books/images_new/10609.jpg",editedByType:null,editors:[{id:"180922",title:"Dr.",name:"Pavel",surname:"Krivenko",slug:"pavel-krivenko",fullName:"Pavel Krivenko"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"10847",title:"Aluminium Alloys",subtitle:null,isOpenForSubmission:!0,hash:"22f9ea5292994565616af16fac347ba0",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/10847.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"10848",title:"Tribology",subtitle:null,isOpenForSubmission:!0,hash:"316ca42e0526e3c775e16b929cc702f9",slug:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/10848.jpg",editedByType:null,editors:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}],filtersByTopic:[{group:"topic",caption:"Agricultural and Biological Sciences",value:5,count:9},{group:"topic",caption:"Biochemistry, Genetics and Molecular Biology",value:6,count:18},{group:"topic",caption:"Business, Management and Economics",value:7,count:2},{group:"topic",caption:"Chemistry",value:8,count:7},{group:"topic",caption:"Computer and Information Science",value:9,count:10},{group:"topic",caption:"Earth and Planetary Sciences",value:10,count:5},{group:"topic",caption:"Engineering",value:11,count:15},{group:"topic",caption:"Environmental Sciences",value:12,count:2},{group:"topic",caption:"Immunology and Microbiology",value:13,count:5},{group:"topic",caption:"Materials Science",value:14,count:4},{group:"topic",caption:"Mathematics",value:15,count:1},{group:"topic",caption:"Medicine",value:16,count:60},{group:"topic",caption:"Nanotechnology and Nanomaterials",value:17,count:1},{group:"topic",caption:"Neuroscience",value:18,count:1},{group:"topic",caption:"Pharmacology, Toxicology and Pharmaceutical Science",value:19,count:6},{group:"topic",caption:"Physics",value:20,count:2},{group:"topic",caption:"Psychology",value:21,count:3},{group:"topic",caption:"Robotics",value:22,count:1},{group:"topic",caption:"Social Sciences",value:23,count:3},{group:"topic",caption:"Technology",value:24,count:1},{group:"topic",caption:"Veterinary Medicine and Science",value:25,count:2}],offset:12,limit:12,total:6},popularBooks:{featuredBooks:[{type:"book",id:"9208",title:"Welding",subtitle:"Modern Topics",isOpenForSubmission:!1,hash:"7d6be076ccf3a3f8bd2ca52d86d4506b",slug:"welding-modern-topics",bookSignature:"Sadek Crisóstomo Absi Alfaro, Wojciech Borek and Błażej Tomiczek",coverURL:"https://cdn.intechopen.com/books/images_new/9208.jpg",editors:[{id:"65292",title:"Prof.",name:"Sadek Crisostomo Absi",middleName:"C. Absi",surname:"Alfaro",slug:"sadek-crisostomo-absi-alfaro",fullName:"Sadek Crisostomo Absi Alfaro"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"9139",title:"Topics in Primary Care Medicine",subtitle:null,isOpenForSubmission:!1,hash:"ea774a4d4c1179da92a782e0ae9cde92",slug:"topics-in-primary-care-medicine",bookSignature:"Thomas F. Heston",coverURL:"https://cdn.intechopen.com/books/images_new/9139.jpg",editors:[{id:"217926",title:"Dr.",name:"Thomas F.",middleName:null,surname:"Heston",slug:"thomas-f.-heston",fullName:"Thomas F. Heston"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"8697",title:"Virtual Reality and Its Application in Education",subtitle:null,isOpenForSubmission:!1,hash:"ee01b5e387ba0062c6b0d1e9227bda05",slug:"virtual-reality-and-its-application-in-education",bookSignature:"Dragan Cvetković",coverURL:"https://cdn.intechopen.com/books/images_new/8697.jpg",editors:[{id:"101330",title:"Dr.",name:"Dragan",middleName:"Mladen",surname:"Cvetković",slug:"dragan-cvetkovic",fullName:"Dragan Cvetković"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"9785",title:"Endometriosis",subtitle:null,isOpenForSubmission:!1,hash:"f457ca61f29cf7e8bc191732c50bb0ce",slug:"endometriosis",bookSignature:"Courtney Marsh",coverURL:"https://cdn.intechopen.com/books/images_new/9785.jpg",editors:[{id:"255491",title:"Dr.",name:"Courtney",middleName:null,surname:"Marsh",slug:"courtney-marsh",fullName:"Courtney Marsh"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"9343",title:"Trace Metals in the Environment",subtitle:"New Approaches and Recent Advances",isOpenForSubmission:!1,hash:"ae07e345bc2ce1ebbda9f70c5cd12141",slug:"trace-metals-in-the-environment-new-approaches-and-recent-advances",bookSignature:"Mario Alfonso Murillo-Tovar, Hugo Saldarriaga-Noreña and Agnieszka Saeid",coverURL:"https://cdn.intechopen.com/books/images_new/9343.jpg",editors:[{id:"255959",title:"Dr.",name:"Mario Alfonso",middleName:null,surname:"Murillo-Tovar",slug:"mario-alfonso-murillo-tovar",fullName:"Mario Alfonso Murillo-Tovar"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7831",title:"Sustainability in Urban Planning and Design",subtitle:null,isOpenForSubmission:!1,hash:"c924420492c8c2c9751e178d025f4066",slug:"sustainability-in-urban-planning-and-design",bookSignature:"Amjad Almusaed, Asaad Almssad and Linh Truong - Hong",coverURL:"https://cdn.intechopen.com/books/images_new/7831.jpg",editors:[{id:"110471",title:"Dr.",name:"Amjad",middleName:"Zaki",surname:"Almusaed",slug:"amjad-almusaed",fullName:"Amjad Almusaed"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"8468",title:"Sheep Farming",subtitle:"An Approach to Feed, Growth and Sanity",isOpenForSubmission:!1,hash:"838f08594850bc04aa14ec873ed1b96f",slug:"sheep-farming-an-approach-to-feed-growth-and-sanity",bookSignature:"António Monteiro",coverURL:"https://cdn.intechopen.com/books/images_new/8468.jpg",editors:[{id:"190314",title:"Prof.",name:"António",middleName:"Cardoso",surname:"Monteiro",slug:"antonio-monteiro",fullName:"António Monteiro"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"8816",title:"Financial Crises",subtitle:"A Selection of Readings",isOpenForSubmission:!1,hash:"6f2f49fb903656e4e54280c79fabd10c",slug:"financial-crises-a-selection-of-readings",bookSignature:"Stelios Markoulis",coverURL:"https://cdn.intechopen.com/books/images_new/8816.jpg",editors:[{id:"237863",title:"Dr.",name:"Stelios",middleName:null,surname:"Markoulis",slug:"stelios-markoulis",fullName:"Stelios Markoulis"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"9376",title:"Contemporary Developments and Perspectives in International Health Security",subtitle:"Volume 1",isOpenForSubmission:!1,hash:"b9a00b84cd04aae458fb1d6c65795601",slug:"contemporary-developments-and-perspectives-in-international-health-security-volume-1",bookSignature:"Stanislaw P. Stawicki, Michael S. Firstenberg, Sagar C. Galwankar, Ricardo Izurieta and Thomas Papadimos",coverURL:"https://cdn.intechopen.com/books/images_new/9376.jpg",editors:[{id:"181694",title:"Dr.",name:"Stanislaw P.",middleName:null,surname:"Stawicki",slug:"stanislaw-p.-stawicki",fullName:"Stanislaw P. Stawicki"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7769",title:"Medical Isotopes",subtitle:null,isOpenForSubmission:!1,hash:"f8d3c5a6c9a42398e56b4e82264753f7",slug:"medical-isotopes",bookSignature:"Syed Ali Raza Naqvi and Muhammad Babar Imrani",coverURL:"https://cdn.intechopen.com/books/images_new/7769.jpg",editors:[{id:"259190",title:"Dr.",name:"Syed Ali Raza",middleName:null,surname:"Naqvi",slug:"syed-ali-raza-naqvi",fullName:"Syed Ali Raza Naqvi"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"9279",title:"Concepts, Applications and Emerging Opportunities in Industrial Engineering",subtitle:null,isOpenForSubmission:!1,hash:"9bfa87f9b627a5468b7c1e30b0eea07a",slug:"concepts-applications-and-emerging-opportunities-in-industrial-engineering",bookSignature:"Gary Moynihan",coverURL:"https://cdn.intechopen.com/books/images_new/9279.jpg",editors:[{id:"16974",title:"Dr.",name:"Gary",middleName:null,surname:"Moynihan",slug:"gary-moynihan",fullName:"Gary Moynihan"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7807",title:"A Closer Look at Organizational Culture in Action",subtitle:null,isOpenForSubmission:!1,hash:"05c608b9271cc2bc711f4b28748b247b",slug:"a-closer-look-at-organizational-culture-in-action",bookSignature:"Süleyman Davut Göker",coverURL:"https://cdn.intechopen.com/books/images_new/7807.jpg",editors:[{id:"190035",title:"Associate Prof.",name:"Süleyman Davut",middleName:null,surname:"Göker",slug:"suleyman-davut-goker",fullName:"Süleyman Davut Göker"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}}],offset:12,limit:12,total:5128},hotBookTopics:{hotBooks:[],offset:0,limit:12,total:null},publish:{},publishingProposal:{success:null,errors:{}},books:{featuredBooks:[{type:"book",id:"9208",title:"Welding",subtitle:"Modern Topics",isOpenForSubmission:!1,hash:"7d6be076ccf3a3f8bd2ca52d86d4506b",slug:"welding-modern-topics",bookSignature:"Sadek Crisóstomo Absi Alfaro, Wojciech Borek and Błażej Tomiczek",coverURL:"https://cdn.intechopen.com/books/images_new/9208.jpg",editors:[{id:"65292",title:"Prof.",name:"Sadek Crisostomo Absi",middleName:"C. Absi",surname:"Alfaro",slug:"sadek-crisostomo-absi-alfaro",fullName:"Sadek Crisostomo Absi Alfaro"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"9139",title:"Topics in Primary Care Medicine",subtitle:null,isOpenForSubmission:!1,hash:"ea774a4d4c1179da92a782e0ae9cde92",slug:"topics-in-primary-care-medicine",bookSignature:"Thomas F. Heston",coverURL:"https://cdn.intechopen.com/books/images_new/9139.jpg",editors:[{id:"217926",title:"Dr.",name:"Thomas F.",middleName:null,surname:"Heston",slug:"thomas-f.-heston",fullName:"Thomas F. Heston"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"8697",title:"Virtual Reality and Its Application in Education",subtitle:null,isOpenForSubmission:!1,hash:"ee01b5e387ba0062c6b0d1e9227bda05",slug:"virtual-reality-and-its-application-in-education",bookSignature:"Dragan Cvetković",coverURL:"https://cdn.intechopen.com/books/images_new/8697.jpg",editors:[{id:"101330",title:"Dr.",name:"Dragan",middleName:"Mladen",surname:"Cvetković",slug:"dragan-cvetkovic",fullName:"Dragan Cvetković"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"9785",title:"Endometriosis",subtitle:null,isOpenForSubmission:!1,hash:"f457ca61f29cf7e8bc191732c50bb0ce",slug:"endometriosis",bookSignature:"Courtney Marsh",coverURL:"https://cdn.intechopen.com/books/images_new/9785.jpg",editors:[{id:"255491",title:"Dr.",name:"Courtney",middleName:null,surname:"Marsh",slug:"courtney-marsh",fullName:"Courtney Marsh"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"9343",title:"Trace Metals in the Environment",subtitle:"New Approaches and Recent Advances",isOpenForSubmission:!1,hash:"ae07e345bc2ce1ebbda9f70c5cd12141",slug:"trace-metals-in-the-environment-new-approaches-and-recent-advances",bookSignature:"Mario Alfonso Murillo-Tovar, Hugo Saldarriaga-Noreña and Agnieszka Saeid",coverURL:"https://cdn.intechopen.com/books/images_new/9343.jpg",editors:[{id:"255959",title:"Dr.",name:"Mario Alfonso",middleName:null,surname:"Murillo-Tovar",slug:"mario-alfonso-murillo-tovar",fullName:"Mario Alfonso Murillo-Tovar"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"8468",title:"Sheep Farming",subtitle:"An Approach to Feed, Growth and Sanity",isOpenForSubmission:!1,hash:"838f08594850bc04aa14ec873ed1b96f",slug:"sheep-farming-an-approach-to-feed-growth-and-sanity",bookSignature:"António Monteiro",coverURL:"https://cdn.intechopen.com/books/images_new/8468.jpg",editors:[{id:"190314",title:"Prof.",name:"António",middleName:"Cardoso",surname:"Monteiro",slug:"antonio-monteiro",fullName:"António Monteiro"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"8816",title:"Financial Crises",subtitle:"A Selection of Readings",isOpenForSubmission:!1,hash:"6f2f49fb903656e4e54280c79fabd10c",slug:"financial-crises-a-selection-of-readings",bookSignature:"Stelios Markoulis",coverURL:"https://cdn.intechopen.com/books/images_new/8816.jpg",editors:[{id:"237863",title:"Dr.",name:"Stelios",middleName:null,surname:"Markoulis",slug:"stelios-markoulis",fullName:"Stelios Markoulis"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7831",title:"Sustainability in Urban Planning and Design",subtitle:null,isOpenForSubmission:!1,hash:"c924420492c8c2c9751e178d025f4066",slug:"sustainability-in-urban-planning-and-design",bookSignature:"Amjad Almusaed, Asaad Almssad and Linh Truong - Hong",coverURL:"https://cdn.intechopen.com/books/images_new/7831.jpg",editors:[{id:"110471",title:"Dr.",name:"Amjad",middleName:"Zaki",surname:"Almusaed",slug:"amjad-almusaed",fullName:"Amjad Almusaed"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"9376",title:"Contemporary Developments and Perspectives in International Health Security",subtitle:"Volume 1",isOpenForSubmission:!1,hash:"b9a00b84cd04aae458fb1d6c65795601",slug:"contemporary-developments-and-perspectives-in-international-health-security-volume-1",bookSignature:"Stanislaw P. Stawicki, Michael S. Firstenberg, Sagar C. Galwankar, Ricardo Izurieta and Thomas Papadimos",coverURL:"https://cdn.intechopen.com/books/images_new/9376.jpg",editors:[{id:"181694",title:"Dr.",name:"Stanislaw P.",middleName:null,surname:"Stawicki",slug:"stanislaw-p.-stawicki",fullName:"Stanislaw P. Stawicki"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}},{type:"book",id:"7769",title:"Medical Isotopes",subtitle:null,isOpenForSubmission:!1,hash:"f8d3c5a6c9a42398e56b4e82264753f7",slug:"medical-isotopes",bookSignature:"Syed Ali Raza Naqvi and Muhammad Babar Imrani",coverURL:"https://cdn.intechopen.com/books/images_new/7769.jpg",editors:[{id:"259190",title:"Dr.",name:"Syed Ali Raza",middleName:null,surname:"Naqvi",slug:"syed-ali-raza-naqvi",fullName:"Syed Ali Raza Naqvi"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}}],latestBooks:[{type:"book",id:"8468",title:"Sheep Farming",subtitle:"An Approach to Feed, Growth and Health",isOpenForSubmission:!1,hash:"838f08594850bc04aa14ec873ed1b96f",slug:"sheep-farming-an-approach-to-feed-growth-and-health",bookSignature:"António Monteiro",coverURL:"https://cdn.intechopen.com/books/images_new/8468.jpg",editedByType:"Edited by",editors:[{id:"190314",title:"Prof.",name:"António",middleName:"Cardoso",surname:"Monteiro",slug:"antonio-monteiro",fullName:"António Monteiro"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9523",title:"Oral and Maxillofacial Surgery",subtitle:null,isOpenForSubmission:!1,hash:"5eb6ec2db961a6c8965d11180a58d5c1",slug:"oral-and-maxillofacial-surgery",bookSignature:"Gokul Sridharan",coverURL:"https://cdn.intechopen.com/books/images_new/9523.jpg",editedByType:"Edited by",editors:[{id:"82453",title:"Dr.",name:"Gokul",middleName:null,surname:"Sridharan",slug:"gokul-sridharan",fullName:"Gokul Sridharan"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9785",title:"Endometriosis",subtitle:null,isOpenForSubmission:!1,hash:"f457ca61f29cf7e8bc191732c50bb0ce",slug:"endometriosis",bookSignature:"Courtney Marsh",coverURL:"https://cdn.intechopen.com/books/images_new/9785.jpg",editedByType:"Edited by",editors:[{id:"255491",title:"Dr.",name:"Courtney",middleName:null,surname:"Marsh",slug:"courtney-marsh",fullName:"Courtney Marsh"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9018",title:"Some RNA Viruses",subtitle:null,isOpenForSubmission:!1,hash:"a5cae846dbe3692495fc4add2f60fd84",slug:"some-rna-viruses",bookSignature:"Yogendra Shah and Eltayb Abuelzein",coverURL:"https://cdn.intechopen.com/books/images_new/9018.jpg",editedByType:"Edited by",editors:[{id:"278914",title:"Ph.D.",name:"Yogendra",middleName:null,surname:"Shah",slug:"yogendra-shah",fullName:"Yogendra Shah"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"8816",title:"Financial Crises",subtitle:"A Selection of Readings",isOpenForSubmission:!1,hash:"6f2f49fb903656e4e54280c79fabd10c",slug:"financial-crises-a-selection-of-readings",bookSignature:"Stelios Markoulis",coverURL:"https://cdn.intechopen.com/books/images_new/8816.jpg",editedByType:"Edited by",editors:[{id:"237863",title:"Dr.",name:"Stelios",middleName:null,surname:"Markoulis",slug:"stelios-markoulis",fullName:"Stelios Markoulis"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9585",title:"Advances in Complex Valvular Disease",subtitle:null,isOpenForSubmission:!1,hash:"ef64f11e211621ecfe69c46e60e7ca3d",slug:"advances-in-complex-valvular-disease",bookSignature:"Michael S. Firstenberg and Imran Khan",coverURL:"https://cdn.intechopen.com/books/images_new/9585.jpg",editedByType:"Edited by",editors:[{id:"64343",title:null,name:"Michael S.",middleName:"S",surname:"Firstenberg",slug:"michael-s.-firstenberg",fullName:"Michael S. Firstenberg"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"10150",title:"Smart Manufacturing",subtitle:"When Artificial Intelligence Meets the Internet of Things",isOpenForSubmission:!1,hash:"87004a19de13702d042f8ff96d454698",slug:"smart-manufacturing-when-artificial-intelligence-meets-the-internet-of-things",bookSignature:"Tan Yen Kheng",coverURL:"https://cdn.intechopen.com/books/images_new/10150.jpg",editedByType:"Edited by",editors:[{id:"78857",title:"Dr.",name:"Tan Yen",middleName:null,surname:"Kheng",slug:"tan-yen-kheng",fullName:"Tan Yen Kheng"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9386",title:"Direct Numerical Simulations",subtitle:"An Introduction and Applications",isOpenForSubmission:!1,hash:"158a3a0fdba295d21ff23326f5a072d5",slug:"direct-numerical-simulations-an-introduction-and-applications",bookSignature:"Srinivasa Rao",coverURL:"https://cdn.intechopen.com/books/images_new/9386.jpg",editedByType:"Edited by",editors:[{id:"6897",title:"Dr.",name:"Srinivasa",middleName:"P",surname:"Rao",slug:"srinivasa-rao",fullName:"Srinivasa Rao"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9139",title:"Topics in Primary Care Medicine",subtitle:null,isOpenForSubmission:!1,hash:"ea774a4d4c1179da92a782e0ae9cde92",slug:"topics-in-primary-care-medicine",bookSignature:"Thomas F. Heston",coverURL:"https://cdn.intechopen.com/books/images_new/9139.jpg",editedByType:"Edited by",editors:[{id:"217926",title:"Dr.",name:"Thomas F.",middleName:null,surname:"Heston",slug:"thomas-f.-heston",fullName:"Thomas F. Heston"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9208",title:"Welding",subtitle:"Modern Topics",isOpenForSubmission:!1,hash:"7d6be076ccf3a3f8bd2ca52d86d4506b",slug:"welding-modern-topics",bookSignature:"Sadek Crisóstomo Absi Alfaro, Wojciech Borek and Błażej Tomiczek",coverURL:"https://cdn.intechopen.com/books/images_new/9208.jpg",editedByType:"Edited by",editors:[{id:"65292",title:"Prof.",name:"Sadek Crisostomo Absi",middleName:"C. Absi",surname:"Alfaro",slug:"sadek-crisostomo-absi-alfaro",fullName:"Sadek Crisostomo Absi Alfaro"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},subject:{topic:{id:"1221",title:"Biomechanics",slug:"optics-and-lasers-biomechanics",parent:{title:"Optics and Lasers",slug:"optics-and-lasers"},numberOfBooks:0,numberOfAuthorsAndEditors:0,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,videoUrl:null,fallbackUrl:null,description:null},booksByTopicFilter:{topicSlug:"optics-and-lasers-biomechanics",sort:"-publishedDate",limit:12,offset:0},booksByTopicCollection:[],booksByTopicTotal:0,mostCitedChapters:[],mostDownloadedChaptersLast30Days:[],onlineFirstChaptersFilter:{topicSlug:"optics-and-lasers-biomechanics",limit:3,offset:0},onlineFirstChaptersCollection:[],onlineFirstChaptersTotal:0},preDownload:{success:null,errors:{}},aboutIntechopen:{},privacyPolicy:{},peerReviewing:{},howOpenAccessPublishingWithIntechopenWorks:{},sponsorshipBooks:{sponsorshipBooks:[{type:"book",id:"10176",title:"Microgrids and Local Energy Systems",subtitle:null,isOpenForSubmission:!0,hash:"c32b4a5351a88f263074b0d0ca813a9c",slug:null,bookSignature:"Prof. Nick Jenkins",coverURL:"https://cdn.intechopen.com/books/images_new/10176.jpg",editedByType:null,editors:[{id:"55219",title:"Prof.",name:"Nick",middleName:null,surname:"Jenkins",slug:"nick-jenkins",fullName:"Nick Jenkins"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter"}}],offset:8,limit:8,total:1},route:{name:"profile.detail",path:"/profiles/138062/fabian-leinisch",hash:"",query:{},params:{id:"138062",slug:"fabian-leinisch"},fullPath:"/profiles/138062/fabian-leinisch",meta:{},from:{name:null,path:"/",hash:"",query:{},params:{},fullPath:"/",meta:{}}}},function(){var m;(m=document.currentScript||document.scripts[document.scripts.length-1]).parentNode.removeChild(m)}()