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International Conference on Symmetry (Symmetry)
Physics in the Complex Domain Carl M. Bender
Published Jan 3, 2018 • 0 citations
Abstract first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing: Column Width: Background: Open AccessAbstract Physics in the Complex Domain † by Carl M. Bender Physics Department, Washington University, St. Louis, MO 63130, USA † Presented at Symmetry 2017—The First International Conference on Symmetry, Barcelona, Spain, 16–18 October 2017. Proceedings 2018, 2(1), 47; https://doi.org/10.3390/proceedings2010047 Published: 3 January 2018 (This article belongs to the Proceedings of The First International Conference on Symmetry) Download Download PDF Download PDF with Cover Download XML Versions Notes The theory of complex variables is extremely useful because it helps to explain the mathematical behavior of functions of a real variable. Complex variable theory also provides insight into the nature of physical theories. For example, it provides a simple and beautiful picture of quantization and it explains the underlying reason for the divergence of perturbation theory. By using complex-variable methods, one can generalize conventional Hermitian quantum theories into the complex domain. The result is a new class of parity-time-symmetric (PT-symmetric) theories whose remarkable physical properties have been studied and verified in many recent laboratory experiments. © 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Share and Cite MDPI and ACS Style Bender, C.M. Physics in the Complex Domain. Proceedings 2018, 2, 47. https://doi.org/10.3390/proceedings2010047 AMA Style Bender CM. Physics in the Complex Domain. Proceedings. 2018; 2(1):47. https://doi.org/10.3390/proceedings2010047 Chicago/Turabian Style Bender, Carl M. 2018. "Physics in the Complex Domain" Proceedings 2, no. 1: 47. https://doi.org/10.3390/proceedings2010047 Find Other Styles Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here. Article Metrics No No Article Access Statistics Multiple requests from the same IP address are counted as one view.
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first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing: Column Width: Background: Open AccessAbstract Physics in the Complex Domain † by Carl M. Bender Physics Department, Washington University, St. Louis, MO 63130, USA † Presented at Symmetry 2017—The First International Conference on Symmetry, Barcelona, Spain, 16–18 October 2017. Proceedings 2018, 2(1), 47;
https://doi.org/10.3390/proceedings2010047 Published: 3 January 2018 (This article belongs to the Proceedings of The First International Conference on Symmetry) Download Download PDF Download PDF with Cover Download XML Versions Notes The theory of complex variables is extremely useful because it helps to explain the mathematical behavior of functions of a real variable. Complex variable theory also provides insight into the nature of physical theories. For example, it provides a simple and beautiful picture of quantization and it explains the underlying reason for the divergence of perturbation theory. By using complex-variable methods, one can generalize conventional Hermitian quantum theories into the complex domain. The result is a new class of parity-time-symmetric (PT-symmetric) theories whose remarkable physical properties have been studied and verified in many recent laboratory experiments. © 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (
https://creativecommons.org/licenses/by/4.0/ ). Share and Cite MDPI and ACS Style Bender, C.M. Physics in the Complex Domain. Proceedings 2018, 2, 47.
https://doi.org/10.3390/proceedings2010047 AMA Style Bender CM. Physics in the Complex Domain. Proceedings. 2018; 2(1):47.
https://doi.org/10.3390/proceedings2010047 Chicago/Turabian Style Bender, Carl M. 2018. "Physics in the Complex Domain" Proceedings 2, no. 1: 47.
https://doi.org/10.3390/proceedings2010047 Find Other Styles Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here. Article Metrics No No Article Access Statistics Multiple requests from the same IP address are counted as one view.
Since the role that 𝓟𝓣-symmetric Hamiltonians could play in quantum mechanics theory was suggested in1998, much work has been conducted on the mathematical analysis of 𝓟𝓣-symmetry. Carl Benderchallenged the perceived wisdom of Quantum Mechanics that the Hamiltonian operator describing anyquantum mechanical system has to be Hermitian. He showed that Hamiltonians that are invariant undercombined parity-time 𝓟𝓣-symmetry transformations likewise can exhibit real eigenvalue spectra.However, 𝓟𝓣-symmetric systems began as an analytic extension of classical and quantum mechanical tothe complex domain. These findings had a particularly profound impact in the field of photonics, where PTsymmetricpotential landscapes can be implemented by introducing dissipative terms appropriatelydistributing gain and loss in open or coupled systems. The refractive index profile plays the role of thecomplex potential. Similar research activity was transferred to other physical settings such as electronicscircuits, microwaves and cavities resonators, acoustic, superconducting wires, and atomic systems.Unconventional characteristics compared with Hermitian ones were found and allow to engineer newdevices.Likewise, topological phases of matter are another new field of physics which have had a rapid growth inthe last years, based on the concept of Geometric or Berry phase. Topological Insulators andSuperconductors, Topological photonics and nanophotonics, Topological Computers are some of theapplications widely described in scientific journals.In Nature all the processes have to be described by a property in a discrete or iterative way instead of acontinuous way which usually is time. Continuous processes, being a mathematical idealization, has beenthe best tool to model observable properties in physics and is the conventional tool used in Classical andQuantum Mechanics. The formal way to model discrete systems is through difference equations which aresimilar to differential equations but not equal.Identification of dynamical discrete system with recurrence equations, give us the possibility to consider ananalytical continuation of its evolution variable, from discrete values to fractional one´s, in the solution ofthe system. Parametric solutions obtained by this procedure takes automatically complex values, avoidingto make any supposition about the possibility of their analytical extension to complex domain. Thisprocedure leads, in a very simple natural way, a rich variety of dynamical behaviors in the complex domainunknown in the real domain. This new mathematical tool is developed to enhance and provide a moresubtle description of the behavior of dynamical systems and provides a more general class of Non-Hermitian Hamiltonians than 𝓟𝓣-symmetric systems with real spectra, keeping also topological propertiesdue to a kind of geometric phase.
P T PT P T symmetry: in quantum and classical physics
"Originated by the author in 1998, the field of PT (parity-time) symmetry has become an extremely active and exciting area of research. PT-symmetric quantum and classical systems have theoretical, experimental, and commercial applications, and have been the subject of many journal articles, PhD theses, conferences, and symposia. Carl Bender's work has influenced major advances in physics and generations of students. This book is an accessible entry point to PT symmetry, ideal for students and scientists looking to begin their own research projects in this field"--
PT-symmetric quantum mechanics (PTQM) has become a hot area of research and investigation. Since its beginnings in 1998, there have been over 1000 published papers and more than 15 international conferences entirely devoted to this research topic. Originally, PTQM was studied at a highly mathematical level and the techniques of complex variables, asymptotics, differential equations and perturbation theory were used to understand the subtleties associated with the analytic continuation of eigenvalue problems. However, as experiments on PT-symmetric physical systems have been performed, a simple and beautiful physical picture has emerged, and a PT-symmetric system can be understood as one that has a balanced loss and gain. Furthermore, the PT phase transition can now be understood intuitively without resorting to sophisticated mathematics. Research on PTQM is following two different paths: at a fundamental level, physicists are attempting to understand the underlying mathematical structure of these theories with the long-range objective of applying the techniques of PTQM to understanding some of the outstanding problems in physics today, such as the nature of the Higgs particle, the properties of dark matter, the matter-antimatter asymmetry in the universe, neutrino oscillations and the cosmological constant; at an applied level, new kinds of PT-synthetic materials are being developed, and the PT phase transition is being observed in many physical contexts, such as lasers, optical wave guides, microwave cavities, superconducting wires and electronic circuits. The purpose of this Theme Issue is to acquaint the reader with the latest developments in PTQM. The articles in this volume are written in the style of mini-reviews and address diverse areas of the emerging and exciting new area of PT-symmetric quantum mechanics.
Complex numbers, i.e., numbers with a and an imaginary part, are essential for mathematical analysis, while their role in other subjects, such as electromagnetism or special relativity, is far less fundamental. Quantum is the only physical theory where these numbers seem to play an indispensible role, as the theory is explicitly formulated in terms of operators acting on complex Hilbert spaces. The occurrence of complex numbers within the quantum formalism has nonetheless puzzled countless physicists, including the fathers of the theory, for whom a version of quantum physics, where states and observables are represented by operators, seemed much more natural. In fact, previous works showed that such real quantum physics can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary quantum states. Thus, are complex numbers really needed for a quantum description of nature? Here, we show this to be case by proving that and complex quantum make different predictions in network scenarios comprising independent quantum state sources. This allows us to devise a Bell-type quantum experiment whose input-output correlations cannot be approximated by any quantum model. The successful realization of such an experiment would disprove quantum physics, in the same way as standard Bell experiments disproved local physics.
first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing: Column Width: Background: Open AccessAbstract Quantum Simulation of SU(4) Symmetric Spin Lattice Models † by Bruno Uchoa Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA † Presented at Symmetry 2017—The First International Conference on Symmetry, Barcelona, Spain, 16–18 October 2017. Proceedings 2018, 2(1), 39; https://doi.org/10.3390/proceedings2010039 Published: 8 January 2018 (This article belongs to the Proceedings of The First International Conference on Symmetry) Download Download PDF Download PDF with Cover Download XML Versions Notes Quantum spin-orbital liquids are strongly correlated states that emerge from quantum frustration between spin and orbital degrees of freedom. Those states are highly entangled, have non-local excitations but do not break any symmetries. A promising route towards observing those elusive states is the creation of artificial Mott insulators, where antiferromagnetic correlations between spins and orbitals can be designed. I will show that Coulomb impurity lattices on the surface of gapped honeycomb substrates, such as graphene on SiC, can be used to simulate SU(4) symmetric spin-orbital lattice models in the Mott regime. The antiferromagnetic correlations follow from super-exchange interactions between Coulomb impurity bound states at quarter filling, with spin and valley degeneracies. I propose that quantum spin-orbital liquids can be engineered in artificially designed solid-state systems at vastly higher temperatures than achievable in optical lattices with cold atoms. © 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Share and Cite MDPI and ACS Style Uchoa, B. Quantum Simulation of SU(4) Symmetric Spin Lattice Models. Proceedings 2018, 2, 39. https://doi.org/10.3390/proceedings2010039 AMA Style Uchoa B. Quantum Simulation of SU(4) Symmetric Spin Lattice Models. Proceedings. 2018; 2(1):39. https://doi.org/10.3390/proceedings2010039 Chicago/Turabian Style Uchoa, Bruno. 2018. "Quantum Simulation of SU(4) Symmetric Spin Lattice Models" Proceedings 2, no. 1: 39. https://doi.org/10.3390/proceedings2010039 Find Other Styles Article Metrics No No Article Access Statistics Multiple requests from the same IP address are counted as one view.
first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing: Column Width: Background: Open AccessAbstract Irreversibility and Quantum Information Flow under Global & Local Gauge Symmetries † by Cristina Cirstoiu 1,* and David Jennings 2 1 Department of Physics, Imperial College London, London SW7 2AZ, UK 2 Department of Physics, Oxford University, Oxford OX1 2JD, UK * Author to whom correspondence should be addressed. † Presented at Symmetry 2017—The First International Conference on Symmetry, Barcelona, Spain, 16–18 October 2017. Proceedings 2018, 2(1), 4; https://doi.org/10.3390/proceedings2010004 Published: 5 January 2018 (This article belongs to the Proceedings of The First International Conference on Symmetry) Download Download PDF Download PDF with Cover Download XML Versions Notes What is the structure of general quantum processes on composite systems that respect a global or local symmetry principle? How does the irreversible use of quantum resources behave under such symmetry principles? Here we develop an information-theoretic framework to address these questions and show that every symmetric quantum process on a system has an essentially unique decomposition in terms of the flow of symmetry-breaking degrees of freedom between each subsystem and its environment. The decomposition has a natural causal structure that can be represented diagrammatically and makes explicit gauge degrees of freedom between subsystems. Our framework also provides a novel quantum information perspective on lattice gauge theories and a method to gauge general quantum processes beyond Lagrangian formulations. This procedure admits a simple resource-theoretic interpretation, and thus offers a natural context in which features such as information flow and entanglement in gauge theories could be studied. The framework also provides a flexible toolkit with which to analyse the structure of general quantum processes. As an application, we make use of a 'polar decomposition' for quantum processes to discuss incompatibility in the use of quantum resources and to provide a novel perspective in terms of the geometry induced on the orbit of a local process under a symmetry action. © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Share and Cite MDPI and ACS Style Cirstoiu, C.; Jennings, D. Irreversibility and Quantum Information Flow under Global & Local Gauge Symmetries. Proceedings 2018, 2, 4. https://doi.org/10.3390/proceedings2010004 AMA Style Cirstoiu C, Jennings D. Irreversibility and Quantum Information Flow under Global & Local Gauge Symmetries. Proceedings. 2018; 2(1):4. https://doi.org/10.3390/proceedings2010004 Chicago/Turabian Style Cirstoiu, Cristina, and David Jennings. 2018. "Irreversibility and Quantum Information Flow under Global & Local Gauge Symmetries" Proceedings 2, no. 1: 4. https://doi.org/10.3390/proceedings2010004 Find Other Styles Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here. Article Metrics No No Article Access Statistics Multiple requests from the same IP address are counted as one view.