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International Conference on Symmetry (Symmetry)

Irreversibility and Quantum Information Flow under Global & Local Gauge Symmetries

Cristina Cîrstoiu, David Jennings

Published Jan 5, 2018• 3 citations
Publisher

Abstract

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.

DOI
10.3390/proceedings2010004

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Global and local gauge symmetries beyond Lagrangian formulations

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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 employ an information-theoretic framework to address these questions and show that every symmetric quantum process on a system has a highly rigid 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. The 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 and quantum thermodynamics 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 the repeatable use of quantum resources and to provide a novel perspective in terms of the coordinates induced on the orbit of a local process under a symmetry action.

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Irreversibility and symmetry principles in quantum information

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Symmetry principles have played a crucial role in the development of modern physics and underpin our most fundamental theories of nature. The present thesis is concerned with the analysis of symmetries in the context of quantum information theory. Whenever a quantum mechanical system interacts with its environment it is subjected to decoherence, a process that is typically irreversible and most generally treated abstractly using the concept of quantum operations. If there is an underlying symmetry principle, additional structures emerge. The central question we address is, What are the consequences of global or local gauge symmetry on the structure of many-body quantum processes?This leads us to a diagrammatic framework of decomposing quantum operations into terms that respond to the symmetry principle in particular ways and respect the causal structures involved. We present two core applications. First, we address the interplay between irreversibility and repeatable use of coherent resources under symmetry constraints. Second, we give an information-theoretic perspective on gauging globally symmetric dynamics to a local symmetry applicable even in the presence of irreversibility and thus it goes beyond the usual Lagrangian formulation. Finally, we analyse the departure from conservation laws under symmetric dynamics subject to decoherence.

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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 Quantum Metrology and Relativistic Symmetries † by Marc-Thierry Jaekel 1 and Serge Reynaud 2,* 1 Laboratoire de Physique Théorique CNRS, ENS-PSL, UPMC-SU, F-75231 Paris, France 2 Laboratoire Kastler Brossel CNRS, ENS-PSL, UPMC-SU, Collège de France, F-75252 Paris, France * 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), 41; https://doi.org/10.3390/proceedings2010041 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 Modern metrology is based on quantum and relativistic physics. The second is defined as a fixed integer number of periods of atomic caesium clocks while the meter is derived from the exact value fixed for the speed of light in vacuum c. Clock synchronization, distance measurement (ranging) or localization in space-time (GNSS localization) are built on the remote distribution of electromagnetic field phases. In 2018, the Planck constant h and elementary charge e should be defined to have exact values, replacing the old definitions of the kilogram and Ampere units.However, the quantum status of time, space and mass observables and its compatibility with relativistic symmetries remain open questions. We address this problem in a quantum algebraic approach where observable positions in space and time are defined so that they obey relativistic and quantum requirements [1]. A mass observable is defined which is no longer a constant as expected from its dimension with respect to dilatation.The commutators of quantum positions involve spin, and frame transformations to accelerated frames differ from their classical counterparts [2]. Relativistic symmetries nevertheless allow one to extend the covariance rules of general relativity to the quantum algebraic framework [3]. It leads to a quantum version of the Einstein equivalence principle identified to the transformation of the mass observable. ReferencesJaekel, M.-T.; Reynaud, S. Quantum Observables associated with Einstein Localisation. arXiv 1998, arXiv:quant-ph/9810027. [Google Scholar]Jaekel, M.-T.; Reynaud, S. Quantum algebraic representation of localization and motion of a Dirac electron. arXiv 2001, arXiv:quant-ph/0101121. [Google Scholar]Jaekel, M.-T.; Reynaud, S. Mass, inertia and gravitation. arXiv 2011, arXiv:0812.3936. [Google Scholar]Publisher's Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. © 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 Jaekel, M.-T.; Reynaud, S. Quantum Metrology and Relativistic Symmetries. Proceedings 2018, 2, 41. https://doi.org/10.3390/proceedings2010041 AMA Style Jaekel M-T, Reynaud S. Quantum Metrology and Relativistic Symmetries. Proceedings. 2018; 2(1):41. https://doi.org/10.3390/proceedings2010041 Chicago/Turabian Style Jaekel, Marc-Thierry, and Serge Reynaud. 2018. "Quantum Metrology and Relativistic Symmetries" Proceedings 2, no. 1: 41. https://doi.org/10.3390/proceedings2010041 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.

International Conference on Symmetry (Symmetry)•Jan 3, 2018•0 citations

Internal Vacuum Gauge Structure as the Physical Origin of Quantum Entanglement

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Quantum entanglement is conventionally characterized as a structural property of tensor-product Hilbert spaces, with limited emphasis on its geometric or gauge-theoretic organization. Within standard quantum field theory on a fixed classical spacetime background, we show that entanglement can be represented as a global compatibility constraint acting on internal degrees of freedom in the gauge-bundle formulation of the Standard Model. This representation is encoded by a vacuum-level internal gauge structure $\Xi(x)$ that is locally pure gauge, dynamically inert, and acts exclusively on internal fibers, leaving all local dynamics and the Standard Model Lagrangian unchanged. We formalize this perspective as a vacuum internal gauge structure (VIGS) and prove a corresponding structural result—the Vacuum Internal Gauge Theorem—which establishes that global compatibility relations associated with $\Xi$ act only on internal Hilbert-space factors and preserve locality and no-signaling within the regime considered here. The framework complements standard Hilbert-space and algebraic descriptions of entanglement by making explicit how global internal correlations can be organized geometrically without invoking nonlocal dynamics. Finally, we identify an experimentally accessible operational signature, based on correlated versus independent internal-frame scrambling, that distinguishes this geometric representation within existing entanglement platforms.

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Quantum information has become a powerful tool for probing the structure of quantum field theories, yet its application to gauge theories remains subtle. On the one hand, quantum information theory assumes subsystem locality, i.e.~the factorization of the total Hilbert space into subsystems. On the other hand, gauge constraints prevent the total Hilbert space to decompose into a spacetime-local tensor product structure. Because the Hilbert space structure of gauge theories does not accommodate the subsystem decomposition used in quantum information theory, standard information-theoretic results, such as the Local Operations and Classical Communication (LOCC) theorem, cannot be used straightforwardly in the context of gauge theories. In this work, we bridge this gap in the case of a two-dimensional lattice gauge model that captures key features of electromagnetism. In particular, we construct gauge-invariant local algebras and derive a physically meaningful decomposition of the Hilbert space, providing an operationally consistent notion of locality in the absence of a local tensor-product structure. We apply this framework to field-mediated entanglement protocols relevant to proposed tests of the quantum nature of gravity. We show that the discretized version of electromagnetism satisfies an analogue of the LOCC theorem: entanglement cannot be generated without genuine quantum field interactions, even in the absence of a spacetime-local tensor product factorization of the Hilbert space. This may point towards an operational way to define a subsystem structure for gauge theories.

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Physics in the Complex Domain

Carl M. Bender

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.

International Conference on Symmetry (Symmetry)•Jan 3, 2018•0 citations
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.
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 employ an information-theoretic framework to address these questions and show that every symmetric quantum process on a system has a highly rigid 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. The 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 and quantum thermodynamics 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 the repeatable use of quantum resources and to provide a novel perspective in terms of the coordinates induced on the orbit of a local process under a symmetry action.
Symmetry principles have played a crucial role in the development of modern physics and underpin our most fundamental theories of nature. The present thesis is concerned with the analysis of symmetries in the context of quantum information theory. Whenever a quantum mechanical system interacts with its environment it is subjected to decoherence, a process that is typically irreversible and most generally treated abstractly using the concept of quantum operations. If there is an underlying symmetry principle, additional structures emerge. The central question we address is, What are the consequences of global or local gauge symmetry on the structure of many-body quantum processes?This leads us to a diagrammatic framework of decomposing quantum operations into terms that respond to the symmetry principle in particular ways and respect the causal structures involved. We present two core applications. First, we address the interplay between irreversibility and repeatable use of coherent resources under symmetry constraints. Second, we give an information-theoretic perspective on gauging globally symmetric dynamics to a local symmetry applicable even in the presence of irreversibility and thus it goes beyond the usual Lagrangian formulation. Finally, we analyse the departure from conservation laws under symmetric dynamics subject to decoherence.
first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing:    Column Width:    Background: Open AccessAbstract Quantum Metrology and Relativistic Symmetries † by Marc-Thierry Jaekel 1 and Serge Reynaud 2,* 1 Laboratoire de Physique Théorique CNRS, ENS-PSL, UPMC-SU, F-75231 Paris, France 2 Laboratoire Kastler Brossel CNRS, ENS-PSL, UPMC-SU, Collège de France, F-75252 Paris, France * 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), 41; https://doi.org/10.3390/proceedings2010041 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 Modern metrology is based on quantum and relativistic physics. The second is defined as a fixed integer number of periods of atomic caesium clocks while the meter is derived from the exact value fixed for the speed of light in vacuum c. Clock synchronization, distance measurement (ranging) or localization in space-time (GNSS localization) are built on the remote distribution of electromagnetic field phases. In 2018, the Planck constant h and elementary charge e should be defined to have exact values, replacing the old definitions of the kilogram and Ampere units.However, the quantum status of time, space and mass observables and its compatibility with relativistic symmetries remain open questions. We address this problem in a quantum algebraic approach where observable positions in space and time are defined so that they obey relativistic and quantum requirements [1]. A mass observable is defined which is no longer a constant as expected from its dimension with respect to dilatation.The commutators of quantum positions involve spin, and frame transformations to accelerated frames differ from their classical counterparts [2]. Relativistic symmetries nevertheless allow one to extend the covariance rules of general relativity to the quantum algebraic framework [3]. It leads to a quantum version of the Einstein equivalence principle identified to the transformation of the mass observable. ReferencesJaekel, M.-T.; Reynaud, S. Quantum Observables associated with Einstein Localisation. arXiv 1998, arXiv:quant-ph/9810027. [Google Scholar]Jaekel, M.-T.; Reynaud, S. Quantum algebraic representation of localization and motion of a Dirac electron. arXiv 2001, arXiv:quant-ph/0101121. [Google Scholar]Jaekel, M.-T.; Reynaud, S. Mass, inertia and gravitation. arXiv 2011, arXiv:0812.3936. [Google Scholar]Publisher's Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. © 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 Jaekel, M.-T.; Reynaud, S. Quantum Metrology and Relativistic Symmetries. Proceedings 2018, 2, 41. https://doi.org/10.3390/proceedings2010041 AMA Style Jaekel M-T, Reynaud S. Quantum Metrology and Relativistic Symmetries. Proceedings. 2018; 2(1):41. https://doi.org/10.3390/proceedings2010041 Chicago/Turabian Style Jaekel, Marc-Thierry, and Serge Reynaud. 2018. "Quantum Metrology and Relativistic Symmetries" Proceedings 2, no. 1: 41. https://doi.org/10.3390/proceedings2010041 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.
Quantum entanglement is conventionally characterized as a structural property of tensor-product Hilbert spaces, with limited emphasis on its geometric or gauge-theoretic organization. Within standard quantum field theory on a fixed classical spacetime background, we show that entanglement can be represented as a global compatibility constraint acting on internal degrees of freedom in the gauge-bundle formulation of the Standard Model. This representation is encoded by a vacuum-level internal gauge structure Ξ(x)\Xi(x)Ξ(x) that is locally pure gauge, dynamically inert, and acts exclusively on internal fibers, leaving all local dynamics and the Standard Model Lagrangian unchanged. We formalize this perspective as a vacuum internal gauge structure (VIGS) and prove a corresponding structural result—the Vacuum Internal Gauge Theorem—which establishes that global compatibility relations associated with Ξ\XiΞ act only on internal Hilbert-space factors and preserve locality and no-signaling within the regime considered here. The framework complements standard Hilbert-space and algebraic descriptions of entanglement by making explicit how global internal correlations can be organized geometrically without invoking nonlocal dynamics. Finally, we identify an experimentally accessible operational signature, based on correlated versus independent internal-frame scrambling, that distinguishes this geometric representation within existing entanglement platforms.
Quantum information has become a powerful tool for probing the structure of quantum field theories, yet its application to gauge theories remains subtle. On the one hand, quantum information theory assumes subsystem locality, i.e.~the factorization of the total Hilbert space into subsystems. On the other hand, gauge constraints prevent the total Hilbert space to decompose into a spacetime-local tensor product structure. Because the Hilbert space structure of gauge theories does not accommodate the subsystem decomposition used in quantum information theory, standard information-theoretic results, such as the Local Operations and Classical Communication (LOCC) theorem, cannot be used straightforwardly in the context of gauge theories. In this work, we bridge this gap in the case of a two-dimensional lattice gauge model that captures key features of electromagnetism. In particular, we construct gauge-invariant local algebras and derive a physically meaningful decomposition of the Hilbert space, providing an operationally consistent notion of locality in the absence of a local tensor-product structure. We apply this framework to field-mediated entanglement protocols relevant to proposed tests of the quantum nature of gravity. We show that the discretized version of electromagnetism satisfies an analogue of the LOCC theorem: entanglement cannot be generated without genuine quantum field interactions, even in the absence of a spacetime-local tensor product factorization of the Hilbert space. This may point towards an operational way to define a subsystem structure for gauge theories.
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.