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Operator Algebra Perspectives on Interacting Quantum Systems
Operator Algebra Perspectives on Interacting Quantum Systems
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20260202105210
- ISBN
- 9798291563854
- DDC
- 510
- 저자명
- Chen, Yidong.
- 서명/저자
- Operator Algebra Perspectives on Interacting Quantum Systems
- 발행사항
- [Sl] : University of Illinois at Urbana-Champaign, 2023
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2023
- 형태사항
- 174 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 87-02, Section: B.
- 주기사항
- Advisor: Junge, Marius;Faulkner, Thomas.
- 학위논문주기
- Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2023.
- 초록/해제
- 요약A central problem in modern mathematical physics is to study interacting systems beyond the reach of perturbation theory. This thesis studies interacting quantum systems using the mathematical theory of operator algebras.In the first part of this thesis, we consider how to construct quantum many-body systems from one-particle systems. This problem has a well-established solution - namely, the second quantization (or the Gaussian functor). However, it seems difficult for the second quantization to go beyond generalized free field constructions without the help of perturbation theory. The second quantization takes real Hilbert spaces as inputs. Using only information from one-particle Hilbert spaces, it is mathematically impossible for the second quantization to produce connected higher-point correlation functions. Moreover, from the perspectives of mathematical quantization and operator algebra theory, it is more natural to have a quantization procedure that takes an algebra along with a functional as the input. Such a pair of mathematical objects contains more information about the underlying one-particle system and includes the Hilbert space of states (via the GNS construction) as part of the data.Inspired by quantum probability theory, we introduce an alternative approach to construct quantum many-body systems. This approach is based on a noncommutative generalization of the classical Poisson random measure. We call this construction Poissonization. Mathematically, Poissonization is a functor from the category of von Neumann algebras with normal semi nite faithful weights to the category of von Neumann algebras with normal faithful states. It is a natural adaptation of the second quantization to the context of von Neumann algebras. This thesis discusses several properties of Poissonization, and uses Poissonization as a tool to construct various toy models of algebraic quantum field theories relevant to high energy physics. This collection of examples aims at demonstrating the versatility of Poissonization.In the second part of this thesis, we study the noisy interactions between an open quantum system and its environment. One of the challenges in quantum information science is to control open quantum systems with a large number of qubits. An important aspect of many-body systems is the emergence of collective phenomena. One collective noise model is an open atomic system in an electromagnetic environment. This model was considered by Dicke in the 50's. In this thesis, we study the entropic decay in Dicke's model and other related collective noise models. Specifically, we develop a general framework to estimate the spectral gap and modified logarithmic Sobolev constant of these collective noise models. In addition, we study the necessary mixing conditions a general Dicke's model must satisfy in order to have a unique equilibrium state.
- 일반주제명
- Mathematics
- 일반주제명
- Theoretical physics
- 일반주제명
- Quantum physics
- 일반주제명
- Theoretical mathematics
- 키워드
- Quantization
- 기타저자
- University of Illinois at Urbana-Champaign Physics
- 기본자료저록
- Dissertations Abstracts International. 87-02B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a510
■1001 ▼aChen, Yidong.
■24510▼aOperator Algebra Perspectives on Interacting Quantum Systems
■260 ▼a[Sl]▼bUniversity of Illinois at Urbana-Champaign▼c2023
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2023
■300 ▼a174 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 87-02, Section: B.
■500 ▼aAdvisor: Junge, Marius;Faulkner, Thomas.
■5021 ▼aThesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2023.
■520 ▼aA central problem in modern mathematical physics is to study interacting systems beyond the reach of perturbation theory. This thesis studies interacting quantum systems using the mathematical theory of operator algebras.In the first part of this thesis, we consider how to construct quantum many-body systems from one-particle systems. This problem has a well-established solution - namely, the second quantization (or the Gaussian functor). However, it seems difficult for the second quantization to go beyond generalized free field constructions without the help of perturbation theory. The second quantization takes real Hilbert spaces as inputs. Using only information from one-particle Hilbert spaces, it is mathematically impossible for the second quantization to produce connected higher-point correlation functions. Moreover, from the perspectives of mathematical quantization and operator algebra theory, it is more natural to have a quantization procedure that takes an algebra along with a functional as the input. Such a pair of mathematical objects contains more information about the underlying one-particle system and includes the Hilbert space of states (via the GNS construction) as part of the data.Inspired by quantum probability theory, we introduce an alternative approach to construct quantum many-body systems. This approach is based on a noncommutative generalization of the classical Poisson random measure. We call this construction Poissonization. Mathematically, Poissonization is a functor from the category of von Neumann algebras with normal semi nite faithful weights to the category of von Neumann algebras with normal faithful states. It is a natural adaptation of the second quantization to the context of von Neumann algebras. This thesis discusses several properties of Poissonization, and uses Poissonization as a tool to construct various toy models of algebraic quantum field theories relevant to high energy physics. This collection of examples aims at demonstrating the versatility of Poissonization.In the second part of this thesis, we study the noisy interactions between an open quantum system and its environment. One of the challenges in quantum information science is to control open quantum systems with a large number of qubits. An important aspect of many-body systems is the emergence of collective phenomena. One collective noise model is an open atomic system in an electromagnetic environment. This model was considered by Dicke in the 50's. In this thesis, we study the entropic decay in Dicke's model and other related collective noise models. Specifically, we develop a general framework to estimate the spectral gap and modified logarithmic Sobolev constant of these collective noise models. In addition, we study the necessary mixing conditions a general Dicke's model must satisfy in order to have a unique equilibrium state.
■590 ▼aSchool code: 0090.
■650 4▼aMathematics
■650 4▼aTheoretical physics
■650 4▼aQuantum physics
■650 4▼aTheoretical mathematics
■653 ▼aVon Neumann algebras
■653 ▼aNoncommutative geometry
■653 ▼aMathematical physics
■653 ▼aQuantization
■653 ▼aOpen quantum systems
■690 ▼a0753
■690 ▼a0405
■690 ▼a0599
■690 ▼a0642
■71020▼aUniversity of Illinois at Urbana-Champaign▼bPhysics.
■7730 ▼tDissertations Abstracts International▼g87-02B.
■790 ▼a0090
■791 ▼aPh.D.
■792 ▼a2023
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17359767▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
