中国物理B ›› 2021, Vol. 30 ›› Issue (11): 110203-110203.doi: 10.1088/1674-1056/ac0529

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Connes distance of 2D harmonic oscillators in quantum phase space

Bing-Sheng Lin(林冰生)1,2,† and Tai-Hua Heng(衡太骅)3   

  1. 1 School of Mathematics, South China University of Technology, Guangzhou 510641, China;
    2 Laboratory of Quantum Science and Engineering, South China University of Technology, Guangzhou 510641, China;
    3 School of Physics and Material Science, Anhui University, Hefei 230601, China
  • 收稿日期:2021-02-23 修回日期:2021-04-21 接受日期:2021-05-26 出版日期:2021-10-13 发布日期:2021-11-03
  • 通讯作者: Bing-Sheng Lin E-mail:sclbs@scut.edu.cn
  • 基金资助:
    Project supported by the Key Research and Development Project of Guangdong Province, China (Grant No. 2020B0303300001), the National Natural Science Foundation of China (Grant No. 11911530750), the Guangdong Basic and Applied Basic Research Foundation, China (Grant No. 2019A1515011703), the Fundamental Research Funds for the Central Universities, China (Grant No. 2019MS109), and the Natural Science Foundation of Anhui Province, China (Grant No. 1908085MA16).

Connes distance of 2D harmonic oscillators in quantum phase space

Bing-Sheng Lin(林冰生)1,2,† and Tai-Hua Heng(衡太骅)3   

  1. 1 School of Mathematics, South China University of Technology, Guangzhou 510641, China;
    2 Laboratory of Quantum Science and Engineering, South China University of Technology, Guangzhou 510641, China;
    3 School of Physics and Material Science, Anhui University, Hefei 230601, China
  • Received:2021-02-23 Revised:2021-04-21 Accepted:2021-05-26 Online:2021-10-13 Published:2021-11-03
  • Contact: Bing-Sheng Lin E-mail:sclbs@scut.edu.cn
  • Supported by:
    Project supported by the Key Research and Development Project of Guangdong Province, China (Grant No. 2020B0303300001), the National Natural Science Foundation of China (Grant No. 11911530750), the Guangdong Basic and Applied Basic Research Foundation, China (Grant No. 2019A1515011703), the Fundamental Research Funds for the Central Universities, China (Grant No. 2019MS109), and the Natural Science Foundation of Anhui Province, China (Grant No. 1908085MA16).

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摘要： We study the Connes distance of quantum states of two-dimensional (2D) harmonic oscillators in phase space. Using the Hilbert-Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional (4D) quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators. We prove that these two-dimensional distances satisfy the Pythagoras theorem. These results are significant for the study of geometric structures of noncommutative spaces, and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.

关键词: Connes distance, noncommutative geometry, harmonic oscillator

Abstract: We study the Connes distance of quantum states of two-dimensional (2D) harmonic oscillators in phase space. Using the Hilbert-Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional (4D) quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators. We prove that these two-dimensional distances satisfy the Pythagoras theorem. These results are significant for the study of geometric structures of noncommutative spaces, and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.

Key words: Connes distance, noncommutative geometry, harmonic oscillator

中图分类号:  (Noncommutative geometry)

  • 02.40.Gh
03.65.-w (Quantum mechanics) 03.65.Fd (Algebraic methods)