Canonical Quantization on a Doubly Connected Space and the Aharonov-Bohm Phase

Masao Hirokawa

doi:10.1006/jfan.2000.3591

Canonical Quantization on a Doubly Connected Space and the Aharonov-Bohm Phase

Masao Hirokawa

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We consider the canonical quantization (Schrödinger representation) on a doubly connected space Ω_R≡R²\{(x, y)x²+y²≤R²} (R>0). We show that, when we employ 2-dimensional orthogonal coordinates Ox₁x₂, there are uncountably many different self-adjoint extensions p_Uj of p_j≡-i∂/∂x_j (j=1, 2), and none of the pairs {p_j, q_j′}_{j, j′=1, 2} (q_j′≡x_j′·) satisfies the Weyl relation. Then, we construct a new canonical pair of canonical momentum and position operators so that the pair can satisfy the Weyl relation by using the streamline coordinates. As its application, in the Weyl relation with respect to the pair of the mv-momentum and position operators by the above new canonical pair, we find the Aharonov-Bohm phase.

Original language	English
Pages (from-to)	322-363
Number of pages	42
Journal	Journal of Functional Analysis
Volume	174
Issue number	2
DOIs	https://doi.org/10.1006/jfan.2000.3591
Publication status	Published - Jul 10 2000
Externally published	Yes

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title = "Canonical Quantization on a Doubly Connected Space and the Aharonov-Bohm Phase",

abstract = "We consider the canonical quantization (Schr{\"o}dinger representation) on a doubly connected space ΩR≡R2\{(x, y)x2+y2≤R2} (R>0). We show that, when we employ 2-dimensional orthogonal coordinates Ox1x2, there are uncountably many different self-adjoint extensions pUj of pj≡-i∂/∂xj (j=1, 2), and none of the pairs {pj, qj′}j, j′=1, 2 (qj′≡xj′·) satisfies the Weyl relation. Then, we construct a new canonical pair of canonical momentum and position operators so that the pair can satisfy the Weyl relation by using the streamline coordinates. As its application, in the Weyl relation with respect to the pair of the mv-momentum and position operators by the above new canonical pair, we find the Aharonov-Bohm phase.",

author = "Masao Hirokawa",

note = "Funding Information: I express my thanks to Professor A. Arai for telling me Reeh{\textquoteright}s result [20] and his result [4] and encouraging me to write this paper. The problem in this paper occurred to me when I was a researcher of the Hitachi Advanced Research Laboratory (HARL) and attended the seminar sponsored by Professor H. Ezawa. I also express my thanks to Professor H. Ezawa for his helpful suggestions and comments, and acknowledge helpful discussions with Professor K. Watanabe, Professor T. Nakamura, and Professor H. Watanabe at the seminar. I thank Professor K. Schmudtgen for insightful advice at Fukuoka University. I thank HARL, where my thoughts about this paper were organized. I also express my gratitude to my friend, Dr. K. Harada, for having discussion with me about the experimental side when I was a researcher in HARL, and Dr. A. Tonomura for sending his books and papers. My research is supported by the Grant-In-Aid 09740092 for Encouragement of Young Scientists from the Ministry of Education, Japan.",

year = "2000",

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doi = "10.1006/jfan.2000.3591",

language = "English",

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journal = "Journal of Functional Analysis",

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T1 - Canonical Quantization on a Doubly Connected Space and the Aharonov-Bohm Phase

AU - Hirokawa, Masao

N1 - Funding Information: I express my thanks to Professor A. Arai for telling me Reeh’s result [20] and his result [4] and encouraging me to write this paper. The problem in this paper occurred to me when I was a researcher of the Hitachi Advanced Research Laboratory (HARL) and attended the seminar sponsored by Professor H. Ezawa. I also express my thanks to Professor H. Ezawa for his helpful suggestions and comments, and acknowledge helpful discussions with Professor K. Watanabe, Professor T. Nakamura, and Professor H. Watanabe at the seminar. I thank Professor K. Schmudtgen for insightful advice at Fukuoka University. I thank HARL, where my thoughts about this paper were organized. I also express my gratitude to my friend, Dr. K. Harada, for having discussion with me about the experimental side when I was a researcher in HARL, and Dr. A. Tonomura for sending his books and papers. My research is supported by the Grant-In-Aid 09740092 for Encouragement of Young Scientists from the Ministry of Education, Japan.

PY - 2000/7/10

Y1 - 2000/7/10

N2 - We consider the canonical quantization (Schrödinger representation) on a doubly connected space ΩR≡R2\{(x, y)x2+y2≤R2} (R>0). We show that, when we employ 2-dimensional orthogonal coordinates Ox1x2, there are uncountably many different self-adjoint extensions pUj of pj≡-i∂/∂xj (j=1, 2), and none of the pairs {pj, qj′}j, j′=1, 2 (qj′≡xj′·) satisfies the Weyl relation. Then, we construct a new canonical pair of canonical momentum and position operators so that the pair can satisfy the Weyl relation by using the streamline coordinates. As its application, in the Weyl relation with respect to the pair of the mv-momentum and position operators by the above new canonical pair, we find the Aharonov-Bohm phase.

AB - We consider the canonical quantization (Schrödinger representation) on a doubly connected space ΩR≡R2\{(x, y)x2+y2≤R2} (R>0). We show that, when we employ 2-dimensional orthogonal coordinates Ox1x2, there are uncountably many different self-adjoint extensions pUj of pj≡-i∂/∂xj (j=1, 2), and none of the pairs {pj, qj′}j, j′=1, 2 (qj′≡xj′·) satisfies the Weyl relation. Then, we construct a new canonical pair of canonical momentum and position operators so that the pair can satisfy the Weyl relation by using the streamline coordinates. As its application, in the Weyl relation with respect to the pair of the mv-momentum and position operators by the above new canonical pair, we find the Aharonov-Bohm phase.

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Canonical Quantization on a Doubly Connected Space and the Aharonov-Bohm Phase

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