TY - JOUR
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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U2 - 10.1006/jfan.2000.3591
DO - 10.1006/jfan.2000.3591
M3 - Article
AN - SCOPUS:0347947913
VL - 174
SP - 322
EP - 363
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 2
ER -