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

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.