Quantum mechanics on the line with two origins
Source: Originally published on arXiv quant-ph on July 30, 2026.
arXiv:2607.26080v1 Announce Type: new Abstract: We study scalar and spinorial quantum dynamics on the standard line with two origins, [ \Ltwo=(\R_1\sqcup\R_2)/!\sim, \qquad (x,1)\sim(x,2)\quad\text{for }x\neq0, ] equipped with its identity-glued smooth structure and flat metric. We first show that the ordinary scalar theory is insensitive to the doubled origin: every continuous map from (\Ltwo) to a Hausdorff space factors through the quotient (q:\Ltwo\to\R), while, for the natural measure, [ C^\infty(\Ltwo)\cong C^\infty(\R), \qquad L^2(\Ltwo)\cong L^2(\R). ] Accordingly, the natural scalar free Hamiltonian is unitarily equivalent to the free Laplacian on (\R). The doubled origin remains visible, however, in the spinor line associated with the nontrivial spin structure. The o...
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