dorsal/arxiv
View SchemaBundle Theory of Improper Spin Transformations
| Authors | D. B. Cervantes, S. L. Quiroga, L. J. Perissinotti, M. Socolovsky |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0410079 |
| URL | https://arxiv.org/abs/quant-ph/0410079 |
| DOI | 10.1007/s10773-005-2061-6 |
Abstract
{\it We first give a geometrical description of the action of the parity operator ($\hat{P}$) on non relativistic spin ${{1}\over{2}}$ Pauli spinors in terms of bundle theory. The relevant bundle, $SU(2)\odot \Z_2\to O(3)$, is a non trivial extension of the universal covering group $SU(2)\to SO(3)$. $\hat{P}$ is the non relativistic limit of the corresponding Dirac matrix operator ${\cal P}=i\gamma_0$ and obeys $\hat{P}^2=-1$. Then, from the direct product of O(3) by $\Z_2$, naturally induced by the structure of the galilean group, we identify, in its double cover, the time reversal operator ($\hat{T}$) acting on spinors, and its product with $\hat{P}$. Both, $\hat{P}$ and $\hat{T}$, generate the group $\Z_4 \times \Z_2$. As in the case of parity, $\hat{T}$ is the non relativistic limit of the corresponding Dirac matrix operator ${\cal T}=\gamma^3 \gamma^1$, and obeys $\hat{T}^2=-1$.}
{
"annotation_id": "a9bc538b-5941-4fa0-9291-73eb6ce5a6ce",
"date_created": "2026-03-02T18:02:10.342000Z",
"date_modified": "2026-03-02T18:02:10.342000Z",
"file_hash": "b67ac86d9bfcb609f5cb93485a979fda6757ca628c242e75664dbac694d81fd4",
"private": false,
"record": {
"abstract": "{\\it We first give a geometrical description of the action of the parity\noperator ($\\hat{P}$) on non relativistic spin ${{1}\\over{2}}$ Pauli spinors in\nterms of bundle theory. The relevant bundle, $SU(2)\\odot \\Z_2\\to O(3)$, is a\nnon trivial extension of the universal covering group $SU(2)\\to SO(3)$.\n$\\hat{P}$ is the non relativistic limit of the corresponding Dirac matrix\noperator ${\\cal P}=i\\gamma_0$ and obeys $\\hat{P}^2=-1$. Then, from the direct\nproduct of O(3) by $\\Z_2$, naturally induced by the structure of the galilean\ngroup, we identify, in its double cover, the time reversal operator ($\\hat{T}$)\nacting on spinors, and its product with $\\hat{P}$. Both, $\\hat{P}$ and\n$\\hat{T}$, generate the group $\\Z_4 \\times \\Z_2$. As in the case of parity,\n$\\hat{T}$ is the non relativistic limit of the corresponding Dirac matrix\noperator ${\\cal T}=\\gamma^3 \\gamma^1$, and obeys $\\hat{T}^2=-1$.}",
"arxiv_id": "quant-ph/0410079",
"authors": [
"D. B. Cervantes",
"S. L. Quiroga",
"L. J. Perissinotti",
"M. Socolovsky"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1007/s10773-005-2061-6",
"title": "Bundle Theory of Improper Spin Transformations",
"url": "https://arxiv.org/abs/quant-ph/0410079"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "de8f63ca-ce6e-4ed5-90a0-eea93b344903",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}