dorsal/arxiv
View SchemaThe Geometry of Consistency: Decohering Histories in Generalized Quantum Theory
| Authors | David Craig |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9704031 |
| URL | https://arxiv.org/abs/quant-ph/9704031 |
Abstract
The geometry of decoherence in generalized "consistent histories" quantum theory is explored, revealing properties of the theory that are independent of any particular application of it. It is shown how the decoherence functional of a closed quantum system may be regarded as an Hermitian form on the space of linear operators on the Hilbert space of the system. This identification makes manifest a number of structural properties of decoherence functionals. For example, a bound on the maximum number of histories in a consistent set is determined. When the decoherence functional is positive -- as in conventional quantum mechanics -- it defines a semi-inner product on the space of history operators. This shows that consistent sets of histories are just orthogonal sets in this inner product. It further implies the existence in general of Cauchy-Schwarz and triangle inequalities for positive decoherence functionals. The geometrical significance of the ILS theorem classifying all possible decoherence functionals is illuminated, and a version of the ILS theorem for decoherence functionals on class operators is given. The class of history operators consistent according to a given decoherence functional is found, and, conversely, it is shown how to construct the decoherence functionals according to which a given set of histories is consistent. More generally, the "geometric" point of view here developed supplies a powerful unified language with which to solve problems in generalized quantum theory.
{
"annotation_id": "ad4465ae-9d61-4369-9e71-f47ac80ee21c",
"date_created": "2026-03-02T18:02:41.149000Z",
"date_modified": "2026-03-02T18:02:41.149000Z",
"file_hash": "2d54e31301aef586cfc1776f0a4f5f7e179a58ef127e505d4eda23cfc9e28512",
"private": false,
"record": {
"abstract": "The geometry of decoherence in generalized \"consistent histories\" quantum\ntheory is explored, revealing properties of the theory that are independent of\nany particular application of it. It is shown how the decoherence functional of\na closed quantum system may be regarded as an Hermitian form on the space of\nlinear operators on the Hilbert space of the system. This identification makes\nmanifest a number of structural properties of decoherence functionals. For\nexample, a bound on the maximum number of histories in a consistent set is\ndetermined. When the decoherence functional is positive -- as in conventional\nquantum mechanics -- it defines a semi-inner product on the space of history\noperators. This shows that consistent sets of histories are just orthogonal\nsets in this inner product. It further implies the existence in general of\nCauchy-Schwarz and triangle inequalities for positive decoherence functionals.\nThe geometrical significance of the ILS theorem classifying all possible\ndecoherence functionals is illuminated, and a version of the ILS theorem for\ndecoherence functionals on class operators is given. The class of history\noperators consistent according to a given decoherence functional is found, and,\nconversely, it is shown how to construct the decoherence functionals according\nto which a given set of histories is consistent. More generally, the\n\"geometric\" point of view here developed supplies a powerful unified language\nwith which to solve problems in generalized quantum theory.",
"arxiv_id": "quant-ph/9704031",
"authors": [
"David Craig"
],
"categories": [
"quant-ph",
"gr-qc"
],
"title": "The Geometry of Consistency: Decohering Histories in Generalized Quantum Theory",
"url": "https://arxiv.org/abs/quant-ph/9704031"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "da709c2c-532c-4ee1-a480-519bfc986bf2",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}