|
| related topics |
| {vol, operators, histories} |
| {operator, operators, space} |
| {observables, space, algebra} |
| {time, decoherence, evolution} |
| {group, space, representation} |
| {let, theorem, proof} |
| {states, state, optimal} |
|
The Geometry of Consistency: Decohering Histories in Generalized Quantum
Theory
David Craig
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.
- oai_identifier:
- oai:arXiv.org:quant-ph/9704031
- categories:
- quant-ph gr-qc
- comments:
- 68 pages, RevTeX, 3 figures. PSNUP -2 strongly recommended
- arxiv_id:
- quant-ph/9704031
- report_no:
- ALBERTA-THY-04-97, UCSB-TH-95-10
- created:
- 1997-04-16
Full article ▸
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