“Solèr's theorem”的意思、由来-开放百科全书

In mathematics, Solèr's theorem is a result concerning certain infinite-dimensional vector spaces. It states that any orthomodular form that has an infinite orthonormal sequence is a Hilbert space over the real numbers, complex numbers or quaternions.^[1]^[2] Originally proved by Maria Pia Solèr, the result is significant for quantum logic^[3]^[4] and the foundations of quantum mechanics.^[5]^[6] In particular, Solèr's theorem helps to fill a gap in the effort to use Gleason's theorem to rederive quantum mechanics from information-theoretic postulates.^[7]^[8]

Statement

Let

be a division ring. That means it is a ring in which one can add, subtract, multiply, and divide but in which the multiplication need not be commutative. Suppose this ring has a conjugation, i.e. an operation

for which

This is called a Hermitian form. Suppose this form is non-degenerate in the sense that

For any subspace S let

be the orthogonal complement of S. Call the subspace "closed" if

Call this whole vector space, and the Hermitian form, "orthomodular" if for every closed subspace S we have that

is the entire space. (The term "orthomodular" derives from the study of quantum logic. In quantum logic, the distributive law is taken to fail due to the uncertainty principle, and it is replaced with the "modular law," or in the case of infinite-dimensional Hilbert spaces, the "orthomodular law."^[6])

References

1. ^{{Cite journal|last=Solèr|first=M. P.|date=1995-01-01|title=Characterization of hilbert spaces by orthomodular spaces|journal=Communications in Algebra|volume=23|issue=1|pages=219–243|doi=10.1080/00927879508825218|issn=0092-7872}}
2. ^{{Cite journal|last=Prestel|first=Alexander|date=1995-12-01|title=On Solèr's characterization of Hilbert spaces|journal=Manuscripta Mathematica|language=en|volume=86|issue=1|pages=225–238|doi=10.1007/bf02567991|issn=0025-2611}}
3. ^{{Cite book|title=Current Research in Operational Quantum Logic|last=Coecke|first=Bob|last2=Moore|first2=David|last3=Wilce|first3=Alexander|date=2000|publisher=Springer, Dordrecht|isbn=978-90-481-5437-1|location=|pages=1–36|language=en|chapter=Operational Quantum Logic: An Overview|arxiv=quant-ph/0008019|doi=10.1007/978-94-017-1201-9_1}}
4. ^{{Cite journal|last=Aerts|first=Diederik|last2=Van Steirteghem|first2=Bart|date=2000-03-01|title=Quantum Axiomatics and a Theorem of M. P. Solèr|journal=International Journal of Theoretical Physics|language=en|volume=39|issue=3|pages=497–502|arxiv=quant-ph/0105107|doi=10.1023/a:1003661015110|issn=0020-7748}}
5. ^{{Cite journal|last=Holland|first=Samuel S.|date=1995|title=Orthomodularity in infinite dimensions; a theorem of M. Solèr|url=http://www.ams.org/bull/1995-32-02/S0273-0979-1995-00593-8/|journal=Bulletin of the American Mathematical Society|volume=32|issue=2|pages=205–234|arxiv=math/9504224|doi=10.1090/s0273-0979-1995-00593-8|issn=0273-0979|via=}}
6. ^¹²{{Cite web|url=https://golem.ph.utexas.edu/category/2010/12/solers_theorem.html|title=Solèr's Theorem|last=Baez|first=John C.|authorlink=John C. Baez|date=1 December 2010|website=The n-Category Café|archive-url=|archive-date=|dead-url=|access-date=2017-07-22}}
7. ^¹{{Cite book|title=Physical Theory and its Interpretation|volume = 72|last=Pitowsky|first=Itamar|date=2006|publisher=Springer, Dordrecht|isbn=978-1-4020-4875-3|location=|pages=213–240|language=en|chapter=Quantum Mechanics as a Theory of Probability|arxiv=quant-ph/0510095|doi=10.1007/1-4020-4876-9_10|series = The Western Ontario Series in Philosophy of Science}}
8. ^{{Cite journal|last=Grinbaum|first=Alexei|date=2007-09-01|title=Reconstruction of Quantum Theory|url=http://philsci-archive.pitt.edu/2703/1/reconstruction2.pdf|journal=The British Journal for the Philosophy of Science|volume=58|issue=3|pages=387–408|doi=10.1093/bjps/axm028|issn=0007-0882|via=}}
{{Cite journal|last=Cassinelli|first=G.|last2=Lahti|first2=P.|date=2017-11-13|title=Quantum mechanics: why complex Hilbert space?|url=http://rsta.royalsocietypublishing.org/content/375/2106/20160393|journal=Philosophical Transactions of the Royal Society A|language=en|volume=375|issue=2106|pages=20160393|doi=10.1098/rsta.2016.0393|issn=1364-503X|pmid=28971945|bibcode=2017RSPTA.37560393C}}