## On characterizing the standard quantum logics

HTML articles powered by AMS MathViewer

- by W. John Wilbur PDF
- Trans. Amer. Math. Soc.
**233**(1977), 265-282 Request permission

## Abstract:

Let $\mathcal {L}$ be a complete projective logic. Then $\mathcal {L}$ has a natural representation as the lattice of $\langle { \cdot , \cdot } \rangle$-closed subspaces of a left vector space*V*over a division ring

*D*, where $\langle {\cdot ,\cdot } \rangle$ is a definite $\theta$-bilinear symmetric form on

*V*, $\theta$ being some involutive antiautomorphism of

*D*. Now a well-known theorem of Piron states that if

*D*is isomorphic to the real field, the complex field or the sfield of quaternions, if $\theta$ is continuous, and if the dimension of $\mathcal {L}$ is properly restricted, then $\mathcal {L}$ is just one of the standard Hilbert space logics. Here we also assume $\mathcal {L}$ is a complete projective logic. Then if every $\theta$-fixed element of

*D*is in the center of

*D*and can be written as $\pm d\theta (d)$, some $d \in D$, and if the dimension of $\mathcal {L}$ is properly restricted, we show that $\mathcal {L}$ is just one of the standard Hilbert space logics over the reals, the complexes, or the quaternions. One consequence is the extension of Piron’s theorem to discontinuous $\theta$. Another is a purely lattice theoretic characterization of the lattice of closed subspaces of separable complex Hilbert space.

## References

- E. G. Beltrametti and G. Cassinelli,
*Quantum mechanics and $p$-adic numbers*, Found. Phys.**2**(1972), 1–7. MR**331996**, DOI 10.1007/BF00708614 - Garrett Birkhoff,
*Lattice Theory*, Revised edition, American Mathematical Society Colloquium Publications, Vol. 25, American Mathematical Society, New York, N. Y., 1948. MR**0029876** - Garrett Birkhoff and John von Neumann,
*The logic of quantum mechanics*, Ann. of Math. (2)**37**(1936), no. 4, 823–843. MR**1503312**, DOI 10.2307/1968621 - Platon C. Deliyannis,
*Vector space models of abstract quantum logics*, J. Mathematical Phys.**14**(1973), 249–253. MR**325052**, DOI 10.1063/1.1666304 - J.-P. Eckmann and Ph. Ch. Zabey,
*Impossibility of quantum mechanics in a Hilbert space over a finite field*, Helv. Phys. Acta**42**(1969), 420–424. MR**246600** - Andrew M. Gleason,
*Measures on the closed subspaces of a Hilbert space*, J. Math. Mech.**6**(1957), 885–893. MR**0096113**, DOI 10.1512/iumj.1957.6.56050 - S. Gudder and C. Piron,
*Observables and the field in quantum mechanics*, J. Mathematical Phys.**12**(1971), 1583–1588. MR**309442**, DOI 10.1063/1.1665777 - Stanley P. Gudder,
*Quantum logics, physical space, position observables and symmetry*, Rep. Mathematical Phys.**4**(1973), 193–202. MR**325030**, DOI 10.1016/0034-4877(73)90024-4 - Samuel S. Holland Jr.,
*Remarks on type $\textrm {I}$ Baer and Baer $^{\ast }$-rings*, J. Algebra**27**(1973), 516–522. MR**330216**, DOI 10.1016/0021-8693(73)90061-6 - A. Kolmogoroff,
*Zur Begründung der projektiven Geometrie*, Ann. of Math. (2)**33**(1932), no. 1, 175–176 (German). MR**1503044**, DOI 10.2307/1968111 - M. J. Mączyński,
*Hilbert space formalism of quantum mechanics without the Hilbert space axiom*, Rep. Mathematical Phys.**3**(1972), no. 3, 209–219. MR**321449**, DOI 10.1016/0034-4877(72)90005-5 - M. J. Mączyński,
*The field of real numbers in axiomatic quantum mechanics*, J. Mathematical Phys.**14**(1973), 1469–1471. MR**329456**, DOI 10.1063/1.1666206 - Ronald P. Morash,
*The orthomodular identity and metric completeness of the coordinatizing division ring*, Proc. Amer. Math. Soc.**27**(1971), 446–448. MR**272689**, DOI 10.1090/S0002-9939-1971-0272689-3 - C. Piron,
*Axiomatique quantique*, Helv. Phys. Acta**37**(1964), 439–468 (French, with English summary). MR**204048** - V. S. Varadarajan,
*Geometry of quantum theory. Vol. I*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. MR**0471674** - Edwin Weiss and Neal Zierler,
*Locally compact division rings*, Pacific J. Math.**8**(1958), 369–371. MR**121432**, DOI 10.2140/pjm.1958.8.369 - W. John Wilbur,
*Quantum logic and the locally convex spaces*, Trans. Amer. Math. Soc.**207**(1975), 343–360. MR**367607**, DOI 10.1090/S0002-9947-1975-0367607-1 - Neal Zierler,
*Axioms for non-relativistic quantum mechanics*, Pacific J. Math.**11**(1961), 1151–1169. MR**140972**, DOI 10.2140/pjm.1961.11.1151 - Neal Zierler,
*On the lattice of closed subspaces of Hilbert space*, Pacific J. Math.**19**(1966), 583–586. MR**202647**, DOI 10.2140/pjm.1966.19.583

## Additional Information

- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**233**(1977), 265-282 - MSC: Primary 81.06
- DOI: https://doi.org/10.1090/S0002-9947-1977-0468710-X
- MathSciNet review: 0468710