nLab
Hilbert lattice

Contents

Idea

A Hilbert lattice is the lattice of closed linear subspaces of a Hilbert spaces (with preorder given by the inclusion) over real, complex or quaternion numbers. This is an orthocomplemented lattice and in fact an orthomodular lattice.

The logic it embodies is Birkhoff-von Neumann quantum logic. See there for more.

References

  • Garrett Birkhoff, John von Neumann, The logic of quantum mechanics, Annals of Mathematics, 37: 823-843 (1936)

  • A. Gleason, Measures on the closed subspaces of a Hilbert space, Journal of Mathematics and Mechanics 6: 885-893 (1957)

  • P. Pták and S. Pulmannová, Orthomodular structures as quantum logics, ser. Fundamental theories of physics. Kluwer Academic Publishers, 1991.

  • Samuel S. Holland Jr., Orthomodularity in infinite dimensions; a theorem of M. Solèr, Bull. Amer. Math. Soc. (N.S.) 32 (1995) 205-234, arXiv:math.RA/9504224

Revised on February 28, 2014 14:16:20 by Urs Schreiber (89.204.139.212)