On the concreteness of quantum logics

Pavel Pták; John David Maitland Wright

Aplikace matematiky (1985)

  • Volume: 30, Issue: 4, page 274-285
  • ISSN: 0862-7940

Abstract

top
It is shown that for any quantum logic L one can find a concrete logic K and a surjective homomorphism f from K onto L such that f maps the centre of K onto the centre of L . Moreover, one can ensure that each finite set of compatible elements in L is the image of a compatible subset of K . This result is “best possible” - let a logic L be the homomorphic image of a concrete logic under a homomorphism such that, if F is a finite subset of the pre-image of a compatible subset of L , then F is compatible. Then L must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.

How to cite

top

Pták, Pavel, and Wright, John David Maitland. "On the concreteness of quantum logics." Aplikace matematiky 30.4 (1985): 274-285. <http://eudml.org/doc/15405>.

@article{Pták1985,
abstract = {It is shown that for any quantum logic $L$ one can find a concrete logic $K$ and a surjective homomorphism $f$ from $K$ onto $L$ such that $f$ maps the centre of $K$ onto the centre of $L$. Moreover, one can ensure that each finite set of compatible elements in $L$ is the image of a compatible subset of $K$. This result is “best possible” - let a logic $L$ be the homomorphic image of a concrete logic under a homomorphism such that, if $F$ is a finite subset of the pre-image of a compatible subset of $L$, then $F$ is compatible. Then $L$ must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.},
author = {Pták, Pavel, Wright, John David Maitland},
journal = {Aplikace matematiky},
keywords = {orthomodular lattice; orthomodular poset; centres; orthocomplemented posets; concrete logics; orthomodular lattice; orthomodular poset; centres; orthocomplemented posets; concrete logics},
language = {eng},
number = {4},
pages = {274-285},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the concreteness of quantum logics},
url = {http://eudml.org/doc/15405},
volume = {30},
year = {1985},
}

TY - JOUR
AU - Pták, Pavel
AU - Wright, John David Maitland
TI - On the concreteness of quantum logics
JO - Aplikace matematiky
PY - 1985
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 30
IS - 4
SP - 274
EP - 285
AB - It is shown that for any quantum logic $L$ one can find a concrete logic $K$ and a surjective homomorphism $f$ from $K$ onto $L$ such that $f$ maps the centre of $K$ onto the centre of $L$. Moreover, one can ensure that each finite set of compatible elements in $L$ is the image of a compatible subset of $K$. This result is “best possible” - let a logic $L$ be the homomorphic image of a concrete logic under a homomorphism such that, if $F$ is a finite subset of the pre-image of a compatible subset of $L$, then $F$ is compatible. Then $L$ must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.
LA - eng
KW - orthomodular lattice; orthomodular poset; centres; orthocomplemented posets; concrete logics; orthomodular lattice; orthomodular poset; centres; orthocomplemented posets; concrete logics
UR - http://eudml.org/doc/15405
ER -

References

top
  1. V. Alda, On 0-1 measures for projectors, Aplikace Matematiky 26, 57-58 (1981). (1981) Zbl0459.28020MR0602402
  2. L. J. Bunce D. M. Wright, Qantum measures and states on Jordan algebras, Comm. Math. Phys. (To appear). Zbl0579.46049MR0786572
  3. J. Brabec P. Pták, 10.1007/BF00736849, Foundations of Physics, Vol. 12, No. 2, 207-212 (1982). (1982) MR0659779DOI10.1007/BF00736849
  4. R. Godowski, Varieties of orthomodular lattices with a strongly full set of states, Demonstration Mathematica, Vol. XIV, No. 3, (1981). (1981) Zbl0483.06007MR0663122
  5. R. Greechie, 10.1016/0097-3165(71)90015-X, J. Comb. Theory 10, 119-132 (1971). (1971) Zbl0219.06007MR0274355DOI10.1016/0097-3165(71)90015-X
  6. S. Gudder, Stochastic Methods in Quantum Mechanics, North-Holland 1979. (1979) Zbl0439.46047MR0543489
  7. P. Pták, 10.1063/1.525758, J. Math. Physics 24 (4), 839-840(1983). (1983) Zbl0508.60006MR0700618DOI10.1063/1.525758
  8. P. Pták V. Rogolewicz, 10.1016/0022-4049(83)90074-9, J. Pure and Applied Algebra 28, 75-85 (1983). (1983) MR0692854DOI10.1016/0022-4049(83)90074-9
  9. S. Pulmannová, Compatibility and partial compatibility in quantum logics, Ann. Inst. Henri Poincaré, Vol. XXXIV, No. 4, 391-403 (1981). (1981) Zbl0469.03045MR0625170
  10. R. Sikorski, Boolean Algebras, Springer-Verlag (1964). (1964) Zbl0123.01303MR0126393
  11. V. Varadarajan, Geometry of Quantum Theory I, Von Nostrand, Princeton (1968). (1968) Zbl0155.56802MR0471674
  12. M. Zierler M. Schlessinger, 10.1215/S0012-7094-65-03224-2, Duke J. Math. 32, 251-262 (1965). (1965) MR0175520DOI10.1215/S0012-7094-65-03224-2

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.