QuantumLogicTopoi 2009-01-11 17:28:10 2009-01-31 23:47:24 Topic generalized quantum topos quantum logics toposes % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners % used for TeXing text within eps files % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} 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\newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \section{Quantum logic topoi} \begin{definition} A \emph{quantum logic topos} (\emph{QLT}) is defined as an extension of the concept of topos in which the Heyting logic algebra (or subobject classifier) of the standard elementary topos is replaced by a \emph{quantum logic}which is axiomatically defined by \emph{non-commutative and non-distributive} lattice structures. \end{definition} \subsection{Remark} Quantum logics topoi are thus generalizations of the Birkhoff and von Neumann definition of quantum state spaces based on their definition of a quantum logic (lattice), as well as a \emph{non-Abelian}, higher dimensional extension of the recently proposed concept of a `quantum' topos which employs the (\emph{commutative}) Heyting logic algebra as a subobject classifier. Some specific examples are considered in the following two recent references. \begin{thebibliography}{9} \bibitem{BIsham1} Butterfield, J. and C. J. Isham: 2001, Space-time and the philosophical challenges of quantum gravity., in C. Callender and N. Hugget (eds. ) \emph{Physics Meets Philosophy at the Planck scale.}, Cambridge University Press,pp.33--89. \bibitem{BIsham2} Butterfield, J. and C. J. Isham: 1998, 1999, 2000--2002, A topos perspective on the Kochen--Specker theorem I - IV, \emph{Int. J. Theor. Phys}, \textbf{37} No 11., 2669--2733 \textbf{38} No 3., 827--859, \textbf{39} No 6., 1413--1436, \textbf{41} No 4., 613--639. \end{thebibliography}