AlgebraicQuantumFieldTheoriesAQFT 2009-01-16 13:45:23 2009-01-16 14:11:29 Topic algebraic quantum field theories (AQFT) % this is the default PlanetPhysics preamble. as your knowledge % there are many more packages, add them here as you need them % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \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%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \section{Algebraic quantum field theories (AQFTs)} Algebraic quantum field theories (AQFTs) are usually described as algebraic (and/or physical-axiomatic) formulations of quantum field theories (QFTs). According to Halvorson and Mueger (ref. \cite{HH-MM2k6}), ``\emph{an algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools: the theory of operator algebras category theory, etc. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT.''} Such tools include: von Neumann, C*-algebras, Hopf algebras and C*- Clifford algebras in quantum operator algebras, algebraic topology ones--such as homotopy groups, groupoids, quantum groups, quantum groupoids, and categorical ones-- such as 2-categories, homotopy functor, 2-Lie group categories, groupoid categories, braided categories, cohomology theories, and extended Tannaka-Krein or Grothendieck duality. In classical logic, an axiom or postulate is a `simple', fundamental proposition that is neither proven nor demonstrated (within a theory) ``but considered to be self-evident''; furthermore, an axiom or system of axioms is `justified' by the large number of consistent consequences derived from such axioms. One needs, however, to distinguish between `physical axioms' (often called `postulates' that apply to various fields of physics), and mathematical axioms that have both a meaning and scope of applicability which is distinct from that of physical postulates (or physical axioms). On the other hand, physical axioms, or postulates, are ultimately also expressed in a mathematical form, albeit without becoming axioms of mathematics, or specific fields of mathematics. (One notes however rare instances of the opinion expressed that `physics is just another area of mathematics, belonging to applied mathematics'). Furthermore, physical postulates, unlike mathematical ones, emerged as a result of numerous experimental studies and crucial physical experiments that can be logically and consistently explained on the basis of such fundamental, physical postulates; often, mathematical formulations of such fundamental physical postulates are referred to as (physical) `axioms', as in the case of axiomatic QFTs. Thus, from a physical standpoint, AQFTs are just as important as from the mathematical viewpoint, because they may include novel approaches which define algebraic structuures over relativistic spaces, either Minkowski, or \PMlinkname{Riemannian manifold or space}{RiemannianMetric}. The recent review of specific AQFT formulations presented in ref. \cite{HH-MM2k6} provides several examples of AQFT approaches in sufficient mathematical detail to be able to evaluate their correctness from a mathematical viewpoint. According to Roberts, a standard AQFT construction defines a \emph{local network, or net of observable algebras} $O_A$; in the case of a Minkowski 4D-space, one assigns to each double lightcone $L_c$ an algebra of observables, such that algebras of subcones $O_S$ are naturally `embedded into those of the lightcones containing them' (ref. \cite{JER2k4}). Moreover, one needs also to postulate that ``algebras of space-like separated double cones always commute with each other''. The latter postulate is sometimes said to ``encode the physical concept of microcausality''. On the other hand, AQFT formulations on Riemannian spaces are much more difficult to formulate and investigate in detail. There are also interesting mathematical and physical connections between AQFTs and topological quantum field theories (TQFT), as the latter have already been studied in much more detail than AQFTs. On the other hand, AQFT already has obtained several important results such as the analysis of superselection rules by S. Doplicher, R. Haag, and J. E. Roberts (DHR), and the S. Doplicher and J.E. Roberts reconstruction of fields and gauge group from the symmetric tensor *-category of physical representations of the observable algebras. \textbf{Remark:} Evidently, such AQFT formulations are not compatible with the Bohm-de Broglie quantum theories. \begin{thebibliography}{9} \bibitem{HH-MM2k6} Hans Halvorson, Michael Mueger. 2006. \PMlinkexternal{Algebraic Quantum Field Theory}{http://arxiv.org/PS_cache/math-ph/pdf/0602/0602036v1.pdf}. arXiv:math-ph/0602036, 202 pages; to appear in \emph{Handbook of the Philosophy of Physics}, North Holland: Amsterdam. \bibitem{JER2k4} John E. Roberts. More lectures on algebraic quantum field theory. In Sergio Doplicher and Roberto Longo, editors, \emph{Noncommutative geometry}, pages 263-342. Springer, Berlin, 2004. \bibitem{JER90} John E. Roberts. Lectures on algebraic quantum field theory. In Daniel Kastler, editor, \emph{The algebraic theory of superselection sectors} (Palermo, 1989), pages 1-112. World Scientific Publishing, River Edge, NJ, 1990. \end{thebibliography}