HaagTheorem 2009-01-29 22:51:16 2009-01-29 23:14:21 Theorem cyclic vector $\mathcal{F}_{Qc}$ CQD Fock representation of quantum free fields canonical quantum dynamics CAR canonical quantum dynamics $\mathcal{F}_{Qc}$ commutation and anti-commutation relationships Fock representation vacuum/ground state canonical quantum dynamics free fields Hamiltonian generator time translations Fock representation of free fields local quantum physics % of TeX increases, you will probably want to edit this, but % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % 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}} \subsection{Introduction} A canonical quantum dynamics (CQD) is determined by the choice of the physical (quantized) `vacuum' state (which is the ground state); thus, the assumption that a field $\mathcal{F}_{Qc}$ shares the ground state with a free field $\mathcal{F}_{0}$, implies that $\mathcal{F}_{Qc}$ is itself free (or admits a Fock representation). This basic assumption is expressed in a mathematically precise form by Haag's theorem in `Local Quantum Physics'. On the other hand, interacting quantum fields generate non-Fock representations of the commutation and anti-commutation relationships (CAR). \subsection{Haag Theorem} \begin{theorem} (The Haag theorem in quantum field theory) Any canonical quantum field, $\mathcal{F}_{Qc}$ that for a fixed value of time $t$ is: \begin{enumerate} \item irreducible, and \item has a cyclic vector, that is \begin{itemize} \item $\mathcal{F}_{Qc}$ has a Hamiltonian generator of time translations, and \item it is unique as a translation-invariant state; \end{itemize} and also, \item is unitarily equivalent to a free field in the Fock representation at the time instant, $t$, \end{enumerate} is itself a \emph{free field}. \end{theorem} \begin{thebibliography}{9} \bibitem{RHaag55} R. Haag, ``On quantum field theories.'', {\em Danske Mat.--Fys. Medd.} , 29 : 12 (1955) pp. 17--112 . \bibitem{GEmch72} [a2] G. Emch, ``Algebraic methods in statistical mechanics and quantum field theory.'' , Wiley (1972) \bibitem{LStreit80} L. Streit, ``Energy forms: Schr\''odinger theory, processes. New stochastic methods in physics.'' Physics reports , 77 : 3 (1980) pp. 363--375. \bibitem{RS-ASW64} R.F. Streater, and A.S. Wightman, ``PCT, spin and statistics, and all that''. , Benjamin (1964) \end{thebibliography}