\documentclass[a4paper,12pt]{article}
\title{Broken symmetry in quantum mechanics}

\author{Walter Thirring}

\date{Universit\"at Wien}

\renewcommand{\baselinestretch}{1.2}
\parskip2ex
\parindent0em

\begin{document}
\maketitle

The name of spontaneous symmetry breaking may hide
one of three entirely different phenomena:
1)  A dynamical system that is an algebra $A$ and an automorphism
group $\tau_t$ (the time evolution) has a time-invariant
state which is not invariant under a symmetry $\alpha$ commuting
with $\tau_t$.
2) An algebra $A$ has a representation $P$ and a symmetry $\alpha$
such that $\alpha$ cannot be extended to the weak closure of $P(A)$.
3) An algebra $A$ has a (not faithful) representation $P$
and a symmetry $\alpha$ such that $\alpha$ cannot be restricted
to $P(A)$.
\end{document}