Observables and StatesObservablesAndStates2009-02-25 15:59:432009-03-11 04:59:21Topicgeometry of state spacesquantum operator algebravon Neumann algebraoperatorsstate vectorsquantum operators% this is the default PlanetPhysics preamble. as your
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\def\D{\mathsf{D}}\section{Introduction}
The notions of observables and states are fundamental
to mechanics. In this entry, we shall begin with the conceptual
background to these ideas, then proceed to examine how these
notions work in classical, statistical, and quantum mechanics.
The basis for these notions lies in
making numerical measurements on physical systems and comparing
the observed values with predicted theoretical values. The
value measured will depend on the quantity being measured and
upon the initial and boundary conditions imposed on the system.
To account for this dependence, we introduce observables and
states --- an observable is a mathematical entity in a theory
which represents a measurement which can be made on the physical
system described by that theory and a state is a mathematical
entity which encodes conditions placed on that system. A
theory of a system will provide the set of observables and
the set of states for that system, describe how they evolve
with time, and specify how to obtain numerical values by
combining states and observables.
To make this discussion concrete, we may consider an elementary
example --- the freely falling body. Here, examples of
observables would include the height and velocity of the object.
The state of the system may be specified by stating the initial
height and velocity or by specifying the height at an initial
time and at a final time. Given such a specification, we can
then compute the values of velocity and position at any time
using these formulae
\begin{align*}
h - h_0 &= {1 \over 2} g (t - t_0)^2 \\
v - v_0 &= g (t - t_0) .
\end{align*}
The values so obtained may then be compared with experiment.
In addition to the height and velocity, there are other observables
such as energy. However, it is possible to express
these observables algebraically in terms of the height and velocity.
(Note that this requires use of the equations of motion.)
\begin{align*}
E &= {1 \over 2} m v^2 + m g h
\end{align*}
\section{Quantum Operators as Observables in Quantum Theories}
\begin{definition} {\em Quantum operator algebras} (QOA) in quantum field theories are defined as the algebras of observable operators, and as such, they are also related to the von Neumann algebra;
quantum operators are usually defined on Hilbert spaces, or in some QFTs on Hilbert space bundles or other similar families of spaces.
\end{definition}
\begin{remark}
{\em Representations of Banach $*$-algebras} (that are defined on Hilbert spaces) are closely related to C* -algebra representations which provide a useful approach to defining quantum space-times.
\end{remark}
\subsection{Quantum operator algebras in quantum field theories: QOA Role in QFTs}
Important examples of quantum operators are: the Hamiltonian operator (or Schr\"odinger operator), the position and momentum operators, Casimir operators, unitary operators and spin operators. The observable operators are also {\em self-adjoint}. More general operators were recently defined, such as Prigogine's superoperators.
Another development in quantum theories was the introduction of Frech\'et nuclear spaces or `rigged' Hilbert spaces (Hilbert bundles).
\subsection{Basic mathematical definitions in QOA: }
\begin{itemize}
\item {\em Von Neumann algebra}
\item {\em Hopf algebra}
\item {\em Groupoids}
\item {\em Haar systems associated to measured groupoids or locally compact groupoids.}
\item C*-algebras and quantum groupoids
\end{itemize}
\textbf{[more to come]}
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