quantum operator algebras
QuantumOperatorAlgebras
2008-10-16 18:15:23
2008-10-16 18:15:23
Topic
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\section{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.
{\em Note:}
Representations of Banach *-algebras, that are also defined on Hilbert spaces, are related to $C^*$-algebra representations which provide a useful approach to defining quantum space-times.
\textbf{Quantum Operator Algebras in Quantum Field Theories: QOA{}s in QFT{}s}
\emph{Examples} of quantum operators are: the Hamiltonian operator (or Schr\"odinger operator), the position and momentum operators, Casimir operators, Unitary operators, spin operators, and so on. The observable operators are also {\em self-adjoint}. More general operators were recently defined, such as Progogine's superoperators. Another development in quantum theories is the introduction of Frech\'et nuclear spaces or `rigged' Hilbert spaces (Hilbert {\em bundles}). The following sections define several types of quantum operator algebras that provide the foundation of modern quantum field theories in Mathematical Physics.
\subsection{Quantum Groups, Quantum Operator Algebras and Related Symmetries.}
Quantum theories adopted a new lease of life post 1955 when von
Neumann beautifully re-formulated Quantum Mechanics (QM) and Quantum theories (QT)
in the mathematically rigorous context of Hilbert spaces and operator
algebras defined over such spaces. From a current physics perspective,
von Neumann' s approach to quantum mechanics has however done much more: it has
not only paved the way to expanding the role of symmetry in
physics, as for example with the Wigner-Eckhart theorem and its
applications, but also revealed the fundamental importance in
Quantum physics of the state space geometry of quantum operator
algebras- Mathematical definitions
{\em Definitions:}
\begin{enumerate}
\item {\em Von Neumann Algebra}
\item {\em Hopf Algebra}
\item {\em Groupoids}
\item {\em Haar Systems associated to Measured Groupoids or Locally Compact Groupoids.}
\end{enumerate}.
\subsection{Von Neumann Algebra}
Let $\H$ denote a complex (separable) Hilbert space. A \emph{von
Neumann algebra} $\A$ acting on $\H$ is a subset of the algebra of
all bounded operators $\cL(\H)$ such that:
\begin{enumerate}
\item (i) $\A$ is closed under the adjoint operation (with the
adjoint of an element $T$ denoted by $T^*$).
\item (ii) $\A$ equals its bicommutant, namely:
\begin{equation}
\A= \{A \in \cL(\H) : \forall B \in \cL(\H), \forall C\in \A,~
(BC=CB)\Rightarrow (AB=BA)\}~.
\end{equation}
\end{enumerate}
If one calls a \emph{commutant} of a set $\A$ the special set of
bounded operators on $\cL(\H)$ which commute with all elements in
$\A$, then this second condition implies that the commutant of the
commutant of $\A$ is again the set $\A$.
\med
On the other hand, a von Neumann algebra $\A$ inherits a
\emph{unital} subalgebra from $\cL(\H)$, and according to the
first condition in its definition $\A$ does indeed inherit a
\emph{*-subalgebra} structure, as further explained in the next
section on C*-algebras. Furthermore, we have notable
\emph{Bicommutant Theorem} which states that $\A$ \emph{is a von
Neumann algebra if and only if $\A$ is a *-subalgebra of
$\cL(\H)$, closed for the smallest topology defined by continuous
maps $(\xi,\eta)\longmapsto (A\xi,\eta)$ for all $<A\xi,\eta)>$
where $<.,.>$ denotes the inner product defined on $\H$}~. For a well-presented
treatment of the geometry of the state spaces of quantum operator algebras, see e.g. Aflsen and Schultz (2003).
\subsubsection{Hopf algebra}
First, a unital associative algebra consists of a linear space
$A$ together with two linear maps
\begin{equation}
\begin{aligned} m &: A \otimes A \lra A~,~(multiplication) \\
\eta &: \bC \lra A~,~ (unity)
\end{aligned}
\end{equation}
satisfying the conditions
\begin{equation}
\begin{aligned}
m(m \otimes \mathbf 1) &= m (\mathbf 1 \otimes m) \\ m(\mathbf 1
\otimes \eta) &= m (\eta \otimes \mathbf 1) = \ID~.
\end{aligned}
\end{equation}
This first condition can be seen in terms of a commuting diagram~:
\begin{equation}
\begin{CD}
A \otimes A \otimes A @> m \otimes \ID>> A \otimes A
\\ @V \ID \otimes mVV @VV m V
\\ A \otimes A @ > m >> A
\end{CD}
\end{equation}
Next suppose we consider `reversing the arrows', and take an
algebra $A$ equipped with a linear homorphisms $\Delta : A \lra A
\otimes A$, satisfying, for $a,b \in A$ :
\begin{equation}
\begin{aligned} \Delta(ab) &= \Delta(a) \Delta(b)
\\ (\Delta \otimes \ID) \Delta &= (\ID \otimes \Delta) \Delta~.
\end{aligned}
\end{equation}
We call $\Delta$ a \emph{comultiplication}, which is said to be
\emph{coasociative} in so far that the following diagram commutes
\begin{equation}
\begin{CD}
A \otimes A \otimes A @< \Delta\otimes \ID<< A \otimes A
\\ @A \ID \otimes \Delta AA @AA \Delta A
\\ A \otimes A @ < \Delta << A
\end{CD}
\end{equation}
There is also a counterpart to $\eta$, the \emph{counity} map
$\vep : A \lra \bC$ satisfying
\begin{equation}
(\ID \otimes \vep) \circ \Delta = (\vep \otimes \ID) \circ \Delta
= \ID~.
\end{equation}
A \emph{bialgebra} $(A, m, \Delta, \eta,
\vep)$ is a linear space $A$ with maps $m, \Delta, \eta, \vep$
satisfying the above properties.
\med
Now to recover anything resembling a group structure, we must
append such a bialgebra with an antihomomorphism $S : A \lra A$,
satisfying $S(ab) = S(b) S(a)$, for $a,b \in A$~. This map is
defined implicitly via the property~:
\begin{equation} m(S \otimes
\ID) \circ \Delta = m(\ID \otimes S) \circ \Delta = \eta \circ
\vep~~.
\end{equation}
We call $S$ the \emph{antipode map}. A \emph{Hopf algebra} is then
a bialgebra $(A,m, \eta, \Delta, \vep)$ equipped with an antipode
map $S$~.
\med
Commutative and noncommutative Hopf algebras form the backbone of
quantum `groups' and are essential to the generalizations of
symmetry. Indeed, in most respects a quantum `group' is identifiable
with a Hopf algebra. When such algebras are actually
associated with proper groups of matrices there is
considerable scope for their representations on both finite
and infinite dimensional Hilbert spaces.
\subsubsection{Groupoids}
Recall that a \emph{groupoid} $\grp$ is, loosely speaking, a small
category with inverses over its set of objects $X = Ob(\grp)$~. One
often writes $\grp^y_x$ for the set of morphisms in $\grp$ from
$x$ to $y$~. \emph{A topological groupoid} consists of a space
$\grp$, a distinguished subspace $\grp^{(0)} = \obg \subset \grp$,
called {\it the space of objects} of $\grp$, together with maps
\begin{equation}
r,s~:~ \xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)} }
\end{equation}
called the {\it range} and {\it source maps} respectively,
together with a law of composition
\begin{equation}
\circ~:~ \grp^{(2)}: = \grp \times_{\grp^{(0)}} \grp = \{
~(\gamma_1, \gamma_2) \in \grp \times \grp ~:~ s(\gamma_1) =
r(\gamma_2)~ \}~ \lra ~\grp~,
\end{equation}
such that the following hold~:~
\begin{enumerate}
\item[(1)]
$s(\gamma_1 \circ \gamma_2) = r(\gamma_2)~,~ r(\gamma_1 \circ
\gamma_2) = r(\gamma_1)$~, for all $(\gamma_1, \gamma_2) \in
\grp^{(2)}$~.
\med
\item[(2)]
$s(x) = r(x) = x$~, for all $x \in \grp^{(0)}$~.
\med
\item[(3)]
$\gamma \circ s(\gamma) = \gamma~,~ r(\gamma) \circ \gamma =
\gamma$~, for all $\gamma \in \grp$~.
\med
\item[(4)]
$(\gamma_1 \circ \gamma_2) \circ \gamma_3 = \gamma_1 \circ
(\gamma_2 \circ \gamma_3)$~.
\med
\item[(5)]
Each $\gamma$ has a two--sided inverse $\gamma^{-1}$ with $\gamma
\gamma^{-1} = r(\gamma)~,~ \gamma^{-1} \gamma = s (\gamma)$~.
Furthermore, only for topological groupoids the inverse map needs be continuous.
\med
It is usual to call $\grp^{(0)} = Ob(\grp)$ {\it the set of objects}
of $\grp$~. For $u \in Ob(\grp)$, the set of arrows $u \lra u$ forms a
group $\grp_u$, called the \emph{isotropy group of $\grp$ at $u$}.
\end{enumerate}
\med
Thus, as is well kown, a topological groupoid is just a groupoid internal to the category of topological spaces and continuous maps. The notion of internal groupoid has proved significant in a number of fields, since groupoids generalise bundles of groups, group actions, and equivalence relations. For a further study of groupoids we refer the reader to Brown (2006).
\med
$>>$
Several examples of groupoids are:
(a) locally compact groups, transformation groups , and any group in general (e.g. [59]
(b) equivalence relations
(c) tangent bundles
(d) the tangent groupoid (e.g. [4])
(e) holonomy groupoids for foliations (e.g. [4])
(f) Poisson groupoids (e.g. [81])
(g) graph groupoids (e.g. [47, 64]).
\med
As a simple, helpful example of a groupoid, consider (b) above. Thus, let \textit{R} be an \textit{equivalence relation} on a set X. Then \textit{R} is a groupoid under the following operations:
$(x, y)(y, z) = (x, z), (x, y)^{-1} = (y, x)$. Here, $\grp^0 = X $, (the diagonal of $X \times X$ ) and $r((x, y)) = x, s((x, y)) = y$.
\med
So $ R^2$ = $\left\{((x, y), (y, z)) : (x, y), (y, z) \in R \right\} $.
When $R = X \times X $, \textit{R} is called a \textit{trivial} groupoid. A special case of a trivial groupoid is
$R = R_n = \left\{ 1, 2, . . . , n \right\}$ $\times $ $\left\{ 1, 2, . . . , n \right\} $. (So every \textit{i} is equivalent to every \textit{j}). Identify $(i,j) \in R_n$ with the matrix unit $e_{ij}$. Then the groupoid $R_n$ is just matrix multiplication except that we only multiply $e_{ij}, e_{kl}$ when $k = j$, and $(e_{ij} )^{-1} = e_{ji}$. We do not really lose anything by restricting the multiplication, since the pairs $e_{ij}, {e_{kl}}$ excluded from groupoid multiplication just give the 0 product in normal algebra anyway.
For a groupoid $\grp_{lc}$ to be a locally compact groupoid means that $\grp_{lc}$ is required to be a (second countable) locally compact Hausdorff space, and the product and also inversion maps are required to be continuous. Each $\grp_{lc}^u$ as well as the unit space $\grp_{lc}^0$ is closed in $\grp_{lc}$. What replaces the left Haar measure on $\grp_{lc}$ is a system of measures $\lambda^u$ ($u \in \grp_{lc}^0$), where $\lambda^u$ is a positive regular Borel measure on $\grp_{lc}^u$ with dense support. In addition, the $\lambda^u$ â~@~Ys are required to vary continuously (when integrated against $f \in C_c(\grp_{lc}))$ and to form an invariant family in the sense that for each x, the map $y \mapsto xy$ is a measure preserving homeomorphism from $\grp_{lc}^s(x)$ onto $\grp_{lc}^r(x)$. Such a system
$\left\{ \lambda^u \right\}$ is called a \textit{left Haar system} for the locally compact groupoid $\grp_{lc}$.
\med
This is defined more precisely next.
\subsubsection{Haar systems for locally compact topological groupoids}
Let
\begin{equation}
\xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)}}=X
\end{equation}
be a locally compact, locally trivial topological groupoid with
its transposition into transitive (connected) components. Recall
that for $x \in X$, the \emph{costar of $x$} denoted
$\rm{CO}^*(x)$ is defined as the closed set $\bigcup\{ \grp(y,x) :
y \in \grp \}$, whereby
\begin{equation}
\grp(x_0, y_0) \hookrightarrow \rm{CO}^*(x) \lra X~,
\end{equation}
is a principal $\grp(x_0, y_0)$--bundle relative to
fixed base points $(x_0, y_0)$~. Assuming all relevant sets are
locally compact, then following Seda (1976), a \emph{(left) Haar
system on $\grp$} denoted $(\grp, \tau)$ (for later purposes), is
defined to comprise of i) a measure $\kappa$ on $\grp$, ii) a
measure $\mu$ on $X$ and iii) a measure $\mu_x$ on $\rm{CO}^*(x)$
such that for every Baire set $E$ of $\grp$, the following hold on
setting $E_x = E \cap \rm{CO}^*(x)$~:
\begin{itemize}
\item[(1)] $x \mapsto \mu_x(E_x)$ is measurable.
\med
\item[(2)]
$\kappa(E) = \int_x \mu_x(E_x)~d\mu_x$ ~.
\med
\item[(3)]
$\mu_z(t E_x) = \mu_x(E_x)$, for all $t \in \grp(x,z)$ and $x, z
\in \grp$~.
\end{itemize}
\med
The presence of a left Haar system on $\grp_{lc}$ has important
topological implications: it requires that the range map $r :
\grp_{lc} \rightarrow \grp_{lc}^0$ is open. For such a $\grp_{lc}$
with a left Haar system, the vector space $C_c(\grp_{lc})$ is a
\textit{convolution} \textit{*--algebra}, where for $f, g \in
C_c(\grp_{lc})$: \\
\med
$f * g(x) = \int f(t)g(t^{-1} x) d \lambda^{r(x)} (t)$, with
f*(x) $ = \overline{f(x^{-1})}$.
\med
One has $C^*(\grp_{lc})$ to be the \textit{enveloping C*--algebra}
of $C_c(\grp_{lc})$ (and also representations are required to be
continuous in the inductive limit topology). Equivalently, it is
the completion of $\pi_{univ}(C_c(\grp_{lc}))$ where $\pi_{univ}$
is the \textit{universal representation} of $\grp_{lc}$. For
example, if $ \grp_{lc} = R_n$ , then $C^*(\grp_{lc})$ is just the
finite dimensional algebra $C_c(\grp_{lc}) = M_n$, the span of the
$e_{ij}$'s.
There exists (e.g.[63, p.91]) a \textit{measurable Hilbert bundle}
$(\grp_{lc}^0, \H, \mu)$ with $\H = \left\{ \H^u_{u \in
\grp_{lc}^0} \right\}$ and a G-representation L on $\H$. Then,
for every pair $\xi, \eta$ of square integrable sections of $\H$,
it is required that the function $x \mapsto (L(x)\xi (s(x)), \eta
(r(x)))$ be $\nu$--measurable. The representation $\Phi$ of
$C_c(\grp_{lc})$ is then given by:\\ $\left\langle \Phi(f) \xi
\vert,\eta \right\rangle = \int f(x)(L(x) \xi (s(x)), \eta (r(x)))
d \nu_0(x)$.
The triple $(\mu, \H, L)$ is called a \textit{measurable
$\grp_{lc}$--Hilbert bundle}.
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