PlanetPhysics/Quantum Operator Algebras

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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.

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.

Quantum Operator Algebras in Quantum Field Theories: \htmladdnormallink{QOAs {http://planetphysics.us/encyclopedia/QAT.html} in QFTs} 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 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 bundles ). The following sections define several types of quantum operator algebras that provide the foundation of modern quantum field theories in mathematical physics.

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

Definitions:

1. Von Neumann Algebra

1. Hopf Algebra

1. Groupoids

1. Haar \htmladdnormallink{systems {http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} associated to Measured Groupoids or Locally Compact Groupoids.}

.

Von Neumann Algebra

Let $\mathbb {H}$ denote a complex (separable) Hilbert space. A \emph{von Neumann algebra} Failed to parse (unknown function "\A"): {\displaystyle \A} acting on $\mathbb {H}$ is a subset of the algebra of all bounded operators Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} such that:

1. (i) Failed to parse (unknown function "\A"): {\displaystyle \A} is closed under the adjoint operation (with the

adjoint of an element $T$ denoted by $T^{*}$).

1. (ii) Failed to parse (unknown function "\A"): {\displaystyle \A} equals its bicommutant, namely:

Failed to parse (unknown function "\A"): {\displaystyle \A= \{A \in \cL(\mathbb{H}) : \forall B \in \cL(\mathbb{H}), \forall C\in \A,~ (BC=CB)\Rightarrow (AB=BA)\}~. }

If one calls a commutant of a set Failed to parse (unknown function "\A"): {\displaystyle \A} the special set of bounded operators on Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} which commute with all elements in Failed to parse (unknown function "\A"): {\displaystyle \A} , then this second condition implies that the commutant of the commutant of Failed to parse (unknown function "\A"): {\displaystyle \A} is again the set Failed to parse (unknown function "\A"): {\displaystyle \A} .

\med On the other hand, a von Neumann algebra Failed to parse (unknown function "\A"): {\displaystyle \A} inherits a unital subalgebra from Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} , and according to the first condition in its definition Failed to parse (unknown function "\A"): {\displaystyle \A} does indeed inherit a *-subalgebra structure, as further explained in the next section on C*-algebras. Furthermore, we have notable Bicommutant Theorem which states that Failed to parse (unknown function "\A"): {\displaystyle \A} \emph{is a von Neumann algebra if and only if Failed to parse (unknown function "\A"): {\displaystyle \A} is a *-subalgebra of Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{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 $\mathbb {H}$}~. For a well-presented treatment of the geometry of the state spaces of quantum operator algebras, see e.g. Aflsen and Schultz (2003).

Hopf algebra

First, a unital associative algebra consists of a linear space $A$ together with two linear maps

Failed to parse (syntax error): {\displaystyle m &: A \otimes A \lra A~,~(multiplication) \\ \eta &: \bC \lra A~,~ (unity) }

satisfying the conditions

Failed to parse (syntax error): {\displaystyle m(m \otimes \mathbf 1) &= m (\mathbf 1 \otimes m) \\ m(\mathbf 1 \otimes \eta) &= m (\eta \otimes \mathbf 1) = \ID~. }

This first condition can be seen in terms of a commuting diagram~:

Failed to parse (unknown function "\begin{CD}"): {\displaystyle \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} }

Next suppose we consider `reversing the arrows', and take an algebra $A$ equipped with a linear homorphisms Failed to parse (unknown function "\lra"): {\displaystyle \Delta : A \lra A \otimes A<math>, satisfying, for } a,b \in A</math> :

Failed to parse (syntax error): {\displaystyle \Delta(ab) &= \Delta(a) \Delta(b) \\ (\Delta \otimes \ID) \Delta &= (\ID \otimes \Delta) \Delta~. }

We call $\Delta$ a comultiplication , which is said to be coasociative in so far that the following diagram commutes

Failed to parse (unknown function "\begin{CD}"): {\displaystyle \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} }

There is also a counterpart to $\eta$, the counity map Failed to parse (unknown function "\vep"): {\displaystyle \vep : A \lra \bC} satisfying

Failed to parse (unknown function "\ID"): {\displaystyle (\ID \otimes \vep) \circ \Delta = (\vep \otimes \ID) \circ \Delta = \ID~. }

A bialgebra Failed to parse (unknown function "\vep"): {\displaystyle (A, m, \Delta, \eta, \vep)<math> is a linear space } A$withmaps$m, \Delta, \eta, \vep</math> satisfying the above properties.

\med Now to recover anything resembling a group structure, we must append such a bialgebra with an antihomomorphism Failed to parse (unknown function "\lra"): {\displaystyle S : A \lra A} , satisfying $S(ab)=S(b)S(a)$, for $a,b\in A$~. This map is defined implicitly via the property~:

Failed to parse (unknown function "\ID"): {\displaystyle m(S \otimes \ID) \circ \Delta = m(\ID \otimes S) \circ \Delta = \eta \circ \vep~~. }

We call $S$ the antipode map . A Hopf algebra is then a bialgebra Failed to parse (unknown function "\vep"): {\displaystyle (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.

Groupoids

Recall that a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} is, loosely speaking, a small category with inverses over its set of objects Failed to parse (unknown function "\grp"): {\displaystyle X = Ob(\grp)} ~. One often writes Failed to parse (unknown function "\grp"): {\displaystyle \grp^y_x} for the set of morphisms in Failed to parse (unknown function "\grp"): {\displaystyle \grp} from $x$ to $y$~. A topological groupoid consists of a space Failed to parse (unknown function "\grp"): {\displaystyle \grp} , a distinguished subspace Failed to parse (unknown function "\grp"): {\displaystyle \grp^{(0)} = \obg \subset \grp} , called {\it the space of objects} of Failed to parse (unknown function "\grp"): {\displaystyle \grp} , together with maps

Failed to parse (unknown function "\xymatrix"): {\displaystyle r,s~:~ \xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)} } }

called the {\it range} and {\it source maps} respectively,

together with a law of composition

Failed to parse (unknown function "\grp"): {\displaystyle \circ~:~ \grp^{(2)}: = \grp \times_{\grp^{(0)}} \grp = \{ ~(\gamma_1, \gamma_2) \in \grp \times \grp ~:~ s(\gamma_1) = r(\gamma_2)~ \}~ \lra ~\grp~, }

such that the following hold~:~

\item[(1)] $s(\gamma _{1}\circ \gamma _{2})=r(\gamma _{2})~,~r(\gamma _{1}\circ \gamma _{2})=r(\gamma _{1})<math>~,forall$(\gamma_1, \gamma_2) \in \grp^{(2)}</math>~.

\med \item[(2)] $s(x)=r(x)=x$~, for all Failed to parse (unknown function "\grp"): {\displaystyle x \in \grp^{(0)}} ~.

\med \item[(3)] $\gamma \circ s(\gamma )=\gamma ~,~r(\gamma )\circ \gamma =\gamma <math>~,forall$\gamma \in \grp</math>~.

\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 Failed to parse (unknown function "\grp"): {\displaystyle \grp^{(0)} = Ob(\grp)} {\it the set of objects} of Failed to parse (unknown function "\grp"): {\displaystyle \grp} ~. For Failed to parse (unknown function "\grp"): {\displaystyle u \in Ob(\grp)} , the set of arrows Failed to parse (unknown function "\lra"): {\displaystyle u \lra u} forms a group Failed to parse (unknown function "\grp"): {\displaystyle \grp_u} , called the isotropy group of Failed to parse (unknown function "\grp"): {\displaystyle \grp} at $u$ .

\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 R be an equivalence relation on a set X. Then R is a groupoid under the following operations: $(x,y)(y,z)=(x,z),(x,y)^{-1}=(y,x)$. Here, Failed to parse (unknown function "\grp"): {\displaystyle \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 $R=X\times X$, R is called a 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 i is equivalent to every 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 Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} to be a locally compact groupoid means that Failed to parse (unknown function "\grp"): {\displaystyle \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 Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u} as well as the unit space Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^0} is closed in Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} . What replaces the left Haar measure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} is a system of measures $\lambda ^{u}$ (Failed to parse (unknown function "\grp"): {\displaystyle u \in \grp_{lc}^0} ), where $\lambda ^{u}$ is a positive regular Borel measure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u} with dense support. In addition, the $\lambda ^{u}$ \^a~@~Ys are required to vary continuously (when integrated against Failed to parse (unknown function "\grp"): {\displaystyle 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 Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^s(x)} onto Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^r(x)} . Such a system $\left\{\lambda ^{u}\right\}$ is called a left Haar system for the locally compact groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} . \med

This is defined more precisely next.

Haar systems for locally compact topological groupoids

Let

Failed to parse (unknown function "\xymatrix"): {\displaystyle \xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)}}=X }

be a locally compact, locally trivial topological groupoid with its transposition into transitive (connected) components. Recall that for $x\in X$, the costar of $x$ denoted ${\rm {{CO}^{*}(x)}}$ is defined as the closed set Failed to parse (unknown function "\grp"): {\displaystyle \bigcup\{ \grp(y,x) : y \in \grp \}} , whereby

Failed to parse (unknown function "\grp"): {\displaystyle \grp(x_0, y_0) \hookrightarrow \rm{CO}^*(x) \lra X~, }

is a principal Failed to parse (unknown function "\grp"): {\displaystyle \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 Failed to parse (unknown function "\grp"): {\displaystyle \grp} } denoted Failed to parse (unknown function "\grp"): {\displaystyle (\grp, \tau)} (for later purposes), is defined to comprise of i) a measure $\kappa$ on Failed to parse (unknown function "\grp"): {\displaystyle \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 Failed to parse (unknown function "\grp"): {\displaystyle \grp} , the following hold on setting $E_{x}=E\cap {\rm {{CO}^{*}(x)}}$~:

 \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)] $\mu _{z}(tE_{x})=\mu _{x}(E_{x})$, for all Failed to parse (unknown function "\grp"): {\displaystyle t \in \grp(x,z)}
 and Failed to parse (unknown function "\grp"): {\displaystyle x, z \in \grp}
~.

\med

The presence of a left Haar system on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} has important topological implications: it requires that the range map Failed to parse (unknown function "\grp"): {\displaystyle r : \grp_{lc} \rightarrow \grp_{lc}^0<math> is open. For such a } \grp_{lc}</math> with a left Haar system, the vector space Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc})} is a convolution *--algebra , where for Failed to parse (unknown function "\grp"): {\displaystyle 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 Failed to parse (unknown function "\grp"): {\displaystyle C^*(\grp_{lc})} to be the enveloping C*--algebra of Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc})} (and also representations are required to be continuous in the inductive limit topology). Equivalently, it is the completion of Failed to parse (unknown function "\grp"): {\displaystyle \pi_{univ}(C_c(\grp_{lc}))} where $\pi _{univ}$ is the universal representation of Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} . For example, if Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc} = R_n} , then Failed to parse (unknown function "\grp"): {\displaystyle C^*(\grp_{lc})} is just the finite dimensional algebra Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc}) = M_n} , the span of the $e_{ij}$'s.

There exists (e.g.[63, p.91]) a measurable Hilbert bundle Failed to parse (unknown function "\grp"): {\displaystyle (\grp_{lc}^0, \mathbb{H}, \mu)} with Failed to parse (unknown function "\grp"): {\displaystyle \mathbb{H} = \left\{ \mathbb{H}^u_{u \in \grp_{lc}^0} \right\}<math> and a G-representation L on } \H</math>. Then, for every pair $\xi ,\eta$ of square integrable sections of $\mathbb {H}$, it is required that the function $x\mapsto (L(x)\xi (s(x)),\eta (r(x)))<math>be$\nu$--measurable.Therepresentation$\Phi</math> of Failed to parse (unknown function "\grp"): {\displaystyle 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 $(\mu ,\mathbb {H} ,L)$ is called a \textit{measurable Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} --Hilbert bundle}.