PlanetPhysics/Quantum Operator Algebras in QFT2

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Introduction

This is a topic entry that introduces [[../Groupoid/|quantum operator algebras]] and presents concisely the important roles they play in quantum field theories.

Quantum operator algebras  (QOA) in quantum field theories are defined as the algebras of [[../QuantumSpinNetworkFunctor2/|observable]] [[../QuantumOperatorAlgebra4/|operators]], and as such, they are also related to the von Neumann algebra;

quantum operators are usually defined on [[../NormInducedByInnerProduct/|Hilbert spaces]], or in some QFTs on [[../HilbertBundle/|Hilbert space bundles]] or other similar families of spaces.

\htmladdnormallink{representations {http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of Banach $*$-algebras}-- that are defined on Hilbert spaces-- are closely related to C* -algebra representations which provide a useful approach to defining [[../SUSY2/|quantum space-times]].

Quantum operator algebras in quantum field theories: QOA Role in QFTs

Important examples of quantum operators are: the [[../HamiltonianOperator3/|Hamiltonian operator]] (or [[../Hamiltonian2/|Schr\"odinger operator]]), the [[../Position/|position]] and [[../Momentum/|momentum]] operators, Casimir operators, unitary operators and [[../QuarkAntiquarkPair/|spin]] operators. The observable operators are also 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 `[[../I3/|rigged' Hilbert spaces]] (Hilbert space bundles ). The following [[../IsomorphicObjectsUnderAnIsomorphism/|sections]] define several [[../Bijective/|types]] of quantum operator algebras that provide the foundation of modern quantum field theories in [[../PhysicalMathematics2/|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 [[../QuantumParadox/|quantum mechanics]] ([[../FTNIR/|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 [[../Formula/|theorem]] and its applications, but also revealed the fundamental importance in quantum physics of the [[../StableAutomaton/|state space]] geometry of quantum operator algebras.

Basic mathematical definitions in QOA:

• Von Neumann algebra
• Hopf algebra
• Groupoids
• Haar systems associated to measured groupoids or \htmladdnormallink{locally compact groupoids {http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html}.}
• [[../VonNeumannAlgebra2/|C*-algebras]] and [[../WeakHopfAlgebra/|quantum groupoids]] entry (attached).

Von Neumann algebra

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

• (i) Failed to parse (unknown function "\A"): {\displaystyle \A} is closed under the adjoint [[../Cod/|operation]] (with the adjoint of an element $T$ denoted by $T^{*}$).
• (ii) Failed to parse (unknown function "\A"): {\displaystyle \A} equals its [[../CoordinateSpace/|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 [[../CoordinateSpace/|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 [[../Commutator/|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} .

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} , it does indeed inherit a $*$-subalgebra structure as further explained in the next section on C* -algebras. Furthermore, one also has available a notable bicommutant theorem which states that: {\em Failed to parse (unknown function "\A"): {\displaystyle \A} 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 $(\xi ,\eta )⟼(A\xi ,\eta )$ 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, the reader is referred to 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 [[../TrivialGroupoid/|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 $\mathrm{\Delta }$ 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)} is a linear space $A$ with maps Failed to parse (unknown function "\vep"): {\displaystyle m, \Delta, \eta, \vep} satisfying the above properties.

Now to recover anything resembling a [[../TrivialGroupoid/|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 $S$.

Commutative and [[../AbelianCategory3/|non-commutative]] Hopf algebras form the backbone of [[../QuantumGroup4/|quantum `groups]]' and are essential to the generalizations of symmetry. Indeed, in most respects a quantum `group' is closely related to its dual Hopf algebra; in the case of a finite, commutative quantum group its dual Hopf algebra is obtained via Fourier transformation of the group elements. When Hopf algebras are actually associated with their dual, proper groups of [[../Matrix/|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 [[../Cod/|small category]] with inverses over its set of [[../TrivialGroupoid/|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 [[../TrivialGroupoid/|morphisms]] in Failed to parse (unknown function "\grp"): {\displaystyle \grp} from $x$ to $y$~. A [[../GroupoidHomomorphism2/|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 [[../SmallCategory/|source maps]]} respectively,

together with a law of [[../Cod/|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>~.

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

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

\item[(4)] $({\gamma }_1\circ {\gamma }_2)\circ {\gamma }_3={\gamma }_1\circ ({\gamma }_2\circ {\gamma }_3)$~.

\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. 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 $u$ .

Thus, as it is well kown, a topological groupoid is just a groupoid internal to the [[../Cod/|category]] of [[../CoIntersections/|topological]] spaces and continuous maps. The notion of internal groupoid has proved significant in a number of [[../CosmologicalConstant/|fields]], since groupoids generalise bundles of groups, group actions, and [[../TrivialGroupoid/|equivalence relations]]. For a further study of groupoids we refer the reader to Brown (2006).

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 [[../MoyalDeformation/|tangent groupoid]]
• (e) holonomy groupoids for foliations
• (f) Poisson groupoids
• (g) [[../Cod/|graph]] groupoids.

As a simple, helpful example of a groupoid, consider (b) above. Thus, let R be an equivalence [[../Bijective/|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 $r((x,y))=x,s((x,y))=y$.

Therefore, $R^2$ = $\{((x,y),(y,z)):(x,y),(y,z)\in R\}$. When $R=X\times X$, R is called a trivial groupoid. A special case of a [[../TrivialGroupoid/|trivial groupoid]] is $R=R_n=\{1,2,...,n\}$ $\times$ $\{1,2,...,n\}$. (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/|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) [[../LocallyCompactHausdorffSpaces/|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 [[../HigherDimensionalQuantumAlgebroid/|Haar measure]] on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} is a [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] of measures ${\lambda }^u$ (Failed to parse (unknown function "\grp"): {\displaystyle u \in \grp_{lc}^0} ), where ${\lambda }^u$ is a positive [[../CoIntersections/|regular]] Borel measure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u} with dense support. In addition, the ${\lambda }^u$ 's 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 [[../TrivialGroupoid/|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}} .

This is defined more precisely in the next subsection.

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 ${\mathrm{C}\mathrm{O}}^{\mathrm{*}}\mathrm{(}\mathrm{x}\mathrm{)}$ 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 ${\mathrm{C}\mathrm{O}}^{\mathrm{*}}\mathrm{(}\mathrm{x}\mathrm{)}$ such that for every Baire set $E$ of Failed to parse (unknown function "\grp"): {\displaystyle \grp} , the following hold on setting $E_x=E\cap {\mathrm{C}\mathrm{O}}^{\mathrm{*}}\mathrm{(}\mathrm{x}\mathrm{)}$~:

 \item[(1)] $x\mapsto {\mu }_x(E_x)$ is measurable. \item[(2)] $\kappa (E)=\underset{x}{\int }{\mu }_x(E_x)d{\mu }_x$ ~. \item[(3)] ${\mu }_z(t{E}_x)={\mu }_x(E_x)$, 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}
~.

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 [[../NormInducedByInnerProduct/|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})} :

$f*g(x)=\int f(t)g(t^{-1}x)d{\lambda }^{r(x)}(t),$

with $f*(x)=\overline{f(x^{-1})}.$

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 a measurable [[../HilbertBundle/|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 [[../PiecewiseLinear/|square]] integrable sections of $ℍ$, it is required that the [[../Bijective/|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:\\ $⟨\mathrm{\Phi }(f)\xi |,\eta ⟩=\int f(x)(L(x)\xi (s(x)),\eta (r(x)))d{\nu }_0(x)$.

The triple $(\mu ,ℍ,L)$ is called a \textit{measurable Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} --Hilbert bundle}.