PlanetPhysics/Quantum Operator Algebras in QFT2

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IntroductionEdit

This is a topic entry that introduces 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 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.

\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 quantum space-times.

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

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 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 space 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.Edit

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.

Basic mathematical definitions in QOA:Edit

Von Neumann algebraEdit

Let denote a complex (separable) Hilbert space. A von Neumann algebra Failed to parse (unknown function "\A"): {\displaystyle \A} acting on 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 operation (with the adjoint of an element denoted by ).
  • (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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\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} .

On the other hand, a von Neumann algebra Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\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 for all where denotes the inner product defined on }~.

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; [1]).

Hopf algebraEdit

First, a unital associative algebra consists of a linear space 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 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 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 , 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 with maps Failed to parse (unknown function "\vep"): {\displaystyle m, \Delta, \eta, \vep} satisfying the above properties.

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 , for ~. 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 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 .

Commutative and non-commutative Hopf algebras form the backbone of 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 matrices, there is considerable scope for their representations on both finite and infinite dimensional Hilbert spaces.

GroupoidsEdit

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 to ~. A topological groupoid consists of a space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\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)] (\gamma_1, \gamma_2) \in \grp^{(2)}</math>~.

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

\item[(3)] \gamma \in \grp</math>~.

\item[(4)] ~.

\item[(5)] Each has a two--sided inverse with ~. 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 .

Thus, as it 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).

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) holonomy groupoids for foliations
  • (f) Poisson groupoids
  • (g) graph groupoids.

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: . Here, Failed to parse (unknown function "\grp"): {\displaystyle \grp^0 = X } , (the diagonal of ) and .

Therefore, = . When , R is called a trivial groupoid. A special case of a trivial groupoid is . (So every i is equivalent to every j ). Identify with the matrix unit . Then the groupoid is just matrix multiplication except that we only multiply when , and . We do not really lose anything by restricting the multiplication, since the pairs 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \grp_{lc}} . What replaces the left Haar measure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} is a system of measures (Failed to parse (unknown function "\grp"): {\displaystyle u \in \grp_{lc}^0} ), where is a positive regular Borel measure on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \grp_{lc}^u} with dense support. In addition, the '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 is a measure preserving homeomorphism from Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^s(x)} onto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \grp_{lc}^r(x)} . Such a system 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 groupoidsEdit

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 , the costar of denoted 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 ~. 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 on Failed to parse (unknown function "\grp"): {\displaystyle \grp} , ii) a measure on and iii) a measure on such that for every Baire set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \grp} , the following hold on setting ~:

 \item[(1)]  is measurable. \item[(2)]  ~. \item[(3)] , 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 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})} :

with

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 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 's.

There exists 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 of square integrable sections of , it is required that the function \nu\Phi</math> of Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc})} is then given by:\\ .

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

All SourcesEdit

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