PlanetPhysics/Quantum Symmetries From Group and Groupoid Representations

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Group and Groupoid representationsEdit

Whereas group representations of quantum unitary operators are extensively employed in standard quantum mechanics, the applications of groupoid representations

are still under development. For example, a description of stochastic quantum mechanics in curved spacetime (Drechsler and Tuckey, 1996) involving a Hilbert bundle is possible in terms of groupoid representations which can indeed be defined on such a Hilbert bundle , but cannot be expressed as the simpler group representations on a Hilbert space . On the other hand, as in the case of group representations, unitary groupoid representations induce associated C*-algebra representations. In the next subsection we recall some of the basic results concerning groupoid representations and their associated groupoid *-algebra representations. For further details and recent results in the mathematical theory of groupoid representations one has also available (\htmladdnormallink{the succint monograph by Buneci (2003) and references cited therein}{}).

Let us consider first the relationships between these mainly algebraic concepts and their extended quantum symmetries, also including relevant computation examples; then let us consider several further extensions of symmetry and algebraic topology in the context of local quantum physics/ quantum field theory, symmetry breaking, quantum chromodynamics and the development of novel supersymmetry theories of quantum gravity. In this respect one can also take spacetime `inhomogeneity' as a criterion for the comparisons between physical, partial or local, symmetries: on the one hand, the example of paracrystals reveals Thermodynamic disorder (entropy) within its own spacetime framework, whereas in spacetime itself, whatever the selected model, the inhomogeneity arises through (super) gravitational effects. More specifically, in the former case one has the technique of the generalized Fourier--Stieltjes transform (along with convolution and Haar measure), and in view of the latter, we may compare the resulting `broken'/paracrystal--type symmetry with that of the supersymmetry predictions for weak gravitational fields (e.g., `ghost' particles) along with the broken supersymmetry in the presence of intense gravitational fields. Another significant extension of quantum symmetries may result from the superoperator algebra/algebroids of Prigogine's quantum superoperators which are defined only for irreversible, infinite-dimensional systems (Prigogine, 1980).

Extended Quantum Groupoid and Algebroid SymmetriesEdit

Quantum groups~ Representations ~ weak Hopf algebras ~ ~quantum groupoids and algebroids Our intention here is to view the latter scheme in terms of a weak Hopf C*--algebroid-- and/or other-- extended symmetries, which we propose to do, for example, by incorporating the concepts of rigged Hilbert spaces and \emph{sectional functions for a small category}. We note, however, that an alternative approach to quantum groupoids has already been reported (Maltsiniotis, 1992), (perhaps also related to noncommutative geometry); this was later expressed in terms of deformation-quantization: the Hopf algebroid deformation of the universal enveloping algebras of Lie algebroids (Xu, 1997) as the classical limit of a quantum `groupoid'; this also parallels the introduction of quantum `groups' as the deformation-quantization of Lie bialgebras. Furthermore, such a Hopf algebroid approach (Lu, 1996) leads to categories of Hopf algebroid modules (Xu, 1997) which are monoidal, whereas the links between Hopf algebroids and monoidal bicategories were investigated by Day and Street (1997).

As defined under the following heading on groupoids, let Failed to parse (unknown function "\grp"): {\displaystyle (\grp_{lc},\tau)} be a locally compact groupoid endowed with a (left) Haar system, and let Failed to parse (unknown function "\grp"): {\displaystyle A= C^*(\grp_{lc},\tau)} be the convolution --algebra (we append with if necessary, so that is unital). Then consider such a \textit{groupoid representation} \\ Failed to parse (unknown function "\grp"): {\displaystyle \Lambda :(\grp_{lc}, \tau) \lra \{\mathcal H_x, \sigma_x \}_{x \in X}} that respects a compatible measure on (cf Buneci, 2003). On taking a state on , we assume a parametrization

Furthermore, each is considered as a \emph{rigged Hilbert space} Bohm and Gadella (1989), that is, one also has the following nested inclusions:

in the usual manner, where is a dense subspace of with the appropriate locally convex topology, and is the space of continuous antilinear functionals of ~. For each , we require to be invariant under and Failed to parse (unknown function "\IM"): {\displaystyle \IM~ \Lambda \vert \Phi_x} is a continuous representation of Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} on ~. With these conditions, representations of (proper) quantum groupoids that are derived for weak C*--Hopf algebras (or algebroids) modeled on rigged Hilbert spaces could be suitable generalizations in the framework of a Hamiltonian generated semigroup of time evolution of a quantum system via integration of Schr\"odinger's equation 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 \iota \hslash \frac{\del \psi}{\del t} = H \psi} as studied in the case of Lie groups (Wickramasekara and Bohm, 2006). The adoption of the rigged Hilbert spaces is also based on how the latter are recognized as reconciling the Dirac and von Neumann approaches to quantum theories (Bohm and Gadella, 1989).

Next, let Failed to parse (unknown function "\grp"): {\displaystyle \grp} be a locally compact Hausdorff groupoid and a locally compact Hausdorff space. (Failed to parse (unknown function "\grp"): {\displaystyle \grp} will be called a \emph{locally compact groupoid, or lc- groupoid} for short). In order to achieve a small C*--category we follow a suggestion of A. Seda (private communication) by using a general principle in the context of Banach bundles (Seda, 1976, 982)). Let Failed to parse (unknown function "\grp"): {\displaystyle q= (q_1, q_2) : \grp \lra X \times X} be a continuous, open and surjective map. For each , consider the fibre Failed to parse (unknown function "\grp"): {\displaystyle \grp_z = \grp (x,y) = q^{-1}(z)} , and set Failed to parse (unknown function "\A"): {\displaystyle \A_z = C_0(\grp_z) = C_0(\grp(x,y))} equipped with a uniform norm ~. Then we set Failed to parse (unknown function "\A"): {\displaystyle \A = \bigcup_z \A_z<math>~. We form a Banach bundle } p : \A \lra X \times X</math> as follows. Firstly, the projection is defined via the typical fibre Failed to parse (unknown function "\A"): {\displaystyle p^{-1}(z) = \A_z = \A_{(x,y)}} ~. Let Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp)} denote the continuous complex valued functions on Failed to parse (unknown function "\grp"): {\displaystyle \grp} with compact support. We obtain a sectional function Failed to parse (unknown function "\wti"): {\displaystyle \wti{\psi} : X \times X \lra \A<math> defined via restriction as } \wti{\psi}(z) = \psi \vert \grp_z = \psi \vert \grp (x,y)\gamma = \{ \wti{\psi} : \psi \in C_c(\grp) \}\{ \wti{\psi}(z) : \wti{\psi} \in \gamma \}\A_z</math>~. For each Failed to parse (unknown function "\wti"): {\displaystyle \wti{\psi} \in \gamma} , the function Failed to parse (unknown function "\wti"): {\displaystyle \Vert \wti{\psi} (z) \Vert_z<math> is continuous on } X\wti{\psi}</math> is a continuous section of Failed to parse (unknown function "\A"): {\displaystyle p : \A \lra X \times X} ~. These facts follow from Seda (1982, theorem 1). Furthermore, under the convolution product , \textit{the space Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp)} forms an associative algebra over Failed to parse (unknown function "\bC"): {\displaystyle \bC} } (cf. Seda, 1982, Theorem 3).


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 (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} from to ~. 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)] (\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 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.g. [4])
  • (e) holonomy groupoids for foliations (e.g. [4])
  • (f) Poisson groupoids (e.g. [81])
  • (g) graph groupoids (e.g. [47, 64]).

As a simple 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 .

So = . 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 (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 (Failed to parse (unknown function "\grp"): {\displaystyle u \in \grp_{lc}^0} ), where is a positive regular Borel measure on Failed to parse (unknown function "\grp"): {\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 (unknown function "\grp"): {\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 next.

Haar systems for locally compact topological groupoidsEdit


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 (unknown function "\grp"): {\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} is open. For such a Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} 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})} :


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 (cf. [1]) 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}.

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