Representations of the Weyl Algebra in Quantum Geometry

The Weyl algebra A is irreducible.

“Additionally, the paper includes also a short and direct proof of the irreducibility of A.”

• Also stated in: § 1.6. Structure of the article. · plain

  “After presenting a pretty short and direct proof for the irreducibility of the Weyl algebra in Sect.5”

For every W ⊆ W_H(μ), the subalgebra spanned by all products f • w with f ∈ C(X) and w ∈ W is dense in A(W, μ).

“For every W ⊆ W H(µ) , the subalgebra spanned by all products f • w with f ∈ C(X ) and w ∈ W is dense in A(W, µ).”

For disjoint quasi-surfaces S1,S2 and functions d1,d2, w_{S1,σS1}^{d1} • w_{S2,σS2}^{d2} = w_{S2,σS2}^{d2} • w_{S1,σS1}^{d1}.

“w S 1 ,σ S 1 d 1 • w S 2 ,σ S 2 d 2 = w S 2 ,σ S 2 d 2 • w S 1 ,σ S 1 d 1 .”

For any complete Q and germ ρ:Q→G, the induced map [ρ]:P→G is a homomorphism, i.e., an element of A.

“• The induced map [ ρ] : P -→ G is a homomorphism, i.e., it is an element of A.”

In the (semi)analytic category for paths, embedded analytic submanifolds that are even semianalytic are quasi-surfaces.

“Examples for quasi-surfaces, in case we are in the (semi)analytic category for the paths, are embedded analytic submanifolds that are even semianalytic.”

If π is -regular for some set of W-functions and for each λ in that set there is a subset Yλ of dom λ such that λ(Yλ) consists of πν-units only and the union of the λ(Yλ) generates W, then every w in W is a πν-unit.

“Proposition 2.22.Let π be -regular for some set of W-functions.Fix for each λ ∈ some subset Y λ in dom λ, such that λ(Y λ ) consists of π ν -units only and λ∈ λ(Y λ ) generates W. Then every w ∈ W is a π ν -unit.”

The Ashtekar-Lewandowski measure μ0 is φ-invariant for all graphomorphisms φ.

“The Ashtekar-Lewandowski measure µ 0 is ϕ-invariant for all graphomorphisms ϕ.”

w_d(f1 f2) = w_d(f1) w_d(f2) for all functions f1,f2 on A.

“w d ( f 1 f 2 ) = w d ( f 1 )w d ( f 2 ) for all functions f 1 , f 2 on A.”

Every Weyl operator is an isometry on C(A).

“Every Weyl operator is an isometry on C(A).”

For any complete Q and germ ρ:Q→G, the extension ρ is given by ρ(γ)=∏_{i=1}^I ρ(γ_i) for each γ in P_gen, where γ1...γn is any Q-decomposition of γ.

“• The map ρ is given by ρ(γ ) = I i=1 ρ(γ i ) for each γ ∈ P gen , where γ 1 • • • γ n is anyfoot_5 Q-decomposition of γ .”

If w is a πν-unit and μν equals μ, then Pν π(w) = π0(w) Pν.

“If w is a π ν -unit and µ ν equals µ, then P ν π(w) = π 0 (w)P ν .”

The inverse of S,σS_d is S,σS_{d^{-1}}.

“Finally, the inverse of S,σ S d is given by S,σ S d -1 .”

For every graphomorphism φ, αφ is a unitary operator on L2(A, μ0).

“α ϕ is a unitary operator on L 2 (A, µ 0 ).”

For every hyph υ, the set Mυ of gauge-variant spin networks on υ is an orthonormal set in L2(A, μ0).

“Lemma 3.1.For every hyph υ, the set M υ of gauge-variant spin networks on υ is an orthonormal set in L 2 (A, µ 0 ).”

A bijection φ on M is a graphomorphism iff φ and φ^{-1} map edges to P_gen.

“A bijection ϕ on M is a graphomorphism iff ϕ and ϕ -1 map edges to P gen .”

• cites [13] Fleischhack, Ch.: Construction of generalized connections. http://arxiv.org/list/math-ph/0601005 (2006)Not checked yet

  Not checked in the free run.

For all υ ≥ υ, Mυ ⊆ span Mυ.

“• We have M υ ⊆ span M υ for all υ ≥ υ .”

For a hyph υ={γ1,...,γn}, w_{S,σS}^d(T1⊗...⊗Tn) = w_{S,σS}^d(T1)⊗...⊗w_{S,σS}^d(Tn) for all Ti∈M_{γi} and all d.

“w S,σ S d (T 1 ⊗ • • • ⊗ T n ) = w S,σ S d (T 1 ) ⊗ • • • ⊗ w S,σ S d (T n )”

For all ψ ∈ H, w ∈ W, and f ∈ C(X), sup_{w'∈W_H(μ)} |⟨ψ, π((w' - 1)(w'(f)))ψ⟩| ≤ 2 ||f||∞ ||ψ|| ||(π(w) - 1)ψ||.

“We have sup w ∈W H(µ) | ψ, π((w -1)(w ( f )))ψ H | ≤ 2 f ∞ ψ H (π(w) -1)ψ H for all ψ ∈ H, w ∈ W and f ∈ C(X ).”

For d: M->g, the one-parameter subgroup t -> w_{S,σS}^{E_d(t)} is strongly continuous w.r.t. L2(A, μ0) for each quasi-surface S with intersection function σS.

“The one-parameter subgroup t -→ w S,σ S E d (t) is strongly continuous w.r.t. to L 2 (A, µ 0 ) for each quasi-surface S with intersection function σ S .”

Let Q be a complete subset of P_gen and κ:Q→G an admissible map. Then there is a unique map :A→A such that h(A)([γ]) = κ(γ)^-1 h_A([γ]) κ(γ^-1) for all γ in Q.

“Theorem 3.12.Let Q be a complete subset of P gen and κ : Q -→ G an admissible map. Then there is a unique map : A -→ A, such that, for all γ ∈ Q, h (A) ([γ ]) = κ(γ ) -1 h A ([γ ]) κ(γ -1 ).”

• cites [13] Fleischhack, Ch.: Construction of generalized connections. http://arxiv.org/list/math-ph/0601005 (2006)Not checked yet

  Not checked in the free run.

Every non-degenerate representation π of A(W, μ) on a Hilbert space H has a first-step decomposition: there are measures μν on X such that π|C(X) is unitarily equivalent to the direct sum of canonical multiplication representations on L2(X, μν), and the measures may be chosen so that equal measures correspond to equivalent representations.

“Then there are measures µ ν on X with ν running over some (not necessarily countable) index set N, such that π | C(X ) is unitarily equivalent to the direct-sum representation ν π µ ν , where π µ ν denotes the canonical representation of C(X ) on L 2 (X, µ ν ) by multiplication operators.”

• cites [12] Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol. 1 (C*- and W*-Algebras, Symmetry Groups, Decomposition of States). Springer-Verlag, New York (1987) sourceNot checked yet

  Not checked in the free run.

• cites [36] Takesaki, M.: Theory of Operator Algebras I (Encyclopaedia of Mathematical Sciences 124). Springer-Verlag, Berlin (2002) sourceNot checked yet

  Not checked in the free run.

Measures, in general, lead to Weyl operators that are ill defined on the L2-functions.

“Note, however, that measures, in general, lead to Weyl operators that are ill defined on the L 2 -functions:”

Let S1 and S2 be quasi-surfaces with intersection functions σS1 and σS2. Then σS1+σS2 is an intersection function for S:=S1∪S2.

“Corollary 3.17.Let S 1 and S 2 be quasi-surfaces with intersection functions σ S 1 and σ S 2 , respectively.Then σ S 1 + σ S 2 is an intersection function for S := S 1 ∪ S 2 .”

Each Weyl operator is both a map on C(A) and on L2(A, μ0).

“Note that each Weyl operator is both a map on C(A) and L 2 (A, µ 0 ).”

For each d ∈ Maps(M,G), the map S,σS_d: A -> A, A ↦ S,σS(A,d), is a homeomorphism.

“Moreover, the map S,σ S d : A -→ A, given by S,σ S d (A) := S,σ S (A, d), is a homeomorphism and preserves the Ashtekar-Lewandowski measure for each d ∈ Maps(M, G).”

For all υ = {γ1, ..., γn} ≥ γ with union of γi equal to γ, Mφγ ⊆ span Mφυ.

“• We have M φ γ ⊆ span M φ υ for all υ = {γ 1 , . . ., γ n } ≥ γ with i γ i = γ .Here, M φ υ := i M φ • π γ i .”

If E ⊆ C(X) is a continuous generating system with respect to two measures μ1 and μ2, then μ1 equals μ2.

“If E ⊆ C(X ) is a continuous generating system w.r.t.two measures µ 1 and µ 2 , then µ 1 equals µ 2 .”

For each d ∈ Maps(M,G), S,σS_d preserves the Ashtekar-Lewandowski measure.

“Moreover, the map S,σ S d : A -→ A, given by S,σ S d (A) := S,σ S (A, d), is a homeomorphism and preserves the Ashtekar-Lewandowski measure for each d ∈ Maps(M, G).”

The set of all edges and trivial paths in P_gen is complete.

“Lemma 3.8.The set of all edges and trivial paths in P gen is complete.”

Fix S and σS. Let 0 be a set of sequential Maps(M,G)-functions such that pr_x • λ0 is sequentially continuous for every x∈M and each λ0∈0. Assign to each λ0 some λ: dom λ0 -> W, y -> w_{S,σS}^{λ0(y)}, and collect all such λ into . Then λ(•)ψ: dom λ0 -> Haux is continuous for all ψ∈Haux and each λ∈.

“Then λ( • )ψ : dom λ 0 -→ H aux ≡ L 2 (A, µ 0 ) is continuous for all ψ ∈ H aux and each λ ∈ .”

Let Q be complete, X a topological space, and λ:X→Germ(Q,G) a map such that λ(•)(γ):X→G is continuous for all γ in Q. Then λ:X→A, x↦[λ(x)] is continuous.

“Proposition 3.10.Let Q be some complete subset of P gen .Let X be some topological space, and let λ : X -→ Germ(Q, G) be some map.Finally, assume that the map λ(•) (γ ) : X -→ G is continuous for all γ ∈ Q. Then λ : X -→ A x -→ [ λ(x) ] is continuous, where • is given as in Proposition 3.9.”

Let S be a subset of M. Then Q_S is complete if every edge has an S-admissible decomposition.

“Lemma 3.14.Let S be a subset of M. Then Q S is complete, if every edge has an S-admissible decomposition.”

For any complete Q and germ ρ:Q→G, the extension ρ is constant on equivalence classes in P_gen.

“• The map ρ is constant on equivalence classes in P gen .”

If Maps(M,G) is given the product topology, then the map S,σS is continuous.

“If Maps(M, G) ∼ = G M is given the product topology, then is continuous.”

The complement of a quasi-surface is a quasi-surface.

“Lemma 3.15. The complement of a quasi-surface is a quasi-surface.”

For a hyph υ and a Weyl operator w for some quasi-surface S, there is a hyph υ' ≥ υ with w(M_υ) ⊆ span M_{υ'}.

“Let υ be a hyph and w be a Weyl operator for some quasi-surface S. Then there is a hyph υ ≥ υ with w(M υ ) ⊆ span M υ .”

The graph S of the function f(x)=e^{-1/x^2} for x≠0 and f(0)=0 is a quasi-surface in R2.

“But, its graph S does not form a semianalytic submanifold in, for simplicity, R 2 .Nevertheless, it is a quasi-surface.”

For d: M->g, pr_x • E_d is continuous for every x∈M.

“pr x • E d is continuous for every x ∈ M.”

w_{d1} w_{d2} = w_{d1 d2} if d1 d2 = d2 d1.

“w d 1 w d 2 = w d 1 d 2 , if d 1 d 2 = d 2 d 1 .”

There is a unique map S,σS: A × Maps(M,G) -> A such that h_{S,σS(A,d)}(γ) = d(γ(0)) σ^-_S(γ) h_A(γ) d(γ(1)) σ^+_S(γ) for S-external γ and h_A(γ) for S-internal γ.

“There is a unique map S,σ S : A × Maps(M, G) -→ A, such that h S,σ S (A,d) (γ ) = d(γ (0)) σ - S (γ ) h A (γ ) d(γ (1)) σ + S (γ ) for S-external γ h A (γ ) for S-internal γ .”

Bγ,φ is orthogonal to its complement in the set of all gauge-variant spin network states, for every edge γ and every irreducible representation φ of G.

“Lemma 3.5.B γ,φ is orthogonal to its complement in the set of all gauge-variant spin network states, for every edge γ and every irreducible representation φ of G.”

For any complete Q and germ ρ:Q→G, there is a unique germ ρ:P_gen→G extending ρ.

“• There is a unique germ ρ : P gen -→ G extending ρ.”

The number of natural intersection functions of quasi-surfaces with higher codimension may be rather large.

“One sees immediately that the number of natural intersection functions of such quasisurfaces with higher codimension may be rather large.”

Any collection of points having no accumulation point is a quasi-surface.

“Therefore, any collection of points having no accumulation point is a quasi-surface.”

There are non-semianalytic submanifolds that are quasi-surfaces.

“On the other hand, there are indeed non-semianalytic submanifolds that are quasisurfaces.”

If an edge γ has any S-admissible decomposition, it has also a minimal S-admissible decomposition.

“Lemma 3.18. If an edge γ has any S-admissible decomposition, it has also a minimal S-admissible decomposition.”

M_SN is a continuous μ-generating system in L2(A, μ0).

“Lemma 3.3. M SN is a continuous µ-generating system in L 2 (A, µ 0 ).”

The minimal S-admissible decomposition of an edge is unique up to the parametrization of its components.

“Moreover, this minimal decomposition is unique up to the parametrization of its components.”

For every graphomorphism φ, αφ is an isometry on C(A).

“α ϕ is an isometry on C(A);”

If W splits W' at 1ν0, then μν0 equals μ.

“If W splits W at 1 ν 0 , then µ ν 0 equals µ.”

Let S1 and S2 be quasi-surfaces, d1,d2: M->G commute on S1∩S2, and d defined piecewise. If σS1 and σS2 coincide for all paths starting in S1∩S2 and vanish for S1- and S2-internal paths, then S1,σS1_d1 • S2,σS2_d2 = S1∪S2,σS1S2_d = S2,σS2_d2 • S1,σS1_d1.

“If σ S 1 and σ S 2 coincide for all paths starting in S 1 ∩ S 2 and vanish both for S 1 -and S 2 -internal paths, then S 1 ,σ S 1 d 1 • S 2 ,σ S 2 d 2 = S 1 ∪S 2 ,σ S 1 S 2 d = S 2 ,σ S 2 d 2 • S 1 ,σ S 1 d 1 .”

If d1,d2: M->G commute pointwise, then S,σS_d1 • S,σS_d2 = S,σS_{d1d2}.

“If d 1 and d 2 commute pointwise, we have S, σ S d 1 • S,σ S d 2 = S,σ S d 1 d 2 .”

If S1 and S2 are quasi-surfaces, then S1∪S2 and S1∩S2 are quasi-surfaces.

“Lemma 3.16. If S 1 and S 2 are quasi-surfaces, then S 1 ∪S 2 and S 1 ∩S 2 are quasi-surfaces.”

If Q ⊆ P_gen is complete, then for every hyph υ there is a hyph υ' ≥ υ with υ' ⊆ Q.

“Lemma 3.7.Let Q ⊆ P gen be complete.Then for every hyph υ there is a hyph υ ≥ υ with υ ⊆ Q.”

For a graphomorphism φ, quasi-surface S, intersection function σ, and function d: M->G, w_{φ(S),φ(σ)}^{φ(d)} = αφ(w_{S,σ}^d) = αφ • w_{S,σ}^d • αφ^{-1}.

“w ϕ(S),ϕ(σ ) ϕ(d) = α ϕ (w S,σ d ) ≡ α ϕ • w S,σ d • α -1 ϕ .”

The map φ -> αφ is a representation of the group of graphomorphisms on L2(A, μ0), with α_{φ1•φ2}=α_{φ1}•α_{φ2} and α_{φ^{-1}}=α^{-1}_φ.

“The map ϕ -→ α ϕ is even a representation of the group of graphomorphisms on L 2 (A, µ 0 ), because α ϕ 1 •ϕ 2 = α ϕ 1 • α ϕ 2 and α ϕ -1 = α -1 ϕ .”

Gauge-variant spin network states do not form an orthonormal basis for L2(A, μ0) even after dropping some subset of them.

“In contrast to the standard, i.e., gauge-invariant spin network states, the gauge-variant ones do not form an orthonormal basis for L 2 (A, µ 0 ) even after dropping some subset of them.”

Every Weyl operator is a unitary operator on L2(A, μ0).

“Every Weyl operator is a unitary operator on L 2 (A, µ 0 ).”

For d: M->g, E_d(t1)E_d(t2)=E_d(t1+t2) for all t1,t2∈R.

“E d (t 1 )E d (t 2 ) = E d (t 1 + t 2 ) for all t 1 , t 2 ∈ R.”

If υ contains S-external and S-internal edges only, then w(M_υ) ⊆ span M_υ.

“If, moreover, υ contains S-external and S-internal edges only, then w(M υ ) ⊆ span M υ .”

Two generalized connections coincide iff they coincide for all equivalence classes of paths of a complete subset of P_gen.

“Lemma 3.11.Two generalized connections coincide iff they coincide for all (equivalence classes of) paths of a complete subset of P gen .”

If 1ν is π(w)-invariant, then Hν and Hν⊥ are π(w)-invariant.

“If 1 ν is π(w)-invariant, then H ν and H ⊥ ν are π(w)-invariant.”

Under the hypotheses of Proposition 6.13, there are products φ0 and φ1 of localized stratified isomorphisms such that R is the disjoint union of φ0R and φ1R and the intersection function of R is the joint intersection function of φ0R and φ1R.

“Then there are products ϕ 0 and ϕ 1 of localized stratified isomorphisms, such that R is the disjoint union of ϕ 0 R and ϕ 1 R and the intersection function of R is the joint intersection function of ϕ 0 R and ϕ 1 R.”

For non-negative integers q<k, every q-simplex and every q-dimensional ball in R^k having a nice orientation is strata equivalent to itself having inverse orientation.

“Then every q-simplex and every q-dimensional ball in R k having a nice orientation, is strata equivalent to itself having inverse orientation.”

In particular, π0 is p,S,σS-regular for all p∈N∪{∞,ω}, S∈S and σS∈(S).

“In particular, π 0 is p,S,σ S -regular for all p ∈ N ∪ {∞, ω}, S ∈ S and σ S ∈ (S).”

Let M be an n-dimensional manifold with n ≥ 2 and S⊆M such that S and ∂S are connected embedded submanifolds without boundary and S is an embedded submanifold with boundary ∂S. If either S or ∂S is an embedded 1-circle S^1, and γ is a graph whose image is this S^1 with vertex m, then there exist a neighbourhood U of m, infinitely many different m_i in S^1∩U, and for each i a stratified diffe

“Proposition 6.23.Let M be some n-dimensional manifold with n ≥ 2 and let S ⊆ M. Assume that S and ∂ S are connected embedded submanifolds in M (without boundary) and that S is an embedded submanifold in M having boundary ∂ S.Moreover, let either S or ∂ S be an embedded 1-circle S 1 .Finally, let γ be a graph whose image is this S 1 and let m be some vertex of γ .”

Let M be an n-dimensional manifold with n ≥ 2 and S⊆M such that S and ∂S are connected embedded submanifolds without boundary and S is an embedded submanifold with boundary ∂S. For any nontrivial graph γ in M whose image is neither S, ∂S, nor S, there exist a nontrivial path γ, a neighbourhood U of some m∈im γ, and infinitely many stratified diffeomorphisms φ_i of M that are the identity outside U

“Proposition 6.22.Let M be some n-dimensional manifold with n ≥ 2 and let S ⊆ M. Assume that S and ∂ S are connected embedded submanifolds in M (without boundary) and that S is an embedded submanifold in M having boundary ∂ S.Moreover, let γ be some nontrivial graph in M, such that the image of γ is neither equal to S, ∂ S nor S.”

The localized (weakly) stratified isomorphisms of M form a subgroup of the group of homeomorphisms of M.

“Corollary 6.18.The localized (weakly) stratified isomorphisms of M form a subgroup of the group of homeomorphisms of M.”

Given the assumptions of Lemma 6.2, there is a stratified C^p diffeomorphism φ mapping S0 to λ0 S1 and R0 to λ0 R1 for some λ−≤λ0≤λ+, such that φ is the identity inside λ−R1 and outside λ+R1.

“Corollary 6.3. Given the assumptions of Lemma 6.2, there is a stratified C p diffeomorphism ϕ mapping S 0 to λ 0 S 1 and R 0 to λ 0 R 1 for some λ -≤ λ 0 ≤ λ + , such that ϕ is the identity inside λ -R 1 and outside λ + R 1 .”

For f∈A⊆L∞, w(f)=f in L∞⊆L2 for all w∈W, by invertibility of w.

“Let now f ∈ A ⊆ L ∞ .Then we have f • w = w • f = w( f ) • w for all w ∈ W, hence w( f ) = f in L ∞ ⊆ L 2 by invertibility of w.”

The algebra A is irreducible, i.e. A = C1.

“Therefore, f ∈ C 1, hence A = C 1.”

• AssumptionRests on an assumption: G is compact and connected.

g -> βg is a representation of G on L2(A, μ0) by unitaries.

“g -→ β g is a representation of G on L 2 (A, µ 0 ) by unitaries.”

1 is a cyclic, diffeomorphism and automorphism invariant vector for π0.

“1 is a cyclic, diffeomorphism and automorphism invariant vector for π 0 .”

Under the hypotheses of Corollary 6.6, for each α∈R there is a stratified isomorphism φα of R^2⊕R^k that is the identity outside B_{1+ε}⊆R^{k+2}, is norm preserving, is homotopic to the identity, satisfies Pφα=P, and maps γ0 to γα.

“Then for each α ∈ R there is a stratified isomorphism ϕ α of R 2 ⊕ R k , such that • ϕ α is the identity outside B 1+ε ⊆ R k+2 ; • ϕ α is norm preserving; • ϕ α is homotopic 19 to the identity; • Pϕ α = P, where P is the canonical projection from R 2 ⊕ R k to R k ; • ϕ α maps γ 0 to γ α .”

For two stratifications M1 and M2 of M, there is a stratification M of M finer than both M1 and M2.

“Then there is a stratification M of M being finer than M 1 and M 2 .”

If φ is abelian, w_S^g(T) = φ(g^2) T = χφ(g^2) T for all g∈G.

“Moreover, if φ is abelian w S g (T ) = φ(g 2 ) T = χ φ (g 2 ) T for all g ∈ G.”

For every nicely oriented 1-dimensional ball S in R^k with k≥3, there are finitely many localized stratified isomorphisms whose product is the identity on S but inverts the orientation of S.

“For every nicely oriented 1-dimensional ball S in R k with k ≥ 3, there are finitely many localized stratified isomorphisms, whose product is the identity on S, but inverts the orientation of S.”

For all nontrivial gauge-variant spin network functions T, T,f=0.

“Altogether, we have proven T, f = 0 for all nontrivial gauge-variant spin network functions T .”

• AssumptionRests on an assumption: G is compact and connected.

For all w∈W, f∈L∞ and ψ∈L2, w(f)w(ψ)=w(fψ).

“Next, one checks immediately that w( f )w(ψ) = w( f ψ) for all w ∈ W, f ∈ L ∞ and ψ ∈ L 2 .”

For positive integers q≤k, all q-simplices and all q-dimensional balls in R^k are strata equivalent.

“Then all q-simplices and all q-dimensional balls in R k are strata equivalent.”

g -> βg is a representation of G on C(A) by isometries.

“g -→ β g is a representation of G on C(A) by isometries.”

For nice enlarged structure data, π0 is regular w.r.t. R.

“1. π 0 is regular w.r.t.R.”

For (γ,γ')-nice surfaces S1,S2 with different punctures and T=(T_{φ,γ})^m_n ⊗ T' with T'∈M_{γ'}, w_{S1}^{g1}(T), w_{S2}^{g2}(T) = χφ(g1^2) χφ(g2^2)/(dim φ)^2 for all g1,g2∈G.

“w S 1 g 1 (T ), w S 2 g 2 (T ) = χ φ (g 2 1 ) χ φ (g 2 2 ) (dim φ) 2 for all g 1 , g 2 ∈ G.”

For nice enlarged structure data, π0 is -regular with given in Proposition 3.28.

“2. π 0 is -regular with given in Proposition 3.28.”

Under the hypotheses of Lemma 6.5, there is a stratified diffeomorphism φ of R^k that coincides with A on B_{r2}, is the identity outside B_{r1}, is norm preserving, is homotopic to the identity, and satisfies Pφ=P.

“Then there is a stratified diffeomorphism ϕ of R k , such that • ϕ coincides with A on B r 2 ; • ϕ is the identity outside of B r 1 ; • ϕ is norm preserving; • ϕ is homotopic to the identity; • P ϕ = P.”

The (weakly) stratified isomorphisms of M form a subgroup of the group of homeomorphisms of M.

“Corollary 6.17.The (weakly) stratified isomorphisms of M form a subgroup of the group of homeomorphisms of M.”

Every (weakly) stratified isomorphism is a graphomorphism.

“Corollary 6.16.Every (weakly) stratified isomorphism is a graphomorphism.”

If π is regular and there exists a cyclic D-invariant vector in H, then there is a first-step decomposition of π such that μ_{ν0} is the Ashtekar-Lewandowski measure μ0 for some ν0 and 1_{ν0} is cyclic and D-invariant.

“Corollary 7.2.Let π be regular, and let there exist a (cyclic) D-invariant vector in H. Then there is a first-step decomposition of π , such that µ ν 0 is the Ashtekar-Lewandowski measure µ 0 for some ν 0 and 1 ν 0 is (cyclic and) D-invariant.”

Under the hypotheses of Lemma 6.4, for all λ>0 and ε>0 there is a stratified C^p isomorphism φ preserving half-rays, such that φ=λ id on R and φ=id outside (1+ε)max(λ,1)R.

“Then for all λ > 0 and all ε > 0, there is a stratified C p isomorphism ϕ preserving half-rays, such that ϕ = λ id on R and ϕ = id outside (1 + ε) max(λ, 1)R.”

Under the hypotheses of Lemma 6.7, there is a finite product of stratified C^p diffeomorphisms of R^k that is the identity outside U and equals the translation by γ(1)−γ(0) on B_r(γ(0)).

“Then there is a finite product of stratified C p diffeomorphisms of R k being the identity outside U and the translation by γ (1) -γ (0) on B r (γ (0)).”

The Weyl algebra A of quantum geometry is irreducible on L2(A, μ0).

“The Weyl algebra A of quantum geometry is irreducible on L 2 (A, µ 0 ).”

• AssumptionThe authors state this as an assumption: The structure data are nice (as in Definition 4.2) for the irreducibility theorem.
• AssumptionThe authors state this as an assumption: S contains at least the closed, oriented hypersurfaces of M.

Under the hypotheses of Lemma 6.9, there is a stratified C^p diffeomorphism φ, identity outside B_{(1+ε)r}, such that R_V is mapped to B_r∩span R_V.

“Then there is a stratified C p diffeomorphism ϕ, being the identity outside of B (1+ε)r , such that R V is mapped to B r ∩ span R V .”

For non-negative integers q<k, all q-simplices and all q-balls in R^k are oriented-strata equivalent, provided they have nice orientations.

“Then all q-simplices and all q-balls in R k are oriented-strata equivalent, provided they have nice orientations.”

Using piecewise analytic diffeomorphisms, it may happen that a surface (including its orientation) is kept invariant, but an originally transversally intersecting path may now be mapped to a tangential one.

“Using piecewise analytic diffeomorphisms, it may happen that a surface (including its orientation) is kept invariant, but an originally transversally intersecting path may now be mapped to a tangential one.”

If π is a D-natural representation of A_Diff such that 1_{ν0} is diffeomorphism-invariant and μ_{ν0}=μ0 for some ν0∈N, then the restriction of π to H_{ν0} is the fundamental representation π0, i.e., P_ν π(a)=π0(a)P_ν for all a∈A_Diff.

“Proposition 7.5.Let π be a D-natural representation of A Diff , such that 1 ν 0 is a diffeomorphism-invariant vector and µ ν 0 equals µ 0 for some ν 0 ∈ N. Then the restriction of π to H ν 0 is the fundamental representation π 0 , i.e., we have P ν π (a) = π 0 (a)P ν for all a ∈ A Diff .”

If S has one of its natural orientations, then each φ_i in Proposition 6.22 may be chosen so that it additionally leaves the orientation of S invariant.

“If we additionally assume, that S has one of its natural orientations, then each ϕ i may be chosen such that, additionally, it leaves the orientation of S invariant.”

For S an open 2-dimensional ball in M with a nice orientation, w:=w_{S,σ_S,d} is a π_{ν0}-unit for every constant d∈(S).

“Corollary 7.12.Let S be an open 2-dimensional ball in M given a nice orientation.Then w := w S,σ S d is a π ν 0 -unit for every constant d ∈ (S).”

If S is a finitely widely triangulizable subset of M with a natural orientation and dim S<dim M, then w:=w_{S,σ_S,d} is a π_{ν0}-unit for every d∈(S) that is constant on S.

“Proposition 7.13.Let S be a finitely widely triangulizable subset in M having a natural orientation with dim S < dim M. Then w := w S,σ S d is a π ν 0 -unit for every d ∈ (S) being constant on S.”

For S an open 1-dimensional ball in M with a nice orientation, w:=w_{S,σ_S,d} is a π_{ν0}-unit for every constant d∈(S).

“Corollary 7.11.Let S be an open 1-dimensional ball in M given a nice orientation.Then w := w S,σ S d is a π ν 0 -unit for every constant d ∈ (S).”

The paper significantly weakened the assumptions on the representation of the diffeomorphisms compared to [35].

“Secondly, we significantly weakened the assumptions on the representation of the diffeomorphisms.”

• cites [35] Sahlmann, H., Thiemann, T.: On the superselection theory of the weyl algebra for diffeomorphism invariant quantum gauge theories. http://arxiv.org/list/gr-qc/0302090 (2003)Not checked yet

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Let S⊆M such that S and ∂S are connected embedded submanifolds without boundary and S is an embedded submanifold with boundary ∂S, and assume S has a natural orientation. For w=w_{S,σ_S,d} with constant d∈(S), P_{ν0}π(w)1_{ν0}∈H_{ν0}≡L^2(A,μ0) is orthogonal to all non-trivial gauge-variant spin network states not based on an edge γ whose image equals S, ∂S, or S.

“Lemma 7.7.Let S some subset of M, such that S and ∂ S are connected embedded submanifolds in M (without boundary) and that S is an embedded submanifold in M having boundary ∂ S.Moreover, assume that S has one of its natural orientations.Finally, let w = w S,σ S d for some constant d ∈ (S).Then P ν 0 π(w)1 ν 0 ∈ H ν 0 ≡ L 2 (A, µ 0 ) is orthogonal to all non-trivial gauge-variant spin network”

Even if the pull-back representation is well defined, it is unitary only if μν is W-invariant.

“Additionally, one sees immediately, that, even if the pull-back representation is well defined, it is unitary only if µ ν is W -invariant.”

Let Y be a sequential topological space, X a Banach space, λ: Y -> B(X) locally bounded, and E ⊆ X such that y -> λ(y)e is continuous for all e in E and span E is dense in X. Then y -> λ(y)x is continuous for all x in X.

“Lemma A.1.Let Y be some sequential topological space.Let X be a Banach space and let λ : Y -→ B(X ) be some map.”

• Also stated in: § Additional information · plain

  “Then y -→ λ(y)x is continuous for all x ∈ X.”

If S has one of its natural orientations, then each φ_m in Proposition 6.23 may be chosen so that it additionally leaves the orientation of S invariant.

“If we additionally assume that S has one of its natural orientations, then each ϕ m may be chosen such that, additionally, it leaves the orientation of S invariant.”

If S1 and S2 are oriented-strata equivalent elements of S with orientations σ_S1 and σ_S2, and d_i:S_i→G are constant functions with value g∈G, then w_{S1,σ_S1,d1} is a π_{ν0}-unit (π_{ν0}-scalar) iff w_{S2,σ_S2,d2} is a π_{ν0}-unit (π_{ν0}-scalar).

“Lemma 7.6.Let S 1 and S 2 be elements in S having orientations σ S 1 and σ S 2 .Assume that they are oriented-strata equivalent.Finally, let g ∈ G be some element and d i : S i -→ G for i = 1, 2 be the constant function with value g.Then w S 1 ,σ S 1 d 1 is a π ν 0 -unit (π ν 0 -scalar) iff w S 2 ,σ S 2 d 2 is a π ν 0 -unit (π ν 0 -scalar).”

If π is a representation of A_Auto on H such that π|A is regular and π|A_Diff is D-natural, there exists a D-E-invariant vector in H cyclic for π, and optimal enlarged structure data are given, then π is unitarily equivalent to the fundamental representation of A_Auto.

“Theorem 7.14.Let π be a representation of A Auto on H, such that π := π | A is regular and π := π | A Diff is D-natural.Moreover, let there exist some D-E-invariant vector in H being cyclic for π .Finally, let optimal enlarged structure data be given.Then π is unitarily equivalent to the fundamental representation of A Auto .”

• AssumptionThe authors state this as an assumption: π restricted to A is regular.
• AssumptionThe authors state this as an assumption: π restricted to A_Diff is D-natural.
• AssumptionThe authors state this as an assumption: There exists a D-E-invariant vector in H that is cyclic for π.
• AssumptionThe authors state this as an assumption: Optimal enlarged structure data are given.

The extension of the pull-back mapping from C(X) to L2(X, μν) is ill defined.

“In other words, the extension of the pull-back mapping from C(X ) to L 2 (X, µ ν ) is ill defined.”

The set of all decompositions of a path γ is directed with respect to ≥.

“It can easily be shown [13] that the set of all decompositions of a path γ is directed w.r.t.≥.”

• cites [13] Fleischhack, Ch.: Construction of generalized connections. http://arxiv.org/list/math-ph/0601005 (2006)Not checked yet

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To make all Weyl operators w_{S,σS_g} well defined as operators on L2(A, μ) for some μ, it is necessary that for each S-external edge γ (having only one end attached to S), the support of the push-forward measure (πγ)*μ equals G.

“To make all these Weyl operators well defined as operators on L 2 (A, µ) for some µ, we have at least to demand that, for each S-external edge γ (having only one end attached to S), the support of the push-forward measure (π γ ) * µ equals G.”

The obstruction to an orthonormal basis is the states arising in the decomposition of an edge into a product of subedges, i.e., having two-valent vertices.

“The problem are the states arising in the decomposition of an edge into a product of subedges, i.e., having two-valent vertices.”

Even if for each t there is an allowed d(t) with d(t1+t2)=d(t1)d(t2), the corresponding maps t -> (d(t))(x) need not be continuous at all.

“But, moreover, even if we might find for each t some allowed d(t) with d(t 1 + t 2 ) = d(t 1 )d(t 2 ), the corresponding maps t -→ (d(t))(x) need not be continuous at all.”

If one enlarges P, one might have to reduce S, because one must guarantee that there are at most finitely many genuine intersections of paths and quasi-surfaces.

“if we would enlarge P, we might have to reduce S, simply because we have to guarantee that there are at most finitely many (genuine) intersections of paths and quasi-surfaces.”

Every semianalytic set is triangulizable.

“Proposition 6.19.Every semianalytic set is triangulizable [29].”

• cites [29] Łojasiewicz S.: Triangulation of semi-analytic sets. Ann. Scuola Norm. Sup. Pisa 18, 449–474 (1964)Not checked yet

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Each nonabelian irreducible character has a zero.

“Moreover, by [20], each nonabelian irreducible character has a zero.”

• cites [20] Gallagher P.X.: Zeros of group characters. Math. Zeitschr. 87, 363–364 (1965) sourceNot checked yet

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It need not be possible to connect any Weyl operator to the identity within the limits of the structure data.

“This shows that it need not be possible to connect any Weyl operator to the identity within the limits of the structure data.”

If the Weyl algebra of quantum geometry is irreducible for some structure data, then it is irreducible for all larger structure data.

“Note that given the irreducibility of the Weyl algebra of quantum geometry for these structure data, we get it immediately for all larger structure data.”

The smooth category for paths is practically excluded because there are analytic submanifolds having an infinite number of isolated transversal intersections with smooth paths.

“this practically excludes the choice of the smooth category for the paths: There are even analytic submanifolds having an infinite number of isolated transversal intersections with smooth paths.”

The full information on any (possibly stratified) C^p bundle automorphism can be encoded in a (possibly stratified) C^p diffeomorphism and a generalized gauge transform.

“the full information on any (possibly stratified) C p bundle automorphism can be encoded in a (again, possibly stratified) C p diffeomorphism and a generalized gauge transform (even of any other bundle).”

If G, M and S are not simply connected, (S) contains continuous functions only, and d: M->G has nontrivial mapping degree, then in general it is not possible to deform d in (S) continuously into the trivial function on G.

“For instance, let G, M and S be not simply connected, allow (S) to contain continuous functions only, and let d : M -→ G have nontrivial mapping degree.Then, in general, it is not possible to deform d in (S) continuously into the trivial function on G.”

The corresponding one-parameter subgroups of Weyl operators are no longer weakly continuous.

“Then the corresponding one-parameter subgroups of Weyl operators are no longer weakly continuous, as one immediately checks.”

Every semianalytic set admits a semianalytic stratification, i.e. a stratification consisting of semianalytic strata only.

“Moreover, it can be shown [21,28] that every semianalytic set admits a semianalytic stratification, i.e., a stratification consisting of semianalytic strata only.”

• cites [21] Hardt R.M.: Stratification of Real Analytic Mappings and Images. Invent. Math. 28, 193–208 (1975) sourceNot checked yet

  Not checked in the free run.

• cites [28] Łojasiewicz, S.: On semi-analytic and subanalytic geometry. In: Panoramas of mathematics (Warszawa, 1992/1994), Banach Center Publications 34, pp. 89–104, Polish Academy of Sciences, Warszawa (1995) sourceNot checked yet

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From the mathematical, technical point of view, one is forced to admit at most (piecewise) analytic paths.

“Therefore, we are -from the mathematical, technical point of view -quite forced to admit at most (piecewise) analytic paths.”

It is sufficient to include all one-parameter subgroups t -> w_t = w_{S,σS}^{d(t)} with d(t):=e^{td} for constant d: M->g.

“it will be sufficient to include that all t -→ w t = w S,σ S d(t) with d(t) := e td ∈ (S) for constant d : M -→ g.”

The results above can be directly extended to semianalytic sets having the same dimension as M.

“We remark that the results above can be directly extended to semianalytic sets having the same dimension as M.”

The paper did not need any assumptions about the domains of the operators.

“First of all, we did not need any assumptions about the domains of the operators.”

Since C(A) is continuous, any representation restricted to C(A) can be decomposed into canonical representations on A with respect to certain measures.

“Moreover, since C(A) is continuous, in any case, we may decompose the restriction of any representation to C(A) into canonical representations on A w.r.t.certain measures.”

The uniqueness result can be extended immediately to the hyperplane example.

“we may extend the uniqueness result immediately to this example.”