Uniqueness Christian Fleischhack Universit at Paderborn Institut - PowerPoint PPT Presentation

Uniqueness Christian Fleischhack Universit¨ at Paderborn Institut f¨ ur Mathematik Jurekfest, Warszawa, September 2019

1 Canonical Quantization Strategy • Given: classical system with first-class constraints 1. Elementary Variables • choose separating space S of phase space functions 2. Quantization • choose “representation” of S on some kinematical Hilbert space H , giving self-adjoint constraints 3. Group Averaging • choose constraint-invariant dense subset Φ in Hilbert space H • solve constraints using Gelfand triple Φ ⊆ H ⊆ Φ ′ � d µ ( Z ) Zφ ∈ Φ ′ η ( φ ) := Z 4. Physical Hilbert Space • inner product: � ηφ 1 , ηφ 2 � phys := ( ηφ 1 )[ φ 2 ] • completion of η (Φ) gives physical Hilbert space, self-adjoint dual representation of observable algebra Ashtekar, Lewandowski, Marolf, Mour˜ ao, Thiemann 1995

1 Canonical Quantization Strategy • Given: classical system with first-class constraints 1. Elementary Variables • choose separating space S of phase space functions 2. Quantization • choose “representation” of S on some kinematical Hilbert space H , giving self-adjoint constraints 3. Group Averaging • choose constraint-invariant dense subset Φ in Hilbert space H • solve constraints using Gelfand triple Φ ⊆ H ⊆ Φ ′ � d µ ( Z ) Zφ ∈ Φ ′ η ( φ ) := Z 4. Physical Hilbert Space • inner product: � ηφ 1 , ηφ 2 � phys := ( ηφ 1 )[ φ 2 ] • completion of η (Φ) gives physical Hilbert space, self-adjoint dual representation of observable algebra thieMann, mourAo, marolF, lewandowskI, ashtekAr 1995

1 Canonical Quantization Strategy • Given: classical system with first-class constraints 1. Elementary Variables • choose separating space S of phase space functions 2. Quantization • choose “representation” of S on some kinematical Hilbert space H , giving self-adjoint constraints

1 Canonical Quantization Strategy • Given: classical system 1. Elementary Variables • choose separating space S of phase space functions

b b 2 Basics Gravity • Given: Ashtekar gravity ( A, E ) 1. Elementary Variables • choose separating space S of phase space functions � γ A • Basic functions h γ ( A ) := P e � := [ ∗ E ]( f ) E S,f S • Cylindrical functions ( γ 1 , . . . , γ n ) γ 1 γ 2 γ 3 γ 4 ψ := ψ γ ◦ π γ → G n h γ 1 × · · · × h γn : A − smooth • Derivations on Cyl := { ψ, E S,f } X S,f ψ • Diffeos act via γ , S , f , e.g.: α Ψ ( f ◦ h γ ) := f ◦ h Ψ( γ )

3 Holonomy-Flux Algebra LOST Theorem • Holonomy-Flux Algebra ∗ -algebra of all words in Cyl and X factorized by the relations H . . . a · X − X · a = i { a, X } (CCR, a ∈ Cyl ∪ X ) + linearity ψ · ψ ′ ψ ψ ′ = ( Cyl -module) • Standard Invariant State ω 0 ω 0 ( a · X ) = 0 ( a ∈ H , X ∈ X ) � ω 0 ( ψ ) = G n ψ γ d µ Haar ( ψ = ψ γ ◦ π γ ∈ Cyl ) Theorem: Lewandowski, Oko� l´ ow, Sahlmann, Thiemann 2005 Assume • dim M ≥ 2 • hypersurfaces – semianalytic • diffeos – semianalytic • smearings with compact support Then ω 0 is the unique state on H that is invariant w.r.t. bundle automorphisms.

b b 4 Basics Gravity • Given: Ashtekar gravity ( A, E ) 1. Elementary Variables • choose separating space S of phase space functions � γ A • Basic functions h γ ( A ) := P e � := [ ∗ E ]( f ) E S,f S • Cylindrical functions ( γ 1 , . . . , γ n ) γ 1 γ 2 γ 3 γ 4 ψ := ψ γ ◦ π γ → G n h γ 1 × · · · × h γn : A − smooth • Derivations on Cyl := { ψ, E S,f } X S,f ψ • Diffeos act via γ , S , f , e.g.: α Ψ ( f ◦ h γ ) := f ◦ h Ψ( γ )

b b 4 Basics Cosmology • Given: Ashtekar gravity ( A, E ) + homogeneity + isotropy 1. Elementary Variables • choose separating space S of phase space functions restricted � γ A • Basic functions h γ ( A ) := P e � := [ ∗ E ]( f ) E S,f S • Cylindrical functions ( γ 1 , . . . , γ n ) γ 1 γ 2 γ 3 γ 4 ψ := ψ γ ◦ π γ → G n h γ 1 × · · · × h γn : A − smooth • Derivations on Cyl := { ψ, E S,f } X S,f ψ • Diffeos act via γ , S , f , e.g.: α Ψ ( f ◦ h γ ) := f ◦ h Ψ( γ )

b b 4 Basics Cosmology • Given: Ashtekar gravity ( A, E ) + homogeneity + isotropy 1. Elementary Variables • choose separating space S of phase space functions restricted � γ A • Basic functions h γ ( A ) := P e � not smeared := [ ∗ E ]( f ) E S,f S • Cylindrical functions ( γ 1 , . . . , γ n ) γ 1 γ 2 γ 3 γ 4 ψ := ψ γ ◦ π γ → G n h γ 1 × · · · × h γn : A − smooth • Derivations on Cyl := { ψ, E S,f } X S,f ψ • Diffeos act via γ , S , f , e.g.: α Ψ ( f ◦ h γ ) := f ◦ h Ψ( γ )

b b 4 Basics Cosmology • Given: Ashtekar gravity ( A, E ) + homogeneity + isotropy 1. Elementary Variables • choose separating space S of phase space functions restricted � γ A • Basic functions h γ ( A ) := P e � not smeared := [ ∗ E ]( f ) E S,f S • Cylindrical functions ( γ 1 , . . . , γ n ) γ 1 γ 2 γ 3 γ 4 dense in A := C 0 ( R ) ⊕ C AP ( R ) restricted to R ψ := ψ γ ◦ π γ → G n h γ 1 × · · · × h γn : A − smooth • Derivations on Cyl := { ψ, E S,f } X S,f ψ • Diffeos act via γ , S , f , e.g.: α Ψ ( f ◦ h γ ) := f ◦ h Ψ( γ )

b b 4 Basics Cosmology • Given: Ashtekar gravity ( A, E ) + homogeneity + isotropy 1. Elementary Variables • choose separating space S of phase space functions restricted � γ A • Basic functions h γ ( A ) := P e � not smeared := [ ∗ E ]( f ) E S,f S • Cylindrical functions ( γ 1 , . . . , γ n ) γ 1 γ 2 γ 3 γ 4 dense in A := C 0 ( R ) ⊕ C AP ( R ) restricted to R ψ := ψ γ ◦ π γ → G n h γ 1 × · · · × h γn : A − smooth d • Derivations on Cyl := { ψ, E S,f } X S,f ψ = t d X • Diffeos act via γ , S , f , e.g.: α Ψ ( f ◦ h γ ) := f ◦ h Ψ( γ )

b b 4 Basics Cosmology • Given: Ashtekar gravity ( A, E ) + homogeneity + isotropy 1. Elementary Variables • choose separating space S of phase space functions restricted � γ A • Basic functions h γ ( A ) := P e � not smeared := [ ∗ E ]( f ) E S,f S • Cylindrical functions ( γ 1 , . . . , γ n ) γ 1 γ 2 γ 3 γ 4 dense in A := C 0 ( R ) ⊕ C AP ( R ) restricted to R ψ := ψ γ ◦ π γ → G n h γ 1 × · · · × h γn : A − smooth d • Derivations on Cyl := { ψ, E S,f } X S,f ψ = t d X residual dilations • Diffeos act via γ , S , f , e.g.: α Ψ ( f ◦ h γ ) := f ◦ h Ψ( γ )

b b 4 Basics Cosmology • Given: Ashtekar gravity ( A, E ) + homogeneity + isotropy 1. Elementary Variables • choose separating space S of phase space functions restricted � γ A • Basic functions h γ ( A ) := P e � not smeared := [ ∗ E ]( f ) E S,f S • Cylindrical functions ( γ 1 , . . . , γ n ) γ 1 γ 2 γ 3 γ 4 dense in A := C 0 ( R ) ⊕ C AP ( R ) restricted to R ψ := ψ γ ◦ π γ → G n h γ 1 × · · · × h γn : A − smooth d • Derivations on Cyl := { ψ, E S,f } X S,f ψ = t d X residual dilations X λ • Diffeos act via γ , S , f , e.g.: α Ψ ( f ◦ h γ ) := f ◦ h Ψ( γ ) = ) X ( α λ

5 Cosmological Holonomy-Flux Algebra LOST Theorem • Holonomy-Flux Algebra ∗ -algebra of all words in Cyl and X factorized by the relations H . . . a · X − X · a = i { a, X } (CCR, a ∈ Cyl ∪ X ) + linearity ψ · ψ ′ ψ ψ ′ = ( Cyl -module) • Standard Invariant State ω 0 ω 0 ( a · X ) = 0 ( a ∈ H , X ∈ X ) � ω 0 ( ψ ) = G n ψ γ d µ Haar ( ψ = ψ γ ◦ π γ ∈ Cyl ) Theorem: Lewandowski, Oko� l´ ow, Sahlmann, Thiemann 2005 Assume • dim M ≥ 2 • hypersurfaces – semianalytic • diffeos – semianalytic • smearings with compact support Then ω 0 is the unique state on H that is invariant w.r.t. bundle automorphisms.

5 Cosmological Holonomy-Flux Algebra THE Theorem • Restricted Holonomy-Flux Algebra ∗ -algebra of all words in B and X factorized by the relations H C . . . a · X − X · a = i { a, X } (CCR, a ∈ Cyl ∪ X ) + linearity ψ · ψ ′ ψ ψ ′ = ( B -module) • Standard Invariant State ω 0 ω 0 ( a · X ) = 0 ( a ∈ H C , X ∈ X ) � ω 0 ( ψ 0 + ψ AP ) = R Bohr ψ AP d µ Bohr Theorem: Thiemann, Hanusch, Engle 2016; Fleischhack 2018 Assume • A := C 0 ( R ) ⊕ C AP ( R ) • B := { ψ ∈ A | ψ ( n ) ∈ A ∀ n } • hypersurfaces – semianalytic • smearings with compact support Then ω 0 is the unique state on H C that is invariant w.r.t. dilations.

5 Cosmological Holonomy-Flux Algebra THE Theorem • Restricted Holonomy-Flux Algebra ∗ -algebra of all words in B and X factorized by the relations H C . . . a · X − X · a = i { a, X } (CCR, a ∈ Cyl ∪ X ) + linearity ψ · ψ ′ ψ ψ ′ = ( B -module) • Standard Invariant State ω 0 ω 0 ( a · X ) = 0 ( a ∈ H C , X ∈ X ) � ω 0 ( ψ 0 + ψ AP ) = R Bohr ψ AP d µ Bohr Theorem: Thiemann, Hanusch, Engle 2016; Fleischhack 2018 Assume • A := C 0 ( R ) ⊕ C AP ( R ) • B := { ψ ∈ A | ψ ( n ) ∈ A ∀ n } • hypersurfaces – semianalytic • smearings with compact support Then ω 0 is the unique state on H C that is invariant w.r.t. dilations. Holds also for A := C AP ( R ) Remark

6 Conclusions Jurek has found, created, inspired strong unique ness results.

6 Conclusions Jurek has found, created, inspired strong unique ness results. Jurek is unique . Theorem: