SLIDE 1
Uniqueness
Christian Fleischhack
Universit¨ at Paderborn Institut f¨ ur Mathematik
Jurekfest, Warszawa, September 2019
SLIDE 2 1 Canonical Quantization Strategy
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′
η(φ) :=
dµ(Z) Zφ ∈ Φ′ 4. Physical Hilbert Space
ηφ1, ηφ2phys := (ηφ1)[φ2]
- completion of η(Φ) gives physical Hilbert space,
self-adjoint dual representation of observable algebra
Ashtekar, Lewandowski, Marolf, Mour˜ ao, Thiemann 1995
SLIDE 3 1 Canonical Quantization Strategy
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′
η(φ) :=
dµ(Z) Zφ ∈ Φ′ 4. Physical Hilbert Space
ηφ1, ηφ2phys := (ηφ1)[φ2]
- completion of η(Φ) gives physical Hilbert space,
self-adjoint dual representation of observable algebra
thieMann, mourAo, marolF, lewandowskI, ashtekAr 1995
SLIDE 4 1 Canonical Quantization Strategy
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
SLIDE 5 1 Canonical Quantization Strategy
classical system 1. Elementary Variables
- choose separating space S of phase space functions
SLIDE 6 2 Basics Gravity
Ashtekar gravity (A, E) 1. Elementary Variables
- choose separating space S of phase space functions
- Basic functions
hγ(A) := Pe
ES,f :=
[∗E](f)
b b
γ1 γ2 γ3 γ4
ψ := ψγ ◦ πγ
smooth (γ1, . . . , γn) hγ1 × · · · × hγn : A − → Gn
XS,fψ := {ψ, ES,f}
- Diffeos act via γ, S, f, e.g.:
αΨ(f ◦ hγ) := f ◦ hΨ(γ)
SLIDE 7 3 Holonomy-Flux Algebra LOST Theorem
H . . . ∗-algebra of all words in Cyl and X factorized by the relations a · X − X · a = i {a, X}
(CCR, a ∈ Cyl ∪ X)
ψ · ψ′ = ψ ψ′
(Cyl-module) + linearity
ω0 ω0(a · X) =
(a ∈ H, X ∈ X)
ω0(ψ) =
(ψ = ψγ ◦ πγ ∈ Cyl)
Theorem:
Lewandowski, Oko l´
- w, Sahlmann, Thiemann 2005
Assume
- dim M ≥ 2
- hypersurfaces – semianalytic
- diffeos – semianalytic
- smearings with compact support
Then ω0 is the unique state on H that is invariant w.r.t. bundle automorphisms.
SLIDE 8 4 Basics Gravity
Ashtekar gravity (A, E) 1. Elementary Variables
- choose separating space S of phase space functions
- Basic functions
hγ(A) := Pe
ES,f :=
[∗E](f)
b b
γ1 γ2 γ3 γ4
ψ := ψγ ◦ πγ
smooth (γ1, . . . , γn) hγ1 × · · · × hγn : A − → Gn
XS,fψ := {ψ, ES,f}
- Diffeos act via γ, S, f, e.g.:
αΨ(f ◦ hγ) := f ◦ hΨ(γ)
SLIDE 9 4 Basics Cosmology
Ashtekar gravity (A, E) + homogeneity + isotropy 1. Elementary Variables
- choose separating space S of phase space functions
- Basic functions
hγ(A) :=
restricted
Pe
ES,f :=
[∗E](f)
b b
γ1 γ2 γ3 γ4
ψ := ψγ ◦ πγ
smooth (γ1, . . . , γn) hγ1 × · · · × hγn : A − → Gn
XS,fψ := {ψ, ES,f}
- Diffeos act via γ, S, f, e.g.:
αΨ(f ◦ hγ) := f ◦ hΨ(γ)
SLIDE 10 4 Basics Cosmology
Ashtekar gravity (A, E) + homogeneity + isotropy 1. Elementary Variables
- choose separating space S of phase space functions
- Basic functions
hγ(A) :=
restricted
Pe
ES,f :=
not smeared
[∗E](f)
b b
γ1 γ2 γ3 γ4
ψ := ψγ ◦ πγ
smooth (γ1, . . . , γn) hγ1 × · · · × hγn : A − → Gn
XS,fψ := {ψ, ES,f}
- Diffeos act via γ, S, f, e.g.:
αΨ(f ◦ hγ) := f ◦ hΨ(γ)
SLIDE 11 4 Basics Cosmology
Ashtekar gravity (A, E) + homogeneity + isotropy 1. Elementary Variables
- choose separating space S of phase space functions
- Basic functions
hγ(A) :=
restricted
Pe
ES,f :=
not smeared
[∗E](f)
b b
γ1 γ2 γ3 γ4
restricted to R dense in A := C0(R) ⊕ CAP(R)
ψ := ψγ ◦ πγ
smooth (γ1, . . . , γn) hγ1 × · · · × hγn : A − → Gn
XS,fψ := {ψ, ES,f}
- Diffeos act via γ, S, f, e.g.:
αΨ(f ◦ hγ) := f ◦ hΨ(γ)
SLIDE 12 4 Basics Cosmology
Ashtekar gravity (A, E) + homogeneity + isotropy 1. Elementary Variables
- choose separating space S of phase space functions
- Basic functions
hγ(A) :=
restricted
Pe
ES,f :=
not smeared
[∗E](f)
b b
γ1 γ2 γ3 γ4
restricted to R dense in A := C0(R) ⊕ CAP(R)
ψ := ψγ ◦ πγ
smooth (γ1, . . . , γn) hγ1 × · · · × hγn : A − → Gn
XS,fψ := {ψ, ES,f}
X =
d d t
- Diffeos act via γ, S, f, e.g.:
αΨ(f ◦ hγ) := f ◦ hΨ(γ)
SLIDE 13 4 Basics Cosmology
Ashtekar gravity (A, E) + homogeneity + isotropy 1. Elementary Variables
- choose separating space S of phase space functions
- Basic functions
hγ(A) :=
restricted
Pe
ES,f :=
not smeared
[∗E](f)
b b
γ1 γ2 γ3 γ4
restricted to R dense in A := C0(R) ⊕ CAP(R)
ψ := ψγ ◦ πγ
smooth (γ1, . . . , γn) hγ1 × · · · × hγn : A − → Gn
XS,fψ := {ψ, ES,f}
X =
d d t
residual dilations
act via γ, S, f, e.g.: αΨ(f ◦ hγ) := f ◦ hΨ(γ)
SLIDE 14 4 Basics Cosmology
Ashtekar gravity (A, E) + homogeneity + isotropy 1. Elementary Variables
- choose separating space S of phase space functions
- Basic functions
hγ(A) :=
restricted
Pe
ES,f :=
not smeared
[∗E](f)
b b
γ1 γ2 γ3 γ4
restricted to R dense in A := C0(R) ⊕ CAP(R)
ψ := ψγ ◦ πγ
smooth (γ1, . . . , γn) hγ1 × · · · × hγn : A − → Gn
XS,fψ := {ψ, ES,f}
X =
d d t
residual dilations
act via γ, S, f, e.g.: αΨ(f ◦ hγ) := f ◦ hΨ(γ)
α
λ
( X ) = λ X
SLIDE 15 5 Cosmological Holonomy-Flux Algebra LOST Theorem
H . . . ∗-algebra of all words in Cyl and X factorized by the relations a · X − X · a = i {a, X}
(CCR, a ∈ Cyl ∪ X)
ψ · ψ′ = ψ ψ′
(Cyl-module) + linearity
ω0 ω0(a · X) =
(a ∈ H, X ∈ X)
ω0(ψ) =
(ψ = ψγ ◦ πγ ∈ Cyl)
Theorem:
Lewandowski, Oko l´
- w, Sahlmann, Thiemann 2005
Assume
- dim M ≥ 2
- hypersurfaces – semianalytic
- diffeos – semianalytic
- smearings with compact support
Then ω0 is the unique state on H that is invariant w.r.t. bundle automorphisms.
SLIDE 16 5 Cosmological Holonomy-Flux Algebra THE Theorem
- Restricted Holonomy-Flux Algebra
HC . . . ∗-algebra of all words in B and X factorized by the relations a · X − X · a = i {a, X}
(CCR, a ∈ Cyl ∪ X)
ψ · ψ′ = ψ ψ′
(B-module) + linearity
ω0 ω0(a · X) =
(a ∈ HC, X ∈ X)
ω0(ψ0 + ψAP) =
Theorem:
Thiemann, Hanusch, Engle 2016; Fleischhack 2018
Assume
- A := C0(R) ⊕ CAP(R)
- B := {ψ ∈ A | ψ(n) ∈ A
∀n}
- hypersurfaces – semianalytic
- smearings with compact support
Then ω0 is the unique state on HC that is invariant w.r.t. dilations.
SLIDE 17 5 Cosmological Holonomy-Flux Algebra THE Theorem
- Restricted Holonomy-Flux Algebra
HC . . . ∗-algebra of all words in B and X factorized by the relations a · X − X · a = i {a, X}
(CCR, a ∈ Cyl ∪ X)
ψ · ψ′ = ψ ψ′
(B-module) + linearity
ω0 ω0(a · X) =
(a ∈ HC, X ∈ X)
ω0(ψ0 + ψAP) =
Theorem:
Thiemann, Hanusch, Engle 2016; Fleischhack 2018
Assume
- A := C0(R) ⊕ CAP(R)
- B := {ψ ∈ A | ψ(n) ∈ A
∀n}
- hypersurfaces – semianalytic
- smearings with compact support
Then ω0 is the unique state on HC that is invariant w.r.t. dilations. Remark Holds also for A := CAP(R)
SLIDE 18
6 Conclusions Jurek has found, created, inspired strong uniqueness results.
SLIDE 19
6 Conclusions Jurek has found, created, inspired strong uniqueness results. Theorem: Jurek is unique.
SLIDE 20
6 Conclusions Jurek has found, created, inspired strong uniqueness results. Theorem: Jurek is unique.
SLIDE 21
6 Conclusions Jurek has found, created, inspired strong uniqueness results. Theorem: Jurek is unique. Proof:
SLIDE 22
6 Conclusions Jurek has found, created, inspired strong uniqueness results. Theorem: Jurek is unique. Proof: Obvious.
SLIDE 23
6 Conclusions Jurek has found, created, inspired strong uniqueness results. Theorem: Jurek is unique. Proof: Obvious. qed
