[PPT] - Loop Quantum Gravity Reduced Phase Space Approach Thomas Thiemann 1 PowerPoint Presentation

SLIDE 1

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook

Loop Quantum Gravity

Reduced Phase Space Approach

Thomas Thiemann1,2

1 Albert Einstein Institut, 2 Perimeter Institute for Theoretical Physics

LQP Workshop, Hamburg, 2008

h G c

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 2

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook

Classical Canonical Formulation

Canonical formulation: M ∼ = R×σ

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 6

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Starting Point: Well posed (causal) initial value formulation for geometry and matter ⇒ Globally hyperbolic spactimes (M, g) ⇒ Topological restriction: M ∼ = R × σ [Geroch, 60’s] No classical topology change, possibly quantum?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 7

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Starting Point: Well posed (causal) initial value formulation for geometry and matter ⇒ Globally hyperbolic spactimes (M, g) ⇒ Topological restriction: M ∼ = R × σ [Geroch, 60’s] No classical topology change, possibly quantum?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 8

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Starting Point: Well posed (causal) initial value formulation for geometry and matter ⇒ Globally hyperbolic spactimes (M, g) ⇒ Topological restriction: M ∼ = R × σ [Geroch, 60’s] No classical topology change, possibly quantum?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 9

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Starting Point: Well posed (causal) initial value formulation for geometry and matter ⇒ Globally hyperbolic spactimes (M, g) ⇒ Topological restriction: M ∼ = R × σ [Geroch, 60’s] No classical topology change, possibly quantum?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 10

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Consequences: [ADM 60’s] Consider arbitrary foliations Y : R × σ → M Require spacelike leaves of foliation Σt := Y(t, σ) Pull all fields on M back to R × σ Obtain velocity phase space of spatial fields (e.g. 3 – metric qab and extrinsic curvature Kab ∝ ∂qab/∂t) Legendre transform Kab → pab singular (due to Diff(M) invariance): Spatial diffeomorphism and Hamiltonian constraints ca, c Canonical Hamiltonian Hcanon = Z

σ

d3x n c + va ca =: c(n)+ c(v)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 11

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Consequences: [ADM 60’s] Consider arbitrary foliations Y : R × σ → M Require spacelike leaves of foliation Σt := Y(t, σ) Pull all fields on M back to R × σ Obtain velocity phase space of spatial fields (e.g. 3 – metric qab and extrinsic curvature Kab ∝ ∂qab/∂t) Legendre transform Kab → pab singular (due to Diff(M) invariance): Spatial diffeomorphism and Hamiltonian constraints ca, c Canonical Hamiltonian Hcanon = Z

σ

d3x n c + va ca =: c(n)+ c(v)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 12

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Consequences: [ADM 60’s] Consider arbitrary foliations Y : R × σ → M Require spacelike leaves of foliation Σt := Y(t, σ) Pull all fields on M back to R × σ Obtain velocity phase space of spatial fields (e.g. 3 – metric qab and extrinsic curvature Kab ∝ ∂qab/∂t) Legendre transform Kab → pab singular (due to Diff(M) invariance): Spatial diffeomorphism and Hamiltonian constraints ca, c Canonical Hamiltonian Hcanon = Z

σ

d3x n c + va ca =: c(n)+ c(v)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 13

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Consequences: [ADM 60’s] Consider arbitrary foliations Y : R × σ → M Require spacelike leaves of foliation Σt := Y(t, σ) Pull all fields on M back to R × σ Obtain velocity phase space of spatial fields (e.g. 3 – metric qab and extrinsic curvature Kab ∝ ∂qab/∂t) Legendre transform Kab → pab singular (due to Diff(M) invariance): Spatial diffeomorphism and Hamiltonian constraints ca, c Canonical Hamiltonian Hcanon = Z

σ

d3x n c + va ca =: c(n)+ c(v)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 14

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Consequences: [ADM 60’s] Consider arbitrary foliations Y : R × σ → M Require spacelike leaves of foliation Σt := Y(t, σ) Pull all fields on M back to R × σ Obtain velocity phase space of spatial fields (e.g. 3 – metric qab and extrinsic curvature Kab ∝ ∂qab/∂t) Legendre transform Kab → pab singular (due to Diff(M) invariance): Spatial diffeomorphism and Hamiltonian constraints ca, c Canonical Hamiltonian Hcanon = Z

σ

d3x n c + va ca =: c(n)+ c(v)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 15

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Consequences: [ADM 60’s] Consider arbitrary foliations Y : R × σ → M Require spacelike leaves of foliation Σt := Y(t, σ) Pull all fields on M back to R × σ Obtain velocity phase space of spatial fields (e.g. 3 – metric qab and extrinsic curvature Kab ∝ ∂qab/∂t) Legendre transform Kab → pab singular (due to Diff(M) invariance): Spatial diffeomorphism and Hamiltonian constraints ca, c Canonical Hamiltonian Hcanon = Z

σ

d3x n c + va ca =: c(n)+ c(v)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 16

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Remarks: Algebraic structure of c, ca Foliation independent Symplectic structure of geometry and matter fields Foliation independent Foliation dependence encoded in lapse, shift n, va Foliation independence (Diff(M) invariance) ⇒ Hcanon ≈ 0 10 Einstein Equations equivalent to ∂tqab = {Hcanon, qab}, ∂tpab = {Hcanon, pab}, c = 0, ca = 0 In particular, building gµν, nµ from qab, n, va one obtains qµν = gµν + nµnν and {Hcanon, qµν} = [Luq]µν, uµ = nnµ+( v)µ recovery of Diff(M) (on shell).

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 17

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Remarks: Algebraic structure of c, ca Foliation independent Symplectic structure of geometry and matter fields Foliation independent Foliation dependence encoded in lapse, shift n, va Foliation independence (Diff(M) invariance) ⇒ Hcanon ≈ 0 10 Einstein Equations equivalent to ∂tqab = {Hcanon, qab}, ∂tpab = {Hcanon, pab}, c = 0, ca = 0 In particular, building gµν, nµ from qab, n, va one obtains qµν = gµν + nµnν and {Hcanon, qµν} = [Luq]µν, uµ = nnµ+( v)µ recovery of Diff(M) (on shell).

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 18

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Remarks: Algebraic structure of c, ca Foliation independent Symplectic structure of geometry and matter fields Foliation independent Foliation dependence encoded in lapse, shift n, va Foliation independence (Diff(M) invariance) ⇒ Hcanon ≈ 0 10 Einstein Equations equivalent to ∂tqab = {Hcanon, qab}, ∂tpab = {Hcanon, pab}, c = 0, ca = 0 In particular, building gµν, nµ from qab, n, va one obtains qµν = gµν + nµnν and {Hcanon, qµν} = [Luq]µν, uµ = nnµ+( v)µ recovery of Diff(M) (on shell).

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 19

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Remarks: Algebraic structure of c, ca Foliation independent Symplectic structure of geometry and matter fields Foliation independent Foliation dependence encoded in lapse, shift n, va Foliation independence (Diff(M) invariance) ⇒ Hcanon ≈ 0 10 Einstein Equations equivalent to ∂tqab = {Hcanon, qab}, ∂tpab = {Hcanon, pab}, c = 0, ca = 0 In particular, building gµν, nµ from qab, n, va one obtains qµν = gµν + nµnν and {Hcanon, qµν} = [Luq]µν, uµ = nnµ+( v)µ recovery of Diff(M) (on shell).

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 20

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Remarks: Algebraic structure of c, ca Foliation independent Symplectic structure of geometry and matter fields Foliation independent Foliation dependence encoded in lapse, shift n, va Foliation independence (Diff(M) invariance) ⇒ Hcanon ≈ 0 10 Einstein Equations equivalent to ∂tqab = {Hcanon, qab}, ∂tpab = {Hcanon, pab}, c = 0, ca = 0 In particular, building gµν, nµ from qab, n, va one obtains qµν = gµν + nµnν and {Hcanon, qµν} = [Luq]µν, uµ = nnµ+( v)µ recovery of Diff(M) (on shell).

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 21

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Remarks: Algebraic structure of c, ca Foliation independent Symplectic structure of geometry and matter fields Foliation independent Foliation dependence encoded in lapse, shift n, va Foliation independence (Diff(M) invariance) ⇒ Hcanon ≈ 0 10 Einstein Equations equivalent to ∂tqab = {Hcanon, qab}, ∂tpab = {Hcanon, pab}, c = 0, ca = 0 In particular, building gµν, nµ from qab, n, va one obtains qµν = gµν + nµnν and {Hcanon, qµν} = [Luq]µν, uµ = nnµ+( v)µ recovery of Diff(M) (on shell).

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 22

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Consistency: First class (Dirac) hypersurface deformation algebra D { c(v), c(v′)} = − c([v, v′]) { c(v), c(n)} = −c(v[n]) {c(n), c(n′)} = − c(q−1[n dn′ − n′ dn]) Universality: purely geometric origin, independent of matter content

[Hojman, Kuchaˇ r, Teitelboim 70’s]

spatial diffeos generate subalgebra but not ideal D no Lie algebra (structure functions)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 23

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Consistency: First class (Dirac) hypersurface deformation algebra D { c(v), c(v′)} = − c([v, v′]) { c(v), c(n)} = −c(v[n]) {c(n), c(n′)} = − c(q−1[n dn′ − n′ dn]) Universality: purely geometric origin, independent of matter content

[Hojman, Kuchaˇ r, Teitelboim 70’s]

spatial diffeos generate subalgebra but not ideal D no Lie algebra (structure functions)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 24

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Consistency: First class (Dirac) hypersurface deformation algebra D { c(v), c(v′)} = − c([v, v′]) { c(v), c(n)} = −c(v[n]) {c(n), c(n′)} = − c(q−1[n dn′ − n′ dn]) Universality: purely geometric origin, independent of matter content

[Hojman, Kuchaˇ r, Teitelboim 70’s]

spatial diffeos generate subalgebra but not ideal D no Lie algebra (structure functions)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 25

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Consistency: First class (Dirac) hypersurface deformation algebra D { c(v), c(v′)} = − c([v, v′]) { c(v), c(n)} = −c(v[n]) {c(n), c(n′)} = − c(q−1[n dn′ − n′ dn]) Universality: purely geometric origin, independent of matter content

[Hojman, Kuchaˇ r, Teitelboim 70’s]

spatial diffeos generate subalgebra but not ideal D no Lie algebra (structure functions)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 26

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Problem of Time

Interpretation: Hcanon constrained to vanish, no true Hamiltonian Hcanon generates gauge transformations, not physical evolution qab, pab, ... not gauge invariant, not observable {Hcanon, O} = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 27

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Problem of Time

Interpretation: Hcanon constrained to vanish, no true Hamiltonian Hcanon generates gauge transformations, not physical evolution qab, pab, ... not gauge invariant, not observable {Hcanon, O} = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 28

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Problem of Time

Interpretation: Hcanon constrained to vanish, no true Hamiltonian Hcanon generates gauge transformations, not physical evolution qab, pab, ... not gauge invariant, not observable {Hcanon, O} = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 29

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Problem of Time

Interpretation: Hcanon constrained to vanish, no true Hamiltonian Hcanon generates gauge transformations, not physical evolution qab, pab, ... not gauge invariant, not observable {Hcanon, O} = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 30

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Problem of Time

Interpretation: Hcanon constrained to vanish, no true Hamiltonian Hcanon generates gauge transformations, not physical evolution qab, pab, ... not gauge invariant, not observable {Hcanon, O} = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 31

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Solution: Deparametrisation [Brown & Kuchaˇ

r 90’s]

In GR, gauge invariant definition of curvature etc. only relative to geodesic test observers [Wald 90’s] Test observers = mathematical idealisation Brown – Kuchaˇ r dust action: 4 scalar fields T, SJ minimially coupled however: geometry backreaction taken seriously Natural: Superposition of ∞ # of point particle actions EL Equations: Dust particles move on unit geodesics, T(x) = proper time along geodesic trough x, SJ(x) labels geodesic Dark matter candidate (NIMP)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 32

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Solution: Deparametrisation [Brown & Kuchaˇ

r 90’s]

In GR, gauge invariant definition of curvature etc. only relative to geodesic test observers [Wald 90’s] Test observers = mathematical idealisation Brown – Kuchaˇ r dust action: 4 scalar fields T, SJ minimially coupled however: geometry backreaction taken seriously Natural: Superposition of ∞ # of point particle actions EL Equations: Dust particles move on unit geodesics, T(x) = proper time along geodesic trough x, SJ(x) labels geodesic Dark matter candidate (NIMP)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 33

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Solution: Deparametrisation [Brown & Kuchaˇ

r 90’s]

In GR, gauge invariant definition of curvature etc. only relative to geodesic test observers [Wald 90’s] Test observers = mathematical idealisation Brown – Kuchaˇ r dust action: 4 scalar fields T, SJ minimially coupled however: geometry backreaction taken seriously Natural: Superposition of ∞ # of point particle actions EL Equations: Dust particles move on unit geodesics, T(x) = proper time along geodesic trough x, SJ(x) labels geodesic Dark matter candidate (NIMP)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 34

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Solution: Deparametrisation [Brown & Kuchaˇ

r 90’s]

In GR, gauge invariant definition of curvature etc. only relative to geodesic test observers [Wald 90’s] Test observers = mathematical idealisation Brown – Kuchaˇ r dust action: 4 scalar fields T, SJ minimially coupled however: geometry backreaction taken seriously Natural: Superposition of ∞ # of point particle actions EL Equations: Dust particles move on unit geodesics, T(x) = proper time along geodesic trough x, SJ(x) labels geodesic Dark matter candidate (NIMP)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 35

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Solution: Deparametrisation [Brown & Kuchaˇ

r 90’s]

In GR, gauge invariant definition of curvature etc. only relative to geodesic test observers [Wald 90’s] Test observers = mathematical idealisation Brown – Kuchaˇ r dust action: 4 scalar fields T, SJ minimially coupled however: geometry backreaction taken seriously Natural: Superposition of ∞ # of point particle actions EL Equations: Dust particles move on unit geodesics, T(x) = proper time along geodesic trough x, SJ(x) labels geodesic Dark matter candidate (NIMP)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 36

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Solution: Deparametrisation [Brown & Kuchaˇ

r 90’s]

In GR, gauge invariant definition of curvature etc. only relative to geodesic test observers [Wald 90’s] Test observers = mathematical idealisation Brown – Kuchaˇ r dust action: 4 scalar fields T, SJ minimially coupled however: geometry backreaction taken seriously Natural: Superposition of ∞ # of point particle actions EL Equations: Dust particles move on unit geodesics, T(x) = proper time along geodesic trough x, SJ(x) labels geodesic Dark matter candidate (NIMP)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 37

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Deparametrisation: c := cD + cND, ca = cD

a + cND a

⇒ ˜ c = P + h, h = q [cND]2 − qabcND

a cND b

For close to flat geometry h ≈ cND ≈ hSM hard to achieve! Remarkably {˜ c(n), ˜ c(n′)} = 0 [Brown & Kuchaˇ

r 90’s]

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt ca [Kuchaˇ

r 90’s] e.g.

qab(x) → qJK(s) := [qab(x)Sa

J(x)Sb K(x)]SJ(x)=sJ, Sa JSJ ,b = δa b, Sa JSK ,a = δK J

For any spatially diffeo inv., dust indep. f get observable Of(τ) := exp({Hτ, .}) · f, Hτ := Z

σ

d3x (τ − T(x)) hND(x) Physical time evolution d dτ Of(τ) = {Hphys, Of(τ)}, Hphys := Z

σ

d3x hND(x)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 38

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Deparametrisation: c := cD + cND, ca = cD

a + cND a

⇒ ˜ c = P + h, h = q [cND]2 − qabcND

a cND b

For close to flat geometry h ≈ cND ≈ hSM hard to achieve! Remarkably {˜ c(n), ˜ c(n′)} = 0 [Brown & Kuchaˇ

r 90’s]

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt ca [Kuchaˇ

r 90’s] e.g.

qab(x) → qJK(s) := [qab(x)Sa

J(x)Sb K(x)]SJ(x)=sJ, Sa JSJ ,b = δa b, Sa JSK ,a = δK J

For any spatially diffeo inv., dust indep. f get observable Of(τ) := exp({Hτ, .}) · f, Hτ := Z

σ

d3x (τ − T(x)) hND(x) Physical time evolution d dτ Of(τ) = {Hphys, Of(τ)}, Hphys := Z

σ

d3x hND(x)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 39

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Deparametrisation: c := cD + cND, ca = cD

a + cND a

⇒ ˜ c = P + h, h = q [cND]2 − qabcND

a cND b

For close to flat geometry h ≈ cND ≈ hSM hard to achieve! Remarkably {˜ c(n), ˜ c(n′)} = 0 [Brown & Kuchaˇ

r 90’s]

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt ca [Kuchaˇ

r 90’s] e.g.

qab(x) → qJK(s) := [qab(x)Sa

J(x)Sb K(x)]SJ(x)=sJ, Sa JSJ ,b = δa b, Sa JSK ,a = δK J

For any spatially diffeo inv., dust indep. f get observable Of(τ) := exp({Hτ, .}) · f, Hτ := Z

σ

d3x (τ − T(x)) hND(x) Physical time evolution d dτ Of(τ) = {Hphys, Of(τ)}, Hphys := Z

σ

d3x hND(x)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 40

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Deparametrisation: c := cD + cND, ca = cD

a + cND a

⇒ ˜ c = P + h, h = q [cND]2 − qabcND

a cND b

For close to flat geometry h ≈ cND ≈ hSM hard to achieve! Remarkably {˜ c(n), ˜ c(n′)} = 0 [Brown & Kuchaˇ

r 90’s]

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt ca [Kuchaˇ

r 90’s] e.g.

qab(x) → qJK(s) := [qab(x)Sa

J(x)Sb K(x)]SJ(x)=sJ, Sa JSJ ,b = δa b, Sa JSK ,a = δK J

For any spatially diffeo inv., dust indep. f get observable Of(τ) := exp({Hτ, .}) · f, Hτ := Z

σ

d3x (τ − T(x)) hND(x) Physical time evolution d dτ Of(τ) = {Hphys, Of(τ)}, Hphys := Z

σ

d3x hND(x)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 41

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Deparametrisation: c := cD + cND, ca = cD

a + cND a

⇒ ˜ c = P + h, h = q [cND]2 − qabcND

a cND b

For close to flat geometry h ≈ cND ≈ hSM hard to achieve! Remarkably {˜ c(n), ˜ c(n′)} = 0 [Brown & Kuchaˇ

r 90’s]

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt ca [Kuchaˇ

r 90’s] e.g.

qab(x) → qJK(s) := [qab(x)Sa

J(x)Sb K(x)]SJ(x)=sJ, Sa JSJ ,b = δa b, Sa JSK ,a = δK J

For any spatially diffeo inv., dust indep. f get observable Of(τ) := exp({Hτ, .}) · f, Hτ := Z

σ

d3x (τ − T(x)) hND(x) Physical time evolution d dτ Of(τ) = {Hphys, Of(τ)}, Hphys := Z

σ

d3x hND(x)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 42

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Deparametrisation: c := cD + cND, ca = cD

a + cND a

⇒ ˜ c = P + h, h = q [cND]2 − qabcND

a cND b

For close to flat geometry h ≈ cND ≈ hSM hard to achieve! Remarkably {˜ c(n), ˜ c(n′)} = 0 [Brown & Kuchaˇ

r 90’s]

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt ca [Kuchaˇ

r 90’s] e.g.

qab(x) → qJK(s) := [qab(x)Sa

J(x)Sb K(x)]SJ(x)=sJ, Sa JSJ ,b = δa b, Sa JSK ,a = δK J

For any spatially diffeo inv., dust indep. f get observable Of(τ) := exp({Hτ, .}) · f, Hτ := Z

σ

d3x (τ − T(x)) hND(x) Physical time evolution d dτ Of(τ) = {Hphys, Of(τ)}, Hphys := Z

σ

d3x hND(x)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 43

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Closed observable algebra due to automorphism property of Hamiltonian flow {Of(τ), Of′(τ)} = O{f,f′}(τ) Reduced phase space Q’ion conceivable since e.g. QJK(s) := OqJK(s)(0), PJK(s) := OpJK(s)(0) ⇒ {PJK(s), QLM(s′)} = δ(J

L δK) M δ(s, s′) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 44

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Closed observable algebra due to automorphism property of Hamiltonian flow {Of(τ), Of′(τ)} = O{f,f′}(τ) Reduced phase space Q’ion conceivable since e.g. QJK(s) := OqJK(s)(0), PJK(s) := OpJK(s)(0) ⇒ {PJK(s), QLM(s′)} = δ(J

L δK) M δ(s, s′) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 45

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Physics of the Dust: Dust = Gravitational Higgs, Non-Dust = Gravitational Goldstone Bosons Conservative Hamiltonian system w/o constraints but true Hamiltonian Hamiltonian EOM wrt Hphys of physical Non-Dust dof agree with Gauge Transformations wrt Hcanon of unphysical Non-Dust dof under proper field substitutions, e.g. qab(x) ↔ Qjk(s) No constraints but energy – momentum current conservation law {Hphys, OhND(s)} = 0, {Hphys, OcND

j

(s)} = 0,

Effectively reduces # of propagating dof by 4, hence in agreement with

bservation (gravitational waves) [Giesel,Hofmann,T.T.,Winkler 00’s]

In terms of ˜ c dust fields are perfect (nowhere singular) clocks

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 46

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Physics of the Dust: Dust = Gravitational Higgs, Non-Dust = Gravitational Goldstone Bosons Conservative Hamiltonian system w/o constraints but true Hamiltonian Hamiltonian EOM wrt Hphys of physical Non-Dust dof agree with Gauge Transformations wrt Hcanon of unphysical Non-Dust dof under proper field substitutions, e.g. qab(x) ↔ Qjk(s) No constraints but energy – momentum current conservation law {Hphys, OhND(s)} = 0, {Hphys, OcND

j

(s)} = 0,

Effectively reduces # of propagating dof by 4, hence in agreement with

bservation (gravitational waves) [Giesel,Hofmann,T.T.,Winkler 00’s]

In terms of ˜ c dust fields are perfect (nowhere singular) clocks

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 47

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Physics of the Dust: Dust = Gravitational Higgs, Non-Dust = Gravitational Goldstone Bosons Conservative Hamiltonian system w/o constraints but true Hamiltonian Hamiltonian EOM wrt Hphys of physical Non-Dust dof agree with Gauge Transformations wrt Hcanon of unphysical Non-Dust dof under proper field substitutions, e.g. qab(x) ↔ Qjk(s) No constraints but energy – momentum current conservation law {Hphys, OhND(s)} = 0, {Hphys, OcND

j

(s)} = 0,

Effectively reduces # of propagating dof by 4, hence in agreement with

bservation (gravitational waves) [Giesel,Hofmann,T.T.,Winkler 00’s]

In terms of ˜ c dust fields are perfect (nowhere singular) clocks

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 48

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Physics of the Dust: Dust = Gravitational Higgs, Non-Dust = Gravitational Goldstone Bosons Conservative Hamiltonian system w/o constraints but true Hamiltonian Hamiltonian EOM wrt Hphys of physical Non-Dust dof agree with Gauge Transformations wrt Hcanon of unphysical Non-Dust dof under proper field substitutions, e.g. qab(x) ↔ Qjk(s) No constraints but energy – momentum current conservation law {Hphys, OhND(s)} = 0, {Hphys, OcND

j

(s)} = 0,

Effectively reduces # of propagating dof by 4, hence in agreement with

bservation (gravitational waves) [Giesel,Hofmann,T.T.,Winkler 00’s]

In terms of ˜ c dust fields are perfect (nowhere singular) clocks

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 49

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Physics of the Dust: Dust = Gravitational Higgs, Non-Dust = Gravitational Goldstone Bosons Conservative Hamiltonian system w/o constraints but true Hamiltonian Hamiltonian EOM wrt Hphys of physical Non-Dust dof agree with Gauge Transformations wrt Hcanon of unphysical Non-Dust dof under proper field substitutions, e.g. qab(x) ↔ Qjk(s) No constraints but energy – momentum current conservation law {Hphys, OhND(s)} = 0, {Hphys, OcND

j

(s)} = 0,

Effectively reduces # of propagating dof by 4, hence in agreement with

bservation (gravitational waves) [Giesel,Hofmann,T.T.,Winkler 00’s]

In terms of ˜ c dust fields are perfect (nowhere singular) clocks

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 50

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Physics of the Dust: Dust = Gravitational Higgs, Non-Dust = Gravitational Goldstone Bosons Conservative Hamiltonian system w/o constraints but true Hamiltonian Hamiltonian EOM wrt Hphys of physical Non-Dust dof agree with Gauge Transformations wrt Hcanon of unphysical Non-Dust dof under proper field substitutions, e.g. qab(x) ↔ Qjk(s) No constraints but energy – momentum current conservation law {Hphys, OhND(s)} = 0, {Hphys, OcND

j

(s)} = 0,

Effectively reduces # of propagating dof by 4, hence in agreement with

bservation (gravitational waves) [Giesel,Hofmann,T.T.,Winkler 00’s]

In terms of ˜ c dust fields are perfect (nowhere singular) clocks

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 51

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Canonical Quantisation Strategies

Objective: Irreducible representation of the ∗−algebra (or C∗) Aphys of Dirac observables supporting b Hphys Strategy 1: Constraint Q’ion (CQ) = Q’ion before reduction Strategy 2: Reduced phase space Q’ion (RQ) = Q’ion after reduction Complementary Advantages and Disadvantages

CQ+: Reps. of Akin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in Akin RQ-: Reps. of Aphys often difficult to find

With dust, reduced phase space q’ion simpler, avoid difficult representation of D

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Canonical Quantisation Strategies

Objective: Irreducible representation of the ∗−algebra (or C∗) Aphys of Dirac observables supporting b Hphys Strategy 1: Constraint Q’ion (CQ) = Q’ion before reduction Strategy 2: Reduced phase space Q’ion (RQ) = Q’ion after reduction Complementary Advantages and Disadvantages

CQ+: Reps. of Akin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in Akin RQ-: Reps. of Aphys often difficult to find

With dust, reduced phase space q’ion simpler, avoid difficult representation of D

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Canonical Quantisation Strategies

Objective: Irreducible representation of the ∗−algebra (or C∗) Aphys of Dirac observables supporting b Hphys Strategy 1: Constraint Q’ion (CQ) = Q’ion before reduction Strategy 2: Reduced phase space Q’ion (RQ) = Q’ion after reduction Complementary Advantages and Disadvantages

CQ+: Reps. of Akin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in Akin RQ-: Reps. of Aphys often difficult to find

With dust, reduced phase space q’ion simpler, avoid difficult representation of D

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 54

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Canonical Quantisation Strategies

Objective: Irreducible representation of the ∗−algebra (or C∗) Aphys of Dirac observables supporting b Hphys Strategy 1: Constraint Q’ion (CQ) = Q’ion before reduction Strategy 2: Reduced phase space Q’ion (RQ) = Q’ion after reduction Complementary Advantages and Disadvantages

CQ+: Reps. of Akin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in Akin RQ-: Reps. of Aphys often difficult to find

With dust, reduced phase space q’ion simpler, avoid difficult representation of D

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 55

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Canonical Quantisation Strategies

Objective: Irreducible representation of the ∗−algebra (or C∗) Aphys of Dirac observables supporting b Hphys Strategy 1: Constraint Q’ion (CQ) = Q’ion before reduction Strategy 2: Reduced phase space Q’ion (RQ) = Q’ion after reduction Complementary Advantages and Disadvantages

CQ+: Reps. of Akin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in Akin RQ-: Reps. of Aphys often difficult to find

With dust, reduced phase space q’ion simpler, avoid difficult representation of D

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 56

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Canonical Quantisation Strategies

Objective: Irreducible representation of the ∗−algebra (or C∗) Aphys of Dirac observables supporting b Hphys Strategy 1: Constraint Q’ion (CQ) = Q’ion before reduction Strategy 2: Reduced phase space Q’ion (RQ) = Q’ion after reduction Complementary Advantages and Disadvantages

CQ+: Reps. of Akin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in Akin RQ-: Reps. of Aphys often difficult to find

With dust, reduced phase space q’ion simpler, avoid difficult representation of D

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 57

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Canonical Quantisation Strategies

Objective: Irreducible representation of the ∗−algebra (or C∗) Aphys of Dirac observables supporting b Hphys Strategy 1: Constraint Q’ion (CQ) = Q’ion before reduction Strategy 2: Reduced phase space Q’ion (RQ) = Q’ion after reduction Complementary Advantages and Disadvantages

CQ+: Reps. of Akin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in Akin RQ-: Reps. of Aphys often difficult to find

With dust, reduced phase space q’ion simpler, avoid difficult representation of D

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 58

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Classical Canonical Formulation Problem of Time Canonical Quantisation Strategies

Canonical Quantisation Strategies

Objective: Irreducible representation of the ∗−algebra (or C∗) Aphys of Dirac observables supporting b Hphys Strategy 1: Constraint Q’ion (CQ) = Q’ion before reduction Strategy 2: Reduced phase space Q’ion (RQ) = Q’ion after reduction Complementary Advantages and Disadvantages

CQ+: Reps. of Akin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in Akin RQ-: Reps. of Aphys often difficult to find

With dust, reduced phase space q’ion simpler, avoid difficult representation of D

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Algebra of Kinematical Functions

Gauge Theory Formulation: Due to fermionic dof need to start with Palatini/Holst action [Ashtekar 80’s],

[Barbero, Holst, Immirzi 90’s]

After solving 2nd class (simplicity) constraints obtain {Ea

j (x), Ak b(y)} = κδa bδk j δ(x, y)

Non-dust, gravitational contributions to the constraints cgeo

j

= DaEa

j

cgeo

a

= Tr ` FabEb´ cgeo =

Tr(Fab[Ea,Eb])

√

| det(E)|

+ ....

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Algebra of Kinematical Functions

Gauge Theory Formulation: Due to fermionic dof need to start with Palatini/Holst action [Ashtekar 80’s],

[Barbero, Holst, Immirzi 90’s]

After solving 2nd class (simplicity) constraints obtain {Ea

j (x), Ak b(y)} = κδa bδk j δ(x, y)

Non-dust, gravitational contributions to the constraints cgeo

j

= DaEa

j

cgeo

a

= Tr ` FabEb´ cgeo =

Tr(Fab[Ea,Eb])

√

| det(E)|

+ ....

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 61

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Algebra of Kinematical Functions

Gauge Theory Formulation: Due to fermionic dof need to start with Palatini/Holst action [Ashtekar 80’s],

[Barbero, Holst, Immirzi 90’s]

After solving 2nd class (simplicity) constraints obtain {Ea

j (x), Ak b(y)} = κδa bδk j δ(x, y)

Non-dust, gravitational contributions to the constraints cgeo

j

= DaEa

j

cgeo

a

= Tr ` FabEb´ cgeo =

Tr(Fab[Ea,Eb])

√

| det(E)|

+ ....

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Algebra of Physical Observables

Simply define (similar for EI

j(s))

Aj

I(s) := Oaj

I(s)(0), aj

I(s) := [Aj aSa I ](x)S(x)=s,

Then {EI

j(s), Ak J(s′)} = κδk j δI J δ(s, s′)

No constraints but phys. Hamiltonian (Σ = S(σ)) H = Z

Σ

p | − ηµν Tr (τµ F ∧ {A, V}) Tr (τν F ∧ {A, V}) | =: Z d3s H(s) Physical total volume V = Z

Σ

p | det(E)| Symmetry group of H: S = N ⋊ Diff(Σ) N: Abelian normal subgroup generated by H(s), active Diff(Σ)

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Algebra of Physical Observables

Simply define (similar for EI

j(s))

Aj

I(s) := Oaj

I(s)(0), aj

I(s) := [Aj aSa I ](x)S(x)=s,

Then {EI

j(s), Ak J(s′)} = κδk j δI J δ(s, s′)

No constraints but phys. Hamiltonian (Σ = S(σ)) H = Z

Σ

p | − ηµν Tr (τµ F ∧ {A, V}) Tr (τν F ∧ {A, V}) | =: Z d3s H(s) Physical total volume V = Z

Σ

p | det(E)| Symmetry group of H: S = N ⋊ Diff(Σ) N: Abelian normal subgroup generated by H(s), active Diff(Σ)

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Algebra of Physical Observables

Simply define (similar for EI

j(s))

Aj

I(s) := Oaj

I(s)(0), aj

I(s) := [Aj aSa I ](x)S(x)=s,

Then {EI

j(s), Ak J(s′)} = κδk j δI J δ(s, s′)

No constraints but phys. Hamiltonian (Σ = S(σ)) H = Z

Σ

p | − ηµν Tr (τµ F ∧ {A, V}) Tr (τν F ∧ {A, V}) | =: Z d3s H(s) Physical total volume V = Z

Σ

p | det(E)| Symmetry group of H: S = N ⋊ Diff(Σ) N: Abelian normal subgroup generated by H(s), active Diff(Σ)

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Algebra of Physical Observables

Simply define (similar for EI

j(s))

Aj

I(s) := Oaj

I(s)(0), aj

I(s) := [Aj aSa I ](x)S(x)=s,

Then {EI

j(s), Ak J(s′)} = κδk j δI J δ(s, s′)

No constraints but phys. Hamiltonian (Σ = S(σ)) H = Z

Σ

p | − ηµν Tr (τµ F ∧ {A, V}) Tr (τν F ∧ {A, V}) | =: Z d3s H(s) Physical total volume V = Z

Σ

p | det(E)| Symmetry group of H: S = N ⋊ Diff(Σ) N: Abelian normal subgroup generated by H(s), active Diff(Σ)

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Algebra of Physical Observables

Simply define (similar for EI

j(s))

Aj

I(s) := Oaj

I(s)(0), aj

I(s) := [Aj aSa I ](x)S(x)=s,

Then {EI

j(s), Ak J(s′)} = κδk j δI J δ(s, s′)

No constraints but phys. Hamiltonian (Σ = S(σ)) H = Z

Σ

p | − ηµν Tr (τµ F ∧ {A, V}) Tr (τν F ∧ {A, V}) | =: Z d3s H(s) Physical total volume V = Z

Σ

p | det(E)| Symmetry group of H: S = N ⋊ Diff(Σ) N: Abelian normal subgroup generated by H(s), active Diff(Σ)

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Algebra of Physical Observables

Simply define (similar for EI

j(s))

Aj

I(s) := Oaj

I(s)(0), aj

I(s) := [Aj aSa I ](x)S(x)=s,

Then {EI

j(s), Ak J(s′)} = κδk j δI J δ(s, s′)

No constraints but phys. Hamiltonian (Σ = S(σ)) H = Z

Σ

p | − ηµν Tr (τµ F ∧ {A, V}) Tr (τν F ∧ {A, V}) | =: Z d3s H(s) Physical total volume V = Z

Σ

p | det(E)| Symmetry group of H: S = N ⋊ Diff(Σ) N: Abelian normal subgroup generated by H(s), active Diff(Σ)

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Physical Hilbert Space

Lattice – inspired canon. gauge theory variables [Gambini & Trias 81], [Jacobson, Rovelli,

Smolin 88]

Magnet. dof.: Holonomy (Wilson – Loop)

A(e) = P exp( Z

e

A)

Electr. dof: flux

Ej(S) = Z

S

ǫabc Ea

j dxb ∧ dxc

Poisson – brackets: {Ej(S), A(e)} = G A(e1) τj A(e2); e = e1 ◦ e2, e1 ∩ e2 = e ∩ S

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Physical Hilbert Space

Lattice – inspired canon. gauge theory variables [Gambini & Trias 81], [Jacobson, Rovelli,

Smolin 88]

Magnet. dof.: Holonomy (Wilson – Loop)

A(e) = P exp( Z

e

A)

Electr. dof: flux

Ej(S) = Z

S

ǫabc Ea

j dxb ∧ dxc

Poisson – brackets: {Ej(S), A(e)} = G A(e1) τj A(e2); e = e1 ◦ e2, e1 ∩ e2 = e ∩ S

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 70

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Physical Hilbert Space

Lattice – inspired canon. gauge theory variables [Gambini & Trias 81], [Jacobson, Rovelli,

Smolin 88]

Magnet. dof.: Holonomy (Wilson – Loop)

A(e) = P exp( Z

e

A)

Electr. dof: flux

Ej(S) = Z

S

ǫabc Ea

j dxb ∧ dxc

Poisson – brackets: {Ej(S), A(e)} = G A(e1) τj A(e2); e = e1 ◦ e2, e1 ∩ e2 = e ∩ S

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

e

S e +

1 2

e S

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 72

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Lattice – inspired gauge theory variables [Gambini & Trias 81], [Jacobson, Rovelli, Smolin 88]

Magnet. dof.: Holonomy (Wilson – Loop)

A(e) = P exp( Z

e

A)

Electr. dof: flux

Ef(S) = Z

S

ǫabc Ea

j dxb ∧ dxc

Poisson – brackets: {Ej(S), A(e)} = G A(e1) τj A(e2); e = e1 ◦ e2, e1 ∩ e2 = e ∩ S Reality conditions: A(e) = [A(e−1)]T, Ej(S) = Ej(S) Defines abstract Poisson∗−algebra Aphys. Bundle automorphisms G ∼ = G ⋊ Diff(Σ) act by Poisson automorphisms

n Aphys e.g. αg = exp({

R λjcj, .}), g = exp(λjτj) αg(A(e)) = g(b(e)) A(e)g(f(e))−1, αϕ(A(e)) = A(ϕ(e))

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 73

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Lattice – inspired gauge theory variables [Gambini & Trias 81], [Jacobson, Rovelli, Smolin 88]

Magnet. dof.: Holonomy (Wilson – Loop)

A(e) = P exp( Z

e

A)

Electr. dof: flux

Ef(S) = Z

S

ǫabc Ea

j dxb ∧ dxc

Poisson – brackets: {Ej(S), A(e)} = G A(e1) τj A(e2); e = e1 ◦ e2, e1 ∩ e2 = e ∩ S Reality conditions: A(e) = [A(e−1)]T, Ej(S) = Ej(S) Defines abstract Poisson∗−algebra Aphys. Bundle automorphisms G ∼ = G ⋊ Diff(Σ) act by Poisson automorphisms

n Aphys e.g. αg = exp({

R λjcj, .}), g = exp(λjτj) αg(A(e)) = g(b(e)) A(e)g(f(e))−1, αϕ(A(e)) = A(ϕ(e))

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 74

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Lattice – inspired gauge theory variables [Gambini & Trias 81], [Jacobson, Rovelli, Smolin 88]

Magnet. dof.: Holonomy (Wilson – Loop)

A(e) = P exp( Z

e

A)

Electr. dof: flux

Ef(S) = Z

S

ǫabc Ea

j dxb ∧ dxc

Poisson – brackets: {Ej(S), A(e)} = G A(e1) τj A(e2); e = e1 ◦ e2, e1 ∩ e2 = e ∩ S Reality conditions: A(e) = [A(e−1)]T, Ej(S) = Ej(S) Defines abstract Poisson∗−algebra Aphys. Bundle automorphisms G ∼ = G ⋊ Diff(Σ) act by Poisson automorphisms

n Aphys e.g. αg = exp({

R λjcj, .}), g = exp(λjτj) αg(A(e)) = g(b(e)) A(e)g(f(e))−1, αϕ(A(e)) = A(ϕ(e))

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 75

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

HS Reps.: In QFT no Stone – von Neumann Theorem!!! Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04] Diff(Σ) inv. states on hol. – flux algebra Aphys unique. wave functions of Hphys ψ(A) = ψγ(A(e1), .., A(eN)), ψγ : SU(2)N → C Holonomy = multiplication – operator [ A(e) ψ](A) := A(e) ψ(A) Flux = derivative – operator [ Ej(S) ψ](A) := i {Ej(S), ψ(A)} Scalar product < ψ, ψ′ >:= Z

SU(2)N dµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′ γ(h1, .., hN) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 76

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

HS Reps.: In QFT no Stone – von Neumann Theorem!!! Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04] Diff(Σ) inv. states on hol. – flux algebra Aphys unique. wave functions of Hphys ψ(A) = ψγ(A(e1), .., A(eN)), ψγ : SU(2)N → C Holonomy = multiplication – operator [ A(e) ψ](A) := A(e) ψ(A) Flux = derivative – operator [ Ej(S) ψ](A) := i {Ej(S), ψ(A)} Scalar product < ψ, ψ′ >:= Z

SU(2)N dµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′ γ(h1, .., hN) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 77

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

HS Reps.: In QFT no Stone – von Neumann Theorem!!! Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04] Diff(Σ) inv. states on hol. – flux algebra Aphys unique. wave functions of Hphys ψ(A) = ψγ(A(e1), .., A(eN)), ψγ : SU(2)N → C Holonomy = multiplication – operator [ A(e) ψ](A) := A(e) ψ(A) Flux = derivative – operator [ Ej(S) ψ](A) := i {Ej(S), ψ(A)} Scalar product < ψ, ψ′ >:= Z

SU(2)N dµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′ γ(h1, .., hN) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 78

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

HS Reps.: In QFT no Stone – von Neumann Theorem!!! Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04] Diff(Σ) inv. states on hol. – flux algebra Aphys unique. wave functions of Hphys ψ(A) = ψγ(A(e1), .., A(eN)), ψγ : SU(2)N → C Holonomy = multiplication – operator [ A(e) ψ](A) := A(e) ψ(A) Flux = derivative – operator [ Ej(S) ψ](A) := i {Ej(S), ψ(A)} Scalar product < ψ, ψ′ >:= Z

SU(2)N dµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′ γ(h1, .., hN) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 79

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

HS Reps.: In QFT no Stone – von Neumann Theorem!!! Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04] Diff(Σ) inv. states on hol. – flux algebra Aphys unique. wave functions of Hphys ψ(A) = ψγ(A(e1), .., A(eN)), ψγ : SU(2)N → C Holonomy = multiplication – operator [ A(e) ψ](A) := A(e) ψ(A) Flux = derivative – operator [ Ej(S) ψ](A) := i {Ej(S), ψ(A)} Scalar product < ψ, ψ′ >:= Z

SU(2)N dµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′ γ(h1, .., hN) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 80

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

HS Reps.: In QFT no Stone – von Neumann Theorem!!! Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04] Diff(Σ) inv. states on hol. – flux algebra Aphys unique. wave functions of Hphys ψ(A) = ψγ(A(e1), .., A(eN)), ψγ : SU(2)N → C Holonomy = multiplication – operator [ A(e) ψ](A) := A(e) ψ(A) Flux = derivative – operator [ Ej(S) ψ](A) := i {Ej(S), ψ(A)} Scalar product < ψ, ψ′ >:= Z

SU(2)N dµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′ γ(h1, .., hN) Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Spin Network ONB Tγ,j,I

v1,I1 v2,I2 v3,I3 v4,I4 v5,I5 v6,I6 v7,I7 v8,I8 v9,I9 v10,I10 vm,Im e1,j1 e2,j2 e3,j3 e4,j4 e5,j5 e6,j6 e7,j7 e8,j8 e9,j9 e10,j10 e11,j11 e12,j12 e13,j13 e14,j14 e15,j15 e16,j16 e17,j17 e18,j18 e19,j19 e20,j20 e21,j21 e22,j22 en,jn Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 82

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I Hphys not separable Hphys = ⊕γ Hγ, Hγ = span{Tγ,j,I; j = 0, I} Diff(Σ) does not downsize it since symmetry group, not gauge group Unitary representation U(ϕ)Tγ,j,I := Tϕ(γ),j,I If U(ϕ) F U(ϕ)−1 = F (e.g. F = H; all operationally defined

bservables) then “superselection” (subgraph preservation)

F Hγ ⊂ Hγ ⇒ F = ⊕γ Fγ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task:

1. Construct

Hγ ∀ γ

2. Compute < ψγ, Hψγ >=< ψγ, Hγψγ > f. semiclass. ψγ

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 83

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I Hphys not separable Hphys = ⊕γ Hγ, Hγ = span{Tγ,j,I; j = 0, I} Diff(Σ) does not downsize it since symmetry group, not gauge group Unitary representation U(ϕ)Tγ,j,I := Tϕ(γ),j,I If U(ϕ) F U(ϕ)−1 = F (e.g. F = H; all operationally defined

bservables) then “superselection” (subgraph preservation)

F Hγ ⊂ Hγ ⇒ F = ⊕γ Fγ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task:

1. Construct

Hγ ∀ γ

2. Compute < ψγ, Hψγ >=< ψγ, Hγψγ > f. semiclass. ψγ

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 84

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I Hphys not separable Hphys = ⊕γ Hγ, Hγ = span{Tγ,j,I; j = 0, I} Diff(Σ) does not downsize it since symmetry group, not gauge group Unitary representation U(ϕ)Tγ,j,I := Tϕ(γ),j,I If U(ϕ) F U(ϕ)−1 = F (e.g. F = H; all operationally defined

bservables) then “superselection” (subgraph preservation)

F Hγ ⊂ Hγ ⇒ F = ⊕γ Fγ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task:

1. Construct

Hγ ∀ γ

2. Compute < ψγ, Hψγ >=< ψγ, Hγψγ > f. semiclass. ψγ

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 85

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I Hphys not separable Hphys = ⊕γ Hγ, Hγ = span{Tγ,j,I; j = 0, I} Diff(Σ) does not downsize it since symmetry group, not gauge group Unitary representation U(ϕ)Tγ,j,I := Tϕ(γ),j,I If U(ϕ) F U(ϕ)−1 = F (e.g. F = H; all operationally defined

bservables) then “superselection” (subgraph preservation)

F Hγ ⊂ Hγ ⇒ F = ⊕γ Fγ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task:

1. Construct

Hγ ∀ γ

2. Compute < ψγ, Hψγ >=< ψγ, Hγψγ > f. semiclass. ψγ

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 86

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I Hphys not separable Hphys = ⊕γ Hγ, Hγ = span{Tγ,j,I; j = 0, I} Diff(Σ) does not downsize it since symmetry group, not gauge group Unitary representation U(ϕ)Tγ,j,I := Tϕ(γ),j,I If U(ϕ) F U(ϕ)−1 = F (e.g. F = H; all operationally defined

bservables) then “superselection” (subgraph preservation)

F Hγ ⊂ Hγ ⇒ F = ⊕γ Fγ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task:

1. Construct

Hγ ∀ γ

2. Compute < ψγ, Hψγ >=< ψγ, Hγψγ > f. semiclass. ψγ

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 87

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I Hphys not separable Hphys = ⊕γ Hγ, Hγ = span{Tγ,j,I; j = 0, I} Diff(Σ) does not downsize it since symmetry group, not gauge group Unitary representation U(ϕ)Tγ,j,I := Tϕ(γ),j,I If U(ϕ) F U(ϕ)−1 = F (e.g. F = H; all operationally defined

bservables) then “superselection” (subgraph preservation)

F Hγ ⊂ Hγ ⇒ F = ⊕γ Fγ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task:

1. Construct

Hγ ∀ γ

2. Compute < ψγ, Hψγ >=< ψγ, Hγψγ > f. semiclass. ψγ

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 88

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I Hphys not separable Hphys = ⊕γ Hγ, Hγ = span{Tγ,j,I; j = 0, I} Diff(Σ) does not downsize it since symmetry group, not gauge group Unitary representation U(ϕ)Tγ,j,I := Tϕ(γ),j,I If U(ϕ) F U(ϕ)−1 = F (e.g. F = H; all operationally defined

bservables) then “superselection” (subgraph preservation)

F Hγ ⊂ Hγ ⇒ F = ⊕γ Fγ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task:

1. Construct

Hγ ∀ γ

2. Compute < ψγ, Hψγ >=< ψγ, Hγψγ > f. semiclass. ψγ

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 89

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I Hphys not separable Hphys = ⊕γ Hγ, Hγ = span{Tγ,j,I; j = 0, I} Diff(Σ) does not downsize it since symmetry group, not gauge group Unitary representation U(ϕ)Tγ,j,I := Tϕ(γ),j,I If U(ϕ) F U(ϕ)−1 = F (e.g. F = H; all operationally defined

bservables) then “superselection” (subgraph preservation)

F Hγ ⊂ Hγ ⇒ F = ⊕γ Fγ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task:

1. Construct

Hγ ∀ γ

2. Compute < ψγ, Hψγ >=< ψγ, Hγψγ > f. semiclass. ψγ

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 90

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Physical coherent states

Choose cell complex γ∗, dual graph γ s.t. e ↔ Se Choose classical field configuration (A0(x), E0(x)), compute ge := exp(iτjEj

0(Se)) A0(e) ∈ GC

Define [Hall 90’s], [Sahlmann, T.T., Winkler 00’s] ψA0,E0 := ⊗e ψe, ψe(he) := X

π

dim(π) e−teλπ χπ(geh−1

e )

Minimal uncertainty states, that is, ∀ e ∈ E(γ)

1. < ψA0,E0, A(e)ψA0,E0 >= A0(e), < ψA0,E0, Ej(Se)ψA0,E0 >= Ej0(Se) 2. < ∆A(e)> <

∆Ej(Se))>= 1

2| < [ A(e), Ej(Se))]> |

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Physical coherent states

Choose cell complex γ∗, dual graph γ s.t. e ↔ Se Choose classical field configuration (A0(x), E0(x)), compute ge := exp(iτjEj

0(Se)) A0(e) ∈ GC

Define [Hall 90’s], [Sahlmann, T.T., Winkler 00’s] ψA0,E0 := ⊗e ψe, ψe(he) := X

π

dim(π) e−teλπ χπ(geh−1

e )

Minimal uncertainty states, that is, ∀ e ∈ E(γ)

1. < ψA0,E0, A(e)ψA0,E0 >= A0(e), < ψA0,E0, Ej(Se)ψA0,E0 >= Ej0(Se) 2. < ∆A(e)> <

∆Ej(Se))>= 1

2| < [ A(e), Ej(Se))]> |

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Physical coherent states

Choose cell complex γ∗, dual graph γ s.t. e ↔ Se Choose classical field configuration (A0(x), E0(x)), compute ge := exp(iτjEj

0(Se)) A0(e) ∈ GC

Define [Hall 90’s], [Sahlmann, T.T., Winkler 00’s] ψA0,E0 := ⊗e ψe, ψe(he) := X

π

dim(π) e−teλπ χπ(geh−1

e )

Minimal uncertainty states, that is, ∀ e ∈ E(γ)

1. < ψA0,E0, A(e)ψA0,E0 >= A0(e), < ψA0,E0, Ej(Se)ψA0,E0 >= Ej0(Se) 2. < ∆A(e)> <

∆Ej(Se))>= 1

2| < [ A(e), Ej(Se))]> |

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 93

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Physical coherent states

Choose cell complex γ∗, dual graph γ s.t. e ↔ Se Choose classical field configuration (A0(x), E0(x)), compute ge := exp(iτjEj

0(Se)) A0(e) ∈ GC

Define [Hall 90’s], [Sahlmann, T.T., Winkler 00’s] ψA0,E0 := ⊗e ψe, ψe(he) := X

π

dim(π) e−teλπ χπ(geh−1

e )

Minimal uncertainty states, that is, ∀ e ∈ E(γ)

1. < ψA0,E0, A(e)ψA0,E0 >= A0(e), < ψA0,E0, Ej(Se)ψA0,E0 >= Ej0(Se) 2. < ∆A(e)> <

∆Ej(Se))>= 1

2| < [ A(e), Ej(Se))]> |

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 94

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Physical coherent states

Choose cell complex γ∗, dual graph γ s.t. e ↔ Se Choose classical field configuration (A0(x), E0(x)), compute ge := exp(iτjEj

0(Se)) A0(e) ∈ GC

Define [Hall 90’s], [Sahlmann, T.T., Winkler 00’s] ψA0,E0 := ⊗e ψe, ψe(he) := X

π

dim(π) e−teλπ χπ(geh−1

e )

Minimal uncertainty states, that is, ∀ e ∈ E(γ)

1. < ψA0,E0, A(e)ψA0,E0 >= A0(e), < ψA0,E0, Ej(Se)ψA0,E0 >= Ej0(Se) 2. < ∆A(e)> <

∆Ej(Se))>= 1

2| < [ A(e), Ej(Se))]> |

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Overlap Function

0.2
0.1

0.1 0.2-0.1

0.05

0.05 0.1 0.25 0.5 0.75 1

0.2
0.1

0.1 0.2

arcsin(ℑ(A(e))) E(S)/L2 E(S)/L2

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Remarks: Notice: Σ just differential manifold, no Riemannian space! No a priori meaning to how densely γ embedded In particular, final operator b H cannot depend on short distance regulator used at intermediate stages of construction Notice: Operator family (b Hγ) defines Continuum operator Expect that good semiclassical states depend on graphs which are very densely embedded wrt background metric to be approximated Choose graph to be countably infinite (for compact Σ large finite graph sufficient)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 97

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Remarks: Notice: Σ just differential manifold, no Riemannian space! No a priori meaning to how densely γ embedded In particular, final operator b H cannot depend on short distance regulator used at intermediate stages of construction Notice: Operator family (b Hγ) defines Continuum operator Expect that good semiclassical states depend on graphs which are very densely embedded wrt background metric to be approximated Choose graph to be countably infinite (for compact Σ large finite graph sufficient)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 98

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Remarks: Notice: Σ just differential manifold, no Riemannian space! No a priori meaning to how densely γ embedded In particular, final operator b H cannot depend on short distance regulator used at intermediate stages of construction Notice: Operator family (b Hγ) defines Continuum operator Expect that good semiclassical states depend on graphs which are very densely embedded wrt background metric to be approximated Choose graph to be countably infinite (for compact Σ large finite graph sufficient)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 99

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Remarks: Notice: Σ just differential manifold, no Riemannian space! No a priori meaning to how densely γ embedded In particular, final operator b H cannot depend on short distance regulator used at intermediate stages of construction Notice: Operator family (b Hγ) defines Continuum operator Expect that good semiclassical states depend on graphs which are very densely embedded wrt background metric to be approximated Choose graph to be countably infinite (for compact Σ large finite graph sufficient)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 100

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Remarks: Notice: Σ just differential manifold, no Riemannian space! No a priori meaning to how densely γ embedded In particular, final operator b H cannot depend on short distance regulator used at intermediate stages of construction Notice: Operator family (b Hγ) defines Continuum operator Expect that good semiclassical states depend on graphs which are very densely embedded wrt background metric to be approximated Choose graph to be countably infinite (for compact Σ large finite graph sufficient)

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 101

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Remarks: Notice: Σ just differential manifold, no Riemannian space! No a priori meaning to how densely γ embedded In particular, final operator b H cannot depend on short distance regulator used at intermediate stages of construction Notice: Operator family (b Hγ) defines Continuum operator Expect that good semiclassical states depend on graphs which are very densely embedded wrt background metric to be approximated Choose graph to be countably infinite (for compact Σ large finite graph sufficient)

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Physical Hamiltonian

Example: Cubic graph

e

nSe

Se

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 103

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Comparison with YM theory on cubic lattice Yang – Mills on (R4, η) [Kogut & Susskind 74] Hγ =

2 g2 ǫ

X

v∈V(γ) 3

X

a=1

Tr “ E(Sa

v)2 + [2 − A(αa v) − A(αa v)−1]

” Gravity on R × Σ [T.T. 96 – 05, Giesel & T.T. 06] Hγ = ℓ4

P

X

v∈V(γ)

v u u u t ˛ ˛ ˛ ˛ ˛ ˛

3

X

µ=0

ηµµ "

3

X

a=1

Tr (τµ A(αa

v) A(ea v) [A(ea v)−1, Vv])

#2˛ ˛ ˛ ˛ ˛ ˛ Volume operator Vv = q |ǫabcTr ` E(Sa

v) E(Sb v) E(Sc v)

´ | Lattice spacing ǫ disappears, automat. UV finite. In a precise sense: ǫ replaced by ℓP

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 104

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Comparison with YM theory on cubic lattice Yang – Mills on (R4, η) [Kogut & Susskind 74] Hγ =

2 g2 ǫ

X

v∈V(γ) 3

X

a=1

Tr “ E(Sa

v)2 + [2 − A(αa v) − A(αa v)−1]

” Gravity on R × Σ [T.T. 96 – 05, Giesel & T.T. 06] Hγ = ℓ4

P

X

v∈V(γ)

v u u u t ˛ ˛ ˛ ˛ ˛ ˛

3

X

µ=0

ηµµ "

3

X

a=1

Tr (τµ A(αa

v) A(ea v) [A(ea v)−1, Vv])

#2˛ ˛ ˛ ˛ ˛ ˛ Volume operator Vv = q |ǫabcTr ` E(Sa

v) E(Sb v) E(Sc v)

´ | Lattice spacing ǫ disappears, automat. UV finite. In a precise sense: ǫ replaced by ℓP

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 105

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Comparison with YM theory on cubic lattice Yang – Mills on (R4, η) [Kogut & Susskind 74] Hγ =

2 g2 ǫ

X

v∈V(γ) 3

X

a=1

Tr “ E(Sa

v)2 + [2 − A(αa v) − A(αa v)−1]

” Gravity on R × Σ [T.T. 96 – 05, Giesel & T.T. 06] Hγ = ℓ4

P

X

v∈V(γ)

v u u u t ˛ ˛ ˛ ˛ ˛ ˛

3

X

µ=0

ηµµ "

3

X

a=1

Tr (τµ A(αa

v) A(ea v) [A(ea v)−1, Vv])

#2˛ ˛ ˛ ˛ ˛ ˛ Volume operator Vv = q |ǫabcTr ` E(Sa

v) E(Sb v) E(Sc v)

´ | Lattice spacing ǫ disappears, automat. UV finite. In a precise sense: ǫ replaced by ℓP

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 106

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Comparison with YM theory on cubic lattice Yang – Mills on (R4, η) [Kogut & Susskind 74] Hγ =

2 g2 ǫ

X

v∈V(γ) 3

X

a=1

Tr “ E(Sa

v)2 + [2 − A(αa v) − A(αa v)−1]

” Gravity on R × Σ [T.T. 96 – 05, Giesel & T.T. 06] Hγ = ℓ4

P

X

v∈V(γ)

v u u u t ˛ ˛ ˛ ˛ ˛ ˛

3

X

µ=0

ηµµ "

3

X

a=1

Tr (τµ A(αa

v) A(ea v) [A(ea v)−1, Vv])

#2˛ ˛ ˛ ˛ ˛ ˛ Volume operator Vv = q |ǫabcTr ` E(Sa

v) E(Sb v) E(Sc v)

´ | Lattice spacing ǫ disappears, automat. UV finite. In a precise sense: ǫ replaced by ℓP

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 107

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Comparison with YM theory on cubic lattice Yang – Mills on (R4, η) [Kogut & Susskind 74] Hγ =

2 g2 ǫ

X

v∈V(γ) 3

X

a=1

Tr “ E(Sa

v)2 + [2 − A(αa v) − A(αa v)−1]

” Gravity on R × Σ [T.T. 96 – 05, Giesel & T.T. 06] Hγ = ℓ4

P

X

v∈V(γ)

v u u u t ˛ ˛ ˛ ˛ ˛ ˛

3

X

µ=0

ηµµ "

3

X

a=1

Tr (τµ A(αa

v) A(ea v) [A(ea v)−1, Vv])

#2˛ ˛ ˛ ˛ ˛ ˛ Volume operator Vv = q |ǫabcTr ` E(Sa

v) E(Sb v) E(Sc v)

´ | Lattice spacing ǫ disappears, automat. UV finite. In a precise sense: ǫ replaced by ℓP

Thomas Thiemann Loop Quantum Gravity (LQG)

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Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Semiclassical Limit

Theorem [Giesel & T.T. 06] For any (A0, E0), suff. large γ

1. Exp. Value < ψA0,E0, b

HψA0,E0 >= H(A0, E0) + O()

2. Fluctuation < ψA0,E0, b

H2ψA0,E0 > − < ψA0,E0, b HψA0,E0 >2= O() Corollary

i. Quantum Hamiltonian correctly implemented
ii. For sufficiently small τ

eiτb

H/ ψA0,E0 ≈ ψA0(τ),E0(τ) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 109

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Semiclassical Limit

Theorem [Giesel & T.T. 06] For any (A0, E0), suff. large γ

1. Exp. Value < ψA0,E0, b

HψA0,E0 >= H(A0, E0) + O()

2. Fluctuation < ψA0,E0, b

H2ψA0,E0 > − < ψA0,E0, b HψA0,E0 >2= O() Corollary

i. Quantum Hamiltonian correctly implemented
ii. For sufficiently small τ

eiτb

H/ ψA0,E0 ≈ ψA0(τ),E0(τ) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 110

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Semiclassical Limit

Theorem [Giesel & T.T. 06] For any (A0, E0), suff. large γ

1. Exp. Value < ψA0,E0, b

HψA0,E0 >= H(A0, E0) + O()

2. Fluctuation < ψA0,E0, b

H2ψA0,E0 > − < ψA0,E0, b HψA0,E0 >2= O() Corollary

i. Quantum Hamiltonian correctly implemented
ii. For sufficiently small τ

eiτb

H/ ψA0,E0 ≈ ψA0(τ),E0(τ) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 111

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Semiclassical Limit

Theorem [Giesel & T.T. 06] For any (A0, E0), suff. large γ

1. Exp. Value < ψA0,E0, b

HψA0,E0 >= H(A0, E0) + O()

2. Fluctuation < ψA0,E0, b

H2ψA0,E0 > − < ψA0,E0, b HψA0,E0 >2= O() Corollary

i. Quantum Hamiltonian correctly implemented
ii. For sufficiently small τ

eiτb

H/ ψA0,E0 ≈ ψA0(τ),E0(τ) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 112

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Algebra of Kinematical Functions Algebra of Physical Observables Physical Hilbert Space Physical coherent states Physical Hamiltonian Semiclassical Limit

Semiclassical Limit

Theorem [Giesel & T.T. 06] For any (A0, E0), suff. large γ

1. Exp. Value < ψA0,E0, b

HψA0,E0 >= H(A0, E0) + O()

2. Fluctuation < ψA0,E0, b

H2ψA0,E0 > − < ψA0,E0, b HψA0,E0 >2= O() Corollary

i. Quantum Hamiltonian correctly implemented
ii. For sufficiently small τ

eiτb

H/ ψA0,E0 ≈ ψA0(τ),E0(τ) Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 113

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Summary

LQG dynamically severly constrained (uniqeness result) correct semiclassical limit of b H established Final picture equivalent to background independent, Hamiltonian “floating” lattice gauge theory

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 114

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Summary

LQG dynamically severly constrained (uniqeness result) correct semiclassical limit of b H established Final picture equivalent to background independent, Hamiltonian “floating” lattice gauge theory

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 115

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Summary

LQG dynamically severly constrained (uniqeness result) correct semiclassical limit of b H established Final picture equivalent to background independent, Hamiltonian “floating” lattice gauge theory

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 116

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Open Questions

Proposal for removing graph dependence (preservation), non separability, controlling fluctuations of all dof: Algebraic Quantum Gravity (AQG) [Giesel, T.T. 06] Implementation of classical N symmetry b H stable coherent states? Better understanding of validity/physics of dust, other types of matter? Scrutinise LQG/AQG by further consistency checks

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 117

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Open Questions

Proposal for removing graph dependence (preservation), non separability, controlling fluctuations of all dof: Algebraic Quantum Gravity (AQG) [Giesel, T.T. 06] Implementation of classical N symmetry b H stable coherent states? Better understanding of validity/physics of dust, other types of matter? Scrutinise LQG/AQG by further consistency checks

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 118

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Open Questions

Proposal for removing graph dependence (preservation), non separability, controlling fluctuations of all dof: Algebraic Quantum Gravity (AQG) [Giesel, T.T. 06] Implementation of classical N symmetry b H stable coherent states? Better understanding of validity/physics of dust, other types of matter? Scrutinise LQG/AQG by further consistency checks

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 119

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Open Questions

Proposal for removing graph dependence (preservation), non separability, controlling fluctuations of all dof: Algebraic Quantum Gravity (AQG) [Giesel, T.T. 06] Implementation of classical N symmetry b H stable coherent states? Better understanding of validity/physics of dust, other types of matter? Scrutinise LQG/AQG by further consistency checks

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 120

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Open Questions

Proposal for removing graph dependence (preservation), non separability, controlling fluctuations of all dof: Algebraic Quantum Gravity (AQG) [Giesel, T.T. 06] Implementation of classical N symmetry b H stable coherent states? Better understanding of validity/physics of dust, other types of matter? Scrutinise LQG/AQG by further consistency checks

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 121

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Outlook

If LQG/AQG pass consistency tests then: All LQG techniques developed so far can be imported to phys. HS level! Physical semiclassical techniques to make contact with standard model

phys. Hamiltonian defines S – Matrix, scattering theory, Feynman rules

conservative system, hence possible improvement of vacuum problem in QFT on time dep. backgrounds (cosmology) Correspondence between ground states of physical Hamiltonian and solutions of classical Einstein equations?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 122

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Outlook

If LQG/AQG pass consistency tests then: All LQG techniques developed so far can be imported to phys. HS level! Physical semiclassical techniques to make contact with standard model

phys. Hamiltonian defines S – Matrix, scattering theory, Feynman rules

conservative system, hence possible improvement of vacuum problem in QFT on time dep. backgrounds (cosmology) Correspondence between ground states of physical Hamiltonian and solutions of classical Einstein equations?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 123

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Outlook

If LQG/AQG pass consistency tests then: All LQG techniques developed so far can be imported to phys. HS level! Physical semiclassical techniques to make contact with standard model

phys. Hamiltonian defines S – Matrix, scattering theory, Feynman rules

conservative system, hence possible improvement of vacuum problem in QFT on time dep. backgrounds (cosmology) Correspondence between ground states of physical Hamiltonian and solutions of classical Einstein equations?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 124

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Outlook

If LQG/AQG pass consistency tests then: All LQG techniques developed so far can be imported to phys. HS level! Physical semiclassical techniques to make contact with standard model

phys. Hamiltonian defines S – Matrix, scattering theory, Feynman rules

conservative system, hence possible improvement of vacuum problem in QFT on time dep. backgrounds (cosmology) Correspondence between ground states of physical Hamiltonian and solutions of classical Einstein equations?

Thomas Thiemann Loop Quantum Gravity (LQG)

SLIDE 125

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook Summary Open Questions Outlook

Outlook

If LQG/AQG pass consistency tests then: All LQG techniques developed so far can be imported to phys. HS level! Physical semiclassical techniques to make contact with standard model

phys. Hamiltonian defines S – Matrix, scattering theory, Feynman rules

conservative system, hence possible improvement of vacuum problem in QFT on time dep. backgrounds (cosmology) Correspondence between ground states of physical Hamiltonian and solutions of classical Einstein equations?

Thomas Thiemann Loop Quantum Gravity (LQG)

Loop Quantum Gravity

Reduced Phase Space Approach

Contents

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook

Contents

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook

Contents

Conceptual Foundations Reduced Phase Space Quantisation Summary, Open Questions & Outlook

Classical Canonical Formulation

Canonical formulation: M ∼ = R×σ

Starting Point: Well posed (causal) initial value formulation for geometry and matter ⇒ Globally hyperbolic spactimes (M, g) ⇒ Topological restriction: M ∼ = R × σ [Geroch, 60’s] No classical topology change, possibly quantum?

Starting Point: Well posed (causal) initial value formulation for geometry and matter ⇒ Globally hyperbolic spactimes (M, g) ⇒ Topological restriction: M ∼ = R × σ [Geroch, 60’s] No classical topology change, possibly quantum?

Starting Point: Well posed (causal) initial value formulation for geometry and matter ⇒ Globally hyperbolic spactimes (M, g) ⇒ Topological restriction: M ∼ = R × σ [Geroch, 60’s] No classical topology change, possibly quantum?

Starting Point: Well posed (causal) initial value formulation for geometry and matter ⇒ Globally hyperbolic spactimes (M, g) ⇒ Topological restriction: M ∼ = R × σ [Geroch, 60’s] No classical topology change, possibly quantum?

d3x n c + va ca =: c(n)+ c(v)

d3x n c + va ca =: c(n)+ c(v)

d3x n c + va ca =: c(n)+ c(v)

d3x n c + va ca =: c(n)+ c(v)

d3x n c + va ca =: c(n)+ c(v)

d3x n c + va ca =: c(n)+ c(v)

Consistency: First class (Dirac) hypersurface deformation algebra D { c(v), c(v′)} = − c([v, v′]) { c(v), c(n)} = −c(v[n]) {c(n), c(n′)} = − c(q−1[n dn′ − n′ dn]) Universality: purely geometric origin, independent of matter content

spatial diffeos generate subalgebra but not ideal D no Lie algebra (structure functions)

Consistency: First class (Dirac) hypersurface deformation algebra D { c(v), c(v′)} = − c([v, v′]) { c(v), c(n)} = −c(v[n]) {c(n), c(n′)} = − c(q−1[n dn′ − n′ dn]) Universality: purely geometric origin, independent of matter content

spatial diffeos generate subalgebra but not ideal D no Lie algebra (structure functions)

Consistency: First class (Dirac) hypersurface deformation algebra D { c(v), c(v′)} = − c([v, v′]) { c(v), c(n)} = −c(v[n]) {c(n), c(n′)} = − c(q−1[n dn′ − n′ dn]) Universality: purely geometric origin, independent of matter content

spatial diffeos generate subalgebra but not ideal D no Lie algebra (structure functions)

Consistency: First class (Dirac) hypersurface deformation algebra D { c(v), c(v′)} = − c([v, v′]) { c(v), c(n)} = −c(v[n]) {c(n), c(n′)} = − c(q−1[n dn′ − n′ dn]) Universality: purely geometric origin, independent of matter content

spatial diffeos generate subalgebra but not ideal D no Lie algebra (structure functions)

Problem of Time

Interpretation: Hcanon constrained to vanish, no true Hamiltonian Hcanon generates gauge transformations, not physical evolution qab, pab, ... not gauge invariant, not observable {Hcanon, O} = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation?

Problem of Time

Interpretation: Hcanon constrained to vanish, no true Hamiltonian Hcanon generates gauge transformations, not physical evolution qab, pab, ... not gauge invariant, not observable {Hcanon, O} = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation?

Problem of Time

Interpretation: Hcanon constrained to vanish, no true Hamiltonian Hcanon generates gauge transformations, not physical evolution qab, pab, ... not gauge invariant, not observable {Hcanon, O} = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation?

Problem of Time

Interpretation: Hcanon constrained to vanish, no true Hamiltonian Hcanon generates gauge transformations, not physical evolution qab, pab, ... not gauge invariant, not observable {Hcanon, O} = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation?

Problem of Time

Interpretation: Hcanon constrained to vanish, no true Hamiltonian Hcanon generates gauge transformations, not physical evolution qab, pab, ... not gauge invariant, not observable {Hcanon, O} = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation?

Solution: Deparametrisation [Brown & Kuchaˇ

Solution: Deparametrisation [Brown & Kuchaˇ

Solution: Deparametrisation [Brown & Kuchaˇ

Solution: Deparametrisation [Brown & Kuchaˇ

Solution: Deparametrisation [Brown & Kuchaˇ

Solution: Deparametrisation [Brown & Kuchaˇ

Deparametrisation: c := cD + cND, ca = cD

⇒ ˜ c = P + h, h = q [cND]2 − qabcND

For close to flat geometry h ≈ cND ≈ hSM hard to achieve! Remarkably {˜ c(n), ˜ c(n′)} = 0 [Brown & Kuchaˇ

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt ca [Kuchaˇ

qab(x) → qJK(s) := [qab(x)Sa

For any spatially diffeo inv., dust indep. f get observable Of(τ) := exp({Hτ, .}) · f, Hτ := Z

d3x (τ − T(x)) hND(x) Physical time evolution d dτ Of(τ) = {Hphys, Of(τ)}, Hphys := Z

d3x hND(x)

Deparametrisation: c := cD + cND, ca = cD

⇒ ˜ c = P + h, h = q [cND]2 − qabcND

For close to flat geometry h ≈ cND ≈ hSM hard to achieve! Remarkably {˜ c(n), ˜ c(n′)} = 0 [Brown & Kuchaˇ

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt ca [Kuchaˇ

qab(x) → qJK(s) := [qab(x)Sa

For any spatially diffeo inv., dust indep. f get observable Of(τ) := exp({Hτ, .}) · f, Hτ := Z

d3x (τ − T(x)) hND(x) Physical time evolution d dτ Of(τ) = {Hphys, Of(τ)}, Hphys := Z

d3x hND(x)

Deparametrisation: c := cD + cND, ca = cD

⇒ ˜ c = P + h, h = q [cND]2 − qabcND

For close to flat geometry h ≈ cND ≈ hSM hard to achieve! Remarkably {˜ c(n), ˜ c(n′)} = 0 [Brown & Kuchaˇ

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt ca [Kuchaˇ

qab(x) → qJK(s) := [qab(x)Sa

For any spatially diffeo inv., dust indep. f get observable Of(τ) := exp({Hτ, .}) · f, Hτ := Z

d3x (τ − T(x)) hND(x) Physical time evolution d dτ Of(τ) = {Hphys, Of(τ)}, Hphys := Z

d3x hND(x)

Deparametrisation: c := cD + cND, ca = cD

⇒ ˜ c = P + h, h = q [cND]2 − qabcND

For close to flat geometry h ≈ cND ≈ hSM hard to achieve! Remarkably {˜ c(n), ˜ c(n′)} = 0 [Brown & Kuchaˇ

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt ca [Kuchaˇ

qab(x) → qJK(s) := [qab(x)Sa

For any spatially diffeo inv., dust indep. f get observable Of(τ) := exp({Hτ, .}) · f, Hτ := Z

d3x (τ − T(x)) hND(x) Physical time evolution d dτ Of(τ) = {Hphys, Of(τ)}, Hphys := Z

d3x hND(x)

Deparametrisation: c := cD + cND, ca = cD

⇒ ˜ c = P + h, h = q [cND]2 − qabcND

For close to flat geometry h ≈ cND ≈ hSM hard to achieve! Remarkably {˜ c(n), ˜ c(n′)} = 0 [Brown & Kuchaˇ

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt ca [Kuchaˇ