Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Remarks: Algebraic structure of c , c a Foliation independent Symplectic structure of geometry and matter fields Foliation independent Foliation dependence encoded in lapse, shift n , v a Foliation independence (Diff(M) invariance) ⇒ H canon ≈ 0 10 Einstein Equations equivalent to ∂ t q ab = { H canon , q ab } , ∂ t p ab = { H canon , p ab } , c = 0 , c a = 0 In particular, building g µν , n µ from q ab , n , v a one obtains q µν = g µν + n µ n ν and { H canon , q µν } = [ L u q ] µν , u µ = nn µ +( � v ) µ recovery of Diff(M) (on shell). Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Remarks: Algebraic structure of c , c a Foliation independent Symplectic structure of geometry and matter fields Foliation independent Foliation dependence encoded in lapse, shift n , v a Foliation independence (Diff(M) invariance) ⇒ H canon ≈ 0 10 Einstein Equations equivalent to ∂ t q ab = { H canon , q ab } , ∂ t p ab = { H canon , p ab } , c = 0 , c a = 0 In particular, building g µν , n µ from q ab , n , v a one obtains q µν = g µν + n µ n ν and { H canon , q µν } = [ L u q ] µν , u µ = nn µ +( � v ) µ recovery of Diff(M) (on shell). Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Remarks: Algebraic structure of c , c a Foliation independent Symplectic structure of geometry and matter fields Foliation independent Foliation dependence encoded in lapse, shift n , v a Foliation independence (Diff(M) invariance) ⇒ H canon ≈ 0 10 Einstein Equations equivalent to ∂ t q ab = { H canon , q ab } , ∂ t p ab = { H canon , p ab } , c = 0 , c a = 0 In particular, building g µν , n µ from q ab , n , v a one obtains q µν = g µν + n µ n ν and { H canon , q µν } = [ L u q ] µν , u µ = nn µ +( � v ) µ recovery of Diff(M) (on shell). Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Remarks: Algebraic structure of c , c a Foliation independent Symplectic structure of geometry and matter fields Foliation independent Foliation dependence encoded in lapse, shift n , v a Foliation independence (Diff(M) invariance) ⇒ H canon ≈ 0 10 Einstein Equations equivalent to ∂ t q ab = { H canon , q ab } , ∂ t p ab = { H canon , p ab } , c = 0 , c a = 0 In particular, building g µν , n µ from q ab , n , v a one obtains q µν = g µν + n µ n ν and { H canon , q µν } = [ L u q ] µν , u µ = nn µ +( � v ) µ recovery of Diff(M) (on shell). Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Consistency: First class (Dirac) hypersurface deformation algebra D c ( v ′ ) } c ([ v , v ′ ]) { � c ( v ) ,� = − � { � c ( v ) , c ( n ) } = − c ( v [ n ]) c ( q − 1 [ n dn ′ − n ′ dn ]) { c ( n ) , c ( n ′ ) } − � = 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)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Consistency: First class (Dirac) hypersurface deformation algebra D c ( v ′ ) } c ([ v , v ′ ]) { � c ( v ) ,� = − � { � c ( v ) , c ( n ) } = − c ( v [ n ]) c ( q − 1 [ n dn ′ − n ′ dn ]) { c ( n ) , c ( n ′ ) } − � = 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)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Consistency: First class (Dirac) hypersurface deformation algebra D c ( v ′ ) } c ([ v , v ′ ]) { � c ( v ) ,� = − � { � c ( v ) , c ( n ) } = − c ( v [ n ]) c ( q − 1 [ n dn ′ − n ′ dn ]) { c ( n ) , c ( n ′ ) } − � = 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)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Consistency: First class (Dirac) hypersurface deformation algebra D c ( v ′ ) } c ([ v , v ′ ]) { � c ( v ) ,� = − � { � c ( v ) , c ( n ) } = − c ( v [ n ]) c ( q − 1 [ n dn ′ − n ′ dn ]) { c ( n ) , c ( n ′ ) } − � = 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)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Problem of Time Interpretation: H canon constrained to vanish, no true Hamiltonian H canon generates gauge transformations, not physical evolution q ab , p ab , ... not gauge invariant, not observable { H canon , O } = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation? Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Problem of Time Interpretation: H canon constrained to vanish, no true Hamiltonian H canon generates gauge transformations, not physical evolution q ab , p ab , ... not gauge invariant, not observable { H canon , O } = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation? Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Problem of Time Interpretation: H canon constrained to vanish, no true Hamiltonian H canon generates gauge transformations, not physical evolution q ab , p ab , ... not gauge invariant, not observable { H canon , O } = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation? Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Problem of Time Interpretation: H canon constrained to vanish, no true Hamiltonian H canon generates gauge transformations, not physical evolution q ab , p ab , ... not gauge invariant, not observable { H canon , O } = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation? Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Problem of Time Interpretation: H canon constrained to vanish, no true Hamiltonian H canon generates gauge transformations, not physical evolution q ab , p ab , ... not gauge invariant, not observable { H canon , O } = 0 for observable, gauge invariant O Problem of time: Dynamical interpretation? Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 r dust action: 4 scalar fields T , S J minimially coupled Brown – Kuchaˇ 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, S J ( x ) labels geodesic Dark matter candidate (NIMP) Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 r dust action: 4 scalar fields T , S J minimially coupled Brown – Kuchaˇ 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, S J ( x ) labels geodesic Dark matter candidate (NIMP) Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 r dust action: 4 scalar fields T , S J minimially coupled Brown – Kuchaˇ 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, S J ( x ) labels geodesic Dark matter candidate (NIMP) Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 r dust action: 4 scalar fields T , S J minimially coupled Brown – Kuchaˇ 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, S J ( x ) labels geodesic Dark matter candidate (NIMP) Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 r dust action: 4 scalar fields T , S J minimially coupled Brown – Kuchaˇ 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, S J ( x ) labels geodesic Dark matter candidate (NIMP) Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 r dust action: 4 scalar fields T , S J minimially coupled Brown – Kuchaˇ 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, S J ( x ) labels geodesic Dark matter candidate (NIMP) Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Deparametrisation: q c := c D + c ND , c a = c D a + c ND [ c ND ] 2 − q ab c ND ⇒ ˜ c = P + h , h = a c ND a b For close to flat geometry h ≈ c ND ≈ h SM hard to achieve! c ( n ′ ) } = 0 [Brown & Kuchaˇ Remarkably { ˜ c ( n ) , ˜ r 90’s] ⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt c a [Kuchaˇ r 90’s] e.g. q ab ( x ) → q JK ( s ) := [ q a b ( x ) S a J ( x ) S b K ( x )] S J ( x )= s J , S a J S J , b = δ a b , S a J S K , a = δ K J For any spatially diffeo inv., dust indep. f get observable Z d 3 x ( τ − T ( x )) h ND ( x ) O f ( τ ) := exp ( { H τ , . } ) · f , H τ := σ Physical time evolution Z d d 3 x h ND ( x ) d τ O f ( τ ) = { H phys , O f ( τ ) } , H phys := σ Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Deparametrisation: q c := c D + c ND , c a = c D a + c ND [ c ND ] 2 − q ab c ND ⇒ ˜ c = P + h , h = a c ND a b For close to flat geometry h ≈ c ND ≈ h SM hard to achieve! c ( n ′ ) } = 0 [Brown & Kuchaˇ Remarkably { ˜ c ( n ) , ˜ r 90’s] ⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt c a [Kuchaˇ r 90’s] e.g. q ab ( x ) → q JK ( s ) := [ q a b ( x ) S a J ( x ) S b K ( x )] S J ( x )= s J , S a J S J , b = δ a b , S a J S K , a = δ K J For any spatially diffeo inv., dust indep. f get observable Z d 3 x ( τ − T ( x )) h ND ( x ) O f ( τ ) := exp ( { H τ , . } ) · f , H τ := σ Physical time evolution Z d d 3 x h ND ( x ) d τ O f ( τ ) = { H phys , O f ( τ ) } , H phys := σ Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Deparametrisation: q c := c D + c ND , c a = c D a + c ND [ c ND ] 2 − q ab c ND ⇒ ˜ c = P + h , h = a c ND a b For close to flat geometry h ≈ c ND ≈ h SM hard to achieve! c ( n ′ ) } = 0 [Brown & Kuchaˇ Remarkably { ˜ c ( n ) , ˜ r 90’s] ⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt c a [Kuchaˇ r 90’s] e.g. q ab ( x ) → q JK ( s ) := [ q a b ( x ) S a J ( x ) S b K ( x )] S J ( x )= s J , S a J S J , b = δ a b , S a J S K , a = δ K J For any spatially diffeo inv., dust indep. f get observable Z d 3 x ( τ − T ( x )) h ND ( x ) O f ( τ ) := exp ( { H τ , . } ) · f , H τ := σ Physical time evolution Z d d 3 x h ND ( x ) d τ O f ( τ ) = { H phys , O f ( τ ) } , H phys := σ Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Deparametrisation: q c := c D + c ND , c a = c D a + c ND [ c ND ] 2 − q ab c ND ⇒ ˜ c = P + h , h = a c ND a b For close to flat geometry h ≈ c ND ≈ h SM hard to achieve! c ( n ′ ) } = 0 [Brown & Kuchaˇ Remarkably { ˜ c ( n ) , ˜ r 90’s] ⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt c a [Kuchaˇ r 90’s] e.g. q ab ( x ) → q JK ( s ) := [ q a b ( x ) S a J ( x ) S b K ( x )] S J ( x )= s J , S a J S J , b = δ a b , S a J S K , a = δ K J For any spatially diffeo inv., dust indep. f get observable Z d 3 x ( τ − T ( x )) h ND ( x ) O f ( τ ) := exp ( { H τ , . } ) · f , H τ := σ Physical time evolution Z d d 3 x h ND ( x ) d τ O f ( τ ) = { H phys , O f ( τ ) } , H phys := σ Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Deparametrisation: q c := c D + c ND , c a = c D a + c ND [ c ND ] 2 − q ab c ND ⇒ ˜ c = P + h , h = a c ND a b For close to flat geometry h ≈ c ND ≈ h SM hard to achieve! c ( n ′ ) } = 0 [Brown & Kuchaˇ Remarkably { ˜ c ( n ) , ˜ r 90’s] ⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt c a [Kuchaˇ r 90’s] e.g. q ab ( x ) → q JK ( s ) := [ q a b ( x ) S a J ( x ) S b K ( x )] S J ( x )= s J , S a J S J , b = δ a b , S a J S K , a = δ K J For any spatially diffeo inv., dust indep. f get observable Z d 3 x ( τ − T ( x )) h ND ( x ) O f ( τ ) := exp ( { H τ , . } ) · f , H τ := σ Physical time evolution Z d d 3 x h ND ( x ) d τ O f ( τ ) = { H phys , O f ( τ ) } , H phys := σ Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Deparametrisation: q c := c D + c ND , c a = c D a + c ND [ c ND ] 2 − q ab c ND ⇒ ˜ c = P + h , h = a c ND a b For close to flat geometry h ≈ c ND ≈ h SM hard to achieve! c ( n ′ ) } = 0 [Brown & Kuchaˇ Remarkably { ˜ c ( n ) , ˜ r 90’s] ⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s] First symplectic reduction wrt c a [Kuchaˇ r 90’s] e.g. q ab ( x ) → q JK ( s ) := [ q a b ( x ) S a J ( x ) S b K ( x )] S J ( x )= s J , S a J S J , b = δ a b , S a J S K , a = δ K J For any spatially diffeo inv., dust indep. f get observable Z d 3 x ( τ − T ( x )) h ND ( x ) O f ( τ ) := exp ( { H τ , . } ) · f , H τ := σ Physical time evolution Z d d 3 x h ND ( x ) d τ O f ( τ ) = { H phys , O f ( τ ) } , H phys := σ Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Closed observable algebra due to automorphism property of Hamiltonian flow { O f ( τ ) , O f ′ ( τ ) } = O { f , f ′ } ( τ ) Reduced phase space Q’ion conceivable since e.g. Q JK ( s ) := O q JK ( s ) ( 0 ) , P JK ( s ) := O p JK ( s ) ( 0 ) ⇒ { P JK ( s ) , Q LM ( s ′ ) } = δ ( J L δ K ) M δ ( s , s ′ ) Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Closed observable algebra due to automorphism property of Hamiltonian flow { O f ( τ ) , O f ′ ( τ ) } = O { f , f ′ } ( τ ) Reduced phase space Q’ion conceivable since e.g. Q JK ( s ) := O q JK ( s ) ( 0 ) , P JK ( s ) := O p JK ( s ) ( 0 ) ⇒ { P JK ( s ) , Q LM ( s ′ ) } = δ ( J L δ K ) M δ ( s , s ′ ) Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 H phys of physical Non-Dust dof agree with Gauge Transformations wrt H canon of unphysical Non-Dust dof under proper field substitutions, e.g. q ab ( x ) ↔ Q jk ( s ) No constraints but energy – momentum current conservation law { H phys , O h ND ( s ) } = 0 , { H phys , O c ND ( s ) } = 0 , j Effectively reduces # of propagating dof by 4, hence in agreement with observation (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)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 H phys of physical Non-Dust dof agree with Gauge Transformations wrt H canon of unphysical Non-Dust dof under proper field substitutions, e.g. q ab ( x ) ↔ Q jk ( s ) No constraints but energy – momentum current conservation law { H phys , O h ND ( s ) } = 0 , { H phys , O c ND ( s ) } = 0 , j Effectively reduces # of propagating dof by 4, hence in agreement with observation (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)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 H phys of physical Non-Dust dof agree with Gauge Transformations wrt H canon of unphysical Non-Dust dof under proper field substitutions, e.g. q ab ( x ) ↔ Q jk ( s ) No constraints but energy – momentum current conservation law { H phys , O h ND ( s ) } = 0 , { H phys , O c ND ( s ) } = 0 , j Effectively reduces # of propagating dof by 4, hence in agreement with observation (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)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 H phys of physical Non-Dust dof agree with Gauge Transformations wrt H canon of unphysical Non-Dust dof under proper field substitutions, e.g. q ab ( x ) ↔ Q jk ( s ) No constraints but energy – momentum current conservation law { H phys , O h ND ( s ) } = 0 , { H phys , O c ND ( s ) } = 0 , j Effectively reduces # of propagating dof by 4, hence in agreement with observation (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)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 H phys of physical Non-Dust dof agree with Gauge Transformations wrt H canon of unphysical Non-Dust dof under proper field substitutions, e.g. q ab ( x ) ↔ Q jk ( s ) No constraints but energy – momentum current conservation law { H phys , O h ND ( s ) } = 0 , { H phys , O c ND ( s ) } = 0 , j Effectively reduces # of propagating dof by 4, hence in agreement with observation (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)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook 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 H phys of physical Non-Dust dof agree with Gauge Transformations wrt H canon of unphysical Non-Dust dof under proper field substitutions, e.g. q ab ( x ) ↔ Q jk ( s ) No constraints but energy – momentum current conservation law { H phys , O h ND ( s ) } = 0 , { H phys , O c ND ( s ) } = 0 , j Effectively reduces # of propagating dof by 4, hence in agreement with observation (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)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Canonical Quantisation Strategies Objective: Irreducible representation of the ∗ − algebra (or C ∗ ) A phys of Dirac observables supporting b H phys 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 A kin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in A kin RQ-: Reps. of A phys often difficult to find With dust, reduced phase space q’ion simpler, avoid difficult representation of D Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Canonical Quantisation Strategies Objective: Irreducible representation of the ∗ − algebra (or C ∗ ) A phys of Dirac observables supporting b H phys 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 A kin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in A kin RQ-: Reps. of A phys often difficult to find With dust, reduced phase space q’ion simpler, avoid difficult representation of D Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Canonical Quantisation Strategies Objective: Irreducible representation of the ∗ − algebra (or C ∗ ) A phys of Dirac observables supporting b H phys 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 A kin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in A kin RQ-: Reps. of A phys often difficult to find With dust, reduced phase space q’ion simpler, avoid difficult representation of D Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Canonical Quantisation Strategies Objective: Irreducible representation of the ∗ − algebra (or C ∗ ) A phys of Dirac observables supporting b H phys 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 A kin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in A kin RQ-: Reps. of A phys often difficult to find With dust, reduced phase space q’ion simpler, avoid difficult representation of D Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Canonical Quantisation Strategies Objective: Irreducible representation of the ∗ − algebra (or C ∗ ) A phys of Dirac observables supporting b H phys 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 A kin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in A kin RQ-: Reps. of A phys often difficult to find With dust, reduced phase space q’ion simpler, avoid difficult representation of D Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Canonical Quantisation Strategies Objective: Irreducible representation of the ∗ − algebra (or C ∗ ) A phys of Dirac observables supporting b H phys 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 A kin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in A kin RQ-: Reps. of A phys often difficult to find With dust, reduced phase space q’ion simpler, avoid difficult representation of D Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Canonical Quantisation Strategies Objective: Irreducible representation of the ∗ − algebra (or C ∗ ) A phys of Dirac observables supporting b H phys 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 A kin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in A kin RQ-: Reps. of A phys often difficult to find With dust, reduced phase space q’ion simpler, avoid difficult representation of D Thomas Thiemann Loop Quantum Gravity (LQG)
Conceptual Foundations Classical Canonical Formulation Reduced Phase Space Quantisation Problem of Time Summary, Open Questions & Outlook Canonical Quantisation Strategies Canonical Quantisation Strategies Objective: Irreducible representation of the ∗ − algebra (or C ∗ ) A phys of Dirac observables supporting b H phys 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 A kin easy to find CQ-: Phys. HS = Kernel(constraints) construction complicated (group averaging) RQ+: Directly phys. HS w/o redundant dof in A kin RQ-: Reps. of A phys often difficult to find With dust, reduced phase space q’ion simpler, avoid difficult representation of D Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 { E a j ( x ) , A k b ( y ) } = κδ a b δ k j δ ( x , y ) Non-dust, gravitational contributions to the constraints c geo D a E a = ` F ab E b ´ j j c geo = Tr a Tr ( F ab [ E a , E b ] ) c geo √ = + .... | det ( E ) | Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 { E a j ( x ) , A k b ( y ) } = κδ a b δ k j δ ( x , y ) Non-dust, gravitational contributions to the constraints c geo D a E a = ` F ab E b ´ j j c geo = Tr a Tr ( F ab [ E a , E b ] ) c geo √ = + .... | det ( E ) | Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 { E a j ( x ) , A k b ( y ) } = κδ a b δ k j δ ( x , y ) Non-dust, gravitational contributions to the constraints c geo D a E a = ` F ab E b ´ j j c geo = Tr a Tr ( F ab [ E a , E b ] ) c geo √ = + .... | det ( E ) | Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Algebra of Physical Observables Simply define (similar for E I j ( s ) ) A j I ( s ) ( 0 ) , a j I ( s ) := [ A j a S a I ( s ) := O a j I ]( x ) S ( x )= s , Then { E I j ( s ) , A k J ( s ′ ) } = κδ k j δ I J δ ( s , s ′ ) No constraints but phys. Hamiltonian ( Σ = S ( σ ) ) Z Z p | − η µν Tr ( τ µ F ∧ { A , V } ) Tr ( τ ν F ∧ { A , V } ) | =: d 3 s H ( s ) H = Σ Physical total volume Z p | det ( E ) | V = Σ 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)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Algebra of Physical Observables Simply define (similar for E I j ( s ) ) A j I ( s ) ( 0 ) , a j I ( s ) := [ A j a S a I ( s ) := O a j I ]( x ) S ( x )= s , Then { E I j ( s ) , A k J ( s ′ ) } = κδ k j δ I J δ ( s , s ′ ) No constraints but phys. Hamiltonian ( Σ = S ( σ ) ) Z Z p | − η µν Tr ( τ µ F ∧ { A , V } ) Tr ( τ ν F ∧ { A , V } ) | =: d 3 s H ( s ) H = Σ Physical total volume Z p | det ( E ) | V = Σ 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)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Algebra of Physical Observables Simply define (similar for E I j ( s ) ) A j I ( s ) ( 0 ) , a j I ( s ) := [ A j a S a I ( s ) := O a j I ]( x ) S ( x )= s , Then { E I j ( s ) , A k J ( s ′ ) } = κδ k j δ I J δ ( s , s ′ ) No constraints but phys. Hamiltonian ( Σ = S ( σ ) ) Z Z p | − η µν Tr ( τ µ F ∧ { A , V } ) Tr ( τ ν F ∧ { A , V } ) | =: d 3 s H ( s ) H = Σ Physical total volume Z p | det ( E ) | V = Σ 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)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Algebra of Physical Observables Simply define (similar for E I j ( s ) ) A j I ( s ) ( 0 ) , a j I ( s ) := [ A j a S a I ( s ) := O a j I ]( x ) S ( x )= s , Then { E I j ( s ) , A k J ( s ′ ) } = κδ k j δ I J δ ( s , s ′ ) No constraints but phys. Hamiltonian ( Σ = S ( σ ) ) Z Z p | − η µν Tr ( τ µ F ∧ { A , V } ) Tr ( τ ν F ∧ { A , V } ) | =: d 3 s H ( s ) H = Σ Physical total volume Z p | det ( E ) | V = Σ 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)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Algebra of Physical Observables Simply define (similar for E I j ( s ) ) A j I ( s ) ( 0 ) , a j I ( s ) := [ A j a S a I ( s ) := O a j I ]( x ) S ( x )= s , Then { E I j ( s ) , A k J ( s ′ ) } = κδ k j δ I J δ ( s , s ′ ) No constraints but phys. Hamiltonian ( Σ = S ( σ ) ) Z Z p | − η µν Tr ( τ µ F ∧ { A , V } ) Tr ( τ ν F ∧ { A , V } ) | =: d 3 s H ( s ) H = Σ Physical total volume Z p | det ( E ) | V = Σ 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)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Algebra of Physical Observables Simply define (similar for E I j ( s ) ) A j I ( s ) ( 0 ) , a j I ( s ) := [ A j a S a I ( s ) := O a j I ]( x ) S ( x )= s , Then { E I j ( s ) , A k J ( s ′ ) } = κδ k j δ I J δ ( s , s ′ ) No constraints but phys. Hamiltonian ( Σ = S ( σ ) ) Z Z p | − η µν Tr ( τ µ F ∧ { A , V } ) Tr ( τ ν F ∧ { A , V } ) | =: d 3 s H ( s ) H = Σ Physical total volume Z p | det ( E ) | V = Σ 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)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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) Z A ( e ) = P exp ( A ) e Electr. dof: flux Z j dx b ∧ dx c ǫ abc E a E j ( S ) = S Poisson – brackets: { E j ( S ) , A ( e ) } = G A ( e 1 ) τ j A ( e 2 ); e = e 1 ◦ e 2 , e 1 ∩ e 2 = e ∩ S Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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) Z A ( e ) = P exp ( A ) e Electr. dof: flux Z j dx b ∧ dx c ǫ abc E a E j ( S ) = S Poisson – brackets: { E j ( S ) , A ( e ) } = G A ( e 1 ) τ j A ( e 2 ); e = e 1 ◦ e 2 , e 1 ∩ e 2 = e ∩ S Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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) Z A ( e ) = P exp ( A ) e Electr. dof: flux Z j dx b ∧ dx c ǫ abc E a E j ( S ) = S Poisson – brackets: { E j ( S ) , A ( e ) } = G A ( e 1 ) τ j A ( e 2 ); e = e 1 ◦ e 2 , e 1 ∩ e 2 = e ∩ S Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit e 2 + e S ��� ��� ��� ��� ��� ��� S e 1 Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Lattice – inspired gauge theory variables [Gambini & Trias 81], [Jacobson, Rovelli, Smolin 88] Magnet. dof.: Holonomy (Wilson – Loop) Z A ( e ) = P exp ( A ) e Electr. dof: flux Z j dx b ∧ dx c ǫ abc E a E f ( S ) = S Poisson – brackets: { E j ( S ) , A ( e ) } = G A ( e 1 ) τ j A ( e 2 ); e = e 1 ◦ e 2 , e 1 ∩ e 2 = e ∩ S Reality conditions: A ( e ) = [ A ( e − 1 )] T , E j ( S ) = E j ( S ) Defines abstract Poisson ∗ − algebra A phys . Bundle automorphisms G ∼ = G ⋊ Diff (Σ) act by Poisson automorphisms R λ j c j , . } ) , g = exp ( λ j τ j ) on A phys e.g. α g = exp ( { α g ( A ( e )) = g ( b ( e )) A ( e ) g ( f ( e )) − 1 , α ϕ ( A ( e )) = A ( ϕ ( e )) Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Lattice – inspired gauge theory variables [Gambini & Trias 81], [Jacobson, Rovelli, Smolin 88] Magnet. dof.: Holonomy (Wilson – Loop) Z A ( e ) = P exp ( A ) e Electr. dof: flux Z j dx b ∧ dx c ǫ abc E a E f ( S ) = S Poisson – brackets: { E j ( S ) , A ( e ) } = G A ( e 1 ) τ j A ( e 2 ); e = e 1 ◦ e 2 , e 1 ∩ e 2 = e ∩ S Reality conditions: A ( e ) = [ A ( e − 1 )] T , E j ( S ) = E j ( S ) Defines abstract Poisson ∗ − algebra A phys . Bundle automorphisms G ∼ = G ⋊ Diff (Σ) act by Poisson automorphisms R λ j c j , . } ) , g = exp ( λ j τ j ) on A phys e.g. α g = exp ( { α g ( A ( e )) = g ( b ( e )) A ( e ) g ( f ( e )) − 1 , α ϕ ( A ( e )) = A ( ϕ ( e )) Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Lattice – inspired gauge theory variables [Gambini & Trias 81], [Jacobson, Rovelli, Smolin 88] Magnet. dof.: Holonomy (Wilson – Loop) Z A ( e ) = P exp ( A ) e Electr. dof: flux Z j dx b ∧ dx c ǫ abc E a E f ( S ) = S Poisson – brackets: { E j ( S ) , A ( e ) } = G A ( e 1 ) τ j A ( e 2 ); e = e 1 ◦ e 2 , e 1 ∩ e 2 = e ∩ S Reality conditions: A ( e ) = [ A ( e − 1 )] T , E j ( S ) = E j ( S ) Defines abstract Poisson ∗ − algebra A phys . Bundle automorphisms G ∼ = G ⋊ Diff (Σ) act by Poisson automorphisms R λ j c j , . } ) , g = exp ( λ j τ j ) on A phys e.g. α g = exp ( { α g ( A ( e )) = g ( b ( e )) A ( e ) g ( f ( e )) − 1 , α ϕ ( A ( e )) = A ( ϕ ( e )) Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 A phys unique. wave functions of H phys ψ ( A ) = ψ γ ( A ( e 1 ) , .., A ( e N )) , ψ γ : SU ( 2 ) N → C Holonomy = multiplication – operator [ � A ( e ) ψ ]( A ) := A ( e ) ψ ( A ) Flux = derivative – operator [ � E j ( S ) ψ ]( A ) := i � { E j ( S ) , ψ ( A ) } Scalar product Z < ψ, ψ ′ > := SU ( 2 ) N d µ H ( h 1 ) .. d µ H ( h N ) ψ γ ( h 1 , .., h N ) ψ ′ γ ( h 1 , .., h N ) Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 A phys unique. wave functions of H phys ψ ( A ) = ψ γ ( A ( e 1 ) , .., A ( e N )) , ψ γ : SU ( 2 ) N → C Holonomy = multiplication – operator [ � A ( e ) ψ ]( A ) := A ( e ) ψ ( A ) Flux = derivative – operator [ � E j ( S ) ψ ]( A ) := i � { E j ( S ) , ψ ( A ) } Scalar product Z < ψ, ψ ′ > := SU ( 2 ) N d µ H ( h 1 ) .. d µ H ( h N ) ψ γ ( h 1 , .., h N ) ψ ′ γ ( h 1 , .., h N ) Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 A phys unique. wave functions of H phys ψ ( A ) = ψ γ ( A ( e 1 ) , .., A ( e N )) , ψ γ : SU ( 2 ) N → C Holonomy = multiplication – operator [ � A ( e ) ψ ]( A ) := A ( e ) ψ ( A ) Flux = derivative – operator [ � E j ( S ) ψ ]( A ) := i � { E j ( S ) , ψ ( A ) } Scalar product Z < ψ, ψ ′ > := SU ( 2 ) N d µ H ( h 1 ) .. d µ H ( h N ) ψ γ ( h 1 , .., h N ) ψ ′ γ ( h 1 , .., h N ) Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 A phys unique. wave functions of H phys ψ ( A ) = ψ γ ( A ( e 1 ) , .., A ( e N )) , ψ γ : SU ( 2 ) N → C Holonomy = multiplication – operator [ � A ( e ) ψ ]( A ) := A ( e ) ψ ( A ) Flux = derivative – operator [ � E j ( S ) ψ ]( A ) := i � { E j ( S ) , ψ ( A ) } Scalar product Z < ψ, ψ ′ > := SU ( 2 ) N d µ H ( h 1 ) .. d µ H ( h N ) ψ γ ( h 1 , .., h N ) ψ ′ γ ( h 1 , .., h N ) Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 A phys unique. wave functions of H phys ψ ( A ) = ψ γ ( A ( e 1 ) , .., A ( e N )) , ψ γ : SU ( 2 ) N → C Holonomy = multiplication – operator [ � A ( e ) ψ ]( A ) := A ( e ) ψ ( A ) Flux = derivative – operator [ � E j ( S ) ψ ]( A ) := i � { E j ( S ) , ψ ( A ) } Scalar product Z < ψ, ψ ′ > := SU ( 2 ) N d µ H ( h 1 ) .. d µ H ( h N ) ψ γ ( h 1 , .., h N ) ψ ′ γ ( h 1 , .., h N ) Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 A phys unique. wave functions of H phys ψ ( A ) = ψ γ ( A ( e 1 ) , .., A ( e N )) , ψ γ : SU ( 2 ) N → C Holonomy = multiplication – operator [ � A ( e ) ψ ]( A ) := A ( e ) ψ ( A ) Flux = derivative – operator [ � E j ( S ) ψ ]( A ) := i � { E j ( S ) , ψ ( A ) } Scalar product Z < ψ, ψ ′ > := SU ( 2 ) N d µ H ( h 1 ) .. d µ H ( h N ) ψ γ ( h 1 , .., h N ) ψ ′ γ ( h 1 , .., h N ) Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Spin Network ONB T γ, j , I v8 , I8 e15 , j15 e14 , j14 e19 , j19 v6 , I6 v10 , I10 e18 , j18 e22 , j22 e17 , j17 e20 , j20 v7 , I7 e16 , j16 e8 , j8 vm , Im e21 , j21 e12 , j12 v9 , I9 e13 , j13 e11 , j11 en , jn v4 , I4 e10 , j10 v5 , I5 e2 , j2 e3 , j3 e6 , j6 v1 , I1 e7 , j7 e9 , j9 e1 , j1 e5 , j5 v3 , I3 e4 , j4 v2 , I2 Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I H phys not separable H phys = ⊕ γ H γ , H γ = span { T γ, j , I ; j � = 0 , I } Diff (Σ) does not downsize it since symmetry group, not gauge group Unitary representation U ( ϕ ) T γ, j , I := T ϕ ( γ ) , j , I If U ( ϕ ) F U ( ϕ ) − 1 = F (e.g. F = H; all operationally defined observables) then “superselection” (subgraph preservation) F H γ ⊂ H γ ⇒ F = ⊕ γ F γ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task: 1. Construct � H γ ∀ γ 2. Compute < ψ γ , H ψ γ > = < ψ γ , H γ ψ γ > f. semiclass. ψ γ Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I H phys not separable H phys = ⊕ γ H γ , H γ = span { T γ, j , I ; j � = 0 , I } Diff (Σ) does not downsize it since symmetry group, not gauge group Unitary representation U ( ϕ ) T γ, j , I := T ϕ ( γ ) , j , I If U ( ϕ ) F U ( ϕ ) − 1 = F (e.g. F = H; all operationally defined observables) then “superselection” (subgraph preservation) F H γ ⊂ H γ ⇒ F = ⊕ γ F γ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task: 1. Construct � H γ ∀ γ 2. Compute < ψ γ , H ψ γ > = < ψ γ , H γ ψ γ > f. semiclass. ψ γ Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I H phys not separable H phys = ⊕ γ H γ , H γ = span { T γ, j , I ; j � = 0 , I } Diff (Σ) does not downsize it since symmetry group, not gauge group Unitary representation U ( ϕ ) T γ, j , I := T ϕ ( γ ) , j , I If U ( ϕ ) F U ( ϕ ) − 1 = F (e.g. F = H; all operationally defined observables) then “superselection” (subgraph preservation) F H γ ⊂ H γ ⇒ F = ⊕ γ F γ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task: 1. Construct � H γ ∀ γ 2. Compute < ψ γ , H ψ γ > = < ψ γ , H γ ψ γ > f. semiclass. ψ γ Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I H phys not separable H phys = ⊕ γ H γ , H γ = span { T γ, j , I ; j � = 0 , I } Diff (Σ) does not downsize it since symmetry group, not gauge group Unitary representation U ( ϕ ) T γ, j , I := T ϕ ( γ ) , j , I If U ( ϕ ) F U ( ϕ ) − 1 = F (e.g. F = H; all operationally defined observables) then “superselection” (subgraph preservation) F H γ ⊂ H γ ⇒ F = ⊕ γ F γ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task: 1. Construct � H γ ∀ γ 2. Compute < ψ γ , H ψ γ > = < ψ γ , H γ ψ γ > f. semiclass. ψ γ Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I H phys not separable H phys = ⊕ γ H γ , H γ = span { T γ, j , I ; j � = 0 , I } Diff (Σ) does not downsize it since symmetry group, not gauge group Unitary representation U ( ϕ ) T γ, j , I := T ϕ ( γ ) , j , I If U ( ϕ ) F U ( ϕ ) − 1 = F (e.g. F = H; all operationally defined observables) then “superselection” (subgraph preservation) F H γ ⊂ H γ ⇒ F = ⊕ γ F γ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task: 1. Construct � H γ ∀ γ 2. Compute < ψ γ , H ψ γ > = < ψ γ , H γ ψ γ > f. semiclass. ψ γ Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I H phys not separable H phys = ⊕ γ H γ , H γ = span { T γ, j , I ; j � = 0 , I } Diff (Σ) does not downsize it since symmetry group, not gauge group Unitary representation U ( ϕ ) T γ, j , I := T ϕ ( γ ) , j , I If U ( ϕ ) F U ( ϕ ) − 1 = F (e.g. F = H; all operationally defined observables) then “superselection” (subgraph preservation) F H γ ⊂ H γ ⇒ F = ⊕ γ F γ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task: 1. Construct � H γ ∀ γ 2. Compute < ψ γ , H ψ γ > = < ψ γ , H γ ψ γ > f. semiclass. ψ γ Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I H phys not separable H phys = ⊕ γ H γ , H γ = span { T γ, j , I ; j � = 0 , I } Diff (Σ) does not downsize it since symmetry group, not gauge group Unitary representation U ( ϕ ) T γ, j , I := T ϕ ( γ ) , j , I If U ( ϕ ) F U ( ϕ ) − 1 = F (e.g. F = H; all operationally defined observables) then “superselection” (subgraph preservation) F H γ ⊂ H γ ⇒ F = ⊕ γ F γ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task: 1. Construct � H γ ∀ γ 2. Compute < ψ γ , H ψ γ > = < ψ γ , H γ ψ γ > f. semiclass. ψ γ Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Does rep. support b H with correct semiclassical limit? Gauss constraint solved by restriction of intertwiners I H phys not separable H phys = ⊕ γ H γ , H γ = span { T γ, j , I ; j � = 0 , I } Diff (Σ) does not downsize it since symmetry group, not gauge group Unitary representation U ( ϕ ) T γ, j , I := T ϕ ( γ ) , j , I If U ( ϕ ) F U ( ϕ ) − 1 = F (e.g. F = H; all operationally defined observables) then “superselection” (subgraph preservation) F H γ ⊂ H γ ⇒ F = ⊕ γ F γ This imposes strong constraints on regularisation of b H and removes most ambiguities usually encountered for b C ! Task: 1. Construct � H γ ∀ γ 2. Compute < ψ γ , H ψ γ > = < ψ γ , H γ ψ γ > f. semiclass. ψ γ Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Physical coherent states Choose cell complex γ ∗ , dual graph γ s.t. e ↔ S e Choose classical field configuration ( A 0 ( x ) , E 0 ( x )) , compute g e := exp ( i τ j E j 0 ( S e )) A 0 ( e ) ∈ G C Define [Hall 90’s], [Sahlmann, T.T., Winkler 00’s] X dim ( π ) e − t e λ π χ π ( g e h − 1 ψ A 0 , E 0 := ⊗ e ψ e , ψ e ( h e ) := e ) π Minimal uncertainty states, that is, ∀ e ∈ E ( γ ) 1. < ψ A 0 , E 0 , � A ( e ) ψ A 0 , E 0 > = A 0 ( e ) , < ψ A 0 , E 0 , � E j ( S e ) ψ A 0 , E 0 > = E j0 ( S e ) 2. ∆ E j ( S e )) > = 1 < � � 2 | < [ � A ( e ) , � ∆ A ( e ) > < E j ( S e ))] > | Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Physical coherent states Choose cell complex γ ∗ , dual graph γ s.t. e ↔ S e Choose classical field configuration ( A 0 ( x ) , E 0 ( x )) , compute g e := exp ( i τ j E j 0 ( S e )) A 0 ( e ) ∈ G C Define [Hall 90’s], [Sahlmann, T.T., Winkler 00’s] X dim ( π ) e − t e λ π χ π ( g e h − 1 ψ A 0 , E 0 := ⊗ e ψ e , ψ e ( h e ) := e ) π Minimal uncertainty states, that is, ∀ e ∈ E ( γ ) 1. < ψ A 0 , E 0 , � A ( e ) ψ A 0 , E 0 > = A 0 ( e ) , < ψ A 0 , E 0 , � E j ( S e ) ψ A 0 , E 0 > = E j0 ( S e ) 2. ∆ E j ( S e )) > = 1 < � � 2 | < [ � A ( e ) , � ∆ A ( e ) > < E j ( S e ))] > | Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Physical coherent states Choose cell complex γ ∗ , dual graph γ s.t. e ↔ S e Choose classical field configuration ( A 0 ( x ) , E 0 ( x )) , compute g e := exp ( i τ j E j 0 ( S e )) A 0 ( e ) ∈ G C Define [Hall 90’s], [Sahlmann, T.T., Winkler 00’s] X dim ( π ) e − t e λ π χ π ( g e h − 1 ψ A 0 , E 0 := ⊗ e ψ e , ψ e ( h e ) := e ) π Minimal uncertainty states, that is, ∀ e ∈ E ( γ ) 1. < ψ A 0 , E 0 , � A ( e ) ψ A 0 , E 0 > = A 0 ( e ) , < ψ A 0 , E 0 , � E j ( S e ) ψ A 0 , E 0 > = E j0 ( S e ) 2. ∆ E j ( S e )) > = 1 < � � 2 | < [ � A ( e ) , � ∆ A ( e ) > < E j ( S e ))] > | Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Physical coherent states Choose cell complex γ ∗ , dual graph γ s.t. e ↔ S e Choose classical field configuration ( A 0 ( x ) , E 0 ( x )) , compute g e := exp ( i τ j E j 0 ( S e )) A 0 ( e ) ∈ G C Define [Hall 90’s], [Sahlmann, T.T., Winkler 00’s] X dim ( π ) e − t e λ π χ π ( g e h − 1 ψ A 0 , E 0 := ⊗ e ψ e , ψ e ( h e ) := e ) π Minimal uncertainty states, that is, ∀ e ∈ E ( γ ) 1. < ψ A 0 , E 0 , � A ( e ) ψ A 0 , E 0 > = A 0 ( e ) , < ψ A 0 , E 0 , � E j ( S e ) ψ A 0 , E 0 > = E j0 ( S e ) 2. ∆ E j ( S e )) > = 1 < � � 2 | < [ � A ( e ) , � ∆ A ( e ) > < E j ( S e ))] > | Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Physical coherent states Choose cell complex γ ∗ , dual graph γ s.t. e ↔ S e Choose classical field configuration ( A 0 ( x ) , E 0 ( x )) , compute g e := exp ( i τ j E j 0 ( S e )) A 0 ( e ) ∈ G C Define [Hall 90’s], [Sahlmann, T.T., Winkler 00’s] X dim ( π ) e − t e λ π χ π ( g e h − 1 ψ A 0 , E 0 := ⊗ e ψ e , ψ e ( h e ) := e ) π Minimal uncertainty states, that is, ∀ e ∈ E ( γ ) 1. < ψ A 0 , E 0 , � A ( e ) ψ A 0 , E 0 > = A 0 ( e ) , < ψ A 0 , E 0 , � E j ( S e ) ψ A 0 , E 0 > = E j0 ( S e ) 2. ∆ E j ( S e )) > = 1 < � � 2 | < [ � A ( e ) , � ∆ A ( e ) > < E j ( S e ))] > | Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook Physical Hamiltonian Semiclassical Limit Overlap Function 1 0.75 0.1 0.5 0.05 0.25 0 -0.2 -0.2 0 arcsin ( ℑ ( A ( e ))) -0.1 -0.1 0 0 -0.05 E ( S ) / L 2 E ( S ) / L 2 0.1 0.1 0.2-0.1 0.2 Thomas Thiemann Loop Quantum Gravity (LQG)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 H γ ) 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)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 H γ ) 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)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 H γ ) 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)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 H γ ) 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)
Algebra of Kinematical Functions Algebra of Physical Observables Conceptual Foundations Physical Hilbert Space Reduced Phase Space Quantisation Physical coherent states Summary, Open Questions & Outlook 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 H γ ) 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)
Recommend
More recommend
