Quantum formalism for systems with temporally varying discretization Philipp H¨ ohn Perimeter Institute FFP14 @ Marseille July 18th, 2014 based on PH arXiv:1401.6062, 1401.7731 and to appear and B. Dittrich, PH, T. Jacobson wip (classical formalism B. Dittrich, PH arXiv:1303.4294, 1108.1974) P. H¨ ohn (Perimeter) Discretization changing dynamics 1 / 15
Discretization changing dynamics Discrete gravity models and lattice field theory (subject to coarse graining/refining dynamics) generically feature temporally varying discretization interpret as dynamical coarse graining/refining operations φ 3 4 see also Dittrich, Steinhaus ’13 φ 2 φ 1 φ 4 4 4 4 leads to varying number of φ 6 φ 2 φ 4 φ 5 φ 1 φ 3 3 degrees of freedom in ‘time’ 3 3 3 3 3 ‘time’ How to treat evolving lattice? φ 3 φ 2 2 φ 4 2 φ 1 need ‘evolving’ phase and 1 2 2 Hilbert spaces unitarity? 2 φ 1 φ 2 φ 3 1 1 1 observables? 3 constraints and symmetries? 4 Goal: understand this systematically! P. H¨ ohn (Perimeter) Discretization changing dynamics 2 / 15
Plan of the talk Classical canonical dynamics 1 Quantum formalism 2 Vacuogenesis and QG dynamics 3 Summary and Outlook 4 P. H¨ ohn (Perimeter) Discretization changing dynamics 3 / 15
Discretization changing dynamics: global moves no Hamiltonian: discrete evolution generated by time evolution moves global time evolution moves: correspond to space-time regions 1 boundary hypersurfaces as discrete time steps 2 evolve entire hypersurface at once 3 discrete time evolution corresponds to gluing regions along common boundaries ⇒ evolves future boundary time step 2 R 2 1 glue 1 R 1 0 P. H¨ ohn (Perimeter) Discretization changing dynamics 4 / 15
Discretization changing dynamics: global moves no Hamiltonian: discrete evolution generated by time evolution moves global time evolution moves: correspond to space-time regions 1 boundary hypersurfaces as discrete time steps 2 evolve entire hypersurface at once 3 discrete time evolution corresponds to gluing regions along common boundaries ⇒ evolves future boundary time step 2 R 0 P. H¨ ohn (Perimeter) Discretization changing dynamics 4 / 15
Discretization changing dynamics: global moves no Hamiltonian: discrete evolution generated by time evolution moves global time evolution moves: correspond to space-time regions 1 boundary hypersurfaces as discrete time steps 2 evolve entire hypersurface at once 3 discrete time evolution corresponds to gluing regions along common boundaries ⇒ evolves future boundary time step 2 R 0 P. H¨ ohn (Perimeter) Discretization changing dynamics 4 / 15
Classical canonical dynamics [Marsden, West ’01; Gambini, Pullin ’03; Dittrich, PH ’11,’13] Associate to every region R k action S k ( x k − 1 , x k ) time step 2 R 2 1 1 R 1 0 P. H¨ ohn (Perimeter) Discretization changing dynamics 5 / 15
Classical canonical dynamics [Marsden, West ’01; Gambini, Pullin ’03; Dittrich, PH ’11,’13] Associate to every region R k action S k ( x k − 1 , x k ) time step 2 S 2 R 2 1 1 R 1 S 1 0 P. H¨ ohn (Perimeter) Discretization changing dynamics 5 / 15
Classical canonical dynamics [Marsden, West ’01; Gambini, Pullin ’03; Dittrich, PH ’11,’13] Associate to every region R k action S k ( x k − 1 , x k ) ⇒ use as generating function (# of x 0 ) � = (# of x 1 ) allowed − p 0 := − ∂ S 1 ( x 0 , x 1 ) + p 1 := ∂ S 1 ( x 0 , x 1 ) , ∂ x 0 ∂ x 1 − p : pre–momenta, + p : post–momenta time step 2 + p 2 S 2 R 2 1 − p 1 1 + p 1 R 1 S 1 − p 0 0 P. H¨ ohn (Perimeter) Discretization changing dynamics 5 / 15
Classical canonical dynamics [Marsden, West ’01; Gambini, Pullin ’03; Dittrich, PH ’11,’13] Associate to every region R k action S k ( x k − 1 , x k ) ⇒ use as generating function (# of x 0 ) � = (# of x 1 ) allowed − p 0 := − ∂ S 1 ( x 0 , x 1 ) + p 1 := ∂ S 1 ( x 0 , x 1 ) , ∂ x 0 ∂ x 1 − p : pre–momenta, + p : post–momenta defines time evolution map time step 2 + p 2 ( x 0 , − p 0 ) �→ ( x 1 , + p 1 ) S 2 R 2 1 − p 1 1 + p 1 R 1 S 1 − p 0 0 P. H¨ ohn (Perimeter) Discretization changing dynamics 5 / 15
Classical canonical dynamics [Marsden, West ’01; Gambini, Pullin ’03; Dittrich, PH ’11,’13] Associate to every region R k action S k ( x k − 1 , x k ) ⇒ use as generating function (# of x 0 ) � = (# of x 1 ) allowed − p 0 := − ∂ S 1 ( x 0 , x 1 ) + p 1 := ∂ S 1 ( x 0 , x 1 ) , ∂ x 0 ∂ x 1 − p : pre–momenta, + p : post–momenta defines time evolution map time step 2 + p 2 ( x 0 , − p 0 ) �→ ( x 1 , + p 1 ) S 2 R 2 1 − p 1 similarly, use S 2 ( x 1 , x 2 ) as gen. fct. 1 + p 1 R 1 S 1 − p 1 = − ∂ S 2 − p 0 0 ∂ x 1 ∂ x 1 = 0 ⇔ + p 1 = − p 1 momentum eom ∂ S 1 ∂ x 1 + ∂ S 2 matching P. H¨ ohn (Perimeter) Discretization changing dynamics 5 / 15
Constraints in the discrete [Dittrich, PH ’11, ’13] evolution 0 → 1 defined by − p 0 := − ∂ S 1 ( x 0 , x 1 ) + p 1 := ∂ S 1 ( x 0 , x 1 ) , ∂ x 0 ∂ x 1 ∂ 2 S 1 ⇒ obtain two types of constraints if 1 has left and right null 0 ∂ x j ∂ x i vectors + C 1 ( x 1 , + p 1 ) = 0 ⇒ post–constraints − C 0 ( x 0 , − p 0 ) = 0 ⇒ pre–constraints time evol. no longer unique: e.g., − C 0 ( x 0 , − p 0 ) = 0 ⇒ x 1 = x 1 ( x 0 , − p 0 , λ m 1 ), λ 1 : a priori free parameter P. H¨ ohn (Perimeter) Discretization changing dynamics 6 / 15
Constraint classification [Dittrich, PH ’13, PH to appear] non-trivialities arise when gluing 2 regions: impose both + C 1 , − C 1 generally, + C 1 � = − C 1 { − C 1 i , − C 1 j } ≈ 0 ≈ { + C 1 i , + C 1 { − C 1 i , + C 1 j } but j } � = 0 possibilities at step 1: x 2 , + p 2 C 1 = − C 1 = + C 1 ⇒ 1st class gauge + C 2 2 1 symmetry generator S 2 2nd class ⇒ fixes free parameters − C 1 2 1 x 1 , − p 1 − C 1 indep. of post–constraints but 1st class match 3 ⇒ non-trivial coarse graining condition for x 1 , + p 1 + C 1 1 data of move 0 → 1 S 1 + C 1 indep. of pre–constraints but 1st class ⇒ 4 − C 0 0 non-trivial coarse graining condition for data x 0 , − p 0 of move 1 → 2 P. H¨ ohn (Perimeter) Discretization changing dynamics 7 / 15
Constraint classification [Dittrich, PH ’13, PH to appear] non-trivialities arise when gluing 2 regions: impose both + C 1 , − C 1 generally, + C 1 � = − C 1 { − C 1 i , − C 1 j } ≈ 0 ≈ { + C 1 i , + C 1 { − C 1 i , + C 1 j } but j } � = 0 possibilities at step 1: C 1 = − C 1 = + C 1 ⇒ 1st class gauge 1 symmetry generator x 2 , + p 2 2nd class ⇒ fixes free parameters 2 + ˜ 2 C 2 − C 1 indep. of post–constraints but 1st class 3 ˛ ˛ ˜ ⇒ non-trivial coarse graining condition for S 02 := S 1 + S 2 ˛ xsol 1 data of move 0 → 1 + C 1 indep. of pre–constraints but 1st class ⇒ 4 − ˜ C 0 0 non-trivial coarse graining condition for data x 0 , − p 0 of move 1 → 2 P. H¨ ohn (Perimeter) Discretization changing dynamics 7 / 15
Coarse graining dynamics and pre–constraints little (coarse) info out + C 2 2 S 2 − C 1 1 ‘anti-drainer’ 1 ∄ + C 1 S 1 0 ∄ − C 0 lots of (fine) info in P. H¨ ohn (Perimeter) Discretization changing dynamics 8 / 15
Coarse graining dynamics and pre–constraints little (coarse) info out + C 2 ‘anti-drainer’ 2 ˛ ˛ ˜ S 02 := S 1 + S 2 ˛ xsol 1 − ˜ 0 C 0 new lots of (fine) info in ⇒ constraints ‘propagate’ and become move/region dependent ⇒ propagation of information becomes move/region dependent! P. H¨ ohn (Perimeter) Discretization changing dynamics 8 / 15
Quantization: general construction [PH ’14] restrict to configuration spaces Q ≃ R N k a la Dirac: ˆ C | ψ phys � = 0 impose constraints in quantum theory ´ quantum pre–/post–constraints: self-adjoint w.r.t. H kin = L 2 ( R N k , dx k ) 1 k have absolutely cont. spectrum 2 orbits non-compact 3 ⇒ proceed by group averaging [Marolf ’95, ’00] : = � I δ ( + ˆ post–physical states: + ψ phys := + P 1 ψ kin C 1 I ) ψ kin 1 1 1 = � I δ ( − ˆ − ψ phys := − P 0 ψ kin C 0 I ) ψ kin pre–physical states: 0 0 0 � C ) := 1 ds e is ˆ δ (ˆ C 2 π physical inner product on pre–/post–physical Hilbert spaces ± H phys � � k � � � ± ψ phys � ± ξ phys � phys = � ψ kin � ± P k ξ kin k � kin k k k P. H¨ ohn (Perimeter) Discretization changing dynamics 9 / 15
Quantum dynamics [PH ’14] H kin H kin 0 1 + P 1 − P 0 ∨ ∨ − H phys + H phys 0 1 P. H¨ ohn (Perimeter) Discretization changing dynamics 10 / 15
Quantum dynamics [PH ’14] H kin H kin 0 1 P 0 → 1 + P 1 − P 0 ∨ ∨ > − H phys + H phys 0 1 P. H¨ ohn (Perimeter) Discretization changing dynamics 10 / 15
Quantum dynamics [PH ’14] H kin H kin 0 1 P 0 → 1 P 1 → 0 + P 1 − P 0 ∨ ∨ < > − H phys + H phys 0 1 P. H¨ ohn (Perimeter) Discretization changing dynamics 10 / 15
Quantum dynamics [PH ’14] H kin H kin 0 1 P 0 → 1 P 1 → 0 + P 1 − P 0 ∨ < > ∨ − H phys + H phys > 0 1 U 0 → 1 P. H¨ ohn (Perimeter) Discretization changing dynamics 10 / 15
Recommend
More recommend
