Quantum formalism for systems with temporally varying discretization - - PowerPoint PPT Presentation

Quantum formalism for systems with temporally varying discretization

Philipp H¨

• hn

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¨
• hn (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

see also Dittrich, Steinhaus ’13

leads to varying number of degrees of freedom in ‘time’ How to treat evolving lattice?

1

need ‘evolving’ phase and Hilbert spaces

2

unitarity?

3

• bservables?

4

constraints and symmetries?

‘time’ φ1

1

φ2

1

φ3

1

φ1

2

φ2

2

φ3

2

φ4

2

φ1

3

φ2

3

φ3

3

φ4

3

φ5

3

φ6

3

φ1

4

φ2

4

φ3

4

φ4

4

Goal: understand this systematically!

• P. H¨
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Discretization changing dynamics 2 / 15

Plan of the talk

1

Classical canonical dynamics

2

Quantum formalism

3

Vacuogenesis and QG dynamics

4

Summary and Outlook

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• hn (Perimeter)

Discretization changing dynamics 3 / 15

Discretization changing dynamics: global moves

no Hamiltonian: discrete evolution generated by time evolution moves global time evolution moves:

1

correspond to space-time regions

2

boundary hypersurfaces as discrete time steps

3

evolve entire hypersurface at once

discrete time evolution corresponds to gluing regions along common boundaries ⇒ evolves future boundary

time step

1 1 2 glue R1 R2

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Discretization changing dynamics 4 / 15

Discretization changing dynamics: global moves

no Hamiltonian: discrete evolution generated by time evolution moves global time evolution moves:

1

correspond to space-time regions

2

boundary hypersurfaces as discrete time steps

3

evolve entire hypersurface at once

discrete time evolution corresponds to gluing regions along common boundaries ⇒ evolves future boundary

time step

2 R

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• hn (Perimeter)

Discretization changing dynamics 4 / 15

Discretization changing dynamics: global moves

no Hamiltonian: discrete evolution generated by time evolution moves global time evolution moves:

1

correspond to space-time regions

2

boundary hypersurfaces as discrete time steps

3

evolve entire hypersurface at once

discrete time evolution corresponds to gluing regions along common boundaries ⇒ evolves future boundary

time step

2 R

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• hn (Perimeter)

Discretization changing dynamics 4 / 15

Classical canonical dynamics [Marsden, West ’01; Gambini, Pullin ’03; Dittrich, PH ’11,’13]

Associate to every region Rk action Sk(xk−1, xk)

time step

1 1 2 R1 R2

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Discretization changing dynamics 5 / 15

Classical canonical dynamics [Marsden, West ’01; Gambini, Pullin ’03; Dittrich, PH ’11,’13]

Associate to every region Rk action Sk(xk−1, xk)

time step

1 1 2 R1 R2 S2 S1

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Discretization changing dynamics 5 / 15

Classical canonical dynamics [Marsden, West ’01; Gambini, Pullin ’03; Dittrich, PH ’11,’13]

Associate to every region Rk action Sk(xk−1, xk) ⇒ use as generating function (# of x0) = (# of x1) allowed

−p0 := −∂S1(x0, x1)

∂x0 ,

+p1 := ∂S1(x0, x1)

∂x1

−p: pre–momenta, +p: post–momenta

time step

1 1 2 R1 R2 S2 S1

−p0 +p1 −p1 +p2

• P. H¨
• hn (Perimeter)

Discretization changing dynamics 5 / 15

Classical canonical dynamics [Marsden, West ’01; Gambini, Pullin ’03; Dittrich, PH ’11,’13]

Associate to every region Rk action Sk(xk−1, xk) ⇒ use as generating function (# of x0) = (# of x1) allowed

−p0 := −∂S1(x0, x1)

∂x0 ,

+p1 := ∂S1(x0, x1)

∂x1

−p: pre–momenta, +p: post–momenta

defines time evolution map (x0, −p0) → (x1, +p1)

time step

1 1 2 R1 R2 S2 S1

−p0 +p1 −p1 +p2

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• hn (Perimeter)

Discretization changing dynamics 5 / 15

Classical canonical dynamics [Marsden, West ’01; Gambini, Pullin ’03; Dittrich, PH ’11,’13]

Associate to every region Rk action Sk(xk−1, xk) ⇒ use as generating function (# of x0) = (# of x1) allowed

−p0 := −∂S1(x0, x1)

∂x0 ,

+p1 := ∂S1(x0, x1)

∂x1

−p: pre–momenta, +p: post–momenta

defines time evolution map (x0, −p0) → (x1, +p1) similarly, use S2(x1, x2) as gen. fct.

−p1 = −∂S2

∂x1 eom ∂S1

∂x1 + ∂S2 ∂x1 = 0 ⇔ +p1 = −p1 momentum

matching

time step

1 1 2 R1 R2 S2 S1

−p0 +p1 −p1 +p2

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Discretization changing dynamics 5 / 15

Constraints in the discrete [Dittrich, PH ’11, ’13]

evolution 0 → 1 defined by

−p0 := −∂S1(x0, x1)

∂x0 ,

+p1 := ∂S1(x0, x1)

∂x1 ⇒ obtain two types of constraints if

∂2S1 ∂xi

0∂xj 1 has left and right null

vectors

+C 1(x1, +p1) = 0

⇒ post–constraints

−C 0(x0, −p0) = 0

⇒ pre–constraints

time evol. no longer unique: e.g., −C 0(x0, −p0) = 0 ⇒ x1 = x1(x0, −p0, λm

1 ),

λ1: a priori free parameter

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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 j }

but {−C 1

i , +C 1 j } = 0

possibilities at step 1:

1

C 1 = −C 1 = +C 1 ⇒ 1st class gauge symmetry generator

2

2nd class ⇒ fixes free parameters

3

−C 1 indep. of post–constraints but 1st class

⇒ non-trivial coarse graining condition for data of move 0 → 1

4

+C 1 indep. of pre–constraints but 1st class ⇒

non-trivial coarse graining condition for data

• f move 1 → 2

1 1 2

+C2 −C0 +C1 −C1

x2, +p2 x0, −p0 x1, +p1 x1, −p1

match

S1 S2

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• hn (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 j }

but {−C 1

i , +C 1 j } = 0

possibilities at step 1:

1

C 1 = −C 1 = +C 1 ⇒ 1st class gauge symmetry generator

2

2nd class ⇒ fixes free parameters

3

−C 1 indep. of post–constraints but 1st class

⇒ non-trivial coarse graining condition for data of move 0 → 1

4

+C 1 indep. of pre–constraints but 1st class ⇒

non-trivial coarse graining condition for data

• f move 1 → 2

2

+ ˜

C2

− ˜

C0 x2, +p2 x0, −p0 ˜ S02 := S1 + S2 ˛ ˛ ˛

xsol 1

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Discretization changing dynamics 7 / 15

Coarse graining dynamics and pre–constraints

1 1 2 S1 S2 ∄−C 0 ∄+C 1

−C 1 +C 2

‘anti-drainer’ lots of (fine) info in little (coarse) info out

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Discretization changing dynamics 8 / 15

Coarse graining dynamics and pre–constraints

2

˜ S02 := S1 + S2 ˛ ˛ ˛

xsol 1

− ˜

C 0

+C 2

‘anti-drainer’ lots of (fine) info in little (coarse) info out new ⇒ constraints ‘propagate’ and become move/region dependent ⇒ propagation of information becomes move/region dependent!

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Discretization changing dynamics 8 / 15

Quantization: general construction [PH ’14]

restrict to configuration spaces Q ≃ RNk impose constraints in quantum theory ´ a la Dirac: ˆ C|ψphys = 0 quantum pre–/post–constraints:

1

self-adjoint w.r.t. Hkin

k

= L2(RNk, dxk)

2

have absolutely cont. spectrum

3

• rbits non-compact

⇒ proceed by group averaging [Marolf ’95, ’00]: post–physical states: +ψphys

1

:= +P1 ψkin

1

=

I δ(+ ˆ

C 1

I ) ψkin 1

pre–physical states:

−ψphys

:= −P0 ψkin =

I δ(− ˆ

C 0

I ) ψkin

δ(ˆ C) := 1 2π

• ds eis ˆ

C

physical inner product on pre–/post–physical Hilbert spaces ±Hphys

k

±ψphys

k

• ±ξphys

k

phys = ψkin

k

• ±Pkξkin

k kin

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Discretization changing dynamics 9 / 15

Quantum dynamics [PH ’14]

Hkin Hkin

1 −Hphys

−P0

∨

+Hphys 1

+P1

∨

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Discretization changing dynamics 10 / 15

Quantum dynamics [PH ’14]

Hkin Hkin

1 P0→1 −Hphys

−P0

∨

+Hphys 1

+P1

∨ >

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Discretization changing dynamics 10 / 15

Quantum dynamics [PH ’14]

Hkin Hkin

1 P1→0 P0→1 −Hphys

−P0

∨ <

+Hphys 1

+P1

∨ >

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Discretization changing dynamics 10 / 15

Quantum dynamics [PH ’14]

Hkin Hkin

1 P1→0 P0→1 −Hphys

−P0

∨

U0→1

> <

+Hphys 1

+P1

∨ >

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Discretization changing dynamics 10 / 15

Quantum dynamics [PH ’14]

propagator ansatz for evolution 0 → 1 K0→1(x0, x1) = M0→1(x0, x1) eiS1(x0,x1) e.g. projector P0→1

+ψphys 1

= P0→1 ψkin =

• dx0 K0→1 ψkin

Hkin Hkin

1 P1→0 P0→1 −Hphys

−P0

∨

U0→1

> <

+Hphys 1

+P1

∨ >

⇒ requires + ˆ C 1 K0→1 = 0 = − ˆ C 0(K0→1)∗ ⇒ K0→1 = +P1 (−P0)∗ κ0→1(x0, x1), κ0→1 kinematical prop.

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• hn (Perimeter)

Discretization changing dynamics 10 / 15

Quantum dynamics [PH ’14]

propagator ansatz for evolution 0 → 1 K0→1(x0, x1) = M0→1(x0, x1) eiS1(x0,x1) e.g. projector P0→1

+ψphys 1

= P0→1 ψkin =

• dx0 K0→1 ψkin

Hkin Hkin

1 P1→0 P0→1 −Hphys

−P0

∨

U0→1

> <

+Hphys 1

+P1

∨ >

⇒ requires + ˆ C 1 K0→1 = 0 = − ˆ C 0(K0→1)∗ ⇒ K0→1 = +P1 (−P0)∗ κ0→1(x0, x1), κ0→1 kinematical prop. ⇒ +ψphys

1

=

• dx0 K0→1 ψkin
• P. H¨
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Discretization changing dynamics 10 / 15

Quantum dynamics [PH ’14]

propagator ansatz for evolution 0 → 1 K0→1(x0, x1) = M0→1(x0, x1) eiS1(x0,x1) e.g. projector P0→1

+ψphys 1

= P0→1 ψkin =

• dx0 K0→1 ψkin

Hkin Hkin

1 P1→0 P0→1 −Hphys

−P0

∨

U0→1

> <

+Hphys 1

+P1

∨ >

⇒ requires + ˆ C 1 K0→1 = 0 = − ˆ C 0(K0→1)∗ ⇒ K0→1 = +P1 (−P0)∗ κ0→1(x0, x1), κ0→1 kinematical prop. ⇒ +ψphys

1

=

• dx0 +P1 (−P0)∗ κ0→1 ψkin
• P. H¨
• hn (Perimeter)

Discretization changing dynamics 10 / 15

Quantum dynamics [PH ’14]

propagator ansatz for evolution 0 → 1 K0→1(x0, x1) = M0→1(x0, x1) eiS1(x0,x1) e.g. projector P0→1

+ψphys 1

= P0→1 ψkin =

• dx0 K0→1 ψkin

Hkin Hkin

1 P1→0 P0→1 −Hphys

−P0

∨

U0→1

> <

+Hphys 1

+P1

∨ >

⇒ requires + ˆ C 1 K0→1 = 0 = − ˆ C 0(K0→1)∗ ⇒ K0→1 = +P1 (−P0)∗ κ0→1(x0, x1), κ0→1 kinematical prop. ⇒ +ψphys

1

=

• dx0 +P1 κ0→1 −P0 ψkin
• P. H¨
• hn (Perimeter)

Discretization changing dynamics 10 / 15

Quantum dynamics [PH ’14]

propagator ansatz for evolution 0 → 1 K0→1(x0, x1) = M0→1(x0, x1) eiS1(x0,x1) e.g. projector P0→1

+ψphys 1

= P0→1 ψkin =

• dx0 K0→1 ψkin

Hkin Hkin

1 P1→0 P0→1 −Hphys

−P0

∨

U0→1

> <

+Hphys 1

+P1

∨ >

⇒ requires + ˆ C 1 K0→1 = 0 = − ˆ C 0(K0→1)∗ ⇒ K0→1 = +P1 (−P0)∗ κ0→1(x0, x1), κ0→1 kinematical prop. ⇒ +ψphys

1

=

• dx0 +P1 κ0→1 −ψphys
• P. H¨
• hn (Perimeter)

Discretization changing dynamics 10 / 15

Quantum dynamics [PH ’14]

propagator ansatz for evolution 0 → 1 K0→1(x0, x1) = M0→1(x0, x1) eiS1(x0,x1) e.g. projector P0→1

+ψphys 1

= P0→1 ψkin =

• dx0 K0→1 ψkin

Hkin Hkin

1 P1→0 P0→1 −Hphys

−P0

∨

U0→1

> <

+Hphys 1

+P1

∨ >

⇒ requires + ˆ C 1 K0→1 = 0 = − ˆ C 0(K0→1)∗ ⇒ K0→1 = +P1 (−P0)∗ κ0→1(x0, x1), κ0→1 kinematical prop. ⇒ +ψphys

1

= U0→1 −ψphys

• P. H¨
• hn (Perimeter)

Discretization changing dynamics 10 / 15

Quantum dynamics [PH ’14]

propagator ansatz for evolution 0 → 1 K0→1(x0, x1) = M0→1(x0, x1) eiS1(x0,x1) e.g. projector P0→1

+ψphys 1

= P0→1 ψkin =

• dx0 K0→1 ψkin

Hkin Hkin

1 P1→0 P0→1 −Hphys

−P0

∨

U0→1

> <

+Hphys 1

+P1

∨ >

⇒ requires + ˆ C 1 K0→1 = 0 = − ˆ C 0(K0→1)∗ ⇒ K0→1 = +P1 (−P0)∗ κ0→1(x0, x1), κ0→1 kinematical prop. ⇒ +ψphys

1

= U0→1 −ψphys U0→1 unitary: +ψphys

1

• +ξphys

1

phys = −ψphys

• −ξphys

phys

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Discretization changing dynamics 10 / 15

Composition of global moves [PH ’14]

1 1 2

R1 R2

−Hphys +Hphys 1 −Hphys 1 +Hphys 2

Σ0 Σ1 Σ2

non-trivialities arise when gluing 2 regions ⇒ amounts to concatenation of propagators K0→2 =

• dx1 K1→2 K0→1
• P. H¨
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Discretization changing dynamics 11 / 15

Composition of global moves [PH ’14]

1 1 2

R1 R2

−Hphys +Hphys 1 −Hphys 1 +Hphys 2

Σ0 Σ1 Σ2

non-trivialities arise when gluing 2 regions ⇒ amounts to concatenation of propagators K0→2 =

• dx1 +P2 κ1→2 −P1 +P1 κ0→1 −P0
• P. H¨
• hn (Perimeter)

Discretization changing dynamics 11 / 15

Composition of global moves [PH ’14]

1 1 2

R1 R2

−Hphys +Hphys 1 −Hphys 1 +Hphys 2

Σ0 Σ1 Σ2

non-trivialities arise when gluing 2 regions recall constraint classification at 1:

1

ˆ C 1 = + ˆ C 1 = − ˆ C 1: 1st class symmetry generators

2

2nd class ⇒ solve classically

3

− ˆ

C 1

A 1st class, but indep. of + ˆ

C 1

4

+ ˆ

C 1

B 1st class, but indep. of − ˆ

C 1

⇒ amounts to concatenation of propagators K0→2 =

• dx1 +P2 κ1→2 −P1 +P1 κ0→1 −P0
• P. H¨
• hn (Perimeter)

Discretization changing dynamics 11 / 15

Composition of global moves [PH ’14]

1 1 2

R1 R2

−Hphys +Hphys 1 −Hphys 1 +Hphys 2

Σ0 Σ1 Σ2

non-trivialities arise when gluing 2 regions recall constraint classification at 1:

1

ˆ C 1 = + ˆ C 1 = − ˆ C 1: 1st class symmetry generators

2

2nd class ⇒ solve classically

3

− ˆ

C 1

A 1st class, but indep. of + ˆ

C 1

4

+ ˆ

C 1

B 1st class, but indep. of − ˆ

C 1

⇒ amounts to concatenation of propagators K0→2 =

• dx1 +P2 κ1→2 −PA

1 (P1)2 +PB 1 κ0→1 −P0

• P. H¨
• hn (Perimeter)

Discretization changing dynamics 11 / 15

Composition of global moves [PH ’14]

1 1 2

R1 R2

−Hphys +Hphys 1 −Hphys 1 +Hphys 2

Σ0 Σ1 Σ2

non-trivialities arise when gluing 2 regions recall constraint classification at 1:

1

ˆ C 1 = + ˆ C 1 = − ˆ C 1: 1st class symmetry generators

2

2nd class ⇒ solve classically

3

− ˆ

C 1

A 1st class, but indep. of + ˆ

C 1

4

+ ˆ

C 1

B 1st class, but indep. of − ˆ

C 1

⇒ amounts to concatenation of propagators K0→2 =

• dx1 +P2 κ1→2 −PA

1 (P1)2 +PB 1 κ0→1 −P0

“(P1)2 → ∞” (integration over non-compact gauge orbit)

• P. H¨
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Discretization changing dynamics 11 / 15

Composition of global moves [PH ’14]

1 1 2

R1 R2

−Hphys +Hphys 1 −Hphys 1 +Hphys 2

Σ0 Σ1 Σ2

non-trivialities arise when gluing 2 regions recall constraint classification at 1:

1

ˆ C 1 = + ˆ C 1 = − ˆ C 1: 1st class symmetry generators

2

2nd class ⇒ solve classically

3

− ˆ

C 1

A 1st class, but indep. of + ˆ

C 1

4

+ ˆ

C 1

B 1st class, but indep. of − ˆ

C 1

⇒ amounts to concatenation of propagators K0→2 =

• dx1 +P2 κ1→2 −PA

1 P1 +PB 1 κ0→1 −P0

“(P1)2 → ∞” (integration over non-compact gauge orbit) ⇒ regularize by dropping one instance of P1

• P. H¨
• hn (Perimeter)

Discretization changing dynamics 11 / 15

Composition of global moves [PH ’14]

1 1 2

R1 R2

−Hphys +Hphys 1 −Hphys 1 +Hphys 2

Σ0 Σ1 Σ2

non-trivialities arise when gluing 2 regions recall constraint classification at 1:

1

ˆ C 1 = + ˆ C 1 = − ˆ C 1: 1st class symmetry generators

2

2nd class ⇒ solve classically

3

− ˆ

C 1

A 1st class, but indep. of + ˆ

C 1

4

+ ˆ

C 1

B 1st class, but indep. of − ˆ

C 1

⇒ amounts to concatenation of propagators K0→2 =

• dx1 +P2 κ1→2 −PA

1 P1 +PB 1 κ0→1 −P0

new effective constraints − ˜ C 0, + ˜ C 2 also arise in QT

• P. H¨
• hn (Perimeter)

Discretization changing dynamics 11 / 15

Composition of global moves [PH ’14]

1 2

R1 R2

− ˜

Hphys

+ ˜

Hphys

2

Σ0 Σ2

non-trivialities arise when gluing 2 regions recall constraint classification at 1:

1

ˆ C 1 = + ˆ C 1 = − ˆ C 1: 1st class symmetry generators

2

2nd class ⇒ solve classically

3

− ˆ

C 1

A 1st class, but indep. of + ˆ

C 1

4

+ ˆ

C 1

B 1st class, but indep. of − ˆ

C 1

⇒ amounts to concatenation of propagators K0→2 =

• dx1 +P2 κ1→2 −PA

1 P1 +PB 1 κ0→1 −P0

new effective constraints − ˜ C 0, + ˜ C 2 also arise in QT ⇒ non-unitary projections physical Hilbert spaces

+ ˜

Hphys

2

:= +˜ P2 (+Hphys

2

) and − ˜ Hphys := −˜ P0(−Hphys ) ⇒ non-trivial dynamical coarse graining of discretization

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Discretization changing dynamics 11 / 15

Composition of global moves [PH ’14]

2

R

− ˜

Hphys

+ ˜

Hphys

2

Σ0 Σ2

non-trivialities arise when gluing 2 regions recall constraint classification at 1:

1

ˆ C 1 = + ˆ C 1 = − ˆ C 1: 1st class symmetry generators

2

2nd class ⇒ solve classically

3

− ˆ

C 1

A 1st class, but indep. of + ˆ

C 1

4

+ ˆ

C 1

B 1st class, but indep. of − ˆ

C 1

⇒ amounts to concatenation of propagators K0→2 =

• dx1 +P2 κ1→2 −PA

1 P1 +PB 1 κ0→1 −P0

new effective constraints − ˜ C 0, + ˜ C 2 also arise in QT ⇒ non-unitary projections physical Hilbert spaces

+ ˜

Hphys

2

:= +˜ P2 (+Hphys

2

) and − ˜ Hphys := −˜ P0(−Hphys ) ⇒ non-trivial dynamical coarse graining of discretization ⇒ physical Hilbert spaces associated to (boundary) of region (rather than time step) as in ‘general boundary formulation’ [Oeckl ’03, ’08]

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Discretization changing dynamics 11 / 15

Physical dofs on varying discretizations [Dittrich, PH ’13; PH ’14]

propagating dofs must commute with constraints, to be well-defined on −Hphys

1

/+Hphys

1

1 pre–observables − ˆ

O1: [− ˆ C 1

i , − ˆ

O1] = 0

2 post–observables + ˆ

O1: [+ ˆ C 1

i , + ˆ

O1 ] = 0

1 1 2

+ ˆ

C2

− ˆ

C0

+ ˆ

C1

− ˆ

C1

+ ˆ

O2(ˆ x2, +ˆ p2)

− ˆ

O0(ˆ x0, −ˆ p0)

+ ˆ

O1(ˆ x1, +ˆ p1)

− ˆ

O1(ˆ x1, −ˆ p1)

integrate

S1 S2

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Discretization changing dynamics 12 / 15

Physical dofs on varying discretizations [Dittrich, PH ’13; PH ’14]

propagating dofs must commute with constraints, to be well-defined on −Hphys

1

/+Hphys

1

1 pre–observables − ˆ

O1: [− ˆ C 1

i , − ˆ

O1] = 0

2 post–observables + ˆ

O1: [+ ˆ C 1

i , + ˆ

O1 ] = 0

2

+ b

˜ C2

−b

˜ C0

+ b

˜ O2(ˆ x2, +ˆ p2)

− b

˜ O0(ˆ x0, −ˆ p0) ˜ S02 := S1 + S2 ˛ ˛ ˛

xsol 1

integrating out 1: − ˜ O0/+ ˜ O2 must commute with new − ˜ C 0, + ˜ C 2 ⇒ ‘too finely grained’ dofs do not commute with new constraints ⇒ dynamical coarse graining projects out dofs irreversibly

• P. H¨
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Discretization changing dynamics 12 / 15

‘Vacuogenesis’ [wip with B. Dittrich and T. Jacobson]

idea: newly added modes ‘Euclideanized’, born in vacuum (almost) trivial toy model: ‘nothing’ → scalar field on single vertex S1 = 1 2φ2

1

→ Seucl

1

= 1 2iφ2

1

post–constraint as annihilation

• perator:

i+ ˆ C 1

eucl = ˆ

a = ˆ φ1 + iˆ p1

• Eucl. time

⇒ post–physical state is unique (Gaussian) vacuum state

+ψphys 1

∼ e− 1

2φ2 1

more complicated for large lattice

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Discretization changing dynamics 13 / 15

Discretization changing dynamics in QG

[Dittrich, Steinhaus ’13, PH ’14]

so far: states evolve in background time (e.g. lattice field theory) in QG: physical states ‘timeless’, do not evolve in external time ⇒ time evolution relational ⇒ discrete state evolution not ‘time evolution’ ⇒ discretization changing dynamics = coarse graining/refining

3D

diffeo sym. perserved ‘evolution’ as change of representation

4D

• sym. broken

non-unitary coarse graining refining non-hyperbolic

⇒ need ‘dynamical cylindrical consistency’ as in Dittrich ’12, Dittrich, Steinhaus ’13

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Discretization changing dynamics 14 / 15

Summary and Outlook

systematic classical and quantum formalism for discretization changing dynamics available constraints, observables, Hilbert spaces,... region dependent non-trivial coarse graining ⇒ non-unitary projections of physical Hilbert spaces and observables analogously with Pachner move dynamics goal: better understand

1

discretization changing dynamics in QG

2

‘vacuogenesis’

further reading: PH arXiv: 1401.6062, 1401.7731, B. Dittrich, PH arXiv:1303.4294, 1108.1974

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• hn (Perimeter)

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