![Page 1: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/1.jpg)
Effective Dynamics in non-perturbative quantum gravity
Timothy Budd
QUIST & Thesis talk, March 15, 2012Institute for Theoretical Physics
![Page 2: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/2.jpg)
Path integral approach to quantum gravity
I Quantum mechanics according toFeynman
x i (t)
I Gravity according toEinstein
gµν(x)
I Can we combine these pillars of modern theoretical physics?
I First try pure gravity: sum over space-time geometries.
I Hope: UV complete model gravity + reproduces general relativity.
I This talk: given a microscopic model (CDT), establish its effectivedynamics.
![Page 3: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/3.jpg)
Path integral approach to quantum gravity
I Quantum mechanics according toFeynman
x i (t)
I Gravity according toEinstein
gµν(x)
I Can we combine these pillars of modern theoretical physics?
I First try pure gravity: sum over space-time geometries.
I Hope: UV complete model gravity + reproduces general relativity.
I This talk: given a microscopic model (CDT), establish its effectivedynamics.
![Page 4: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/4.jpg)
Path integral approach to quantum gravity
I Quantum mechanics according toFeynman
x i (t)
I Gravity according toEinstein
gµν(x)
I Can we combine these pillars of modern theoretical physics?
I First try pure gravity: sum over space-time geometries.
I Hope: UV complete model gravity + reproduces general relativity.
I This talk: given a microscopic model (CDT), establish its effectivedynamics.
![Page 5: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/5.jpg)
Path integral approach to quantum gravity
I Quantum mechanics according toFeynman
x i (t)
I Gravity according toEinstein
gµν(x)
I Can we combine these pillars of modern theoretical physics?
I First try pure gravity: sum over space-time geometries.
I Hope: UV complete model gravity + reproduces general relativity.
I This talk: given a microscopic model (CDT), establish its effectivedynamics.
![Page 6: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/6.jpg)
Path integral approach to quantum gravity
I Quantum mechanics according toFeynman
x i (t)
I Gravity according toEinstein
gµν(x)
I Can we combine these pillars of modern theoretical physics?
I First try pure gravity: sum over space-time geometries.
I Hope: UV complete model gravity + reproduces general relativity.
I This talk: given a microscopic model (CDT), establish its effectivedynamics.
![Page 7: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/7.jpg)
Path integral approach to quantum gravity
I Quantum mechanics according toFeynman
x i (t)
I Gravity according toEinstein
gµν(x)
I Can we combine these pillars of modern theoretical physics?
I First try pure gravity: sum over space-time geometries.
I Hope: UV complete model gravity + reproduces general relativity.
I This talk: given a microscopic model (CDT), establish its effectivedynamics.
![Page 8: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/8.jpg)
Outline
I Introduction to CDT in 2+1 dimensions
I Previous results for topology S2 × R: spatial volume
I Effective action for CDT: conformal model problem
I Alternative to Einstein–Hilbert
I Moduli as observables for CDT with topology T 2 × RI Monte Carlo measurements vs ansatz
I Summary and outlook
![Page 9: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/9.jpg)
CDT in 2+1 dimensionsI Causal Dynamical Triangulation is a regularization of the Euclidean
path integral over geometries
Z =
∫ Dg
Diffe−SEH [g ] → ZCDT =
∑triangulationsT
1
CTe−SCDT [T ].
→
t=0
t=T
I Triangulations T are built from equilateral tetrahedra. The sum isover inequivalent ways of putting them together.
I “Causal” in CDT means that we only allow triangulations that arefoliated by 2D triangulations with constant topology.
I The Euclidean Einstein–Hilbert action S [g ] = −κ∫
d3x√
g(R − 2Λ)evaluated on the piecewise linear geometry leads to
SCDT [T ] = k3N3 − k0N0.
![Page 10: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/10.jpg)
CDT in 2+1 dimensionsI Causal Dynamical Triangulation is a regularization of the Euclidean
path integral over geometries
Z =
∫ Dg
Diffe−SEH [g ] → ZCDT =
∑triangulationsT
1
CTe−SCDT [T ].
→
t=0
t=T
I Triangulations T are built from equilateral tetrahedra. The sum isover inequivalent ways of putting them together.
I “Causal” in CDT means that we only allow triangulations that arefoliated by 2D triangulations with constant topology.
I The Euclidean Einstein–Hilbert action S [g ] = −κ∫
d3x√
g(R − 2Λ)evaluated on the piecewise linear geometry leads to
SCDT [T ] = k3N3 − k0N0.
![Page 11: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/11.jpg)
CDT in 2+1 dimensionsI Causal Dynamical Triangulation is a regularization of the Euclidean
path integral over geometries
Z =
∫ Dg
Diffe−SEH [g ] → ZCDT =
∑triangulationsT
1
CTe−SCDT [T ].
→
t=0
t=T
I Triangulations T are built from equilateral tetrahedra. The sum isover inequivalent ways of putting them together.
I “Causal” in CDT means that we only allow triangulations that arefoliated by 2D triangulations with constant topology.
I The Euclidean Einstein–Hilbert action S [g ] = −κ∫
d3x√
g(R − 2Λ)evaluated on the piecewise linear geometry leads to
SCDT [T ] = k3N3 − k0N0.
![Page 12: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/12.jpg)
CDT in 2+1 dimensionsI Causal Dynamical Triangulation is a regularization of the Euclidean
path integral over geometries
Z =
∫ Dg
Diffe−SEH [g ] → ZCDT =
∑triangulationsT
1
CTe−SCDT [T ].
→
t=0
t=T
I Triangulations T are built from equilateral tetrahedra. The sum isover inequivalent ways of putting them together.
I “Causal” in CDT means that we only allow triangulations that arefoliated by 2D triangulations with constant topology.
I The Euclidean Einstein–Hilbert action S [g ] = −κ∫
d3x√
g(R − 2Λ)evaluated on the piecewise linear geometry leads to
SCDT [T ] = k3N3 − k0N0.
![Page 13: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/13.jpg)
Monte Carlo methods for CDTI The fixed volume partition function reads
Z (N3) =∑
T
1
CTe−k0N0 .
I The expectation value of an observableO : T → O(T ) is given by
〈O〉N3 =1
Z
∑T
O(T )
CTe−k0N0 .
I We use Monte Carlo methods to approximate these:
〈O〉N3 ≈1
n
n∑i=1
O(Ti ),
where the Ti is a large set of random triangulations generated byapplying a large number of random update moves satisfying adetailed balance condition.
![Page 14: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/14.jpg)
Monte Carlo methods for CDTI The fixed volume partition function reads
Z (N3) =∑
T
1
CTe−k0N0 .
I The expectation value of an observableO : T → O(T ) is given by
〈O〉N3 =1
Z
∑T
O(T )
CTe−k0N0 .
I We use Monte Carlo methods to approximate these:
〈O〉N3 ≈1
n
n∑i=1
O(Ti ),
where the Ti is a large set of random triangulations generated byapplying a large number of random update moves satisfying adetailed balance condition.
![Page 15: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/15.jpg)
Monte Carlo methods for CDTI The fixed volume partition function reads
Z (N3) =∑
T
1
CTe−k0N0 .
I The expectation value of an observableO : T → O(T ) is given by
〈O〉N3 =1
Z
∑T
O(T )
CTe−k0N0 .
I We use Monte Carlo methods to approximate these:
〈O〉N3 ≈1
n
n∑i=1
O(Ti ),
where the Ti is a large set of random triangulations generated byapplying a large number of random update moves satisfying adetailed balance condition.
![Page 16: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/16.jpg)
CDT with spherical spatial topologyI Take periodic time and
spatial topology S2
I 〈V (t)〉 ∝ cos2(√
c1t) plus minimal“stalk”
-30 -20 -10 0 10 20 30t'
300
600
900
1200VHt'L
I 〈V (t)〉 is classical solution to
Seff [V ] = c0
∫dt
(V 2
V− 2c1V
).
I Also correlations
〈V (t)V (t ′)〉 − 〈V (t)〉〈V (t ′)〉 ∝(
δ2Seff
δV (t)δV (t′)
∣∣∣〈V (t)〉
)−1
well-described by this action for suitable values c0, c1 > 0.
![Page 17: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/17.jpg)
CDT with spherical spatial topologyI Take periodic time and
spatial topology S2I 〈V (t)〉 ∝ cos2(
√c1t) plus minimal
“stalk”
-30 -20 -10 0 10 20 30t'
300
600
900
1200VHt'L
I 〈V (t)〉 is classical solution to
Seff [V ] = c0
∫dt
(V 2
V− 2c1V
).
I Also correlations
〈V (t)V (t ′)〉 − 〈V (t)〉〈V (t ′)〉 ∝(
δ2Seff
δV (t)δV (t′)
∣∣∣〈V (t)〉
)−1
well-described by this action for suitable values c0, c1 > 0.
![Page 18: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/18.jpg)
CDT with spherical spatial topologyI Take periodic time and
spatial topology S2I 〈V (t)〉 ∝ cos2(
√c1t) plus minimal
“stalk”
-30 -20 -10 0 10 20 30t'
300
600
900
1200VHt'L
I 〈V (t)〉 is classical solution to
Seff [V ] = c0
∫dt
(V 2
V− 2c1V
).
I Also correlations
〈V (t)V (t ′)〉 − 〈V (t)〉〈V (t ′)〉 ∝(
δ2Seff
δV (t)δV (t′)
∣∣∣〈V (t)〉
)−1
well-described by this action for suitable values c0, c1 > 0.
![Page 19: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/19.jpg)
CDT with spherical spatial topologyI Take periodic time and
spatial topology S2I 〈V (t)〉 ∝ cos2(
√c1t) plus minimal
“stalk”
-30 -20 -10 0 10 20 30t'
300
600
900
1200VHt'L
I 〈V (t)〉 is classical solution to
Seff [V ] = c0
∫dt
(V 2
V− 2c1V
).
I Also correlations
〈V (t)V (t ′)〉 − 〈V (t)〉〈V (t ′)〉 ∝(
δ2Seff
δV (t)δV (t′)
∣∣∣〈V (t)〉
)−1
well-described by this action for suitable values c0, c1 > 0.
![Page 20: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/20.jpg)
CDT with spherical spatial topology (continued)
I Seff [V ] = c0
∫dt(
V 2
V − 2c1V)
is of the same form as the
minisuperspace action obtained by evaluating Einstein–Hilbert onspherical cosmology ds2 = dt2 + V (t)dΩ2,
SEH = κ
∫d3x√
g(−R + 2Λ) = −κ∫
dt
(V 2
V− 2ΛV
)
I This is non-trivial! We did not impose homogeneity/isotropy fromthe start. A (Euclidean) de Sitter space seems to emerge.
I How much does this tell us about the “full” effective dynamics ofCDT? Is it close to EH? Is the emerging geometry really isotropic?
I To find out we could study deviations from 〈V (t)〉 ∝ cos2(√
c1t)due to anisotropies, or introduce observables that go beyond spatialvolume.
![Page 21: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/21.jpg)
CDT with spherical spatial topology (continued)
I Seff [V ] = c0
∫dt(
V 2
V − 2c1V)
is of the same form as the
minisuperspace action obtained by evaluating Einstein–Hilbert onspherical cosmology ds2 = dt2 + V (t)dΩ2,
SEH = κ
∫d3x√
g(−R + 2Λ) = −κ∫
dt
(V 2
V− 2ΛV
)
I This is non-trivial! We did not impose homogeneity/isotropy fromthe start. A (Euclidean) de Sitter space seems to emerge.
I How much does this tell us about the “full” effective dynamics ofCDT? Is it close to EH? Is the emerging geometry really isotropic?
I To find out we could study deviations from 〈V (t)〉 ∝ cos2(√
c1t)due to anisotropies, or introduce observables that go beyond spatialvolume.
![Page 22: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/22.jpg)
CDT with spherical spatial topology (continued)
I Seff [V ] = c0
∫dt(
V 2
V − 2c1V)
is of the same form as the
minisuperspace action obtained by evaluating Einstein–Hilbert onspherical cosmology ds2 = dt2 + V (t)dΩ2,
SEH = κ
∫d3x√
g(−R + 2Λ) = −κ∫
dt
(V 2
V− 2ΛV
)
I This is non-trivial! We did not impose homogeneity/isotropy fromthe start. A (Euclidean) de Sitter space seems to emerge.
I How much does this tell us about the “full” effective dynamics ofCDT? Is it close to EH? Is the emerging geometry really isotropic?
I To find out we could study deviations from 〈V (t)〉 ∝ cos2(√
c1t)due to anisotropies, or introduce observables that go beyond spatialvolume.
![Page 23: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/23.jpg)
CDT with spherical spatial topology (continued)
I Seff [V ] = c0
∫dt(
V 2
V − 2c1V)
is of the same form as the
minisuperspace action obtained by evaluating Einstein–Hilbert onspherical cosmology ds2 = dt2 + V (t)dΩ2,
SEH = κ
∫d3x√
g(−R + 2Λ) = −κ∫
dt
(V 2
V− 2ΛV
)
I This is non-trivial! We did not impose homogeneity/isotropy fromthe start. A (Euclidean) de Sitter space seems to emerge.
I How much does this tell us about the “full” effective dynamics ofCDT? Is it close to EH? Is the emerging geometry really isotropic?
I To find out we could study deviations from 〈V (t)〉 ∝ cos2(√
c1t)due to anisotropies, or introduce observables that go beyond spatialvolume.
![Page 24: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/24.jpg)
CDT with spherical spatial topology (continued)
I Seff [V ] = c0
∫dt(
V 2
V − 2c1V)
is of the same form as the
minisuperspace action obtained by evaluating Einstein–Hilbert onspherical cosmology ds2 = dt2 + V (t)dΩ2,
SEH = κ
∫d3x√
g(−R + 2Λ) = −κ∫
dt
(V 2
V− 2ΛV
)
I This is non-trivial! We did not impose homogeneity/isotropy fromthe start. A (Euclidean) de Sitter space seems to emerge.
I How much does this tell us about the “full” effective dynamics ofCDT? Is it close to EH? Is the emerging geometry really isotropic?
I To find out we could study deviations from 〈V (t)〉 ∝ cos2(√
c1t)due to anisotropies, or introduce observables that go beyond spatialvolume.
![Page 25: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/25.jpg)
I Seff [V ] should have a minimum at its classical solution, since weidentify semiclassically
ZCDT =∑
T
1
CTe−SCDT [T ] ≈
∫ ( T∏t=0
dV (t)
)e−Seff [V (t)].
I This is true for Seff [V ] = c0
∫dt(
V 2
V − 2c1V)
for fixed 3-volume
but not for Einstein–Hilbert action in general: conformal modeproblem.
I Metric in proper-time form, ds2 = dt2 + gab(t, x)dxadxb. Then
SEH [gab] = κ
∫dt
∫d2x√
g
(1
4gabGabcd gcd − R + 2Λ
)(1)
where Gabcd is the Wheeler–DeWitt metric,
Gabcd =1
2
(g ac g bd + g ad g bc
)− g abg cd . (2)
I Indefinite metric! Positive definite on traceless directions, negativedefinite on trace/conformal directions in superspace.
![Page 26: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/26.jpg)
I Seff [V ] should have a minimum at its classical solution, since weidentify semiclassically
ZCDT =∑
T
1
CTe−SCDT [T ] ≈
∫ ( T∏t=0
dV (t)
)e−Seff [V (t)].
I This is true for Seff [V ] = c0
∫dt(
V 2
V − 2c1V)
for fixed 3-volume
but not for Einstein–Hilbert action in general: conformal modeproblem.
I Metric in proper-time form, ds2 = dt2 + gab(t, x)dxadxb. Then
SEH [gab] = κ
∫dt
∫d2x√
g
(1
4gabGabcd gcd − R + 2Λ
)(1)
where Gabcd is the Wheeler–DeWitt metric,
Gabcd =1
2
(g ac g bd + g ad g bc
)− g abg cd . (2)
I Indefinite metric! Positive definite on traceless directions, negativedefinite on trace/conformal directions in superspace.
![Page 27: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/27.jpg)
I Seff [V ] should have a minimum at its classical solution, since weidentify semiclassically
ZCDT =∑
T
1
CTe−SCDT [T ] ≈
∫ ( T∏t=0
dV (t)
)e−Seff [V (t)].
I This is true for Seff [V ] = c0
∫dt(
V 2
V − 2c1V)
for fixed 3-volume
but not for Einstein–Hilbert action in general: conformal modeproblem.
I Metric in proper-time form, ds2 = dt2 + gab(t, x)dxadxb. Then
SEH [gab] = κ
∫dt
∫d2x√
g
(1
4gabGabcd gcd − R + 2Λ
)(1)
where Gabcd is the Wheeler–DeWitt metric,
Gabcd =1
2
(g ac g bd + g ad g bc
)− g abg cd . (2)
I Indefinite metric! Positive definite on traceless directions, negativedefinite on trace/conformal directions in superspace.
![Page 28: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/28.jpg)
I Seff [V ] should have a minimum at its classical solution, since weidentify semiclassically
ZCDT =∑
T
1
CTe−SCDT [T ] ≈
∫ ( T∏t=0
dV (t)
)e−Seff [V (t)].
I This is true for Seff [V ] = c0
∫dt(
V 2
V − 2c1V)
for fixed 3-volume
but not for Einstein–Hilbert action in general: conformal modeproblem.
I Metric in proper-time form, ds2 = dt2 + gab(t, x)dxadxb. Then
SEH [gab] = κ
∫dt
∫d2x√
g
(1
4gabGabcd gcd − R + 2Λ
)(1)
where Gabcd is the Wheeler–DeWitt metric,
Gabcd =1
2
(g ac g bd + g ad g bc
)− g abg cd . (2)
I Indefinite metric! Positive definite on traceless directions, negativedefinite on trace/conformal directions in superspace.
![Page 29: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/29.jpg)
Alternative ansatz for effective actionI Fixed distance between initial and final boundary and preferred time
foliation suggest Seff [g ] only needs to be invariant under foliationpreserving diffeomorphisms.
I The Einstein–Hilbert action (with N = 1 and Na = 0)
SEH = κ
∫ T
0
dt
∫d2x√
g(gabGabcd gab + R − 2Λ)
generalizes naturally to
Sansatz = κ
∫ T
0
dt
∫d2x√
g(gabGabcdλ gab − U[g ]),
in which the most general ultralocal supermetric is
Gabcdλ =
1
2
(g ac g bd + g ad g bc
)− λg abg cd .
I Gλ is positive definite for λ < 1/2; λ = 1 in EH.I We have ended up with an ansatz in the realm of Euclidean
(projectable) Horava–Lifshitz gravity.
![Page 30: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/30.jpg)
Alternative ansatz for effective actionI Fixed distance between initial and final boundary and preferred time
foliation suggest Seff [g ] only needs to be invariant under foliationpreserving diffeomorphisms.
I The Einstein–Hilbert action (with N = 1 and Na = 0)
SEH = κ
∫ T
0
dt
∫d2x√
g(gabGabcd gab + R − 2Λ)
generalizes naturally to
Sansatz = κ
∫ T
0
dt
∫d2x√
g(gabGabcdλ gab − U[g ]),
in which the most general ultralocal supermetric is
Gabcdλ =
1
2
(g ac g bd + g ad g bc
)− λg abg cd .
I Gλ is positive definite for λ < 1/2; λ = 1 in EH.I We have ended up with an ansatz in the realm of Euclidean
(projectable) Horava–Lifshitz gravity.
![Page 31: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/31.jpg)
Alternative ansatz for effective actionI Fixed distance between initial and final boundary and preferred time
foliation suggest Seff [g ] only needs to be invariant under foliationpreserving diffeomorphisms.
I The Einstein–Hilbert action (with N = 1 and Na = 0)
SEH = κ
∫ T
0
dt
∫d2x√
g(gabGabcd gab + R − 2Λ)
generalizes naturally to
Sansatz = κ
∫ T
0
dt
∫d2x√
g(gabGabcdλ gab − U[g ]),
in which the most general ultralocal supermetric is
Gabcdλ =
1
2
(g ac g bd + g ad g bc
)− λg abg cd .
I Gλ is positive definite for λ < 1/2; λ = 1 in EH.
I We have ended up with an ansatz in the realm of Euclidean(projectable) Horava–Lifshitz gravity.
![Page 32: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/32.jpg)
Alternative ansatz for effective actionI Fixed distance between initial and final boundary and preferred time
foliation suggest Seff [g ] only needs to be invariant under foliationpreserving diffeomorphisms.
I The Einstein–Hilbert action (with N = 1 and Na = 0)
SEH = κ
∫ T
0
dt
∫d2x√
g(gabGabcd gab + R − 2Λ)
generalizes naturally to
Sansatz = κ
∫ T
0
dt
∫d2x√
g(gabGabcdλ gab − U[g ]),
in which the most general ultralocal supermetric is
Gabcdλ =
1
2
(g ac g bd + g ad g bc
)− λg abg cd .
I Gλ is positive definite for λ < 1/2; λ = 1 in EH.I We have ended up with an ansatz in the realm of Euclidean
(projectable) Horava–Lifshitz gravity.
![Page 33: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/33.jpg)
New observables
I Need observable f : 2d triangulations → R. Whole journals arededicated to shape recognition in medical imaging, computergraphics, etc. However the random geometries in CDT are muchwilder.
I The conformal structure of a torus! Moduli τi (Teichmullerparameters) .
![Page 34: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/34.jpg)
New observables
I Need observable f : 2d triangulations → R. Whole journals arededicated to shape recognition in medical imaging, computergraphics, etc. However the random geometries in CDT are muchwilder.
I The conformal structure of a torus! Moduli τi (Teichmullerparameters) .
![Page 35: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/35.jpg)
New observables
I Need observable f : 2d triangulations → R. Whole journals arededicated to shape recognition in medical imaging, computergraphics, etc. However the random geometries in CDT are muchwilder.
I The conformal structure of a torus! Moduli τi (Teichmullerparameters) .
![Page 36: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/36.jpg)
Moduli on a triangulation
I Replace every edge by a perfectspring: periodic discrete harmonicembedding.
I Invariant under linear transformations.Moduli τi correspond to energeticallypreferred configuration with unitvolume. “Most equilateralconfiguration”.
Τ
I The functions τi :
→ R are well-defined.
![Page 37: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/37.jpg)
Moduli on a triangulation
I Replace every edge by a perfectspring: periodic discrete harmonicembedding.
I Invariant under linear transformations.Moduli τi correspond to energeticallypreferred configuration with unitvolume. “Most equilateralconfiguration”.
Τ
I The functions τi :
→ R are well-defined.
![Page 38: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/38.jpg)
Moduli on a triangulation
I Replace every edge by a perfectspring: periodic discrete harmonicembedding.
I Invariant under linear transformations.Moduli τi correspond to energeticallypreferred configuration with unitvolume. “Most equilateralconfiguration”.
Τ
I The functions τi :
→ R are well-defined.
![Page 39: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/39.jpg)
CDT with spatial torus topologyI Try periodic boundary conditions T 2 × S1. No breaking of time
translation symmetry:
-30 -20 -10 0 10 20 30t'
300
600
900
1200VHt'L
0 10 20 30 40 50 60 70t
300
600
900
1200VHtL
4 8 12 16t
500
1000
1500
2000
VHtL
I We need different boundary conditions to get non-trivial dynamics.Use fixed boundaries at t = 0 and t = T .
I Hopf foliation of S3! Maximally different moduli: τ(0) = 0 andτ(T ) = i∞.
Hopf Link
![Page 40: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/40.jpg)
CDT with spatial torus topologyI Try periodic boundary conditions T 2 × S1. No breaking of time
translation symmetry:
-30 -20 -10 0 10 20 30t'
300
600
900
1200VHt'L
0 10 20 30 40 50 60 70t
300
600
900
1200VHtL
4 8 12 16t
500
1000
1500
2000
VHtL
I We need different boundary conditions to get non-trivial dynamics.Use fixed boundaries at t = 0 and t = T .
I Hopf foliation of S3! Maximally different moduli: τ(0) = 0 andτ(T ) = i∞.
Hopf Link
![Page 41: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/41.jpg)
CDT with spatial torus topologyI Try periodic boundary conditions T 2 × S1. No breaking of time
translation symmetry:
-30 -20 -10 0 10 20 30t'
300
600
900
1200VHt'L
0 10 20 30 40 50 60 70t
300
600
900
1200VHtL
4 8 12 16t
500
1000
1500
2000
VHtL
I We need different boundary conditions to get non-trivial dynamics.Use fixed boundaries at t = 0 and t = T .
I Hopf foliation of S3! Maximally different moduli: τ(0) = 0 andτ(T ) = i∞.
Hopf Link
![Page 42: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/42.jpg)
CDT with spatial torus topologyI Try periodic boundary conditions T 2 × S1. No breaking of time
translation symmetry:
-30 -20 -10 0 10 20 30t'
300
600
900
1200VHt'L
0 10 20 30 40 50 60 70t
300
600
900
1200VHtL
4 8 12 16t
500
1000
1500
2000
VHtL
I We need different boundary conditions to get non-trivial dynamics.Use fixed boundaries at t = 0 and t = T .
I Hopf foliation of S3! Maximally different moduli: τ(0) = 0 andτ(T ) = i∞.
Hopf Link
![Page 43: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/43.jpg)
CDT with spatial torus topologyI Try periodic boundary conditions T 2 × S1. No breaking of time
translation symmetry:
-30 -20 -10 0 10 20 30t'
300
600
900
1200VHt'L
0 10 20 30 40 50 60 70t
300
600
900
1200VHtL
4 8 12 16t
500
1000
1500
2000
VHtL
I We need different boundary conditions to get non-trivial dynamics.Use fixed boundaries at t = 0 and t = T .
I Hopf foliation of S3! Maximally different moduli: τ(0) = 0 andτ(T ) = i∞.
Hopf Link
![Page 44: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/44.jpg)
CDT with spatial torus topologyI Try periodic boundary conditions T 2 × S1. No breaking of time
translation symmetry:
-30 -20 -10 0 10 20 30t'
300
600
900
1200VHt'L
0 10 20 30 40 50 60 70t
300
600
900
1200VHtL
4 8 12 16t
500
1000
1500
2000
VHtL
I We need different boundary conditions to get non-trivial dynamics.Use fixed boundaries at t = 0 and t = T .
I Hopf foliation of S3! Maximally different moduli: τ(0) = 0 andτ(T ) = i∞.
Hopf Link
![Page 45: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/45.jpg)
CDT with spatial torus topologyI Try periodic boundary conditions T 2 × S1. No breaking of time
translation symmetry:
-30 -20 -10 0 10 20 30t'
300
600
900
1200VHt'L
0 10 20 30 40 50 60 70t
300
600
900
1200VHtL
4 8 12 16t
500
1000
1500
2000
VHtL
I We need different boundary conditions to get non-trivial dynamics.Use fixed boundaries at t = 0 and t = T .
I Hopf foliation of S3! Maximally different moduli: τ(0) = 0 andτ(T ) = i∞.
Hopf Link
![Page 46: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/46.jpg)
Comparison with ansatz
I Evaluating
Sansatz = κ
∫ T
0
dt
∫d2x√
g(gabGabcdλ gcd − U[g ]),
on spatially flat cosmology ds2 = dt2 + V (t)gab(τ)dxadxb,
gab(τ, x) = 1τ2
(1 τ1
τ1 τ 21 + τ 2
2
)gives
S [V , τ ] = κ
∫dt(
(1
2− λ)
V 2
V+
V
2
τ 21 + τ 2
2
τ 22
− U(V , τ)).
I Seff [V , τ ] should describe all 〈V (t)〉, 〈τi (t)〉, 〈τi (t)τj (t ′)〉,. . .. Tooambitious. What is U(V , τ)? Focus on kinetic term.
I Prefactors in kinetic term related to inverse of correlators:
(〈τi (t)τj (t ′)〉 − 〈τi (t)〉〈τj (t ′)〉)−1 ∝(
Pi (t)d2
dt2+ · · ·
)δ(t − t ′)δij
![Page 47: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/47.jpg)
Comparison with ansatz
I Evaluating
Sansatz = κ
∫ T
0
dt
∫d2x√
g(gabGabcdλ gcd − U[g ]),
on spatially flat cosmology ds2 = dt2 + V (t)gab(τ)dxadxb,
gab(τ, x) = 1τ2
(1 τ1
τ1 τ 21 + τ 2
2
)gives
S [V , τ ] = κ
∫dt(
(1
2− λ)
V 2
V+
V
2
τ 21 + τ 2
2
τ 22
− U(V , τ)).
I Seff [V , τ ] should describe all 〈V (t)〉, 〈τi (t)〉, 〈τi (t)τj (t ′)〉,. . .. Tooambitious. What is U(V , τ)? Focus on kinetic term.
I Prefactors in kinetic term related to inverse of correlators:
(〈τi (t)τj (t ′)〉 − 〈τi (t)〉〈τj (t ′)〉)−1 ∝(
Pi (t)d2
dt2+ · · ·
)δ(t − t ′)δij
![Page 48: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/48.jpg)
Comparison with ansatz
I Evaluating
Sansatz = κ
∫ T
0
dt
∫d2x√
g(gabGabcdλ gcd − U[g ]),
on spatially flat cosmology ds2 = dt2 + V (t)gab(τ)dxadxb,
gab(τ, x) = 1τ2
(1 τ1
τ1 τ 21 + τ 2
2
)gives
S [V , τ ] = κ
∫dt(
(1
2− λ)
V 2
V+
V
2
τ 21 + τ 2
2
τ 22
− U(V , τ)).
I Seff [V , τ ] should describe all 〈V (t)〉, 〈τi (t)〉, 〈τi (t)τj (t ′)〉,. . .. Tooambitious. What is U(V , τ)? Focus on kinetic term.
I Prefactors in kinetic term related to inverse of correlators:
(〈τi (t)τj (t ′)〉 − 〈τi (t)〉〈τj (t ′)〉)−1 ∝(
Pi (t)d2
dt2+ · · ·
)δ(t − t ′)δij
![Page 49: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/49.jpg)
I Compare Pi (t) and 〈V (t)/τ2(t)2〉in CDT simulation.
I Ansatz wrong or homogeneity is abad approximation.
I We can do better!
t/T0 10.2 0.4 0.6 0.8
0
1000
2000
3000
4000V/τ 2
2Pi (t)
S [V , τ ] = κ
∫dt(
(1
2− λ)
V 2
V+
V
2︸︷︷︸Depends sensitively on the flatness.
τ 21 + τ 2
2
τ 22
− U(V , τ)).
I A[g ] is the expected change in τ under a random deformation δgab
(normalized w.r.t. G)
A[g ] =δij
4τ 22
∫d2x√
gδτi
δgabGabcd
δτj
δgcd.
![Page 50: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/50.jpg)
I Compare Pi (t) and 〈V (t)/τ2(t)2〉in CDT simulation.
I Ansatz wrong or homogeneity is abad approximation.
I We can do better!
t/T0 10.2 0.4 0.6 0.8
0
1000
2000
3000
4000V/τ 2
2Pi (t)
S [V , τ ] = κ
∫dt(
(1
2− λ)
V 2
V+
V
2︸︷︷︸Depends sensitively on the flatness.
τ 21 + τ 2
2
τ 22
− U(V , τ)).
I A[g ] is the expected change in τ under a random deformation δgab
(normalized w.r.t. G)
A[g ] =δij
4τ 22
∫d2x√
gδτi
δgabGabcd
δτj
δgcd.
![Page 51: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/51.jpg)
I Compare Pi (t) and 〈V (t)/τ2(t)2〉in CDT simulation.
I Ansatz wrong or homogeneity is abad approximation.
I We can do better!t/T
0 10.2 0.4 0.6 0.80
1000
2000
3000
4000V/τ 2
2Pi (t)
S [V , τ ] = κ
∫dt(
(1
2− λ)
V 2
V+
V
2︸︷︷︸Depends sensitively on the flatness.
τ 21 + τ 2
2
τ 22
− U(V , τ)).
I A[g ] is the expected change in τ under a random deformation δgab
(normalized w.r.t. G)
A[g ] =δij
4τ 22
∫d2x√
gδτi
δgabGabcd
δτj
δgcd.
![Page 52: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/52.jpg)
I Compare Pi (t) and 〈V (t)/τ2(t)2〉in CDT simulation.
I Ansatz wrong or homogeneity is abad approximation.
I We can do better!t/T
0 10.2 0.4 0.6 0.80
1000
2000
3000
4000V/τ 2
2Pi (t)
S [V , τ ] = κ
∫dt(
(1
2− λ)
V 2
V+
1
2A︸︷︷︸Depends sensitively on the flatness.
τ 21 + τ 2
2
τ 22
− U(V , τ)).
I A[g ] is the expected change in τ under a random deformation δgab
(normalized w.r.t. G)
A[g ] =δij
4τ 22
∫d2x√
gδτi
δgabGabcd
δτj
δgcd.
![Page 53: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/53.jpg)
I A[g ] can be worked out explicitly for a general metric gab on thetorus
A[g ] =δij
4τ 22
∫d2x√
gδτi
δgabGλabcd
δτj
δgcd
=
∫d2x√
g exp(2∆−1R)(∫d2x√
g exp(∆−1R))2
≥ 1
V
I A natural discretization of A[g ] to triangulations T isA[T ] =
∑σ∈T area(σ)2, where area(σ) is the area of the triangle σ
in the harmonic embedding of T in the plane.
VA[T ] = 1 VA[T ] ≈ 1.9 VA[T ] ≈ 4.9
![Page 54: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/54.jpg)
I A[g ] can be worked out explicitly for a general metric gab on thetorus
A[g ] =δij
4τ 22
∫d2x√
gδτi
δgabGλabcd
δτj
δgcd
=
∫d2x√
g exp(2∆−1R)(∫d2x√
g exp(∆−1R))2 ≥
1
V
I A natural discretization of A[g ] to triangulations T isA[T ] =
∑σ∈T area(σ)2, where area(σ) is the area of the triangle σ
in the harmonic embedding of T in the plane.
VA[T ] = 1 VA[T ] ≈ 1.9 VA[T ] ≈ 4.9
![Page 55: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/55.jpg)
I A[g ] can be worked out explicitly for a general metric gab on thetorus
A[g ] =δij
4τ 22
∫d2x√
gδτi
δgabGλabcd
δτj
δgcd
=
∫d2x√
g exp(2∆−1R)(∫d2x√
g exp(∆−1R))2 ≥
1
V
I A natural discretization of A[g ] to triangulations T isA[T ] =
∑σ∈T area(σ)2, where area(σ) is the area of the triangle σ
in the harmonic embedding of T in the plane.
VA[T ] = 1 VA[T ] ≈ 1.9 VA[T ] ≈ 4.9
![Page 56: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/56.jpg)
I Retry: compare Pi and〈1/(A(t)τ2(t)2)〉.
l0
PSfrag
A(1−|τ |2)2
l0 = 180
l0 = 40l0 = 5
l0 = 80
t/Tt/T
t/Tt/T
00
00
11
11
0.20.2
0.20.2
0.40.4
0.40.4
0.60.6
0.60.6
0.80.8
0.80.8
00
00
100100
100100
200200
200200
300300
300300
400400
400400
500500
500500
1/(Aτ 22 )
Pi (t)
I The fitted overall factor corresponds to κ,
S [V , τ ] = κ
∫dt(
(1
2− λ)
V 2
V+
1
2A
τ 21 + τ 2
2
τ 22
− U(V , τ)).
I Similar analysis for spatial volume V (t) leads to a measurement ofκ( 1
2 − λ). Combining both we get a value λ ≈ 0.22 (for couplingk0 = 2.5).
![Page 57: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/57.jpg)
I Retry: compare Pi and〈1/(A(t)τ2(t)2)〉.
l0
PSfrag
A(1−|τ |2)2
l0 = 180
l0 = 40l0 = 5
l0 = 80
t/Tt/T
t/Tt/T
00
00
11
11
0.20.2
0.20.2
0.40.4
0.40.4
0.60.6
0.60.6
0.80.8
0.80.8
00
00
100100
100100
200200
200200
300300
300300
400400
400400
500500
500500
1/(Aτ 22 )
Pi (t)
I The fitted overall factor corresponds to κ,
S [V , τ ] = κ
∫dt(
(1
2− λ)
V 2
V+
1
2A
τ 21 + τ 2
2
τ 22
− U(V , τ)).
I Similar analysis for spatial volume V (t) leads to a measurement ofκ( 1
2 − λ). Combining both we get a value λ ≈ 0.22 (for couplingk0 = 2.5).
![Page 58: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/58.jpg)
I Retry: compare Pi and〈1/(A(t)τ2(t)2)〉.
l0
PSfrag
A(1−|τ |2)2
l0 = 180
l0 = 40l0 = 5
l0 = 80
t/Tt/T
t/Tt/T
00
00
11
11
0.20.2
0.20.2
0.40.4
0.40.4
0.60.6
0.60.6
0.80.8
0.80.8
00
00
100100
100100
200200
200200
300300
300300
400400
400400
500500
500500
1/(Aτ 22 )
Pi (t)
I The fitted overall factor corresponds to κ,
S [V , τ ] = κ
∫dt(
(1
2− λ)
V 2
V+
1
2A
τ 21 + τ 2
2
τ 22
− U(V , τ)).
I Similar analysis for spatial volume V (t) leads to a measurement ofκ( 1
2 − λ). Combining both we get a value λ ≈ 0.22 (for couplingk0 = 2.5).
![Page 59: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/59.jpg)
I We can perform this analysis for variouscouplings k0 (≈ bare Newton’sconstant).
I k0 → k∗0 ≈ 5.6 is CDT phase transition:number of 22-simplices drops to zero.
I Plot shows that as k0 → k∗0 the spatialvolumes V (t),V (t + 1) decouple, whilethe moduli τi (t), τi (t + 1) remaincoupled!
I Possible explanation: moduli aretopological degrees of freedom.
I However, seems to hold more generallyfor trace vs traceless d.o.f. in gab. Seeextrinsic curvature measurements inchapter 6 of my thesis.
k0
λ
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
−0.1
−0.2
![Page 60: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/60.jpg)
I We can perform this analysis for variouscouplings k0 (≈ bare Newton’sconstant).
I k0 → k∗0 ≈ 5.6 is CDT phase transition:number of 22-simplices drops to zero.
I Plot shows that as k0 → k∗0 the spatialvolumes V (t),V (t + 1) decouple, whilethe moduli τi (t), τi (t + 1) remaincoupled!
I Possible explanation: moduli aretopological degrees of freedom.
I However, seems to hold more generallyfor trace vs traceless d.o.f. in gab. Seeextrinsic curvature measurements inchapter 6 of my thesis.
k0
λ
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
−0.1
−0.2
![Page 61: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/61.jpg)
I We can perform this analysis for variouscouplings k0 (≈ bare Newton’sconstant).
I k0 → k∗0 ≈ 5.6 is CDT phase transition:number of 22-simplices drops to zero.
I Plot shows that as k0 → k∗0 the spatialvolumes V (t),V (t + 1) decouple, whilethe moduli τi (t), τi (t + 1) remaincoupled!
I Possible explanation: moduli aretopological degrees of freedom.
I However, seems to hold more generallyfor trace vs traceless d.o.f. in gab. Seeextrinsic curvature measurements inchapter 6 of my thesis.
k0
λ
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
−0.1
−0.2
![Page 62: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/62.jpg)
I We can perform this analysis for variouscouplings k0 (≈ bare Newton’sconstant).
I k0 → k∗0 ≈ 5.6 is CDT phase transition:number of 22-simplices drops to zero.
I Plot shows that as k0 → k∗0 the spatialvolumes V (t),V (t + 1) decouple, whilethe moduli τi (t), τi (t + 1) remaincoupled!
I Possible explanation: moduli aretopological degrees of freedom.
I However, seems to hold more generallyfor trace vs traceless d.o.f. in gab. Seeextrinsic curvature measurements inchapter 6 of my thesis.
k0
λ
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
−0.1
−0.2
![Page 63: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/63.jpg)
Summary
I In order to describe the measurements of the moduli in CDT withtopology T 2 × R we have introduced a non-covariant ansatz for aneffective action.
I The correlations in V and τ are compatible with a kinetic term givenby a generalized Wheeler–DeWitt metric Gλ with λ < 1/2 (recallthat λ = 1 in GR).
I This strengthens an earlier observed connection between CDT andHorava–Lifshitz gravity, a potentially renormalizable non-covariantgeneralization of GR.
![Page 64: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/64.jpg)
Outlook
I Construct full effective action S [V (t), τ(t)]. What is U[V , τ ] andhow does A[g ] scale with V ?
I Can we better understand CDT close to the phase transition?
I Is there any way to get rid of the spurious conformal degree offreedom?
I Having (more) large scale observables helps when studyingrenormalization. Outcomes of measurements should be invariant.
Don’t forget: Tuesday at 2:30 PM, my thesis defense!
Thanks!
![Page 65: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/65.jpg)
Outlook
I Construct full effective action S [V (t), τ(t)]. What is U[V , τ ] andhow does A[g ] scale with V ?
I Can we better understand CDT close to the phase transition?
I Is there any way to get rid of the spurious conformal degree offreedom?
I Having (more) large scale observables helps when studyingrenormalization. Outcomes of measurements should be invariant.
Don’t forget: Tuesday at 2:30 PM, my thesis defense!
Thanks!
![Page 66: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/66.jpg)
Outlook
I Construct full effective action S [V (t), τ(t)]. What is U[V , τ ] andhow does A[g ] scale with V ?
I Can we better understand CDT close to the phase transition?
I Is there any way to get rid of the spurious conformal degree offreedom?
I Having (more) large scale observables helps when studyingrenormalization. Outcomes of measurements should be invariant.
Don’t forget: Tuesday at 2:30 PM, my thesis defense!
Thanks!
![Page 67: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/67.jpg)
Outlook
I Construct full effective action S [V (t), τ(t)]. What is U[V , τ ] andhow does A[g ] scale with V ?
I Can we better understand CDT close to the phase transition?
I Is there any way to get rid of the spurious conformal degree offreedom?
I Having (more) large scale observables helps when studyingrenormalization. Outcomes of measurements should be invariant.
Don’t forget: Tuesday at 2:30 PM, my thesis defense!
Thanks!
![Page 68: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/68.jpg)
Outlook
I Construct full effective action S [V (t), τ(t)]. What is U[V , τ ] andhow does A[g ] scale with V ?
I Can we better understand CDT close to the phase transition?
I Is there any way to get rid of the spurious conformal degree offreedom?
I Having (more) large scale observables helps when studyingrenormalization. Outcomes of measurements should be invariant.
Don’t forget: Tuesday at 2:30 PM, my thesis defense!
Thanks!
![Page 69: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/69.jpg)
Appendices
![Page 70: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/70.jpg)
Measurement of τ in the continuumI Any metric gab on the torus is conformally flat and up to
diffeomorphisms the flat unit-volume metrics are given by
gab(τ, x) =1
τ2
(1 τ1
τ1 τ 21 + τ 2
2
).
I How do we find the “periodic” coordinates x1, x2 ∈ [0, 1) such thatds2 = Ω2(x)gabdxadxb?
I The 1-forms α1 = dx1 and α2 = dx2 arespecial: they form a basis of the space ofharmonic forms, i.e. dαi = δαi = 0 or∆αi = 0 with
∆ = dδ + δd (Hodge Laplacian)
d exterior derivative, δ its adjoint w.r.t.standard inner-product 〈φ, ψ〉 =
∫φ ∧ ∗ψ.
I Given generators γj , the αi are uniquely determined by∫γjαi = δi
j .
I τ = − 〈α1,α2〉
〈α2,α2〉 + i
√〈α1,α1〉〈α2,α2〉 −
(〈α1,α2〉〈α2,α2〉
)2
.
![Page 71: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/71.jpg)
Measurement of τ in the continuumI Any metric gab on the torus is conformally flat and up to
diffeomorphisms the flat unit-volume metrics are given by
gab(τ, x) =1
τ2
(1 τ1
τ1 τ 21 + τ 2
2
).
I How do we find the “periodic” coordinates x1, x2 ∈ [0, 1) such thatds2 = Ω2(x)gabdxadxb?
I The 1-forms α1 = dx1 and α2 = dx2 arespecial: they form a basis of the space ofharmonic forms, i.e. dαi = δαi = 0 or∆αi = 0 with
∆ = dδ + δd (Hodge Laplacian)
d exterior derivative, δ its adjoint w.r.t.standard inner-product 〈φ, ψ〉 =
∫φ ∧ ∗ψ.
I Given generators γj , the αi are uniquely determined by∫γjαi = δi
j .
I τ = − 〈α1,α2〉
〈α2,α2〉 + i
√〈α1,α1〉〈α2,α2〉 −
(〈α1,α2〉〈α2,α2〉
)2
.
![Page 72: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/72.jpg)
Measurement of τ in the continuumI Any metric gab on the torus is conformally flat and up to
diffeomorphisms the flat unit-volume metrics are given by
gab(τ, x) =1
τ2
(1 τ1
τ1 τ 21 + τ 2
2
).
I How do we find the “periodic” coordinates x1, x2 ∈ [0, 1) such thatds2 = Ω2(x)gabdxadxb?
I The 1-forms α1 = dx1 and α2 = dx2 arespecial: they form a basis of the space ofharmonic forms, i.e. dαi = δαi = 0 or∆αi = 0 with
∆ = dδ + δd (Hodge Laplacian)
d exterior derivative, δ its adjoint w.r.t.standard inner-product 〈φ, ψ〉 =
∫φ ∧ ∗ψ.
I Given generators γj , the αi are uniquely determined by∫γjαi = δi
j .
I τ = − 〈α1,α2〉
〈α2,α2〉 + i
√〈α1,α1〉〈α2,α2〉 −
(〈α1,α2〉〈α2,α2〉
)2
.
![Page 73: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/73.jpg)
Measurement of τ in the continuumI Any metric gab on the torus is conformally flat and up to
diffeomorphisms the flat unit-volume metrics are given by
gab(τ, x) =1
τ2
(1 τ1
τ1 τ 21 + τ 2
2
).
I How do we find the “periodic” coordinates x1, x2 ∈ [0, 1) such thatds2 = Ω2(x)gabdxadxb?
I The 1-forms α1 = dx1 and α2 = dx2 arespecial: they form a basis of the space ofharmonic forms, i.e. dαi = δαi = 0 or∆αi = 0 with
∆ = dδ + δd (Hodge Laplacian)
d exterior derivative, δ its adjoint w.r.t.standard inner-product 〈φ, ψ〉 =
∫φ ∧ ∗ψ.
Τ
Γ1
Γ2
-0.5 0 0.5 1
0.5
1
1.5
I Given generators γj , the αi are uniquely determined by∫γjαi = δi
j .
I τ = − 〈α1,α2〉
〈α2,α2〉 + i
√〈α1,α1〉〈α2,α2〉 −
(〈α1,α2〉〈α2,α2〉
)2
.
![Page 74: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/74.jpg)
Measurement of τ in the continuumI Any metric gab on the torus is conformally flat and up to
diffeomorphisms the flat unit-volume metrics are given by
gab(τ, x) =1
τ2
(1 τ1
τ1 τ 21 + τ 2
2
).
I How do we find the “periodic” coordinates x1, x2 ∈ [0, 1) such thatds2 = Ω2(x)gabdxadxb?
I The 1-forms α1 = dx1 and α2 = dx2 arespecial: they form a basis of the space ofharmonic forms, i.e. dαi = δαi = 0 or∆αi = 0 with
∆ = dδ + δd (Hodge Laplacian)
d exterior derivative, δ its adjoint w.r.t.standard inner-product 〈φ, ψ〉 =
∫φ ∧ ∗ψ.
Τ
Γ1
Γ2
-0.5 0 0.5 1
0.5
1
1.5
I Given generators γj , the αi are uniquely determined by∫γjαi = δi
j .
I τ = − 〈α1,α2〉
〈α2,α2〉 + i
√〈α1,α1〉〈α2,α2〉 −
(〈α1,α2〉〈α2,α2〉
)2
.
![Page 75: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/75.jpg)
Measurement of τ for torus triangulations
I Recipe: [Ambjørn, Barkley, TB, arXiv:1110.4649]
I Determine pair of curves γj that generate fundamental group.I Find the 2-dimensional kernel of ∆.I Determine basis αj such that
∫γjαi = δi
j .
I Compute τ = − 〈α1,α2〉〈α2,α2〉 + i
√〈α1,α1〉〈α2,α2〉 −
(〈α1,α2〉〈α2,α2〉
)2
.
I We need discrete differential forms! We will borrow them from thetheory of simplicial complexes.
I Once we have these ingredients we can construct discrete conformalmaps:
![Page 76: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/76.jpg)
Measurement of τ for torus triangulations
I Recipe: [Ambjørn, Barkley, TB, arXiv:1110.4649]
I Determine pair of curves γj that generate fundamental group.I Find the 2-dimensional kernel of ∆.I Determine basis αj such that
∫γjαi = δi
j .
I Compute τ = − 〈α1,α2〉〈α2,α2〉 + i
√〈α1,α1〉〈α2,α2〉 −
(〈α1,α2〉〈α2,α2〉
)2
.
I We need discrete differential forms! We will borrow them from thetheory of simplicial complexes.
I Once we have these ingredients we can construct discrete conformalmaps:
![Page 77: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/77.jpg)
Measurement of τ for torus triangulations
I Recipe: [Ambjørn, Barkley, TB, arXiv:1110.4649]
I Determine pair of curves γj that generate fundamental group.I Find the 2-dimensional kernel of ∆.I Determine basis αj such that
∫γjαi = δi
j .
I Compute τ = − 〈α1,α2〉〈α2,α2〉 + i
√〈α1,α1〉〈α2,α2〉 −
(〈α1,α2〉〈α2,α2〉
)2
.
I We need discrete differential forms! We will borrow them from thetheory of simplicial complexes.
I Once we have these ingredients we can construct discrete conformalmaps:
![Page 78: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/78.jpg)
Discrete differential forms
I In 2d triangulations we have
I Vertices: 0-simplices denoted by i ,
I Edges: 1-simplices denoted by (ij),
I Triangles: 2-simplices denoted by (ijk).
i
jk
HijLHkiL
HjkL
HijkL
I A discrete p-form φ assigns a real number φσ to each (oriented)p-simplex σ.
I Exterior derivative on 1-forms: (dφ)(ijk) = φ(ij) + φ(jk) + φ(ki)
I Divergence on 1-forms: (δφ)j =∑
edges (ij) φ(ij)
I More generally: (dψ)(σp+1) =∑σp∈σp+1
(−1)σpψ(σp).
I δ adjoint of d w.r.t. 〈φ, ψ〉 =∑σ φ(σ)ψ(σ).
I ∆ = dδ + δd becomes a matrix of which we can determine thenullspace ∆α = 0 (⇔ dα = 0 and δα = 0)
![Page 79: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/79.jpg)
Discrete differential forms
I In 2d triangulations we have
I Vertices: 0-simplices denoted by i ,
I Edges: 1-simplices denoted by (ij),
I Triangles: 2-simplices denoted by (ijk).
i
jk
HijLHkiL
HjkL
HijkL
I A discrete p-form φ assigns a real number φσ to each (oriented)p-simplex σ.
I Exterior derivative on 1-forms: (dφ)(ijk) = φ(ij) + φ(jk) + φ(ki)
I Divergence on 1-forms: (δφ)j =∑
edges (ij) φ(ij)
I More generally: (dψ)(σp+1) =∑σp∈σp+1
(−1)σpψ(σp).
I δ adjoint of d w.r.t. 〈φ, ψ〉 =∑σ φ(σ)ψ(σ).
I ∆ = dδ + δd becomes a matrix of which we can determine thenullspace ∆α = 0 (⇔ dα = 0 and δα = 0)
![Page 80: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/80.jpg)
Discrete differential forms
I In 2d triangulations we have
I Vertices: 0-simplices denoted by i ,
I Edges: 1-simplices denoted by (ij),
I Triangles: 2-simplices denoted by (ijk).
i
jk
HijLHkiL
HjkL
HijkL
I A discrete p-form φ assigns a real number φσ to each (oriented)p-simplex σ.
I Exterior derivative on 1-forms: (dφ)(ijk) = φ(ij) + φ(jk) + φ(ki)
I Divergence on 1-forms: (δφ)j =∑
edges (ij) φ(ij)
I More generally: (dψ)(σp+1) =∑σp∈σp+1
(−1)σpψ(σp).
I δ adjoint of d w.r.t. 〈φ, ψ〉 =∑σ φ(σ)ψ(σ).
I ∆ = dδ + δd becomes a matrix of which we can determine thenullspace ∆α = 0 (⇔ dα = 0 and δα = 0)
![Page 81: E ective Dynamics in non-perturbative quantum gravity ...tbudd/docs/quistphdtalk.pdf · Path integral approach to quantum gravity I Quantum mechanics according to Feynman xi(t) I](https://reader033.vdocuments.pub/reader033/viewer/2022052012/602954ff7c6e8837247f24df/html5/thumbnails/81.jpg)
Discrete differential forms
I In 2d triangulations we have
I Vertices: 0-simplices denoted by i ,
I Edges: 1-simplices denoted by (ij),
I Triangles: 2-simplices denoted by (ijk).
i
jk
HijLHkiL
HjkL
HijkL
I A discrete p-form φ assigns a real number φσ to each (oriented)p-simplex σ.
I Exterior derivative on 1-forms: (dφ)(ijk) = φ(ij) + φ(jk) + φ(ki)
I Divergence on 1-forms: (δφ)j =∑
edges (ij) φ(ij)
I More generally: (dψ)(σp+1) =∑σp∈σp+1
(−1)σpψ(σp).
I δ adjoint of d w.r.t. 〈φ, ψ〉 =∑σ φ(σ)ψ(σ).
I ∆ = dδ + δd becomes a matrix of which we can determine thenullspace ∆α = 0 (⇔ dα = 0 and δα = 0)
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Discrete differential forms
I In 2d triangulations we have
I Vertices: 0-simplices denoted by i ,
I Edges: 1-simplices denoted by (ij),
I Triangles: 2-simplices denoted by (ijk).
i
jk
HijLHkiL
HjkL
HijkL
I A discrete p-form φ assigns a real number φσ to each (oriented)p-simplex σ.
I Exterior derivative on 1-forms: (dφ)(ijk) = φ(ij) + φ(jk) + φ(ki)
I Divergence on 1-forms: (δφ)j =∑
edges (ij) φ(ij)
I More generally: (dψ)(σp+1) =∑σp∈σp+1
(−1)σpψ(σp).
I δ adjoint of d w.r.t. 〈φ, ψ〉 =∑σ φ(σ)ψ(σ).
I ∆ = dδ + δd becomes a matrix of which we can determine thenullspace ∆α = 0 (⇔ dα = 0 and δα = 0)