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CCCC – p. 1
Covariant Canonical Cwantum Cosmology
Martin Bojowald
The Pennsylvania State University
Institute for Gravitation and the Cosmos
University Park, PA

CCCC – p. 2
Background independence
Background independent quantizations of gravity must show that
some form of space-time “emerges” at sufficiently low curvature.
Not guaranteed.
Canonical approaches: constraints are quantized, receive
quantum corrections. Not just dynamics but also space-time
structure (gauge content) may change.
S = 1
16πG
∫
d 4 x √ |detg|
(
)
R+αR 2 +βR ab R ab +···
?
Must know space-time structure before deciding which covariant
combinations of metric derivatives can appear.
Not just R may receive quantum corrections but also “d 4 x,” or
nature of covariance and space-time tensors.

CCCC – p. 3
Loop quantum gravity
−→ Holonomies h e = exp( ∫ e dλAi aė a τ i ) along curves e in space.
Represented as operators, but not continuous in connection.
No connection operator.
−→ Holonomies as ladder operators raise excitation level of
geometry by discrete steps.
Densitized triad quantized as flux ∫ S d2 yn a Ê a i for surfaces S in
space, with discrete spectrum containing zero.
−→ Spatially diffeomorphism invariant: sum over all deformed
graphs. Background independent: No metric used in basic
operators.
More tricky: Space-time covariance, Hamiltonian constraint.
Background independence often forsaken at this stage.

CCCC – p. 4
Quantum corrections
−→ Inverse-triad corrections from quantizing [T Thiemann 1996]
{ } √|detE|d
Aa,∫ i 3 x
= 2πGǫ ijk ǫ abc
E b j Ec k
√
|detE|
2.5
Automatic cut-off of
1/E-divergences.
α (r)
−→ Higher-order corrections:
holonomies for A i a
−→ Quantum back-reaction
2
1.5
1
0.5
0
0 0.5 1 1.5 2
µ
flux eigenvalues
r=1/2
r=3/4
r=1
r=3/2
r=2

CCCC – p. 5
Hypersurface-deformation algebra
Diffeomorphism constraint D[w a ] and Hamiltonian constraint H[N].
Classical algebra
{D[w a 1],D[w a 2]} = −D[L w2 w a 1]
{H[N],D[w a ]} = −H[w a ∇ a N]
{H[N 1 ],H[N 2 ]} = D[h ab (N 1 ∇ b N 2 −N 2 ∇ b N 1 )]
ensures that theory respects
→ gauge transformations for coordinate changes: space-time
Lie derivative {f,H[ǫ]+D[ξ a ]} = L (ǫ/N,ξa +ǫN a /N)f if
constraints satisfied and time direction t a = Nn a +N a . [Dirac]
→ slicing independence: algebra amounts to deformations of
hypersurfaces
[Kuchar, Teiltelboim]
Anomaly-free quantum-corrected constraints generate modified
gauge transformations. Space-time?

CCCC – p. 6
Background independence?
Status of background independence in loop quantum gravity:
(+) No spatial background metric used to define states and
operators, summed over spatial diffeomorphisms.
Uniqueness of representation.
(−) Hamiltonian constraint quantized in much more messy way.
Modify classical expression by holonomy corrections
(quantum corrections mixed with regulator).
(−) Diffeomorphism constraint and Hamiltonian constraint
contain curvature of A i a, but treated differently.
(−) Off-shell algebra of Hamiltonian constraints often ignored.
[see also H Nicolai, K Peeters, M Zamaklar: hep-th/0501114]
Can this procedure result in any consistent space-time picture?
Energy conservation?

CCCC – p. 7
Energy conservation
[with M Kagan, G Hossain, C Tomlin: arXiv:1302.5695]
N √ deth∇ µ T µ 0 = −N ∂H matter
∂t
−N a∂Dmatter a
∂t
+L ⃗N C matter [N,N a ]+ ∂h ab
∂t
+∂ b
(
N 2 h ab D matter
a
δH matter
δh ab
)
+2N c h baδH matter
δh ac
Classical off-shell algebra: ∂H matter /∂t = {H matter ,H[N,N a ]}
cancels ∂ a (N 2 D matter
a ), only one term from ∂ b T b 0.
→ Deformed algebra cannot be taken care of by modified
coefficients in ∇ µ T µ ν = 0.
→ No energy conservation if off-shell algebra broken
(or unchecked in gauge-fixed/deparameterized models).

CCCC – p. 8
Consistent loop quantum gravity
→ Off-shell closed operator algebra of constraints likely?
→ Semiclassical states (at dynamical level)?
More realistic (at least in particle-physics analogy):
Effective equations.
→ Ease subtleties of factor ordering and quantum
representation.
→ Describe general classes of semiclassical states without
requiring full wave function.
(Vacuum problem of quantum gravity.)
→ Can be generalized to constrained systems. Allow local
internal times and time changes by gauge transformations.
→ Simpler derivation of closed constraint algebras.
But still incomplete.

CCCC – p. 9
Effective canonical dynamics
Parameterize state by expectation values 〈ˆq〉 and 〈ˆp〉 of basic
operators and moments
(Includes mixed states.)
G a,n = 〈(ˆq −〈ˆq〉) a (ˆp−〈ˆp〉) n−a 〉 symm
Commutator of operators determines Poisson bracket of
moments. (Symplectic space, but not after restriction to finite n.)
Expectation value of Hamiltonian, interpreted as function of 〈ˆq〉,
〈ˆp〉 and G a,n , provides canonical equations of motion.
Effective equations: Correct classical equations for 〈ˆq〉, 〈ˆp〉 by
quantum back-reaction from moments.
[with A Skirzewski: math-ph/0511043]

CCCC – p. 10
Quantum equations of motion
Anharmonic oscillator:
˙q = p m
ṗ = −mω 2 q −U ′ (q)− ∑ n
1
n!
( ) n/2
U (n+1) (q)˜G 0,n
mω
˙˜G a,n = −aω ˜G a−1,n +(n−a)ω ˜G a+1,n −a U′′ (q)
mω
√
aU ′′′ (q)
+
2(mω) 3 2
(√
U ′′′ (q)
− a 2
(mω) 3 2
˜G
a−1,n
˜G a−1,n−1 ˜G0,2 (q)
+ aU′′′′ a−1,n−1 ˜G ˜G0,3
3!(mω) 2
˜G a−1,n+1 + U′′′′ (q)
3(mω) 2 ˜G a−1,n+2 )
∞ly many coupled equations for ∞ly many variables.
+···

CCCC – p. 11
Low energy effective action
To second adiabatic order, as second order equation of motion:
( )
as it results from
Γ eff [q(t)] =
m+
U ′′′ (q) 2
32m 2 ω 5 (1+ U′′ (q)
mω 2 ) 5 2
+ ˙q2 (
4mω 2 U ′′′ (q)U ′′′′ (q)
+mω 2 q +U ′ (q)+
(
¨q
128m 3 ω 7 (1+ U′′ (q)
mω 2 ) 7 2
) )
1+ U′′ (q)
mω 2 −5U ′′′ (q) 3
U ′′′ (q)
4mω(1+ U′′ (q)
mω 2 ) 1 2
= 0.
∫ ( (
)
1 U ′′′ (q) 2
dt m+
2 2 5 m 2 (ω 2 +m −1 U ′′ (q)) 5 2
− 1 2 mω2 q 2 −U(q)− ω ( )
)1
1+ U′′ (q)
2
2 mω 2
˙q 2

Higher time derivatives
Higher adiabatic order:
[with S Brahma, E Nelson: arXiv:1208.1242]
¨q = −ω 2 q −U ′ (q)/m
where
−
2m 2 ω U′′′ (q) ( f(q, ˙q)+f 1 (q, ˙q)¨q +f 2 (q)¨q 2 +f 3 (q, ˙q)˙¨q +f 4 (q)¨q )
+···
( ) −1/2 ( ) −5/2
f(q, ˙q) = 1 2
1+ U′′ (q)
mω +
U ′′′′ (q)˙q 2
2 16mω
1+ U′′ (q)
4 mω −
5(U ′′′ (q)) 2 ˙q 2
2 64m 2 ω
(1+ 6
( ) −7/2 ( ) −9/2
− U′′′′′′ (q)˙q 4
64mω
1+ U′′ (q)
6 mω +
21(U ′′′′ (q)) 2 ˙q 4
2 256m 2 ω
1+ U′′ (q)
8 mω 2
( ) −9/2 (
+ 7U′′′′′ (q)U ′′′ (q)˙q 4
64m 2 ω
1+ U′′ (q)
8 mω −
231U ′′′′ (q)(U ′′′ (q)) 2 ˙q 4
2 512m 3 ω
1+ U′′ (q
10 mω 2
( ) −13/2
+ 1155(U′′′ (q)) 4 ˙q 4
4096m 4 ω
1+ U′′ (q)
12 mω 2 CCCC – p. 12

CCCC – p. 13
Effective constraints
Dynamics by effective Hamiltonian
constraints 〈̂polĈ〉(〈·〉,∆(·)).
→ Captures dynamics
of physical states.
→ Allows local internal times.
[arXiv:1009.5953]
[Figure from arXiv:0911.4950]
Α 0
〈Ĥ〉(〈·〉,∆(·)) or effective
p Α
Α 0
Α 0
.5Α 0
.5Α 0
.5Α 0
Α 0
.5Α 0
Α
[Cosmological model in P Höhn, E Kubalova, A Tsobanjan: arXiv:1111.5193]

CCCC – p. 14
Effective constraint algebras
Systems with several classical constraints C I :
→ Effective constraints C I,pol = 〈̂polĈI〉(〈·〉,∆(·)) with ̂pol
polynomial in basic operators.
→ Compute effective constraint algebra by Poisson brackets.
Easier than commutators of operators. Parameterizable.
→ Check off-shell consistency of first-class constraints.
Example: Hamiltonian and diffeomorphism constraints obey
hypersurface-deformation algebra of classical space-time.
{D[w a 1],D[w a 2]} = −D[L w2 w a 1]
{H[N],D[w a ]} = −H[w a ∇ a N]
{H[N 1 ],H[N 2 ]} = D[h ab (N 1 ∇ b N 2 −N 2 ∇ b N 1 )]

CCCC – p. 15
Relevance of effective methods
→ “Big-bang state” largely unknown.
Consider generic states in effective parameterization.
→ Local internal times: Do not require artificial matter contents
such as free, massless scalar or dust.
→ No gauge-fixing or deparameterization, or solution of
classical constraints required before quantization:
Check off-shell consistency of constraint algebra and
corresponding space-time structure.
Quantum space-time structure derived, not presupposed.
Surprises . . .

CCCC – p. 16
Homogeneous models
Dynamics poorly understood even in homogeneous models.
(Even kinematics [arXiv:1206.6088].)
In most cases, restricted not just by symmetry but also, and
crucially, by matter ingredients.
→ Usual choices of matter (for deparameterization) or states
(Gaussian) eliminate most quantum back-reaction, too
restrictive.
→ Symmetry eliminates control on space-time structure.
Homogeneous models trivialize the constraint algebra,
crucial issues overlooked.
Need realistic matter and at least perturbative inhomogeneity to
obtain propagation equations and see if and how structure
evolves through high density.

CCCC – p. 17
Deformed deformations
−→ Inverse-triad corrections in Hamiltonian
1
16πG
∫
d 3 xNα ǫ ijkF i ab Ea j Eb k
√
|detE|
+···
−→ Poisson-bracket algebra modified.
Deform but do not violate covariance:
{D[w a 1],D[w a 2]} = −D[L w2 w a 1]
{H[N],D[w a ]} = −H[w a ∇ a N]
{H[N 1 ],H[N 2 ]} = D[α 2 h ab (N 1 ∇ b N 2 −N 2 ∇ b N 1 )]
[with G Hossain, M Kagan, S Shankaranarayanan: arXiv:0806.3929]

CCCC – p. 18
Space-time?
Deformed gauge algebra implies that theory is no longer
invariant under space-time diffeomorphisms.
Same number of gauge generators: theory remains consistent.
Can be evaluated with canonical methods (Hamiltonian
equations of motion, gauge transformations, observables, . . . ).
But no analog of space-time metric or line element:
ds 2 = g ab dx a dx b
not invariant since transformations of g ab and dx a no longer
match.
(Warning for homogeneous models or perturbations:
Often, “effective” FLRW metric −dt 2 +a(t) 2 (dx 2 +dy 2 +dz 2 )
assumed with a(t) subject to modified Friedmann equation.)

CCCC – p. 19
Cosmological perturbation equations
[with G Calcagni: arXiv:1011.2779]
Dynamics of density perturbations u, gravitational waves w:
−u ′′ +s(α) 2 ∆u+(˜z ′′ /˜z)u = 0
−w ′′ +α 2 ∆w +(ã ′′ /ã)w = 0
Wave equations. Corrections to tensor-to-scalar ratio.
2.5
2
Do not need high density.
Crucial for falsifiability:
α−1 large for small lattice spacing.
Two-sided bounds on
discreteness scale.
α (r)
1.5
1
0.5
0
0 0.5 1 1.5 2
µ
r=1/2
r=3/4
r=1
r=3/2
r=2

Scalar mode
−u ′′ +s(α) 2 ∆u+(˜z ′′ /˜z)u = 0 [with G Calcagni, S Tsujikawa 2011]
1 x 10−4
0.9
0.8
10 −8 < δ = α−1 < 10 −4
0.7
0.6
much closer than
δ (k 0
)
0.5
0.4
0.3
0.2
0.1
l P and l H
0
0 0.005 0.01
ε (k ) V 0
0.015 0.02
CCCC – p. 20

CCCC – p. 21
Holonomy corrections
Hypersurface-deformation algebra with pointwise holonomy
corrections: H −→ l −1 sin(lH) in perturbative cosmology.
Incomplete: Corrections in series expansion cannot be
separated from higher space and time derivatives. (R ∼ ∂Γ+Γ 2 )
with β = cos(2lH).
[S(⃗w 1 ),S(⃗w 2 )] = −S(L ⃗w2 ⃗w 1 )
[T(N),S(⃗w)] = −T(⃗w · ⃗∇N)
[T(N 1 ),T(N 2 )] = S(β(N 1
⃗ ∇N2 −N 2
⃗ ∇N1 ))
−w ′′ +β∆w +(ã ′′ /ã)w = 0
β = −1 at maximal density (maximum of sin(lH), “bounce”).
[J Reyes 2009; A Barrau, T Cailleteau, J Grain, J Mielczarek 2011]

CCCC – p. 22
Signature change
β ≈ −1 at high density.
Space-time signature Euclidean.
[with G Paily: arXiv:1112.1899]
c∆t
v/c
Minkowski geometry
Euclidean geometry

CCCC – p. 23
Ill-posed bounce
Bounce models require high density, where signature turns
Euclidean.
→ Quantum space-time: no metric/line element, but deformed
constraint algebra determines space(-time) structure.
→ Physical consequence: elliptic rather than hyperbolic partial
differential equations for physical modes.
No deterministic evolution. No initial-value problem.
→ Signature change not a consequence of small
inhomogeneity:
Inhomogeneity only used to probe space-time structure
because homogeneous models are too restrictive.
→ Does not rely on subtleties of perturbation theory:
Same effects in spherically symmetric models.

CCCC – p. 24
Some recent results
Off-shell constraint algebra.
Operator results encouraging and fully consistent with effective
calculations.
[A Perez, D Pranzetti: arXiv:1001.3292]
[A Henderson, A Laddha, C Tomlin: arXiv:1204.0211, arXiv:1210.3960]
[C Tomlin, M Varadarajan: arXiv:1210.6869]
Derivative expansion for integrated holonomies.
Work in progress. [with G Paily and J Reyes]
Combination of inverse-triad and holonomy corrections.
Work in progress. [T Cailleteau]

CCCC – p. 25
Models of quantum space-time
(Weak) relation with doubly special relativity.
[MB, G Paily: arXiv:1212.4773 ]
Holonomy-deformation interpreted as version of κ-Poincaré:
−→ Linear N and ⃗w and flat space produce Poincaré if β = 1.
−→ For β ≠ 1, dependence on extrinsic curvature takes the form
of non-linear energy dependence at boundaries.
−→ However, this version does not act on standard κ-Minkowski.
Background independence and form of quantum space-time
remain unclear in loop quantum gravity.
Homogeneous models and most extensions do not grasp
quantum space-time structure.
Not an approximation: just anomalous.