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Effective Constraints of Loop Quantum Gravity
Mikhail Kagan
Institute for Gravitational Physics and Geometry,
Pennsylvania State University
in collaboration with
M. Bojowald, G. Hossain,
(IGPG, Penn State)
H.H.Hernandez, A. Skirzewski
(Max-Planck-Institut für Gravitationsphysik,
Albert-Einstein-Institut, Potsdam, Germany
1
Outline
1. Motivation
2. Effective approximation. Overview.
3. Effective constraints.
• Implications.
5. Summary
2
Motivation.
Test semi-classical limit of LQG.
Evolution of inhomogeneities is expected to explain
cosmological structure formation and lead to observable
results.
Effective approximation allows to extract predictions of
the underlying quantum theory without going into
consideration of quantum states.
3
Effective Approximation. Strategy.
Classical Theory
Classical Constraints & { , }PB
Quantization
Quantum Operators & [ , ]
Effective Theory
Quantum variables:
expectation values, spreads, deformations, etc.
Truncation
Expectation Values
classically well
behaved expressions
Classical
Expressions
classically diverging
expressions
Classical
Correction
x
Expressions Functions
4
Effective Approximation. Summary.
Classical Constraints & Poisson Algebra
Constraint Operators & Commutation Relations
Effective Constraints & Effective Poisson Algebra
(differs from classical constraint algebra)
Effective Equations of Motion
(Bojowald, Hernandez, MK, Singh, Skirzewski
Phys. Rev. D, 74, 123512, 2006;
Phys. Rev. Lett. 98, 031301, 2007
for scalar mode in longitudinal gauge)
Anomaly Issue
(Non-anomalous algebra implies possibility of
writing consistent system of equations of motion in
terms of gauge invariant perturbation variables)
Non-trivial non-anomalous corrections found
5
Source of Corrections.
Densitized triad
Basic Variables
Ashtekar connection
Scalar field
Field momentum
Diffeomorphism Constraint
intact
Hamiltonian constraint
a(E)
D(E)
s(E)6
Effective Constraints. Lattice formulation.
(scalar mode/longitudinal gauge, Bojowald, Hernandez, MK, Skirzewski, 2007)
J
K
-labels
Fluxes
Holonomies
I
v
(integrated over Sv,I)
(integrated over ev,I)
Basic operators:
7
Effective Constraints. Types of corrections.
1. Discretization corrections.
2. Holonomy corrections (higher curvature corrections).
3. Inverse triad corrections.
8
Effective Constraints. Hamiltonian.
Curvature
0
Hamiltonian
Inverse triad
operator
+higher curvature corrections
Effective Constraints. Inverse triad corrections.
Asymptotics:
Effective Constraints. Inverse triad corrections.
Generalization
for
and
Implications. Corrections to Newton’s potential.
Corrected Poisson Equation
2  k2 

2ab


3
p   H (  P )
a


On Hubble scales:
classically
0,
1
12
Implications. Inflation.
(P = w)
Corrected Raychaudhuri Equation
  3(1  w e1)H  (w e2 )2  e3 H 2  0
_
where e32ap2/a< 0
Large-scale Fourier Modes
  (1  v ) /  e3 / 2  0
Two Classical Modes
decaying ( < 0) const (_=0)
 ( )    , with   
(
v
1  1  4e3 / v
2
)
With Quantum Corrections
decaying ( < 0)
growing (_e3/n)
(_ mode describes measure of inhomogeneity)
classically
0,
1
13
Implications. Inflation.
Conformal time
(changes by e60)
Effective corrections
_
e32ap2/a~
Conservative bound (particle physics)
Energy scale during Inflation
Metric perturbation
corrected by factor
(Phys. Rev. Lett. 98, 031301, 2007)
14
Summary.
1. There is a consistent set of corrected
constraints which are first class.
2. Cosmology:
• can formulate equations of motion in terms
of gauge invariant variables.
• potentially observable predictions.
3. Indications that quantization ambiguities are
restricted.