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Fundamentals of
Quantum Cosmology
Ljubisa Nesic
Department of Physics,
University of Nis, Serbia
ISC2008, Nis, Serbia, August 26 - 31,
2008
1
Fundamentals of Quantum Cosmology
1.
2.
Basic Ideas of Quantum Cosmology
Minisuperspace Models in Quantum
Cosmology
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2008
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Basic Ideas of Quantum Cosmology

Introduction



Hamiltonian Formulation of General Relativity



Quantum cosmology and quantum gravity
A brief history of quantum cosmology
The 3+1 decomposition
The action
Quantization




Superspace
Canonical quantization
Path integral quantization
Minisuperspace
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Introduction
The status of QC: dangerous field to work
in if you hope to get a permanent job
 “Quantum” and “Cosmology” – inherently
incompatible?




“cosmology” – very large structure of the
universe
“quantum phenomena” – relevant in the
microscopic regime
If the hot big bang is the correct
description of the universe, it must have
been an such (quantum) epoch
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Formulations of QM









wavefunction (Schrodinger), state
matrix (Heisenberg), June 1925, measurable
quantity
path integral-sum over histories (Feynman) –
transition amplitude from (xi,ti) to (xf,tf ) is
proportional to exp(2piS/h)
phase space (Wigner)
density matrix
second quantization
variational
pilot wave (de Broglie-Bohm)
Hamilton-Jacobi (Hamilton’s principal function),
1983-Robert Leacock and Michael Pagdett
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Interpretation of QM
The many world interpretation (Everett)
 The transactional interpretation (Cramer)
…

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Standard Copenhagen interpretation of
quantum mechanics – classical world in
which the quantum one is embedded.
 Quantum mechanics is a universal theory
– some form of “quantum cosmology” was
important at the earliest of conceivable
times
 conceivable times?

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
Planck time
G
 44
t Planck 

5
,
4

10
s
5
c
lPlanck 


c 5
EPlanck 
 1,22 1019 GeV
G
G
35

1
,
6

10
m
3
c
At Planck scale, Compton wavelength is roughly
equal to its gravitational (Shwarzschild) radius.
classical concept of time and space is
meaningless
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Quantum Cosmology (QC) and
Quantum Gravity (QG)
Gravity is dominant interaction at large
scales – QC must be based on the theory
of QG.
 Quantization of gravity?




quantum general relativity (GR)
string theory
Quantization of GR?


GR is not perturbatively renormalisable
reason: GR is a theory of space-time – we
have to quantize spacetime itself (other fields
are the fields IN spacetime)
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String theory

Drasctically different approach to quantum
gravity – the idea is to first construct a
quantum theory of all interactions (a
‘theory of everything’) from which
separate quantum effects of the
gravitational field follow in some
appropriate limit
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Quantization of Gravity

Two main motivations


QFT – unification of all fundamental
interactions is an appealing aim
GR – quantization of gravity is necessary to
supersede GR – GR (although complete
theory) predicts its own break-down
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Quantization of GR: Two main
approaches

Covariant

examples:



path-integral approach
perturbation theory (Feynman diagrams)
Canonical


starts with a split of spacetime into space and
time – (Hamiltonian formalism) 4-metric as an
evolution of 3-metric in time.
examples:


quantum geometrodinamics
loop quantum gravity
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Hamiltonian Formulation of GR : 3+1
decomposition

3+1 split of the 4-dimensional spacetime
manifold M
M , g 
Metric
Differentiable
Manifold
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3+1 decomposition

spatial hypersurfaces St labeled by a global
time function t
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3+1 decomposition

4-dimensional metric
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3+1 decomposition
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3+1 decomposition

In components
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3+1 decomposition




Intrinsic curvature tensor (3)Rijkl – from the intrinsic metric
alone – describes the curvature intrinsic to the
hypersurfaces St
Extrinsic curvature (second fundamental form), Kij –
describes how the spatial hypersurfaces curve with respect
to the 4-dimensional spacetime manifold within which they
are embedded.
semicolon – covariant differentiation with respect to the 4metric,
vertical bar – covariant differentiation with respect to the
induced 3-metric.
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The action

Einstein-Hilbert action

Matter – single scalar field
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Gibbons-Hawking-York boundary term


Term that needs to be added to the Einstein-Hilbert action
when the underlying spacetime manifold has a boundary
Varying the action with respect to the metric g  gives the
Einstein equations
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The action in 3-1 decomposition

The action
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Canonical momenta

Canonical momenta for the basic variables

Last two equations – primary constraints in Dirac’s terminology
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Hamiltonian

Hamiltonian

Action

If we vary S with respect to pij and pF we obtain their
defining relations
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Hamiltonian




Variation S with respect laps function and shift vector,
yields the Hamiltonian and momentum constraints
(00) and (0i) parts of the Einstein equations
In Dirac’s terminology these are the secondary or
dynamical constraints
The laps and shift functions acts as Lagrange
multipliers
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Quantization


Relevant configuration space for the definition
of quantum dynamics
Superspace




space of all Riemannian 3-metrics and matter
configurations on the spatial hypersurfaces S
infinite-dimensional space, with finite number
degrees of freedom (hij(x), F(x)) at each point, x in S
This infinite-dimensional space will be configuration
space of quantum cosmology.
Metric on superspace-DeWitt metric
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Canonical Quantization


Wavefunction (WF) of the universe Y[hij,F] - functional on superspace
Unlike ordinary QM, WF does not depend explicitly on time


GR is “already parametrised” theory - GR (EH action) is timereparametrisation invariant
Time is contained implicitly in the dynamical variables, hij and F

Dirac’s quantization procedure (h/2p=1)

The WF is annihilated by the operator version of the
constraint
For the primary constraints we have

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Canonical Quantization

For the momentum constraint we have

WF is the same for configurations {hij(x), F(x)} which
are related by a coordinate transformation in the
spatial hypersurface.
Finally, the Hamiltonian constraint yields


  Gijkl    h

hij hkl


( 3)

1 ˆ
R  2 
H matter Y  0

16pG

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
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Canonical Quantization: WheelerDeWitt equation

  Gijkl    h

hij hkl




( 3)

1 ˆ
R  2 
H matter Y  0

16pG


It is not single equation – one equation at each point
xS
second order hyperbolic differential equation on
superspace
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Covariant Quantization - summary
•Canonical variables are the hij(x), and its conjugate
momentum. Wheeler-DeWitt equation, H Y=0.
•Some characteristics of this approach:
• Wave functional Y depends on the three-dimensional
metric. It is invariant under coordinate transformation on
three-space.
• No external time parameter is present anymore – theory
is “timeless”
•Wheeler-DeWitt equation is hyperbolic
•this approach is good candidate for a non-perturbative
quantum theory of gravity. It should be valid away the
Planck scale. The reason is that GR is then approximately
valid, and the quantum theory from which it emerges in
the WKB limit is
quantum
ISC2008,
Nis, Serbia,geometrodinamics
August 26 - 31,
2008
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Path Integral Quantization


An alternative to canonical quantization
The starting point: the amplitude to go from one state
with intrinsic metric hij and matter configuration F on an
initial hypersurface S to another with metric h’ij and
matter configuration F’ on a final hypersurface S’ is given
by a functional integral exp(2piS/h)=exp(iS) over all 4geometries gmn and matter configurations f which
interpolate between initial and final configurations
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Path Integral Quantization

Ordinary QFT





QG




For the real lorentzian metrics gmn and real fields f, action S is a real.
Integral oscillates and do not converge.
Wick rotation to “imaginary time” t=-it
Action is a “Euclidean”, I=-iS
The action is positive-definite, path integral is exponentially damped and
should converge.
I [gmn ,f] = -iS [gmn ,f]
sum in the integral to be over all metrics with signature (++++) which
induce the appropriate 3-metrics
Successes
 thermodynamics properties of the black holes
 gravitational instantons
Problems
 gravitational action is not positive definite – path integral does not
converge if one restricts the sum to real Euclidean-signature metric
 to make the path integral converge it is necessary to include complex
metrics in the sum.
 there is not unique contour to integrate - the results depends crucially
on the contour that
is chosen
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Nis, Serbia, August 26 - 31,
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2008
Minisuperspace



Superspace – infinite-dimensional space, with finite number
degrees of freedom (hij(x), F(x)) at each point, x in S
In practice to work with inf.dim. is not possible
One useful approximation – to truncate inf. degrees of freedom
to a finite number – minisuperspace model.



Homogeneity
isotropy or anisotropy
Homogeneity and isotropy


instead of having a separate Wheeler-DeWitt equation for each point of the
spatial hypersurface S, we then simply have a SINGLE equation for all of S.
metrics (shift vector is zero)

ds   N (t )dt  hij (q (t ))dx dx ,   1,2,..., n
2
2
2
i
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j
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Minisuperspace – isotropic model

The standard FRW metric

Model with a single scalar field simply has TWO
minisuperspace coordinates {a, F} (the cosmic scale factor and
the scalar field)
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Minisuperspace – anisotropic model

All anisotropic models



Kantowski-Sachs models
Bianchi
Kantowski-Sachs models, 3-metric


THREE minisuperspace coordinates {a, b, F} (the cosmic scale
factors and the scalar field) (topology is S1xS2)
Bianchi, most general homogeneous 3-metric with a 3dimensional group of isometries (these are in 1-1
correspondence with nine 3-dimensional Lie algebras-there are
nine types of Bianchi cosmology)
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Minisuperspace – anisotropic model




Bianchi, most general homogeneous 3-metric with a 3-dimensional
group of isometries (these are in 1-1 correspondence with nine 3dimensional Lie algebras-there are nine types of Bianchi cosmology)
The 3-metric of each of these models can be written in the form
wi are the invariant 1-forms associated with a isometry group
The simplest example is Bianchi 1, corresponds to the Lie group R3
(w1=dx, w2=dy, w3=dz)

FOUR minisuperspace coordinates {a, b, c, F} (the cosmic scale
factors and the scalar field)
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Minisuperspace propagator


For the minisuperspace models path (functional) integral is reduced
to path integral over 3-metric and configuration of matter fields, and
to another usual integration over the lapse function N.
For the boundary condition q(t1)=q’, q(t2)=q’’, in the gauge,
N=const, we have
q "; q '   dNK (q " , N ; q ' ,0)

where
K (q " , N ; q ' ,0)   Dq exp(  I [q ])


ordinary (euclidean) QM propagator between fixed minisuperspace
coordinates (q’, q’’ ) in a fixed time N
S (I) is the action of the minisuperspace model
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K (q " , N ; q ' ,0)   Dq exp(  I [q ])
Minisuperspace propagator


ordinary QM propagator between fixed minisuperspace coordinates
(q’, q’’ ) in a fixed time N
S is the action of the minisuperspace model
 1

I [q ]   dtN  2 f (q)q  q   U (q)
 2N

0
1

f is a minisuperspace metric
dsm2  f dq dq 

with an indefinite signature (-+++…)
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K (q " , N ; q ' ,0)   Dq exp(  I [q ])
Minisuperspace propagator

for the quadratic action path integral is
1
K (q " , N ; q ' ,0) 
2p


I (q " , N ; q ' ,0)

1/ 2

 I 
  det 

 
q " q ' 

2
exp(  I (q " , N ; q ' ,0))
euclidean classical action for the solution of
classical equation of motion for the q
Minisuperspace propagator is
1
q "; q ' 
2p


1/ 2

 I 
 dN   det q " q ' 
2
exp(  I (q " , N ; q ' ,0))
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Minisuperspace propagator

Procedure







metric
action
Lagrangian
equation of motion
classical action
path integral
minisuperspace propagator
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Hartle Hawking instanton

The dominating
contribution to the
Euclidean path
integral is assumed
to be half of a foursphere attached to
a part of de Sitter
space.
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Quantum Cosmology (QC)
Application of quantum theory to the
universe as a whole.
 Gravity is dominating interaction on
cosmic scales – quantum theory of gravity
is needed as a formal prerequisite for QC.
 Most work is based on the Wheeler–
DeWitt equation of quantum
geometrodynamics.

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Quantum Cosmology (QC)



The method is to restrict first the configuration space to a finite
number of variables (scale factor, inflaton field, . . . ) and then to
quantize canonically.
Since the full configuration space of three-geometries is called
‘superspace’, the ensuing models are called ‘minisuperspace
models’.
The following issues are typically addressed within quantum
cosmology:







How does one have to impose boundary conditions in quantum
cosmology?
Is the classical singularity being avoided?
How does the appearance of our classical universe emerge from
quantum cosmology?
Can the arrow of time be understood from quantum cosmology?
How does the origin of structure proceed?
Is there a high probability for an inflationary phase?
Can quantum cosmological results be justified from full quantum
gravity?
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Literature
B. de Witt, “Quantum Theory of Gravity. I.
The canonical theory”, Phys. Rev. 160,
113 (1967)
 C. Mysner, “Feynman quantization of
general relativity”, Rev. Mod. Phys, 29,
497 (1957).
 D. Wiltshire, “An introduction to Quantum
Cosmology”, lanl archive

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