isc2008, nis, serbia, august 26 - 31, 2008 1 fundamentals of quantum cosmology ljubisa nesic...

ISC2008, Nis, Serbia, August 26 - 31, 2008 1

Fundamentals of Quantum

CosmologyLjubisa Nesic

Department of Physics, University of Nis, Serbia


Fundamentals of Quantum Cosmology1. Basic Ideas of Quantum Cosmology2. Minisuperspace Models in Quantum

Cosmology


Basic Ideas of Quantum Cosmology Introduction

Quantum cosmology and quantum gravity A brief history of quantum cosmology

Hamiltonian Formulation of General Relativity The 3+1 decomposition The action

Quantization Superspace Canonical quantization Path integral quantization Minisuperspace


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


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(2iS/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


Interpretation of QM The many world interpretation (Everett) The transactional interpretation (Cramer) …


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?


mc

GlPlanck

353

106,1

Planck time

At Planck scale, Compton wavelength is roughly equal to its gravitational (Shwarzschild) radius.

classical concept of time and space is meaningless

sc

GtPlanck

445

104,5

GeVG

cEPlanck

195

1022,1


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)


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


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


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


Hamiltonian Formulation of GR : 3+1 decomposition 3+1 split of the 4-dimensional spacetime

manifold M

gM ,

Differentiable Differentiable

ManifoldManifold

MetricMetric


3+1 decomposition spatial hypersurfaces t labeled by a global

time function t


3+1 decomposition 4-dimensional metric


3+1 decomposition


3+1 decomposition

In components


3+1 decomposition

semicolon – covariant differentiation with respect to the 4-metric,

vertical bar – covariant differentiation with respect to the induced 3-metric.

Intrinsic curvature tensor (3)Rijkl – from the intrinsic metric

alone – describes the curvature intrinsic to the hypersurfaces t

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.


The action

Matter – single scalar field

Einstein-Hilbert action


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 gαβ gives the Einstein equations


The action in 3-1 decomposition

The action


Canonical momenta

Canonical momenta for the basic variables

Last two equations – primary constraints in Dirac’s terminology


Hamiltonian Hamiltonian

If we vary S with respect to ij and we obtain their defining relations

Action


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


Quantization Relevant configuration space for the definition

of quantum dynamics Superspace

space of all Riemannian 3-metrics and matter configurations on the spatial hypersurfaces

infinite-dimensional space, with finite number degrees of freedom (hij(x), (x)) at each point, x in

This infinite-dimensional space will be configuration space of quantum cosmology.

Metric on superspace-DeWitt metric


Canonical Quantization Wavefunction (WF) of the universe [hij - functional on superspace Unlike ordinary QM, WF does not depend explicitly on time

GR is “already parametrised” theory - GR (EH action) is time-reparametrisation invariant

Time is contained implicitly in the dynamical variables, hij and

The WF is annihilated by the operator version of the constraint For the primary constraints we have

Dirac’s quantization procedure (h/2=1)


Canonical Quantization

WF is the same for configurations {hij(x), (x)} which are related by a coordinate transformation in the spatial hypersurface.

Finally, the Hamiltonian constraint yields

For the momentum constraint we have

0ˆ16

12)3(

matter

klijijkl H

GRh

hhG


Canonical Quantization: Wheeler-DeWitt equation

It is not single equation – one equation at each point x

second order hyperbolic differential equation on superspace

0ˆ16

12)3(

matter

klijijkl H

GRh

hhG


Covariant Quantization - summary

•Canonical variables are the hij(x), and its conjugate momentum. Wheeler-DeWitt equation, =0.

•Some characteristics of this approach:

• Wave functional 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 geometrodinamics


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 on an initial hypersurface to another with metric h’ij and matter configuration ’ on a final hypersurface ’ is given by a functional integral exp(2iS/h)=exp(iS) over all 4-geometries g and matter configurations which interpolate between initial and final configurations


Path Integral Quantization

QG I [g,] = -iS [g,] 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

Ordinary QFT For the real lorentzian metrics g and real fields , action S is a real.

Integral oscillates and do not converge. Wick rotation to “imaginary time” t=-i Action is a “Euclidean”, I=-iS The action is positive-definite, path integral is exponentially damped and

should converge.


Minisuperspace Superspace – infinite-dimensional space, with finite number degrees of freedom (hij(x), (x)) at each point, x

in 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 , we then simply have a SINGLE

equation for all of . metrics (shift vector is zero)

ndxdxtqhdttNds jiij ,...,2,1,))(()( 222 αα


Minisuperspace – isotropic model

The standard FRW metric

Model with a single scalar field simply has TWO minisuperspace coordinates {a, } (the cosmic scale factor and the scalar field)


Minisuperspace – anisotropic model

All anisotropic models Kantowski-Sachs models Bianchi

THREE minisuperspace coordinates {a, b, } (the cosmic scale factors and the scalar field) (topology is S1xS2)

Bianchi, most general homogeneous 3-metric with a 3-dimensional group of isometries (these are in 1-1 correspondence with nine 3-dimensional Lie algebras-there are nine types of Bianchi cosmology)

Kantowski-Sachs models, 3-metric


Minisuperspace – anisotropic model

i are the invariant 1-forms associated with a isometry group The simplest example is Bianchi 1, corresponds to the Lie group R3

(1=dx, 2=dy, 3=dz)

Bianchi, most general homogeneous 3-metric with a 3-dimensional group of isometries (these are in 1-1 correspondence with nine 3-dimensional Lie algebras-there are nine types of Bianchi cosmology)

The 3-metric of each of these models can be written in the form

FOUR minisuperspace coordinates {a, b, c, } (the cosmic scale factors and the scalar field)


Minisuperspace propagator

ordinary (euclidean) QM propagator between fixed minisuperspace coordinates (qα’, qα’’ ) in a fixed time N S (I) is the action of the minisuperspace model

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 qα(t1)=qα’, qα(t2)=qα’’, in the gauge, =const, we have

)0,';,"(';" αααα qNqdNKqq

where

])[exp()0,';,"( αααα qIDqqNqK



with an indefinite signature (-+++…)

βααβ dqdqfdsm 2

1

02

)()(2

1][ qUqqqf

NdtNqI βα

αβα

ordinary QM propagator between fixed minisuperspace coordinates (qα’, qα’’ ) in a fixed time N

S is the action of the minisuperspace model

fαβ is a minisuperspace metric




Minisuperspace propagator is

)0,';,"( αα qNqI

for the quadratic action path integral is

euclidean classical action for the solution of classical equation of motion for the qα


))0,';,"(exp('"

det2

1';"

2/12αα

αααα

qNqI

qq

IdNqq

))0,';,"(exp('"

det2

1)0,';,"(

2/12αα

αααα

qNqI

qq

IqNqK


Minisuperspace propagator Procedure

metric action Lagrangian equation of motion classical action path integral minisuperspace propagator


Hartle Hawking instanton The dominating

contribution to the Euclidean path integral is assumed to be half of a four-sphere attached to a part of de Sitter space.


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.


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?


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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