Upload
jaylin-gamble
View
219
Download
4
Tags:
Embed Size (px)
Citation preview
ISC2008, Nis, Serbia, August 26 - 31, 2008 1
Fundamentals of Quantum
CosmologyLjubisa Nesic
Department of Physics, University of Nis, Serbia
ISC2008, Nis, Serbia, August 26 - 31, 2008 2
Fundamentals of Quantum Cosmology1. Basic Ideas of Quantum Cosmology2. Minisuperspace Models in Quantum
Cosmology
ISC2008, Nis, Serbia, August 26 - 31, 2008 3
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 4
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 5
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 6
Interpretation of QM The many world interpretation (Everett) The transactional interpretation (Cramer) …
ISC2008, Nis, Serbia, August 26 - 31, 2008 7
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?
ISC2008, Nis, Serbia, August 26 - 31, 2008 8
ISC2008, Nis, Serbia, August 26 - 31, 2008 9
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 10
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)
ISC2008, Nis, Serbia, August 26 - 31, 2008 11
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 12
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 13
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 14
Hamiltonian Formulation of GR : 3+1 decomposition 3+1 split of the 4-dimensional spacetime
manifold M
gM ,
Differentiable Differentiable
ManifoldManifold
MetricMetric
ISC2008, Nis, Serbia, August 26 - 31, 2008 15
3+1 decomposition spatial hypersurfaces t labeled by a global
time function t
ISC2008, Nis, Serbia, August 26 - 31, 2008 16
3+1 decomposition 4-dimensional metric
ISC2008, Nis, Serbia, August 26 - 31, 2008 17
3+1 decomposition
ISC2008, Nis, Serbia, August 26 - 31, 2008 18
3+1 decomposition
In components
ISC2008, Nis, Serbia, August 26 - 31, 2008 19
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.
ISC2008, Nis, Serbia, August 26 - 31, 2008 20
The action
Matter – single scalar field
Einstein-Hilbert action
ISC2008, Nis, Serbia, August 26 - 31, 2008 21
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 22
The action in 3-1 decomposition
The action
ISC2008, Nis, Serbia, August 26 - 31, 2008 23
Canonical momenta
Canonical momenta for the basic variables
Last two equations – primary constraints in Dirac’s terminology
ISC2008, Nis, Serbia, August 26 - 31, 2008 24
Hamiltonian Hamiltonian
If we vary S with respect to ij and we obtain their defining relations
Action
ISC2008, Nis, Serbia, August 26 - 31, 2008 25
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 26
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 27
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)
ISC2008, Nis, Serbia, August 26 - 31, 2008 28
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 29
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 30
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 31
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 32
ISC2008, Nis, Serbia, August 26 - 31, 2008 33
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.
ISC2008, Nis, Serbia, August 26 - 31, 2008 34
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 35
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)
ISC2008, Nis, Serbia, August 26 - 31, 2008 36
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 37
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)
ISC2008, Nis, Serbia, August 26 - 31, 2008 38
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
ISC2008, Nis, Serbia, August 26 - 31, 2008 39
Minisuperspace propagator
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
])[exp()0,';,"( αααα qIDqqNqK
ISC2008, Nis, Serbia, August 26 - 31, 2008 40
Minisuperspace propagator
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
])[exp()0,';,"( αααα qIDqqNqK
))0,';,"(exp('"
det2
1';"
2/12αα
αααα
qNqI
IdNqq
))0,';,"(exp('"
det2
1)0,';,"(
2/12αα
αααα
qNqI
IqNqK
ISC2008, Nis, Serbia, August 26 - 31, 2008 41
Minisuperspace propagator Procedure
metric action Lagrangian equation of motion classical action path integral minisuperspace propagator
ISC2008, Nis, Serbia, August 26 - 31, 2008 42
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.
ISC2008, Nis, Serbia, August 26 - 31, 2008 43
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.
ISC2008, Nis, Serbia, August 26 - 31, 2008 44
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?
ISC2008, Nis, Serbia, August 26 - 31, 2008 45
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