ISC2008, Nis, Serbia, August , Fundamentals of Quantum Cosmology Ljubisa Nesic Department of Physics, University of Nis, Serbia

ISC2008, Nis, Serbia, August , Fundamentals of Quantum Cosmology 1. Basic Ideas of Quantum Cosmology 2. Minisuperspace Models in Quantum Cosmology

ISC2008, Nis, Serbia, August , 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 , 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 , Formulations of QM  wavefunction (Schrodinger), state  matrix (Heisenberg), June 1925, measurable quantity  path integral-sum over histories (Feynman) – transition amplitude from (x i,t i ) to (x f,t f ) is proportional to exp(2iS/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 , Interpretation of QM  The many world interpretation (Everett)  The transactional interpretation (Cramer)  …

ISC2008, Nis, Serbia, August ,  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 ,

9  Planck time  At Planck scale, Compton wavelength is roughly equal to its gravitational (Shwarzschild) radius.  classical concept of time and space is meaningless

ISC2008, Nis, Serbia, August , 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 , 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 , 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 , 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 , Hamiltonian Formulation of GR : 3+1 decomposition  3+1 split of the 4-dimensional spacetime manifold M Differentiable Manifold Metric

ISC2008, Nis, Serbia, August , decomposition  spatial hypersurfaces  t labeled by a global time function t

ISC2008, Nis, Serbia, August , decomposition  4-dimensional metric

ISC2008, Nis, Serbia, August , decomposition

ISC2008, Nis, Serbia, August , decomposition  In components

ISC2008, Nis, Serbia, August , 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) R i jkl – from the intrinsic metric alone – describes the curvature intrinsic to the hypersurfaces  t  Extrinsic curvature (second fundamental form), K ij – describes how the spatial hypersurfaces curve with respect to the 4-dimensional spacetime manifold within which they are embedded.

ISC2008, Nis, Serbia, August , The action  Matter – single scalar field  Einstein-Hilbert action

ISC2008, Nis, Serbia, August , 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

ISC2008, Nis, Serbia, August , The action in 3-1 decomposition  The action

ISC2008, Nis, Serbia, August , Canonical momenta  Canonical momenta for the basic variables  Last two equations – primary constraints in Dirac’s terminology

ISC2008, Nis, Serbia, August , Hamiltonian  Hamiltonian  If we vary S with respect to  ij and   we obtain their defining relations  Action

ISC2008, Nis, Serbia, August , 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 , 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 ( h ij (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 , Canonical Quantization  Wavefunction (WF) of the universe  [h ij  - 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, h ij 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 , Canonical Quantization  WF is the same for configurations { h ij (x),  (x)} which are related by a coordinate transformation in the spatial hypersurface.  Finally, the Hamiltonian constraint yields  For the momentum constraint we have

ISC2008, Nis, Serbia, August , Canonical Quantization: Wheeler- DeWitt equation  It is not single equation – one equation at each point x   second order hyperbolic differential equation on superspace

ISC2008, Nis, Serbia, August , Covariant Quantization - summary Canonical variables are the h ij (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 , Path Integral Quantization  An alternative to canonical quantization  The starting point: the amplitude to go from one state with intrinsic metric h ij 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(2iS/h)=exp(iS) over all 4- geometries g  and matter configurations  which interpolate between initial and final configurations

ISC2008, Nis, Serbia, August ,

ISC2008, Nis, Serbia, August , 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 , Minisuperspace  Superspace – infinite-dimensional space, with finite number degrees of freedom ( h ij (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)

ISC2008, Nis, Serbia, August , 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 , 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 S 1 xS 2 )  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 , 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 R 3 ( 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 , Minisuperspace propagator  ordinary (euclidean) QM propagator between fixed minisuperspace coordinates ( q  ’, q  ’’ ) 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 q  ( t 1 )= q  ’, q  ( t 2 )= q  ’’, in the gauge, =const, we have  where

ISC2008, Nis, Serbia, August , Minisuperspace propagator  with an indefinite signature (-+++…)  ordinary QM propagator between fixed minisuperspace coordinates ( q  ’, q  ’’ ) in a fixed time N  S is the action of the minisuperspace model  f  is a minisuperspace metric

ISC2008, Nis, Serbia, August , Minisuperspace propagator  Minisuperspace propagator is  for the quadratic action path integral is euclidean classical action for the solution of classical equation of motion for the q 

ISC2008, Nis, Serbia, August , Minisuperspace propagator  Procedure metric action Lagrangian equation of motion classical action path integral minisuperspace propagator

ISC2008, Nis, Serbia, August , 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 , 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 , 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 , 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