The first models of quantum cosmology appear in the sixties with the work of John Archibald Wheeler and Bryce DeWitt in canonical quantum gravity. However, the detailed models were formulated in the eigthties. At that time a lot of debate on quantum cosmology took place, especially driven by people like Hartle, Hawking, Vilenkin, Linde, etc.

Canonical Quantum Gravity

The start to set-up a theory of canonical quantum gravity is the ADM formalism. This is a hamiltonian formulation of general relativity. Space-time M is assumed to be globally hyperbolic (without closed timelike curves and allowing well-defined initial value problems) and is foliated in spatial hypersurfaces for each value of time M = S x t. It has to be noted that this is a coodinate dependent procedure, because different observers may foliate space-time in different ways without parallel time coordinates and even without the same time parametrization.

The three-dimensional riemannian metric hij (i, j = 1, 2, 3) of the spatial hypersurfaces is taken as the new dynamic variable instead of the four-dimensional lorentzian metric guv (u, v = 0, 1, 2, 3). The set of metrics hij is not a physical space as two elements that are related by spatial diffeomorphisms correspond to the same physical situation. The physical space is therefore the equivalence classes hij which are related by means of spatial diffeomorphisms. This space is called superspace.

The equations of motion for hij and its conjugate momentum follow from Einstein's equations for guv. Additionally to these dynamical variables there exists two parameters that describe how the different spatial hypersurfaces are glued toghether. This parameters are not dynamical and give rise to two constraints. On the one hand a constraint for the freedom of performing spatial diffeomorphisms within a spatial hypersurface, called diffeomorphism constraint, and on the other hand a constraint for the freedom of selecting and parametrizing the time coordinate, called hamiltonian constraint. The hamiltonian constraint is equivalent to an indentically zero hamiltonian H = 0.

For the hij and their canonically conjugate momenta the Dirac quantization procedure is applied. This speficies that the dynamical variables are promoted to operators acting on states in a Hilbert space. The states are functionals of metrics F(hij), similar to NRQM where the states are wavefunctions of positions. The constraints have to be implemented as operator equations in Hilbert space.

Canonical Quantum Cosmology and the Wheeler-deWitt equation

Now, lets move to quantum cosmology. The way to proceed is basically the same, but easier. The space of dynamical variables is reduced to only those metrics that fulfil the cosmological principle (i.e. only FRW metrics). This space is called minisuperspace. The single degree of freedom in the hij becomes therefore the cosmological scale-factor a. The reformulation of general relativity into a hamiltonian formulation of classical cosmology is much more easy than in the full theory. The basic result we are interested in is that the first classical Friedmann equation is nothing else than the hamiltonian constraint.

The Dirac quantization procedure promotes the scale-factor and its conjugate momentum to operators a -> a and p -> - i d/da (same as for NRQM). They act on the wavefunctional F(a) that is called wavefunction of the universe. The imposition of the hamiltonian constraint as an operator equation leads to H F(a) = 0. This is the Wheeler-deWitt equation that encondes a great deal of the dynamics of the universe, same as the first Friedmann equation does for classical cosmology.

For the simplest models, like for example an empty model with positive cosmological constant, the equation is formally equivalent to the equation of motion of a point particle in a (effective) potential. The hamiltonian for such a model with cosmological constant L is

H = 1/2 (p� / a - a� L)

which leads to the Wheeler-deWitt equation

[d�/da� - V(a)] F(a) = 0

To solve this Wheeler-DeWitt equation it is possible to make use of the semiclassical WKB approximation, same than for the point particle. It is assumed that F(a) is a linear combination of basic states and that each of them fulfils the Wheeler-deWitt equation mentioned before. For each of these basic states the WKB approximation can be used. This splits the solutions for the basic states into two regimes: (i) the oscillating regime of the classical solution (ii) the exponentially decreasing regime of the classically prohibited solution, that corresponds to a tunneling process.

What is the physical meaning of such solutions? Specific solutions are only determined after imposing boundary conditions, but we can already mention the consequence of having a tunneling process in the semiclassical regime of an empty model with positive cosmological constant. The empty model with positive cosmological constant is isotropic in space as well as in time and does not contain a singularity in its past. It is usually called de-Sitter model. The scale factor in a de-Sitter model evolves as depicted below. In this model the universe is eternal in past, which does not fit with observations.

deSitter model

However, the surprising thing in canonical quantum cosmology is that it allows for an universe that starts classically at t = 0 behaving as a de-Sitter model. The mechanism that makes this possible is quantum tunneling from a state without any classical analogue, across the effective potential V(a), to the classical de-Sitter phase. This behaviour is described by the Wheeler-deWitt equation.

Quantum tunneling of the universe wavefunction

In order to determine a unique solution for the wavefunction of the universe from the Wheeler-DeWitt equation it is necessary to impose initial conditions. Two proposals can be considered to be the most important ones:

• The Vilenkin proposal or tunneling proposal.
• The Hartle-Hawking proposal or no-boundary proposal.

From each proposal there is a solution for the wavefunction of the universe that is respectively know as Vilenkin wavefunction and Hartle-Hawking wavefunction.

Same as in non-relativistic quantum mechanics, canonical quantum gravity can be reformulated in terms of a path integral. This is a way of calculating transition probabilities between two spatial configurations hij (i, i = 1, 2, 3), summing over all possible space-time geometries guv (u, v = 0, 1, 2, 3) that fit between the initial and final hij's. Same as in non-relativistic quantum mechanics, in the limit of very small h, non-classical geometries will have a fast oscilating contribution to the path integral and will interfere and cancel their contributions.

The Vilenkin proposal

In the proposal by Vilenkin it is assumed that the initial condition is a vanishing spatial geometry, hij = 0. This state was called �nothing� by Vilenkin, but it would rather correspond to the vacuum state of a theory more fundamental than general relativity. Vilenkin has shown that for a model of a universe with a scalar field with self-interacting potential (instead ot the empty model with cosmological constant), the transition from hij = 0, over some set of geometries, leads to most probable final states in which the scalar field is at a high potential energy. A high potential energy of the scalar field is a necessary condition for inflation.

The evolution in the Vilenkin model

The Hartle-Hawking proposal

In the proposal by Hartle-Hawking it is assumed that the initial condition is the set of all possible compact euclidean geometries, like for example a 4-sphere. Intuitively, this it means connecting space-time with all geometries that close it smoothly below the Planck time, without tips or singularities at the beginning. Quoting Steffen Hawking "The boundary condition of the universe is that it has no boundary".

The Hartle-Hawking approach is supported by the following. In the path integral formulation in quantum field theory it is usual to perform a Wick rotation to improve the mathematical behaviour of the integral. A Wick rotation consists of changing time to imaginary time t -> - i t and changing therefore the lorentzian metric with signature (-+++) to an euclidean metric of signature (++++). This approach is known as euclidean quantum gravity and has several advantages: (i) it makes the path integral solvable (ii) the procedure can be used successfully to explain the thermodynamic properties of black holes (iii) it makes it possible to analyze gravitational instantons (stable solutions to the euclidean equations of motion). In summary, the use of euclidean geometries as initial conditions is supported by the success of the Wick rotation to solve some of the problems of canonical quantum gravity.

The evolution in the Hartle-Hawking model

Note that in both proposals the behaviour of the universe is given by the Wheeler-DeWitt equation. The solutions for the wavefunction can be split into the classical regime and the tunneling regime that takes place through the effective potential. There are other proposals for the initial conditions of the universe that lead to other wavefunctions peaking on other classical trayectories. These are however the most important ones. The need to impose initial conditions to the universe can be regarded as a very unpleasant feature of the theory. Any initial condition adquires the status of a fundamental law, since it cannot be deduced from first principles. The hope that initial conditions may result from the dynamics of the theory is not realistic in canonical quantum cosmology nor in string cosmology, but, as we will see, can be realized in loop quantum cosmology.

Both, the no-boundary proposal and the tunneling psoposal, provide consistent conditions for the inflationary regime as well as for the creation of the first density perturbations from quantum fluctuations of the cosmological vacuum.

Other models

Both models discussed above make it possible to compute transition probabilities for the very early universe. However, the very definition of probabilities is not well defined. First, there is an issue about interpretation and possible measurement of absolute probabilities in a single universe. Some alternative approaches (see e.g. the work of D. N. Page) try to make use of conditional probabilities. There is another issue about the fact that the Wheeler-DeWitt equation does not lead to positive definite probabilities, in the same way than the Klein-Gordon equation in quantum mechanics. In quantum field theory this problem is solved by means of the second quantization, i.e. promoting the wavefunction to a field operator that creates and annihilates basis states in the particle representation. In canonical quantum cosmology one may try a similar approach, sometimes called "third quantization", expressing the wavefunctional of the universe as creation and annihilation operators on basic states (baby universes...?). But as we do not perform measurements over a statistical ensemble of universes it is not clear how one can arrive at sensible probabilities using such a formulation.

Problems and unsolved issues

The main problems of canonical quantum cosmology are the following:

• The singularity. The Wheeler-DeWitt equation does not remove the singularity, although in some cases the solutions avoid the singularity due to the effective potential as we have seen. In short, canonical quantum gravity is not well defined. The models of canonical quantum cosmology do only apply on a semiclassical approximation and there is no way to make predictions about the dynamics of the singularity.
• Decoherence. If the wavefunction of the universe is a linear combination of basic states, then these can interfere with each other. However, we observe a classical universe without interference on the macrocospic world. This requires of a mechanism to allow for decoherence of the wavefunction of the universe.
• The problem of time. This is more a general problem in the canonical approach of quantum gravity, but its implications are important in cosmology. The Hamiltonian constraint indicates that the theory does not have a predefined notion of time. Reparametrizations of the time evolution are gauge transformations without physical content. To single out the real time evolution of the universe one has to fix a specific gauge. This breaks the original symmetry of the classical equations under diffeomorphisms and, moreover, different choices of different time variables lead to different quantum theories. The question that arises is related to the next issue: are we allowed to select the scale factor as a parameter that scans time evolution already in the quantum regime?
• The minisuperspace approximation. The minisuperspace approximation does fix simultaneously canonically conjugate variables violating the uncertainty principle (due to symmetry the dynamic variable and its conjugate momentum are both zero). It leaves the scale factor as the single degree of freedom in the models. Although our universe is currently homogenous and isotropic, this assumption need not to be valid at the beginning. Note that the inflationary phase, already in the classical regime, might have created homogeneity out of an inhomogeneous and random initial state.
• Inhomogeneities. There exist models that describe inhomogeneities, but they are poorly understood.
• Initial conditions. Initial conditions that are imposed to the universe cannot be derived from any principle and become the same status than fundamental laws. Bryce DeWitt envisioned a theory in which the requirement of mathematical consistency should be sufficient to guarantee a unique solution to the Wheeler-deWitt equation. However, this cannot be realized in canonical quatum cosmology.
• Formalism. The theory has diverse mathematical and consistency problems, like factor ordering and other ambiguities, and the definition of a proper notion of probabilites in a single universe.
Nevertheless, one would expect that some aspects of the mentioned canonical quantum cosmology models should be present in a complete and consistent model of quantum cosmology. There is no clear reason to expect the description of the origin of the universe beyond the singularity to be correct, but the calculations for transition probabilities and set-up of inflation near the classical regime may be a good approximation to reality.

Some further reading

• Wikipedia, ADM formalism, http://en.wikipedia.org/wiki/ADM_formalism
• R. M. Wald, General Relativity, University of Chicago Press, Chicago 1984 (Appendix E)
• Wikipedia, WKB approximation, http://en.wikipedia.org/wiki/WKB_approximation
• D. L. Wiltshire, Quantum Cosmology, gr-qc/0101003
• A. Vilenkin, Appoaches to Quantum Cosmology, gr-qc/9403010
• A. Vilenkin, The quantum cosmology debate, gr-qc/9812027
• A. Vilenkin, Predictions from Quantum Cosmology, gr-qc/9507018
• A. Mostafazadeh, Wave function of the universe and its meaning, gr-qc/0308029
• D. N. Page, Clock Time and Entropy, gr-qc/9303020