Cambridge Relativity: Quantum Gravity

Quantum Cosmology

The physical laws that govern the universe prescribe how an initial state evolves with time. In classical physics, if the initial state of a system is specified exactly then the subsequent motion will be completely predictable. In quantum physics, specifying the initial state of a system allows one to calculate the probability that it will be found in any other state at a later time. Cosmology attempts to describe the behaviour of the entire universe using these physical laws. In applying these laws to the universe one immediately encounters a problem. What is the initial state that the laws should be applied to? In practice, cosmologists tend to work backwards by using the observed properties of the universe now to understand what it was like at earlier times. This approach has proved very successful. However it has led cosmologists back to the question of the initial conditions.

Inflation (a period of accelerating expansion in the very early universe) is now accepted as the standard explanation of several cosmological problems. In order for inflation to have occurred, the universe must have been formed containing some matter in a highly excited state. Inflationary theory does not address the question of why this matter was in such an excited state. Answering this demands a theory of the pre-inflationary initial conditions. There are two serious candidates for such a theory. The first, proposed by Andrei Linde of Stanford University, is called chaotic inflation. According to chaotic inflation, the universe starts off in a completely random state. In some regions matter will be more energetic than in others and inflation could ensue, producing the observable universe.

The second contender for a theory of initial conditions is quantum cosmology, the application of quantum theory to the entire universe. At first this sounds absurd because typically large systems (such as the universe) obey classical, not quantum, laws. Einstein's theory of general relativity is a classical theory that accurately describes the evolution of the universe from the first fraction of a second of its existence to now. However it is known that general relativity is inconsistent with the principles of quantum theory and is therefore not an appropriate description of physical processes that occur at very small length scales or over very short times. To describe such processes one requires a theory of quantum gravity.

In non-gravitational physics the approach to quantum theory that has proved most successful involves mathematical objects known as path integrals. Path integrals were introduced by the Nobel prizewinner Richard Feynman, of CalTech. In the path integral approach, the probability that a system in an initial state A will evolve to a final state B is given by adding up a contribution from every possible history of the system that starts in A and ends in B. For this reason a path integral is often referred to as a `sum over histories'. For large systems, contributions from similar histories cancel each other in the sum and only one history is important. This history is the history that classical physics would predict.

For mathematical reasons, path integrals are formulated in a background with four spatial dimensions rather than three spatial dimensions and one time dimension. There is a procedure known as `analytic continuation' which can be used to convert results expressed in terms of four spatial dimensions into results expressed in terms of three spatial dimensions and one time dimension. This effectively converts one of the spatial dimensions into the time dimension. This spatial dimension is sometimes referred to as `imaginary' time because it involves the use of so-called imaginary numbers, which are well defined mathematical objects used every day by electrical engineers.

The success of path integrals in describing non-gravitational physics naturally led to attempts to describe gravity using path integrals. Gravity is rather different from the other physical forces, whose classical description involves fields (e.g. electric or magnetic fields) propagating in spacetime. The classical description of gravity is given by general relativity, which says that the gravitational force is related to the curvature of spacetime itself i.e. to its geometry. Unlike for non-gravitational physics, spacetime is not just the arena in which physical processes take place but it is a dynamical field. Therefore a sum over histories of the gravitational field in quantum gravity is really a sum over possible geometries for spacetime.

The gravitational field at a fixed time can be described by the geometry of the three spatial dimensions at that time. The history of the gravitational field is described by the four dimensional spacetime that these three spatial dimensions sweep out in time. Therefore the path integral is a sum over all four dimensional spacetime geometries that interpolate between the initial and final three dimensional geometries. In other words it is a sum over all four dimensional spacetimes with two three dimensional boundaries which match the initial and final conditions. Once again, mathematical subtleties require that the path integral be formulated in four spatial dimensions rather than three spatial dimensions and one time dimension.

The path integral formulation of quantum gravity has many mathematical problems. It is also not clear how it relates to more modern attempts at constructing a theory of quantum gravity such as string/M-theory. However it can be used to correctly calculate quantities that can be calculated independently in other ways e.g. black hole temperatures and entropies.

We can now return to cosmology. At any moment, the universe is described by the geometry of the three spatial dimensions as well as by any matter fields that may be present. Given this data one can, in principle, use the path integral to calculate the probability of evolving to any other prescribed state at a later time. However this still requires a knowledge of the initial state, it does not explain it.

Quantum cosmology is a possible solution to this problem. In 1983, Stephen Hawking and James Hartle developed a theory of quantum cosmology which has become known as the `No Boundary Proposal'. Recall that the path integral involves a sum over four dimensional geometries that have boundaries matching onto the initial and final three geometries. The Hartle-Hawking proposal is to simply do away with the initial three geometry i.e. to only include four dimensional geometries that match onto the final three geometry. The path integral is interpreted as giving the probability of a universe with certain properties (i.e. those of the boundary three geometry) being created from nothing.

In practice, calculating probabilities in quantum cosmology using the full path integral is formidably difficult and an approximation has to be used. This is known as the semiclassical approximation because its validity lies somewhere between that of classical and quantum physics. In the semiclassical approximation one argues that most of the four dimensional geometries occuring in the path integral will give very small contributions to the path integral and hence these can be neglected. The path integral can be calculated by just considering a few geometries that give a particularly large contribution. These are known as instantons. Instantons don't exist for all choices of boundary three geometry; however those three geometries that do admit the existence of instantons are more probable than those that don't. Therefore attention is usually restricted to three geometries close to these.

Remember that the path integral is a sum over geometries with four spatial dimensions. Therefore an instanton has four spatial dimensions and a boundary that matches the three geometry whose probability we wish to compute. Typical instantons resemble (four dimensional) surfaces of spheres with the three geometry slicing the sphere in half. They can be used to calculate the quantum process of universe creation, which cannot be described using classical general relativity. They only usually exist for small three geometries, corresponding to the creation of a small universe. Note that the concept of time does not arise in this process. Universe creation is not something that takes place inside some bigger spacetime arena - the instanton describes the spontaneous appearance of a universe from literally nothing. Once the universe exists, quantum cosmology can be approximated by general relativity so time appears.

People have found different types of instantons that can provide the initial conditions for realistic universes. The first attempt to find an instanton that describes the creation of a universe within the context of the `no boundary' proposal was made by Stephen Hawking and Ian Moss. The Hawking-Moss instanton describes the creation of an eternally inflating universe with `closed' spatial three-geometries.

It is presently an unsolved question whether our universe contains closed, flat or open spatial three-geometries. In a flat universe, the large-scale spatial geometry looks like the ordinary three-dimensional space we experience around us. In contrast to this, the spatial sections of a realistic closed universe would look like three-dimensional (surfaces of) spheres with a very large but finite radius. An open geometry would look like an infinite hyperboloid. Only a closed universe would therefore be finite. There is, however, nowadays strong evidence from cosmological observations in favour of an infinite open universe. It is therefore an important question whether there exist instantons that describe the creation of open universes.

The idea behind the Coleman-De Luccia instanton, discovered in 1987, is that the matter in the early universe is initially in a state known as a false vacuum. A false vacuum is a classically stable excited state which is quantum mechanically unstable. In the quantum theory, matter which is in a false vacuum may `tunnel' to its true vacuum state. The quantum tunnelling of the matter in the early universe was described by Coleman and De Luccia. They showed that false vacuum decay proceeds via the nucleation of bubbles in the false vacuum. Inside each bubble the matter has tunnelled. Surprisingly, the interior of such a bubble is an infinite open universe in which inflation may occur. The cosmological instanton describing the creation of an open universe via this bubble nucleation is known as a Coleman-De Luccia instanton.

The Coleman-De Luccia Instanton

Remember that this scenario requires the existence of a false vacuum for the matter in the early universe. Moreover, the condition for inflation to occur once the universe has been created strongly constrains the way the matter decays to its true vacuum. Therefore the creation of open inflating universes appears to be rather contrived in the absence of any explanation of these specific pre-inflationary initial conditions.

Recently, Stephen Hawking and Neil Turok have proposed a bold solution to this problem. They constructed a class of instantons that give rise to open universes in a similar way to the instantons of Coleman and De Luccia. However, they did not require the existence of a false vacuum or other very specific properties of the excited matter state. The price they pay for this is that their instantons have singularities: places where the curvature becomes infinite. Since singularities are usually regarded as places where the theory breaks down and must be replaced by a more fundamental theory, this is a quite controversial feature of their work.

The Hawking-Turok Instanton

The question of course arises which of these instantons describes correctly the creation of our own universe. The way one might hope to distinguish between different theories of quantum cosmology is by considering quantum fluctuations about these instantons. The Heisenberg uncertainty principle in quantum mechanics implies that vacuum fluctuations are present in every quantum theory. In the full quantum picture therefore, an instanton provides us just with a background geometry in the path integral with respect to which quantum fluctuations need to be considered.

During inflation, these quantum mechanical vacuum fluctuations are amplified and due to the accelerating expansion of the universe they are stretched to macroscopic length scales. Later on, when the universe has cooled, they seed the growth of large scale structures (e.g. galaxies) like those we see today. One sees the imprint of these primordial fluctuations as small temperature perturbations in the cosmic microwave background radiation.

Since different types of instantons predict slightly different fluctuation spectra, the temperature perturbations in the cosmic microwave background radiation will depend on the instanton from which the universe was created. In the next decade the satellites MAP and PLANCK will be launched to measure the temperature of the microwave background radiation in different directions on the sky to a very high accuracy. The observations will not only provide us with a very important test of inflation itself but may also be the first possibility to observationally distinguish between different theories for quantum cosmology.

The observations made by MAP and PLANCK will therefore turn the `no boundary' proposal and instanton cosmology into real testable science!

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