## Quantum gravity

### Path integrals

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 Prize winner 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.

### Issues

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.

### No boundary proposal

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.

### Instantons

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 occurring 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.

There are many theorized instantons that explain how the Universe could come about. Click here to read about them.