### Consistent Histories

Consistent Histories is an interpretation of quantum mechanics that started in 1984 with a statistical physicist Robert Griffiths, who discovered a set of criteria designed to assign conditional probabilities based on classical rules of probability. The interpretation was picked up by Roland Omnes in 1988 who gave the theory a formal logical systemisation and later by James Hartle and Murray Gell-Mann, who emphasise the role of decoherence in replacing collapse of the wave function in 1990.

The interpretation starts with with subspaces of the Hilbert space and applies to closed systems by treating them as a sequence of events or "histories". An event specificies the properties of a system as a projection operator on the Hilbert space, where:

A "homogenous history" is taken as a series of such operators. The evolution of these histories from event-to-event is stochastic and evolves according to the Schrodinger equation in Griffiths' original formulation. Only histories which satisfy a set of consistency conditions are assigned probabilities and given an interpretation in the theory. These histories do not represent real features of reality but are a useful frame work for discussing time sequences of possibilities, they behave as classical histories only to the extent that they are non-interfering. The weight of a history is given by

E represents the event or its operator. If two histories have weights which are orthogonal such that interference is negligible

Then their weights can be added together. This may be extended to include so called "families" which is a space to include all non-interfering histories. For such families we can consider a generalised coordinate description of any classical field that is considered fundamental to physics, like the gravitational field, if the projection operators up to n include all the possible field variables, Hartle and Gell-Mann call this set exhaustive. We can then go on to define "alternative histories" as an exhaustive set of histories containing an alternative range. The extreme case is to give a complete set of operators at all times, a completely fine-grained history. However it is essential to the Histories approach that fine grained histories cannot be assigned probabilities.

A consistent account of probability will involve a suitable set of coarse grainned histories. The Hartle Gell-Mann approach further specifies specific conditions which apply to the early universe for projection operators. Not only must they commute but their product must be equal to zero. We can later abandon this restrictive condition for later times for an alternative set of histories. It is not necessary to apply Consitent Histories to the entire universe but if one does then the wave function is treated as pre-probability but plays no role in the ontology of the interpretation.

When one integrates the projection operators

You obtain a decoherence function

Once decoherence is introduced the two projection operators will diverge and represent possible histories of the system. Griffiths and Omnes began considering small levels of decoherence whereas Hartle and Gell-Mann are interested in strong decoherence. In either case decoherence replaces measurement of the system and the interpretation retains locality. The decoherence function depends on both the density matrix and Hamiltonian of the early universe, Hartle and Gell-Mann-Mann follow the time evolution of the operator back to the early universe, which is intended to solve the preferred basis problem.

What history actually occurs? According the formalism, histories are treated similar to the wave function when calculating the path of a particle, a central tendency is located along classical trajectories when the projection operator is coarse grained and the wave function very slowly spreads out. Only histories which are sufficiently coarse grained and therefore close to the classical path will have a high probability of occurring.

In order for a quasi-classical realm to correspond to the set of histories however, one must avoid a completely fine grained description but mustn't have a description which is grained too coarse such that no classically unambiguous description is any longer available.

### Three Things William Lane Craig Gets Wrong About the Hartle-Hawking No Boundary Proposal

Classical standard hot Big Bang cosmology represents the universe as beginning from a singular dense point, with no prior description or explanation of classical spacetime. Quantum cosmology is different in that it replaces the initial singularity with a description in accord with some law the "quantum mechanical wave function of the universe", different approaches to quantum cosmology differ in their appeal either to describe the origin of the material content of the universe e.g., Tyron 1973, Linde 1983a, Krauss 2012 or the origin of spacetime itself e.g., Vilenkin 1982, Linde 1983b, Hartle-Hawking 1983, Vilenkin 1984.

These last few proposals by Vilenkin, Hartle-Hawking and others are solutions to the Wheeler-DeWitt equation and exist in a category of proposals called "quantum gravity cosmologies" which make cosmic applications of an approach to quantum gravity called "closed dynamic triangulation" or CDT (also known as Euclidean quantum gravity). I&#…

### How Should Thatcherites Remember the '80s?

Every now and again, when I talk to people about the '80s I'm told that it was a time of unhinged selfishness, that somehow or other we learned the price of everything but the value of nothing. I can just remember that infamous line from Billy Elliot; 'Merry Christmas Maggie Thatcher. We all celebrate today because its one day closer to your death'. If it reflected the general mood of the time, one might wonder how it is she won, not one but three elections.

In an era when a woman couldn't be Prime Minister, her launch into power was accidental owing in part to Manchester liberals and the Winter of Discontent. Yet I'm convinced her election victory in '79 was the only one that ever truly mattered. Simply consider the calamity of what preceded it, the 1970s was a decade of double-digit inflation, power cuts, mass strikes, price and income controls, and the three day week. Britain was sick, it needed fundamental restructuring but no one seemed to quite under…

### Can inflation be eternal into the past?

Back in 2003 a paper appeared on the arXiv titled "Inflationary spacetimes are not past complete" that was published by Arvind Borde, Alan Guth and Alexander Vilenkin which has had considerable amounts of attention online. The theorem is rather uninteresting but simple and doesn't require a very complicated understanding of math. So I thought I'd explain the result here.

It's purpose is to demonstrate that inflationary models are geodesically incomplete into the past which they take as "synonymous to a beginning" but Vilenkin stresses that the theorem can be extended to non inflationary models so long as the condition of the theorem that the average rate of expansion is never below zero is met. These models too then are incomplete into the past. Consider the metric for an FRW universe with an exponential expansion

Where the scale factor is

Since the eternal inflation model is a "steady state cosmology" the mass density and the Hubble paramet…