### Consistent Histories and the PBR theorem

Several years ago now a much discussed theorem in the foundations of physics was discovered by three physicists, Matthew F. PuseyJonathan Barrett and Terry Rudolph. I had meant to discuss their paper some time ago, and realized I couldn't find anywhere, where anyone was talking about the theorem in the context of the Consistent Histories approach. So I've decided now to correct that.

The theorem states that under certain conditions a model which affirms both that (a) there are underlying ontological states of a system and (b) the wave function is given an epistemic interpretation, cannot be consistent with quantum theory, and so must be discarded.

Right off the bat, we know the Consistent Histories approach affirms (b), so the most straight forward response could be to reject (a) and some Consistent Historians are perfectly willing to do that, e.g. Lubos Motl and Roland Omnes. As for (b) in the CH approach, the wave function is treated as "pre-probability" in the sense that, it's used to express an observers subjective knowledge of an underlying state, which they then use to calculate probability.

As far as the CH goes, (b) is non-negotiable and we know this for two reasons. The wave function describes non-commuting operators, which in this approach means it describes physical properties of a system which are incompatible, like definite momentum and position. The second is that, while in quantum theory you may obtain the wave function by integrating the Schrodinger equation, you can obtain the same probabilities using a method that makes no mention of that wave function. By integrating in a different time direction (a different wave function is obtained).

With that in mind, here's a more interesting question. Can one be a Consistent Historian and affirm both (a) and (b) without any inconsistency? In other words, can one be both a Consistent Historian and a scientific realist? I think the answer is, obviously, a yes, but we'll need to learn a bit more about the PBR theorem first.

In quantum theory having underlying ontic states but an epistemic interpretation of the wave function means you have a probability distribution over those states, which may overlap over some of those states.

The PBR result shows that for a tensor product corresponding to say, two elections, being prepared in any state, any possible outcome of measurement should have a zero probability of occurring (see the original paper for technical details). Which is clearly false. So there's a tension created between (a) and (b).

I already stated that there were conditions of the PBR theorem that needed to be assumed in order to get the argument to work, so we're going to have deny one or more of these assumptions if we want to maintain both (a) and (b). As it turns out, we don't need to postulate anything new, because the CH approach already, denies the "separability" assumption made by PBR (it was also assumed in Bell's theorem), that ontic properties are localized in spacetime.

That is to say that the properties of a quantum system may be correlated with another, very far away system. When two systems are considered, along with a measuring device there is going to be a certain correlation between them. What happens to the measured properties of P is not irrelevant if a statement about the properties of P' has already been made.

This tells us that systems with no direct interaction can still be correlated, even if they're separated at distances greater than what the speed of light can reach (in a given time). This is only true in some special cases and is determined by our preparation device. This is hardly surprising to a Consistent Historian, in this approach there is only one wave function, the wave function of the universe, that never collapses, only parts of it decohere with respect to other pats. It makes sense to talk about the properties of a whole system rather than individual parts.

As one quantum theorist put it "a wave function for several particles is not a product of the wave function for individual particles".

This isn't the only possible way of 'getting around' the PBR theorem but its the one I find most intuitive and appealing. It's compatible with relativity arguments and doesn't postulate anything like retro-causality (though a CH may avail themselves of that too). If it isn't entirely clear, try not worry, hopefully this blog post will become more understandable when I talk about 'Consistent Histories and Bell's theorem' and it might be a good idea, to come back and re-read this post at a future time.

### Set Theory

This post is a very brief introduction to some of the basic concepts of set theory. Set theory is a branch of mathematical-logic, that has wide applications across disciplines. Its not just used in the obvious way of studying the foundations of mathematics by mathematicians but also in physics, social science, and even by philosophers as a theory of semantics for predicate logic (although you can do propositional logic without set theory).

A set is a collection of elements, or members; the notation for a set is specified by listing its components. So the set of even numbers can be represented a
$E: \left \{ 2,4,6,8 ... \right \}$$E: \left \{ x: x > 0 \wedge even\right \}$ Either of these notations is valid. Further, elements of a set can only be in that set, once. So   $E: \left \{ 2,2,2,4,4,6,8 ... \right \} = E: \left \{ 2,4,6,8 ... \right \}$ The notation used to indicate that something is an element of a set, is using the Greek symbol "epsilon". That is: $4 \epsilon S$…

### William Lane Craig and 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&#…

### 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…