## An unchanging quantum universe

This another one of those bizarre thoughts that cosmology throws up which manages to be both simple and profound.

Imagine the wave function for the whole universe.

By its nature the universe cannot change its quantum state: it’s the ultimate closed system. Of course there is a  probabilistic distribution of energy inside the system but the total energy of the system does not change and therefore its quantum state cannot change either.

So, in quantum terms the universe is unchanging over time.

## A (final) speculation from “The Hidden Reality”

I have just finished Brian Greene’s The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos, so want to take this opportunity to once again endorse it and also to restate or reprise one of his speculations in the book: this time one grounded firmly in computing rather than string theory or conventional cosmology.

The key question at the heart of this speculation is this: do you think that, given we have seen computational power double every 18 – 24 months or so for around a century now, we will reach the point where we will be able to convince computer-based agents that they are in a physically real world when they are, in fact, in a computer-based simulation?

And if you answer yes to that question, do you realise that you are more or less forced to accept that you are one such agent and what you see around you as real is, in fact, a computer-generated “virtual reality”?

The reasoning is simple: once the technology to build a sufficiently convincing virtual world exists then such worlds are likely to rapidly grow in number and there will be far many more ‘people’ who are agents than people who are physically real. By simple averaging you and I are far more likely to be such agents than to be real.

It’s a truly mind-bending thought.

Saying ‘no’ to the question requires putting a hard limit on human progress in the field of computation. How can that accord with our experience of ever faster, more powerful and ubiquitous computing?

Well, there are some reasons – but even they are open to attack.

Reality appears smooth and continuous, but smooth and continuous numbers are not computable. Computable numbers are of finite extent – as otherwise the computer would never finish computing them. For instance no computer can ever compute $\pi$, only render an approximation. Indeed most numbers are “transcendental numbers” and inherently not computable.

But this argument – which sounds like a sure-fire winner for knocking down the idea that our reality is, in fact, a simulation, has a weakness: we cannot measure smoothness either. Our measurements are discrete and while it appears to us that the world is smooth and continuous, maybe it is not – it is just that the computed approximation is beyond our ability to (presently) measure it.

If, at some point in the future, we discovered a finite limit to measurement that was beyond physical explanation it would surely confirm we were in a simulation.

## Why isn’t the universe of infinite density?

Another speculation produced by Brian Greene’s The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos.

Imagine the universe was infinite (along the “quilted multiverse” pattern – namely that it streched on and on and we could only see a part). That would imply, assuming that the “cosmological principle” that one bit of the universe looked like any other, applied, that there were an infinite number of hydrogen atoms out there.

So, why is the universe not of infinite density? Because surely Shroedinger’s Equation means that there is a finite probability that electrons could be in any given region of space? (Doesn’t it?)

For any given electron the probability in “most” regions of space is zero in any measurable sense. But if there are an infinite number of electrons then the probability at a given point that there is an electron there is infinite, isn’t it?

OK, I have obviously got something wrong here because nobody is dismissing the “quilted multiverse” idea so simply – but could someone explain what it is I have got wrong?

Update: Is this because space-time is a continuum and the number of electrons a countable infinity?

## Cosmologists’ problems with aleph-null and the multiverse

This is another insight from Brian Greene’s book The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos – well worth reading.

Aleph-null ($\aleph_0$) is the order (size) of the set of countably infinite objects. The counting numbers are the obvious example: one can start from one and keep on going. But any infinite set where one can number the members has the order of $\aleph_0$. (There are other infinities – eg that of the continuum, which have a different size of infinity.)

It is the nature of $\aleph_0$ that proportions of it are also infinite with the same order. So 1% of a set with the order $\aleph_0$ is also of order $\aleph_0$. To understand why, think of the counting numbers. If we took a set of 1%, then the first member would be 1, the second 101, the third 201 and so on. It would seem this set is $\frac{1}{100}^{th}$ of the size of the counting numbers, but it is also the case that because the counting number set is infinite with order $\aleph_0$, the 1% set must also be infinite and have the same order. In other words, if paradoxically, the sets are in fact of the same order (size) – $\aleph_0$.

The problem for cosmologists comes when considering the whether we can use observations of our “universe” to point to the experimental provability of theories of an infinite number of universes – the multiverse.

The argument runs like this: we have a theory that predicts a multiverse. Such a theory also predicts that certain types of universe are more typical, perhaps much more typical than others. Applying the Copernican argument we would expect that we, bog-standard observers of the universe – nothing special in other words – are likely to be in one of those typical universes. If we were in a universe that was atypical it would weaken the case for the theory of the multiverse.

But what if there were an infinite number of universes in the multiverse? Then, no matter how atypical any particular universe was (as measured by the value of various physical constants) then there would be an infinite number of such a typical universes. It would hardly weaken the case of the multiverse theory if it turned out we were stuck inside one of these highly atypical universes: because there were an infinite number of them.

This “measure problem” is a big difficulty for cosmologists who, assuming we cannot build particle accelerators much bigger than the Large Hadron Collider, are stuck with only one other “experiment” to observe – the universe. If all results of that experiment are as likely as any other, it is quite difficult to draw conclusions.

Greene seems quite confident that the measure problem can be overcome. I am not qualified to pass judgement on that, though it is not going to stop me from saying it seems quite difficult to imagine how.

## The second law of thermodynamics and the history of the universe

I had to go on quite a long plane journey yesterday and I bought a book to read – Roger Penrose‘s work on cosmology: Cycles of Time: An Extraordinary New View of the Universe

I bought it on spec – it was on the popular science shelves: somewhere I usually avoid at least for the physical sciences, as I know enough about them to make hand waving more annoying than illuminating, but it seemed to have some maths in it so I thought it might be worthwhile.

I have only managed the first 100 pages of it so far, so have not actually reached his new cosmology, but already feel it was worth every penny.

Sometimes you are aware of a concept for many years but never really understand it, until some book smashes down the door for you. “Cycles of Time” is just such a book when it comes to the second law of thermodynamics. At ‘A’ level and as an undergraduate we were just presented with Boltzmann’s constant and told it was about randomness. If anybody talked about configuration space or phase space in any meaningful sense it passed me by.

Penrose gives both a brilliant exposition of what entropy is all about in both intuitive and mathematical form but also squares the circle by saying that, at heart, there is an imprecision in the law. And his explanation of why the universe moves from low entropy to high entropy is also brilliantly simple but also (to me at least) mathematically sound: as the universe started with such a low entropy in the big bang a random walk process would see it move to higher entropy states (volumes of phase space).

There are some frustrating things about the book – but overall it seems great. I am sure I will be writing more about it here, if only to help clarify my own thoughts.

In the meantime I would seriously recommend it to any undergraduate left wondering what on earth entropy really is. In doing so I am also filled with regret at how I wasted so much time as an undergrad: university really is wasted on the young!

(On breakthrough books: A few years ago I had this experience with Diarmaid MacCulluch’s Reformation and protestantism. People may think that the conflict in the North of Ireland is about religion – but in reality neither ‘side’ really knows much about the religious views of ‘themuns’. That book ought to be compulsory reading in all Ireland’s schools – North and South. Though perhaps the Catholic hierarchy would have some issues with that!)