Causality

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```Causality

I heard a news article on the radio recently. It reported a survey which had shown that people who smoke are more likely to suffer from tooth decay than those who don't. The news presenter concluded that smoking contributed to tooth decay, another good reason to give up cigarettes.
But how right was he? The survey would have looked at the rates of tooth decay among people who smoke and among those who don't. It must have found that there was a statistically significant positive correlation between how many fillings people have in their teeth and how many cigarettes they smoke each day. Given such data, can we conclude that smoking causes tooth decay? Would it not have been equally valid to conclude that having fillings causes people to smoke?
The news presenter applied his prejudices and drew a conclusion which sounds reasonable without realising that the converse was also a possible explanation of the survey results. In fact the real reason behind the correlation was probably a third factor such as general variations in health awareness among the people surveyed. Those who care about their good health are more likely to brush their teeth and not to smoke.
The difference between the possible conclusions from the survey is not just one of semantics. People listening to the radio might have thought that if they gave up smoking then they would have less tooth decay. They would be wrong. The correct way for them to prevent tooth decay is to brush their teeth more often. In this case the false conclusion is not very dangerous, but such false conclusions about causal relationships drawn from surveys are common. If it is a survey which shows a correlation between race and crime rates then the wrong conclusion could lead to increased racism. In truth it is probably the consequences of racism, not race, which are the real cause of the crime problem identified by the survey in the first place.

Causality in Physics

If your mind is opened a little by my story of the survey in the news article, then now would be a good time to ask yourself if you are drawing the wrong conclusion about causality in physics. Suppose you saw your child bump into a table and an expensive vase fell off smashing into pieces on the floor. Would you conclude that her carelessness caused the vase to be broken? Probably you would. Why would you not conclude that the vase falling off the table caused her to bump, quite innocently, into the table?
Your response might be that, for one thing, the vase was broken after her collision with the table so the direction of the causal link is incontestable. Do the laws of physics support such a stance?
To keep things simple, let's start by considering just classical Newtonian mechanics. The form which the laws of physics take is crucial to our understanding of causality. Newton's laws take the form of a set of differential equations describing the motion of particles under forces that act between them. If we know the initial positions and velocities of all the particles at an initial time then their positions are determined at any future time. So does this form for the laws of physics allow us to justify our concept of causality. It would seem so because the initial conditions seem to be causing all that happens in the future.
There is a catch. The laws of physics in this form can be made to work identically in reverse. If we know the final state of a system we can just as easily determine its past. Newton's laws do not explain why past events are the cause of future events.
How about the laws of thermodynamics? If we have a system of many particles then we can not determine all their positions and velocities exactly. When we know only some statistical information about them they obey laws which seem not to be reversible. The second laws of thermodynamics says that entropy must always increase. Could this be linked to causality?
Indeed, the continual increase of entropy is intimately linked to our perception of causality. Entropy is a measure of disorder in a system and defines an arrow of time which can be linked to the psychological arrow. There is, however, a catch. The second law of thermodynamics is inexplicable in terms of the underlying laws of physics which, as far as we know, are reversible. This is enshrined in a theorem of relativistic quantum field theory which proves the necessity of CPT conservation.
The increase of entropy can be understood in certain idealised experiments. For example, if we take two closed containers filled with gases which are each in thermal and chemical equilibrium, and allow them to mix by connecting the two systems without allowing any energy to escape or enter, then when the system comes back into equilibrium the entropy of the final state can be shown theoretically to be higher than the combined entropies in the two original systems. This seems to be theoretical evidence for increasing entropy and it is confirmed by experiment, but we must not be missled. The assumption that systems tend towards equilibrium has been justified. We are victims of our prejudices about causality again and have devised an argument with circular reasoning to support it.
Physicists have devised many other arguments for why entropy always increases, trying to get round the problem of CPT symmetry. Here are a few possibilities:

*	CPT symmetry exchanges matter for antimatter so perhaps entropy would decrease for antimatter.
*	Fault: Electromagnetic radiation can not be distinguished from its antimatter image, and yet it obeys the second law of thermodynamics.

*	CPT symmetry does not apply to the collapse of the wavefunction in quantum mechanics which is a time asymmetric process.
*	Query: Does this mean that the third law of thermodynamics is not valid for classical statistical mechanics?

*	CPT conservation is violated by quantum gravity
*	This could be true but can the laws of thermodynamics be a result of quantum gravity who's effects are normally thought to be irrelevant except in the most extreme physical regimes.

*	Entropy increases on account of the fact that it started very low at the beginning of time. Thus it is due to the initial conditions being set in a special way. and from then on it could only increase.
*	But then why were initial conditions set rather than final or mixed boundary conditions.

When I was an undergraduate I naively thought that physicists understood entropy. Some have produced arguments based on any or all of the above possibilities. In retrospect I think now that I should be no more convinced by any of those arguments than I should if I heard someone arguing that smoking causes tooth decay based on the correlation reported in the survey.
One of the difficulties is that we don't really have an ideal definition of entropy. We can understand it as a measure of disorder in a closed equilibrium system. More generally we have to resort to some kind of coarse graining process in which we imagine that a non-equilibrium system can be seen as made of small sub-systems, or grains, which are in equilibrium themselves but not in equilibrium with each other.
Entropy might be better understood in terms of information. It can be linked to the number of bits which are needed to describe a system accurately. In a hot disordered system you need to specify the individual state of each particle, while a cold lattice can be described in terms of its lattice shape, size and orientation. Far less information is needed for the low entropy system.
The claim that entropy increases because it started low in the big bang is perhaps the one which has fallen into conventional wisdom, even if it is admitted that we don't understand why it started low. Perhaps it is because of some huge unknown symmetry which was valid at the high temperatures of the big bang and broken later. It is not really clear why it should increase all the time either. Why can't it just go up and down?
In a completely deterministic system the evolution of the system is equally well determined by its final state as by its initial so we could argue that the amount of information in the system must be constant. The difficulty there is that we are assuming an exact knowledge of state which is impossible. In any case, quantum mechanics is not deterministic. If we make a perfect crystal with an unstable isotope, as time passes some of the atoms will decay. The amount of information needed to track the decayed atoms increases. Perhaps, then, it really is quantum mechanics and the collapse of the wave function which is responsible.
If physicists used to think they understood entropy then their faith was deeply shaken when Hawking and Bekenstein discovered that the laws of thermodynamics could be extended to the quantum mechanics of black holes. The entropy is given by the area of the black hole but its temperature can only be understood through quantum mechanical effects. This shows that classical understanding of thermodynamics is indeed incomplete and perhaps only a complete theory of quantum gravity can explain the laws fully.
We might try to understand the quantum state of the entire universe by using Feynman's path integral formulation of quantum mechanics. We must form a sum over all possible space-time manifolds allowed in general relativity. Hawking has argued that we can understand entropy in this way if the universe is an entirely closed system, bounded in both time and space. He claims that there are two possible ways a universe could start or end. One has low entropy the other high. The only consistent picture is one in which it is low at one end and high at the other hence temporal symmetry is broken.
If this argument could be made solid then it would be a powerful one. The path integral formulation avoids problems of time since it is a sum over all possible universes rather than an evolutions with separate boundary conditions. The argument can only be made complete when we understand quantum gravity better.
At this point our belief in causality seems to rest in our faith that the universe has a simple topology as described by standard big bang models. This rests on little more than some limited observations and an application of Occam's razor. Our measurements of the cosmic microwave backgrounds show a high degree of isotropy and the universe seems to be homogenous on large scales in so far as we can tell. Our observation is limited by a horizon defined by the age of the universe and the speed of light. Thus we cannot observe anything beyond about 15 billion light years distance. Why should we imagine that the size of the universe is a similar order of magnitude to its current age? We have been unable to measure the extent to which space is curved and can not place limits on its size, or even be sure it is finite.
It seems to be only an application of Occam's razor which justifies the assumption that space is homogenous on scales hundreds of orders of magnitude larger than the observable horizon. It is quite possible, as far as we can tell, that the big bang is actually just a huge white hole which formed in a larger universe. Perhaps on some huge scale there are a population of black and white holes of vastly different sizes. What does that say about the laws of thermodynamics?
Apart from entropy there are other aspects of causality. We know that in general relativity causal effects are limited by the light cones which are part of the geometry of space-time. But the geometry is itself dynamic. In general relativity it is possible to construct space-time models which have closed time-like paths. If such things really exist in the universe we would be able to travel back to our past.
Traditionally physicists have simply said that such universes must be ruled out because if we could travel back to our past we could change our history, which seems to raise contradictions. Recently some physicists have started to question this assumption. It seems possible that quantum mechanics may allow closed time like curves through space-time wormholes to be constructed, at least in principle. The contradictions which were thought to be a consequence of time travel do not stand up to close examination.
Perhaps it would be possible to travel back to the past and see our parents but some chance event would prevent us from being able to change their lives in ways which we know never happened. If that is a correct interpretation then it attacks our faith in our own free will.
There is perhaps little that we can conclude reliably about causality from our current understanding of physics. Only when we have found and understood the correct theory for quantum gravity will we be able to know the truth. We may be prevented from finding that theory if we hold fast to our prejudices.
Occam's razor does not have a very good track record in cosmology. Usually space turns out to contain more complexity than we imagined before we looked. It will be billions of years before we are able to see beyond the current horizon defined by the speed of light . In the nearer term theory is our only hope to know what the structure of space-time is like on very large scales.
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