Friday, January 2, 2009

Why shouldn't the rules change?

I have become fond of games which allow you to modify the rules by which the game is played while you are playing the game. I just bought a game called Fluxx and as it so happens basically the whole idea is to alter the rules of the game to make yourself win. Obviously the rule changes are themselves operating within a larger framework which one might arguably say are the real rules of the game. For instance the new rule cards can't contradict one another if a new rule card would contradict an old one then the old one is discarded and the new one takes its place. There are a few other such sub-rules which constitute an underlying framework to the world of rules that can change but saying what they are won't further the discussion.

What I really want to talk about is the assumption that the rules don't change in the physical world. Believe it or not this is the principle of relativity on which Einstein built his special (and later general) theory of relativity. So Einstein was kind of saying we don't want anyone changing the rules, they have to be the same everywhere at all times. In this case "same" means more or less that you have to be able to apply the formalism in the same way regardless of if you are moving or not or what time it is or if you are on mars or earth. The first question I want to ask is whether or not this is more like saying that the underlying framework rules cannot change or is this saying that we can't ever lay any new rule cards?

The framework which underlies physics is mathematics and the fundamental rule of the framework of mathematics is consistency. There are arguably infinitely many different ways in which the game of physics could be played. Possibly there are only a finite (but vast number) of ways as some string theorists think. The particular number isn't important but the thought that there could be other ways of doing physics that are still logically consistent is basically the same as saying ok well you could play the game with different rule cards on the table and still not break the framework. Einstein is saying that the cards never get mixed up. If the big bang played the f=ma card then that's how it is end of story. I find it kind of ironic that later in life while fighting with quantum mechanics Einstein would propose the hidden variables idea which basically equates to saying that physics acts differently at different times and different places because we see random stuff happening but physics must be deterministic. Of course for Einstein the idea was that eventually a deeper theory would be uncovered which would replace these hidden variables and keep physics as being both deterministic and free of time and place.

This is an interesting idea, and an old one. The game that we think we are playing is almost certainly not the game we actually are playing. In fact at the moment since we are sure we are playing a game within a framework that requires consistency we know that the model of the rules that we have is wrong because general relativity and quantum mechanics disagree with each other. There could be many many possible "real" rule sets that we are working towards but what if the rules don't exist at all? What if the laws of physics are actually constantly changing and what we think we are trying to model is just some sort of crazy long term average? This actually has sort of happened to physicists already they found out that just about every law of physics was in the quantum world just a long term average. So we reformulated the laws of physics so that anything that was subject to fluctuation was no longer part of the rules but merely part of the specific situation. But there is a fundamental difference between considering f=ma to be a long term average accounting for the complicated random interactions of things and thinking that the interactions themselves occur according to random rules. Take the good old time independent schroedinger equation Hp = Ep where p is the wave function H is the Hamiltonian and E is the energy. Usually we think of the randomness as inherent in p not in H but in a way revisiting the hidden variables of Einstein we could try to hide the randomness in H instead. I'm sure you could come up with a completely equivalent description of quantum mechanics this way making the interactions of particles be totally deterministic but acting according to random rules. I imagine it would be a lot more difficulty than its worth but I would also be willing to bet some fun stuff would come out of it.

As a final side note I often wonder if logical consistency is actually a fundamental requirement of the laws of the universe. Human minds work in worlds where inconsistency is rampant. We find consistency incredibly useful but not absolutely necessary. This is in fact the way that the laws of physics have worked out as well we know that they are inconsistent but that they are consistent within small domains or at least relatively consistent and we find that consistency useful. Ultimately though utility outweighs consistency and we are free to use inconsistent and blatantly incorrect models and theories just so long as in the end we get an answer and can have some confidence in it. Of course even though quantum mechanics and relativity are inconsistent with each other they are operating in a larger realm where the people using them are able to evaluate whether the disagreement matters at the moment. So people impose consistency when necessary which puts us again back into the world where everything must mesh logically. All I can say I suppose is that it would be a stunning blow to the world of philosophy and physics were someone somehow to prove that physical laws which have consistency as a requirement cannot produce certain behavior that is in fact observed. In other words if a Godel came along and proved a very general theorem about how physical theories can act and then proved that if there is consistency then there is "incompleteness" in some sense. I wonder if that lies somewhere down the road for us.

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