I don't follow why the restriction to relational degrees of freedom necessarily implies that as we get closer to a fundamental theory there should be fewer symmetries.

Because diffeomorphism invariance is the lack of symnmetries. When one compares GR and SR, both have diffeomorphism invariance, but SR additionally has global Lorentz invariance (let's take GR with a cosmological constant). So GR has less symmetry than SR.

I first encountered this argument of his in http://arxiv.org/abs/hep-th/0507235: "One way to formulate the argument against background spacetime is through a second principle of Leibniz, the identity of the indiscernible. This states that any two entities which share the same properties are to be identified. Leibniz argues that were this not the case, the first principle would be violated, as there would be a distinction between two entities in nature without a rational basis. If there is no experiment that could tell the difference between the state in which the universe is here, and the state in which it is translated 10 feel to the left, they cannot be distinguished. The principle says that they must then be identified. In modern terms, this is something like saying that a cosmological theory should not have global symmetries, for they generate motions and charges that could only be measured by an observer at infinity, who is hence not part of the universe. In fact, when we impose the condition that the universe is spatially compact without boundary, general relativity tells us there are no global spacetime symmetries ..."

There are many other passages expanding on that in the article.

Harsh words -- though I think I know what you mean. I often find Lee Smolin's papers to be like a meal in a Chinese restaurant: it seems tasty and interesting, but half an hour later I'm hungry again.

Interesting, that he shows interest and like the idea in the holographic String Model of Susskind and Maldacena. On the physics forum, where he is writing too, there is an interesting idea of a holographic string model with backrunning time which should give dynamic spacetime in background and solved energyproblem which is not negative anymore he asked for. Maybe Smolin will publish next time a new Model. I'm looking forward to read it then.

Yes,... but (in principle) that's possible more generally.

But I think I see it now: restricting to relational degrees of freedom is analogous to (eg) decomposing the Kepler/Hydrogen problem into CoM dof's and relative dof's. The former are essentially "background", so we forget about them. The interesting physical features emerge by analysis of the latter.

I assume by "any statement" you mean any demonstrated fact, not just some imaginative utterance. In other words, without any symmetries, a fact such as "the Moon is orbiting the Earth" might be called a natural law, removing the distinction between laws of nature and specific circumstances. But I do not think anyone has suggested that all symmetries were suddenly going to disappear, only that symmetries might no longer be the keystones of discovery they have been in the past.

What I understand is that any symmetry a theory has, is actually a choice for that theory because we could make other theories for the same thing with other symmetries. Now if we accept that every choice in our theories should have a reason, any choice we make for the symmetry of our theory among a possible set of symmetries should have a reason, which means we're asking for an underlying theory. So it seems to me that Smolin is arguing that if our theories always have symmetries, we have to keep going down for ever and can never stop. So the fundamental theory has no symmetry.
But I don't think it follows. One caveat to the above argument is that maybe we can show that there is only one symmetry possible for the theories of a particular aspect of nature, or in this case, we can show that there is only one symmetry possible for any theory that is going to give us the current physics in some limit.
Also our theories do satisfy Leibniz's principle of sufficient reason even if they have symmetries: The sufficient reason can be that they correctly describe nature!

Crazy over-large Question that keeps bugging me: What kind of system design must display a mixture of symmetry and asymmetry? Why should a system with one have any of the other at all?

Any entropic system has asymmetry... I think the symmetry belongs in the models, nothing in nature is perfectly symmetrical. The more symmetry the models "encompass" the less likely "other" symmetries present themselves...

Smolin may have a point. Our own work, which comes from a very different direction, indicates that both the asymmetrical leptogenesis and baryogenesis processes can be conceptually explained as consequences of a single deeper symmetry. That is the matter-antimatter species differentiation. That also explains asymmetries in decay rates.

I do wonder whether the 'symmetries' idea is overloaded. The basic concept is that some attribute of the system should be preserved when transformed about some dimension. Even if it is possible to represent this mathematically, we should still be prudent about which attributes, transformations, and dimensions to accept. Actual physics does not necessarily follow mathematical representation. There is generally a lack of critical evaluation of the validity of specific attributes, transformations, and dimensions for the proposed symmetries.

The *time* variable is a case in point. Mathematical treatments invariably consider it to be a dimension, yet empirical evidence overwhelmingly shows this not to be the case.Logically, we should therefore discard any mathematical symmetry that has a time dimension to it. That reduces the field considerably, since many symmetries have a temporal component. This may be uncomfortable from some, but is something we need to be open to considering. When something is not working, then it appropriate to go back and question the premises. Our own work suggests that time could be an emergent property of matter, rather than a dimension. This makes it much easier to explain the origins of the arrow of time and of irreversibility. So it can be fruitful, in an ontological way, to be sceptical of the idea that mathematical formalisms of symmetry are necessarily valid representations of actual physics.

"Mathematical treatments invariably consider [time] to be a dimension, yet empirical evidence overwhelmingly shows this not to be the case."

This is a very interesting point. I want to make sure I understand you. You say that time is not a dimension. Is this because, for example, objects cannot be moved to an arbitrary coordinate in time, as they can in a spatial dimension? Or because the time coordinate must be multiplied by i before it can be manipulated like a spatial coordinate?

Perhaps, like Ralph above, I too do not understand you. Superficially, your words would rule out Lorentz boost transformations, as they mix space and time.

You refer to your "own work". Please give a reference to a arxiv paper, or (preferably) a paper published in a peer-reviewed journal. (Note that the PF rules are quite strong concerning unpublished personal theories.)

That's a useful clarification question. I propose the former.

I'm saying that empirical evidence, e.g.irreversibility, shows that time does not evidence symmetry. That includes the observation that 'objects cannot be moved to an arbitrary coordinate in time'. Consequently *time* cannot be considered to be a dimension about which it is valid to apply a symmetry transformation even when one exists mathematically. Alternatively, if we are to continue to rely on temporal symmetries, it will be necessary to understand how the mechanics of irreversibility arises, and why those symmetries are exempt therefrom.

I accept that relativity considers time to be a dimension, and has achieved significant theoretical advances with that premise. However relativity is also a theory of macroscopic interactions, and it is possible that assuming time to be a dimension is a sufficiently accurate premise at this scale, but not at others.

I must say I'm still not grasping this idea of Smolin. I have to think about it, but if someone would like to elaborate even more, I'd be happy to read it :P