Lee Smolin has a new preprint discussing the “anthropic principle”. He argues that one standard form of the anthropic principle that has been invoked by proponents of the “Landscape” is not falsifiable and he gives an eloquent explanation of the importance of falsifiability for a shared scientific enterprise. He also discusses the “prediction” of the rough magnitude of the cosmological constant that supposedly uses the anthropic principle and is due to Weinberg. He points out that this argument really isn’t an anthropic one, since it is independent of the existence of intelligent life. It just relies on showing that there is a relation between the cosmological constant and the existence of gravitationally bound structures. Then, since we see galaxies, we know something about the cosmological constant.

One of Smolin’s concerns is to show that his theory of “cosmological natural selection” (discussed in his book “The Life of the Cosmos”), while being a theory of a “multiverse” just like the string theory Landscape, is different in that it is potentially falsifiable, unlike some recent anthropic arguments.

He states well the predicament that theoretical physics finds itself in, with the tactic that worked so well throughout the 20th century, that of searching for unification by exploiting symmetry, no longer having much success. While I agree with most of what he has to say in this preprint, I’m more optimistic than him that future progress through new ideas about unification and the exploitation of symmetry is still possible. My point of view is more that the reason the last twenty years have seen no progress of this kind is that virtually all the field’s effort has gone into pursuing one very speculative and not very promising idea about unification, ignoring other possible lines of research.

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Mathematically, exploitation of symmetry=representation theory. There are several techniques for constructing and studying representations that mathematicians have developed over the last 50 years, but physicists have never learned. In addition, for some of the groups that are experimentally known to be physically relevant (4d gauge and diffeomorphism groups), very little is known about their representations.Actually, the classical representation theory of these groups is well known – it is called

differential geometry.This is pretty obvious once you think about it. Differential geometry is the theory of well-defined objects, which are precisely those that transform consistently under diffeomorphisms. Thus, tensor densities are (often irreducible) modules, the exterior derivative is an intertwining operator, a connection is a central extension of the module of (1,2) tensor fields, etc. Special geometries, like symplectic or contact geometry, can likewise be regarded as the representation theories of the appropriate subgroups of symplectomorphisms and contactomorphisms. To treat spinors, you need to consider groups of local frame rotations and vielbeine, etc.

However, we know from conformal field theory that the Virasoro algebra has two qualitatively different kinds of reps: classical modules, which are called primary and secondary fields, and modules of lowest-weight type (Verma, Fock, vertex operator, coset, minimal models, etc.) which are relevant to quantum theory. Differential geometry only deals with the higher-dimensional analogues of the classical modules. This is easy to see, because all interesting (= non-trivial and unitary) quantum irreps always have a positive central charge.

The Virasoro algebra is a central extension of the diffeomorphism algebra in 1D, so generalizing it to several dimensions would be a first step towards constructing quantum reps of the diffeomorphism group in several dimensions. At first sight this seems hopeless, since a no-go theorem tells us that the diff algebra has no central extension except in 1D. Nevertheless, a multi-dimensional Virasoro algebra does exist, although the extension is not central in higher dimensions. The first quantum reps were found in the seminal paper

S.E. Rao and R.V. Moody,

Vertex representations for N-toroidal Lie algebras and a generalization of the Virasoro algebra,

Commun. Math. Phys. 159, 239-264 (1994)

The underlying geometry was clarified in http://www.arxiv.org/abs/math-ph/9810003, which led to considerable generalization. We can now a construct quantum rep for each tensor density and each non-negative integer.

Much remains to be done, of course. In particular, I don’t understand how to generalize the minimal models, which are needed to make physical predictions in CFT. One must expect that the full representation theory is much more complicated than for the Virasoro algebra. This is true already on the classical level, where the differential geometry of a circle is not very exciting. But the construction of a quantum analogue of differential geometry seems to me to be a quite interesting result in its own right. Anyway, I cannot imagine that I will ever discover anything as important again.

So, it seems to me that the kind of new mathematics that is needed is clear and there are plenty of techniques to try and attack the problem with. But no physicists want to think about this, their reaction generally is “What does this have to do with M-theory?”Hear, hear! When I started to search for a multi-dimensional Virasoro algebra in 1987, one of my main motivations was that I wanted to apply it to quantum gravity. The fact that such an algebra exists and that it has good quantum representations (other anomalous algebras like Mickelsson-Faddeev apparently have not) is to me a strong indication that my physical motivation was not completely wrong.

Whereupon I will disappear into a computer-free zone for a week.

Crackpots are much easier to deal with in mathematics since mathematicians insist that things be well defined and logically coherent before publishing them or taking them seriously. Coming up with math that hasn’t been done before that satisfies this criterion is hard, much too hard for the kind of crackpots that infest physics.

Mathematically, exploitation of symmetry=representation theory. There are several techniques for constructing and studying representations that mathematicians have developed over the last 50 years, but physicists have never learned. In addition, for some of the groups that are experimentally known to be physically relevant (4d gauge and diffeomorphism groups), very little is known about their representations. So, it seems to me that the kind of new mathematics that is needed is clear and there are plenty of techniques to try and attack the problem with. But no physicists want to think about this, their reaction generally is “What does this have to do with M-theory?”

Peter,

If Smolin’s scenario of symmetry principles becoming less and less useful in physics turns out to be true, what do you think physicists will do in place of exploiting symmetries?

Naively if I didn’t know any better, I would be tempted to search through the math literature looking for mathematical structures that have not been used extensively in physics previously. Albeit, this would seem like a haphazard “brain dead” way of doing theoretical physics that could easily lead to numerous dead ends and wasted effort. This would definitely be an act of desperation, when everybody else has run out of brilliant ideas and are floundering around.

Only other thing I can think of offhand, would be to “invent” some new mathematics. Though inventing new mathematics out of “nothing” seems like it would be a lot harder than it looks. At times I wonder what the crackpot to non-crackpot ratio is these days in the area of “inventing” new mathematics.