Saturday, October 21, 2006 ... /////

The temptation of rigor

Jacques Distler wrote a nice text about

in physics. He explains that rigor can't replace physical input and insights: very rigorous theorems may lead to very misleading physical conclusions. A reader could ask: how could a physicist ever question the unlimited power of a rigorous proof of a mathematical theorem? The technical answer is hidden in the assumptions of the theorem.

A proof of a theorem can be completely correct but the theorem can still be physically worthless. This occurs when the assumptions of the theorem are not satisfied by the relevant physical systems. When you start with incorrect or naive assumptions, your reasoning is likely to follow the GIGO rule: garbage in, garbage out. Rigor simply can't save you from errors in physical reasoning. There exists no systematic or rigorous method to find the correct physical theories.

Jacques mentions two examples:
1. algebraic holography
2. uniqueness of the polymer representation of the spatial diffeomorphism constraints
In both cases, rigorous proofs about systems called "quantum field theory" or "quantum gravity" have been constructed. These theorems are proven by valid proofs so what's the problem? The problem is that it is misleading to use the terms "quantum field theory" and "quantum gravity" for the theories addressed by these theorems.

In the case of algebraic holography, a rigorous proof due to Karl-Henning Rehren may be given to show that a local "quantum field theory in d+1-dimensional anti de Sitter space" is equivalent to a local "quantum field theory defined on its d-dimensional boundary". In this case, the definition of a "quantum field theory in d dimensions" is far too loose and includes structures that a physicist would never count as quantum field theories in d dimensions.

For example, a four-dimensional interacting Klein-Gordon theory can be included among three-dimensional quantum field theories according to this definition because the field "phi(x,y,z,t)" can be written as "phi_z(x,y,t)" with an index "z". There are many good reasons why continuous indices for fields are never treated as indices by the physicists and why the theoretical physicists would declare the conclusion of the theorem to be manifestly incorrect physically: local theories in d dimensions have free energy density that scales like "T^d" for high temperatures "T"; theories in different dimensions therefore can't be equivalent because you can always determine the dimension from the exponent.

However, with the unusual definitions of quantum field theories and their dimensionality used in the theorem, it is not surprising that one can prove a statement that sounds like Maldacena's correspondence. But in reality, it has nothing to do with the real essence of holography in quantum gravity. The theories discussed by algebraic holography are not holographic in any useful sense and they cannot be equivalent to lower-dimensional theories as long as you compute the spacetime dimension of a theory as the physicists do.

The second example - the polymer representations - is analogous but in some sense it suffers from the opposite flaw. In this case, the definition of "quantum gravity" is too narrow-minded. Too many things are assumed to be true about the structure that is called "quantum gravity": it is essentially assumed that quantum gravity must be constructed in the most naive way one can imagine.

These assumptions seem to be invalidated in the actual working theories of quantum gravity because of many unexpected twists and turns. For example, the definition of quantum gravity in terms of the dual CFT directly constructs the physical Hilbert space of quantum gravity - a superselection sector of string theory - without any intermediate steps where the diffeomorphism constraints would have to be imposed by hand. In this construction, a whole dimension of space - the holographic dimension - emerges unexpectedly: it was not used as a starting point at all. Still, all the facts that are normally derived as consequences of the diffeomorphism symmetry hold in this setup.

On the other hand, the polymer theorem is based on the assumption that neither of these "miracles" ever occurs. But these miracles and many other miracles that would be shocking for the thinkers in 2000 B.C. and other thinkers in the past have been found and they represent what we really mean by progress in science. The expectation that we already know all the right assumptions in physics is equivalent to the expectation that there will no longer be any substantial progress in the research of a given class of questions. This expectation is usually incorrect although it is valid whenever someone is smart and lucky enough to find the right assumptions.

The physical conclusion is obvious: Maldacena's correspondence is highly non-trivial, fascinating, and true - it is even more fascinating because rather reasonably sounding but naive and flawed arguments could lead us to believe, together with Roger Penrose, that it can't be right. And in some sense, quantum gravity and holography as painted by string theory is so valuable not only despite but because the structure is so rich that we can't yet fully squeeze these ideas into a small mobile rigorous box that is fully understood.

The fact that we don't yet know everything about string theory and its universal definition is actually one of the reasons why people can't resist and they continue to study it.

On the other hand, algebraic holography and polymer representations of quantum gravity are not so interesting because what is hiding in these fancy rigorous clothes is a physically flawed content that has nothing to do with the important principles of observable physics that are incorporated in the state-of-the-art theories. It's a content that has no relations with the actual surprises that we learned in the recent decades, centuries, and millenia. This content can perhaps be fully understood and the science behind it is falsifiable. Indeed, it is falsifiable in less than an hour. But it is a huge disadvantage, not an advantage, of such a system of ideas.

Rigor can sometimes be useful for physics, especially when physicists are making an error that is caused by a somewhat sloppy reasoning. But the physicists typically reach the right conclusions without insisting on all the formal features of a rigorous proof: they prefer the content over the form. If someone prefers the form over content, it doesn't save him or her from deep physical errors or from naivite. That's why we have so many examples of pairs of answers among which the more rigorous one is obviously the wrong one physically.

The previous articles related to the relation of rigor and physics: