December 10, 2000

This Week's Finds in Mathematical Physics (Week 161)

John Baez

I'm in the middle of reading this book, so I don't know how it ends yet, but it's good:

1) Dava Sobel, Galileo's Daughter, Penguin Books, London, 2000.

Galileo had two daughters and a son with a beautiful woman whom never married - Marina Gamba of Venice. The son was a wastrel, and the younger daughter was very shy, but the older daughter, Virginia, loved Galileo very much and wrote him many letters. Of these, 124 have been preserved, which serve as the basis of this book. At the age of 13 she was sent to a convent, and she later became a nun. She took on the name Suor Maria Celeste - Sister Mary of the Heavens. Unfortunately, all of Galileo's letters to her were destroyed by her abbess after his trial by the Inquisition. Thus, what was really a dialog has come down to us as a monolog. Nonetheless it is fascinating, especially since Sobel elegantly fills in many of the holes using other sources.

Since I haven't read much about Galileo, I didn't know that this man, often considered the father of experimental physics and telescope-aided astronomy, was officially the "Chief Mathematician of the University of Pisa". Now I can add him to my list of mathematicians who have done good physics.

Two later figures standing on the border of math and physics are Kelvin and Stokes:

2) David B. Wilson, Kelvin and Stokes: A Comparative Study in Victorian Physics, Adam Hilger, Bristol, 1987.

One thing I like about this book is the debunking of the popular image of quantum mechanics and relativity as "bolts from the blue" shattering the complacent serenity of 19th-century physics. In physics, the 19th century was also a century of drastic change! To quote:

Science in Victorian Britain underwent revolutionary conceptual and institutional changes. Together, thermodynamics and the electromagnetic theory of light, for example, transformed a bundle of only partially linked, largely experimental sciences into a coherent, unified, mathematical physics of energy and ether. In the 1890s one could contemplate reducing the phenomena of matter, electricity, magnetism, heat and light to an underlying reality of potential and kinetic energy in an all-pervading ether. The pursuit of scientific research, largely avocational early in the century, was a full-fledged profession by the century's end. Science became important to university curricula, and the universities expanded their science faculties. Institutions like the British Association for the Advancement of Science, founded in 1831, and Royal Society of London, reformed at mid-century, provided organizational support for a growing community of scientists. And that community of late- Victorian scientists resided in a community which, on balance, was much more scientific and less religious than it had been only two or three generations earlier. In sum late-Victorian society endorsed the imporance of scientific knowledge and research, and late-Victorian physics affirmed the primary significance of the ideal of unification and the language of mathematics. In these respects, there was an essential similarity between late-Victorian Britain and both the "big science" and the modern physics of the twentieth century. The metamorphosis that created this state of affairs was the context of the the careers of G G Stokes and William Thomson, Lord Kelvin.

Marching forwards into the 20th century, we find Einstein as another physicist with a special tie to mathematics. Certainly he was no mathematician, but his search for a theory of general relativity was a curious combination of philosophical and mathematical reasoning, with very little support from experiment. How did he really figure it out? This book is a good place to learn the details:

3) Don Howard and John Stachel eds., Einstein and the History of General Relativity, Birkhauser, Boston, 1989.

There are a number of essays exploring the interesting period between 1912, when Einstein recognized that gravity was caused by spacetime curvature, and 1915, when he found his field equations and used them to compute the anomalous precession of the perihelion of Mercury. Why did it take him so long? According to Einstein himself, "The main reason lies in the fact that it is not easy to free oneself from the idea that co-ordinates must have an immediate metrical significance".

Indeed, in 1913 he noticed that generally covariant field equations could not uniquely determine the gravitational field generated by a fixed mass distribution. The reason - apart from the existence of gravitational waves, which he was not concerned with here - is that one can take any solution, apply an arbitrary change of coordinates, and get a new solution. This seemed to suggest a conflict between general covariance and the principle that every effect should have a sufficient cause.

Before he solved it, this conceptual problem aggravated the technical problem of getting the right field equations: there aren't that many good candidates for these equations if one demands general covariance, but during the period when he distrusted this principle, Einstein and his collaborator Grossman put a lot of work into other candidates. The main one they tried gave Mercury an anomalous precession of 18" per century instead of the correct value of 45" per century. Einstein only discarded this theory in November, 1915.

On November 11th he tried a theory where the Ricci tensor was proportional to the stress-energy tensory. On November 25th he tried a better one, where what we now call the Einstein tensor is proportional to the stress-energy tensor. He quickly used this to derive the correct precession for Mercury. And so general relativity was born! In January 1916 he explained in letters to Ehrenfest and Besso how he had reconciled general covariance with causality: two solutions of the field equations that differ only by a change of coordinates should be regarded as physically the same.

Now I'd like to switch to something else: a couple of emails I got. A while back I wrote up a webpage about the end of the universe:

4) John Baez, The end of the universe,

I got a lot of the numbers out of a book I bet you've already read:

5) John D. Barrow and Frank J. Tipler, The Cosmological Anthropic Principle, Oxford U. Press, Oxford, 1988.

What - you haven't read it? Yikes! Hurry up and give it to a friend for Christmas - and then make them lend it to you. Regardless of what you think about the anthropic principle, you're bound to enjoy the cool facts this book is stuffed with! Anyway, I got an email from Barrow saying that he's coming out with a new book. Like the previous one, it's sure to be full of interesting things. You can tell from the title:

6) John D. Barrow, The Book of Nothing, to be published.

My other email was from Bert Schroer, an expert on the C*-algebraic approach to quantum field theory. He has written a paper about the "AdS-CFT correspondence" which is bound to stir up controversy:

7) Bert Schroer, Facts and fictions about Anti de Sitter spacetimes with local quantum matter, available as hep-th/9911100.

Let me just quote the beginning:

There has been hardly any problem in particle physics which has has attracted as much attention as the problem if and in what way quantum matter in the Anti de Sitter spacetime and the one dimension lower conformal field theories are related and whether this could possibly contain clues about the meaning of quantum gravity.

In more specific quantum physical terms the question is about a conjectured (and meanwhile in large parts generically and rigorously understood) correspondence between two quantum field theories in different spacetime dimensions; the lower-dimensional conformal one being the "holographic image" or projection of the AdS theory.

The entire globalized community of string physicists has placed this problem in the centre of their interest and treated it as the dominating problem of theoretical particle physics with the result that there have been approximately around 100-150 papers per month during a good part of 1999. Even if one takes into account the increase in the number of particle physicists during the last decades and compares it with the relative number of participants in previous fashionable topics (the S-matrix bootstrap, Regge theory, the SU(6) - U(12) symmetric and the so-called relativistic quark theory, to name some of them) which also led to press-conferences, interviews and articles in the media (but not to awards and prizes), it remains still an impressive sociological phenomenon. Just imagine yourslef working on this kind of problem and getting up every morning turning nervously to the hep-th server in order to check that nobody has beaten you to similar results. What a life in an area which used to required a contemplative critical attitude!

This is clearly a remarkable situation in the exact sciences which warrants an explanation. This is particularly evident to somebody old enough to have experienced theoretical particle physics at times of great conceptual and calculational achievements, e.g. the derivation of scattering theory and dispersion theory from local fields, achievements with which the name of Harry Lehmann (to whose memory this article is dedicated) is inexorably linked. In those times the acceptance of a theoretical proposal in particle physics was primarily coupled to its experimental verifiability and its conceptual standing within physics and not yet to the beauty of its differential-geometric content. There were also fashions, but if they did not deliver what they promised they were allowed to die.

In the opinion of Roger Penrose, the new totalitarian attitude in particle physics is the result of the rapid and propagandistic communication through the new electronic media which favors speedy calculations with no or only insufficient superficial physical interpretation to more contemplative and not instantly profitable conceptual investments. He cites supersymmetry and inflation cosmology as examples of theories which achieved a kind of monopolistic dominance despite a total lack of experimental fact (or even convincing theoretical arguments). It seems to me that this phenomenon receives an even stronger illustration from string theory, and I am not the only one who thinks this way [here he cites a paper by I. Todorov].

Leaving the final explanation of this phenomenon to historians or sociologists of the exact sciences, I will limit myself to analyzing the particle physics content of the so-called Anti de Sitter - conformal QFT correspondence from the conservative point of view of a quantum field theorist with a 30 year professional experience who, although having no active ambitions outside QFT, still nourishes a certain curiosity about present activities in particle physics, e.g. string theory or the use of noncommutative geometry. Some of the consistency calculations one finds there are really surprising and if one could consider them in the critical Bohr-Sommerfeld spirit as ciphers encoding possibly new principles in fundamental physics and not as a theory (let alone a theory of everything), these observations may have an enigmatic use. But for this to be successful one would have to make a much more serious attempt at confronting the new mathematical consistency observations with local quantum physics on a more conceptual level beyond the standard formalism. Only in this way can one be sure to confront something new and not just a new formalism which implements the same principles in a different way.

The AdS model of a curved spacetime has a long history as a theoretical laboratory of what can happen with particle physics in a universe which is the extreme opposite of globally hyperbolic in that it possesses a self-closing time, whereas the proper de Sitter spacetime was once considered among the more realistic models of the universe. The recent surge of interest about AdS came from string theory and is different in motivation and more related to the hope (or dream) to attribute a meaning to "Quantum Gravity" from a string theory viewpoint.

Fortunately for the curious outsider (otherwise I would have to quit right here), this motivation has no bearing on the conceptual and mathematical problems posed by the would-be AdS-conformal QFT correspondence, which turned out to be one of those properties discovered in the setting of string theory which allow an interesting and rigorous formulation in QFT which confirms some but not all of the conjectured properties. The rigorous treatment however requires a reformulation of (conformal) QFT. The standard formalism based on pointlike "field coordinatizations" which underlies the Lagrangian (and Wightman) formulations does not provide a natural setting for the study of isomorphisms between models in different spacetime dimensions, even though the underlying principles are the same. One would have to introduce too many additional concepts and auxiliary tricks into the standard framework. The important aspects in this isomorphism are related to space and time-like (Einstein, Huyghens) causality, localization of corresponding objects and problems of degree of freedom counting. All these issues belong to real-time physics and in most cases their meaning in terms of Euclidean continuation (statistical mechanics) remains obscure; but this of course does not make them less physical.

This note is organized as follows. In the next section I elaborate on the kinematical aspects of the AdSd+1-CQFTd situation as a collateral of the old (1974/75) compactification formalism for the "conformalization" of the d-dimensional Minkowski spacetime. For this reason the seemingly more demanding problem of studying QFT directly in AdS within a curved spacetime formalism can be bypassed. The natural question whose answer would have led directly from CQFT4 to AdS5 in the particle physics setting (without string theory as a midwife) is: Does there exist a quantum field theory which has the same SO(4,2) symmetry and just reprocesses the CQFT4 matter content in such a way that the "conformal hamiltonian" (the timelike generator of rotations of conformally compactified Minkowski space) becomes the true hamiltonian? The theory exists and is an AdS theory with a specific local matter content computable from the CQFT matter content. The answer is unique, but as a result of the different dimensionality one cannot describe this unique relation between matter contents in terms of pointlike fields. This will be treated in Section 3, where we will also compare the content of Rehren's isomorphism with Maldacena, Witten et al conjectures and notice some subtle but potentially serious differences. Whoever is aware of the fact that subtle differences have often been the enigmatic motor of progress in good physics times will not dismiss such observations.

The last section presents some results of algebraic QFT on degrees of freedom counting and holography. Closely connected is the idea of "chiral scanning", i.e. the encoding of the full content of a higher dimensional (massive) QFT into a finite number of copies of one chiral theory in a carefully selected position within a common Hilbert space. In this case the price one has to pay for this more generic holography (light-front holography) is that some of the geometrically acting spacetime symmetry transformations become "fuzzy" in the holographic projection and some of the geometrically acting symmetries on the holographic image are not represented by diffeomorphisms if pulled back to the original QFT.

As you can see, there is some interesting mathematical physics in here, as well as some serious criticism of how particle physics is done these days.

By the way, Schroer has recently written a paper about the braid group and quantum field theory. Everyone knows how the braid group shows up in 3d quantum field theory, but this is about 4d quantum field theory:

8) Bert Schroer, Braided structure in 4-dimensional conformal quantum field theory, available as hep-th/0012021.

© 2000 John Baez