General discussion

So far we have talked about energy transport by radiation only. We may also have energy transport by mass motions. If these occur hot material may rise to the top, where it cools and then falls down as cold material. The net energy transport is given by the difference of the upward transported energy and the amount which is transported back down. Such mass motions are also called convection. Our first question is: when and where do these mass motions exist, or in other words where do we find instability to convection? When will a gas bubble which is accidentally displaced upwards continue to move upwards and when will a gas bubble which is accidentally displaced downwards continue to move downwards? Due to the buoyancy force a volume of gas will be carried upwards if its density is lower than the density of the surroundings and it will fall downwards if its density is larger than that of the surroundings.

From our daily experience we know that convection occurs at places of large temperature gradients, for instance over a hot asphalt street in the sunshine in the summer, or over a radiator in the winter. The hot air over the hot asphalt, heated by the absorption of solar radiation, has a lower density than the overlying or surrounding air. As soon as the hot air starts rising by an infinitesimal amount, it gets into cooler and therefore higher density surroundings and keeps rising due to the buoyancy force like a hot air balloon in the cooler surrounding air. This always occurs if a rising gas bubble is hotter than its surroundings.

The hydrostatic equilibrium equation

What information can we use to determine the interior structure of the stars? All we see is a faint dot of light from which we have to deduce everything. We saw in Volume 2 that the light we receive from main sequence stars comes from a surface layer which has a thickness of the order of 100 to 1000 km, while the radii of main sequence stars are of the order of 105 to 107 km. Any light emitted in the interior of the stars is absorbed and re-emitted in the star, very often before it gets close enough to the surface to escape without being absorbed again. For the sun it actually takes a photon 107 years to get from the interior to the surface, even though for a radius of 700 000 km a photon would need only 2.5 seconds to get out in a straight line. There is only one kind of radiation that can pass straight through the stars – these are the neutrinos whose absorption cross-sections are so small that the chances of being absorbed on the way out are essentially zero. Of course, the same property makes it very difficult to observe them because they hardly interact with any material on Earth either. We shall return to this problem later. Except for neutrinos we have no radiation telling us directly about the stellar interior. We have, however, a few basic observations which can inform us indirectly about stellar structure.

For most stars, we observe that neither their brightness nor their color changes measurably in centuries. This basic observation tells us essentially everything about the stellar interior.

Evolution along the giant branch

Just as for low mass stars, the evolution of high mass stars is caused by the change in chemical composition when hydrogen fuses to helium. These stars, however, have a convective core such that the newly formed helium is evenly mixed throughout the core. When hydrogen is consumed, the convective core contracts and also shrinks in mass (because the κ + σ per gram decreases and therefore ∇r decreases); the mixing then occurs over a smaller mass fraction, while some material, which was originally part of the convective region, is left in a stable region but with a slightly enriched helium abundance and also a slight increase in the N14/C12 and C13/C12 ratios. (See Figs. 13.2 and 13.4.) When the convective core mass reduces further, another region with still higher helium abundance and higher N14/C12 and C13/C12 is left outside the convection zone. The remaining convective core becomes hydrogen exhausted homogeneously while it contracts to a smaller volume and becomes hotter. The stars also develop hydrogen burning shell sources around the helium core. Again the core acts like a helium star with a very high temperature; the temperature at the bottom of the hydrogen envelope becomes too high to sustain hydrostatic equilibrium in the hydrogen envelope. The envelope expands and the stellar surface becomes cooler, moving the star in the HR diagram towards the red giant region. Again an outer hydrogen convection zone develops and reaches into deeper and deeper layers. Finally it dredges up some of the material which was originally in the convective core when it included a rather large mass-fraction of the star.

Schwarzschild's method

Before we can discuss the detailed structure of the stars on the main sequence we have to outline the methods by which it can be calculated. In Chapter 10 we have compared homologous stars on the main sequence. While we were able to see how temperatures and pressures in the stars change qualitatively with changing mass and chemical composition, we have never calculated what the radius and effective temperature of a star with a given mass really is. In order to do this we need to integrate the basic differential equations, which determine the stellar structure as outlined in Chapter 9. Two methods are in use: Schwarzschild's method and Henyey's method.

Schwarzschild's method is described in his book on stellar structure and evolution (1958). The basic differential equations are integrated both from the inside out and from the outside in. In the dimensionless form the differential equations for the integration from the outside in contain the unknown constant C (see Chapter 9), for the integration from the inside out the differential equations also contain the unknown constant D. A series of integrations from both sides of the star is performed for different values of these constants. The problem then is to find the correct values for the constants C and D and thereby the correct solutions for the stellar structure. At some fitting point Xf = (r/R)f we have to fit the exterior and the interior solutions together in order to get the solution for the whole star. At this fitting point we must of course require that pressure and temperature are continuous.

Completely degenerate stars, white dwarfs

In Chapter 14 we saw that low mass stars apparently lose their hydrogen envelope when they reach the tip of the asymptotic giant branch. What is left is a degenerate carbon–oxygen core surrounded by a helium envelope. The mass of this remnant is approximately 0.5 to 0.7 solar masses depending perhaps slightly on the original mass and metal abundances. The density is so high that the electrons are partly or completely degenerate except in the outer envelope. We also saw that central stars of planetary nebulae seem to outline the evolutionary track of these remnants which decrease in radius, still losing mass and increasing their surface temperature. Their luminosities do not seem to change much until they reach the region below the main sequence (see Fig. 14.14). In the interiors these remnants are not hot enough to start any new nuclear reactions. When they started to lose their hydrogen envelope they still had a helium burning and a hydrogen burning shell source. When the hydrogen envelope is lost the hydrogen burning shell source comes so close to the surface that it soon becomes too cool and is extinguished. The helium burning shell source survives longer but finally is also extinguished, when the star gets close to the white dwarf region. The remnant ends up as a degenerate star with no nuclear energy source in its interior but which still has very high temperatures. This is the beginning of the evolution of a white dwarf. It loses energy at the surface, which is replenished by energy from the interior, i.e. by thermal energy from the heavy particles.

Introduction

The fact that we see massive, luminous stars which cannot be older than about 106 years tells us that stars must have been formed within the last million years. In association with these luminous young stars we often see some peculiar stars with emission lines, called the T Tauri stars (see Volume 1). These can therefore be assumed to be young stars also. They have lower luminosities and are more red than the massive O and B stars but are considerably more luminous than main sequence stars of the same color. Because of their lower luminosities they must have lower masses than the O and B stars. For the lower mass stars the contraction times are longer, as we have seen in Chapter 2, because these stars cannot radiate away the surplus gravitational energy as fast as the more luminous, massive stars. If these lower mass T Tauri stars were formed at the same time as their more massive associates they have not had enough time to contract to the main sequence during the main sequence lifetime of the massive stars. Lower mass stars must therefore still be in the contraction phase. It is then reasonable to assume that these T Tauri stars are young stars still in the contraction phase.

Both kinds of stars, the massive O main sequence stars and the less massive young T Tauri stars, appear in association with large dust complexes, i.e. regions of high density where many interstellar molecules are formed. It thus appears that new stars may be born in regions of high density interstellar material.

Introduction

Supersymmetry is the first real extension of space-time symmetry. It has given us great hope that we should be able to generalize ordinary geometry into a super-geometry and in this process obtain more unique and consistent models of physics. In some cases this has been achieved, but in most cases we still lack a natural and unique extention into a superspace.

The concept of superspace, i.e., a space with fermionic coordinates as well as bosonic coordinates, was introduced first in dual models by Montonen in an attempt to construct multiloops in the Ramond-Neveu-Schwarz model. This led eventually to the superconformal algebras and super-Riemannian spaces. When supersymmetric field theories were discovered, it was soon realized that a super-space is the natural space in which to describe these models. However, these descriptions, although in the end quite successful in establishing renormalization properties, always lacked a certain sense of naturalness. For each supermultiplet different ideas had to be used.

In supergravity theories, being extensions of truly geometric theories, the hopes were even higher and the results more discouraging. So far one has only managed to write superspace actions for the N=1 theory, and none of them is a natural extension of the Hilbert action. Superspace techniques were though eventually useful in describing the classical theories and led to the really important result that any supergravity theory has infinitely many possible counterterms.

This essay is dedicated to Murray Gell-Mann upon his sixtieth birthday. I assume he will be particularly gratified to read that the inspiration for a new look at the 1964 work came during a visit to the College of Judea and Samaria at Ariel and to the “Nir” Yeshiva in Hebron. Also, while travelling in the area, I came across one of the five types of pheasant-like birds (Hebrew “HoGLaH”) that he had enumerated to me on our 1967 tour of the Negev. There were no Malayalam-speaking immigrant Cochin Jews around this time as there had been at Nabatean Mamshit (Roman “Mempsis”, and in Arabic “Kurnub”) but I did visit Tat-speaking Daghestani “mountain Jews” in the Dothan valley in northern Shomron (Samaria). It is interesting that they have become important producers of goose-liver which is exported to France. Aside from Tat proper, they use “Judeo-Tat”, a Tat-based “Yiddish”.

QUANTUM FIELD THEORY, STRONG INTERACTIONS AND SU(3) IN 1964

In 1963–65, I was at Caltech as Murray's guest. We had met at CERN during the 1962 conference, at the end of the rapporteur session on strange particles. I have related elsewhere the events of that day and Gerson Goldhaber has added a witness-participant account. Anyhow, the outcome was a two-year stay at Caltech and the beginning of a lasting friendship.

ABSTRACT

Recently Friedberg, Lee and Ren have pointed out that at low density the ideal charged boson system turns out not to be a superconductor, but becomes a type II superconductor at high density. This conclusion differs from the well-known Schafroth solution of superconductivity at any density for the same problem. Schafroth's analysis is found to contain a mistake due to the neglect of the electrostatic exchange energy Eex. Based on the Schafroth solution, Eex is shown to be +∞ in the normal phase, but 0 in the condensed phase (at T = 0). Of course, the correct solution has to give a finite Eex.

This research was supported in part by the U.S. Department of Energy.

SCHAFROTH'S SOLUTION

Schafroth's superconductivity solution of an ideal charged boson system published 35 years ago has always been considered to be the definitive work, comparable in depth to the analysis made by Landau on the diamagnetism of an ideal charged fermion system. However, recently it was found that the Schafroth solution contains a serious mistake due to the neglect of the electrostatic exchange energy Eex. It turns out that based on the Schafroth solution, Eex is +∞ in the normal phase, but 0 in the condensed phase (at T = 0). Of course, the correct solution has to give a finite Eex.

For ideal charged particles, bosons or fermions, there is only the electromagnetic interaction.

ABSTRACT

We study the algebra of normal ordered and reparametrization invariant operators of the open bosonic string field theory. These, besides the Poincaré group generators, include the ghost number operator and two translationally invariant symmetric second-rank space-time tensors. The BRST operator of string field theory is the trace of the fermionic one, and the second is the BRST transform of the former. Their algebra closes only when certain Lorentz non-invariant projections over the fermionic tensor are taken. There are many inequivalent such algebras, corresponding to manifest Lorentz invariance in lower dimensions. Some of these contain, besides the Lorentz invariant BRST, another nilpotent operator. We provide an example where Lorentz invariance is manifest in 1 + 1 dimensions.

I recall that the first time Murray asked me to come and talk at Caltech sixteen years ago, the subject was one we both shared an interest in: covariant string field theory. After all these years, things have not changed so much as to force me to change the subject matter: Plus ça change, plus c'est la même chose. I recall Murray inquiring at the time as to the underlying principle behind string theory. His penetrating question is still valid and unanswered today.

In a series of publications, we set about to list normal-ordered operators which are invariant under complex reparametrizations. In open string field theory, these include familiar operators such as the ghost number operator, the Poincaré generators, and the BRST charge.

I first met Murray Gell-Mann when he popped up in my office at the Institute for Advanced Study, and described to me the isospin-strangeness rule he had discovered. He pronounced my name correctly and interpreted its meaning correctly. That was September 1953.

The post-War decades have been the Golden Age of particle physics. Theory and experiment went hand in hand to make amazing advances. What we know now about the world of elementary particles is incredibly richer than what we did forty years ago, and we owe this to Murray above all. Looking beyond the Baroque period we are in now, I hope Murray's spirit will come back alive again.

Recently I have been taking a renewed interest in the BCS mechanism as a model for spontaneous generation of fermion mass and associated Goldstone (G) and Higgs (H) collective bosons. Here I mean by a BCS mechanism the formation of fermion pair condensates due to a short range attraction. In an idealized situation, this may be represented by a four-fermion interaction, and the dynamics is essentially determined by the properties of fermion bubble diagrams. A characteristic feature of the bubble approximation is that the Bogoliubov–Valatin (BV) fermion and the Higgs boson have the simple mass ratio 1:2. Such modes are known to exist in superconductors.

These low energy modes can be represented by an effective Ginzburg–Landau– Gell-Mann-Lévy Hamiltonian in which the boson self-coupling and the boson-fermion Yukawa coupling are related so as to satisfy the mass ratios.

I would like to correct a misrepresentation made by several of the preceding speakers. We are not celebrating the sixtieth anniversary of Murray's birthday. We are celebrating the sixtieth anniversary of his conception. His actual birthday is in September.

We are, of course, celebrating Murray Gell–Mann, whom I've known now since 1951. We joined the University of Chicago the same year, a few months apart. And before I go into the more scientific part of my lecture, I think you might be interested in the origin of the name “Murray.” Presumably it was, like some other first names, derived from a surname. And these, in turn, often come from geographical names, in the present case, from a Scottish province, “Muraih.” Already in 1203 we find a William de Moravia, and in 1317 an Orland de Morris, and in 1327, an Andrew Muraih. [This does not prove the point, because these family names could well have come from “Murie,” the Middle English form of Merry.] As a first name, it has also been surmised, as I see from a book by Partridge, Name This Child, that “Murray” comes from “Murrey” a word for dark red or eggplant colored, an adjective which in turn presumably comes from mulberry, in turn connected to maroon. Which brings us back to Murray's favorite color of corduroy jackets at the University of Chicago.

Now having explained the word Murray, I cannot refrain from giving you the origin of the name Gell–Mann.

It is an honor to have been selected among Murray's many friends and colleagues to speak to you this afternoon. No doubt I have been chosen because of the mathematical component present in high-energy theory today. Before concentrating on the interface of elementary particle physics and modern geometry, I'd like to record my own pleasure in knowing Murray this past decade. Perhaps it's just as well we didn't get acquainted earlier; I think he would have frightened me to death. You all are aware of Murray's great intellectual powers; but to me, equally amazing, is his enthusiasm for all creative endeavors, large and small. More than anyone, he firmly believes that the human mind and the human spirit can cure the ills of society. This birthday celebration expresses his personality in several ways. The diversity of topics reflects his many interests. And the theme stresses his positive view of the future.

Our charge was to pick some subject — mine is mathematics and physics — and discuss its present status and future prospects. Like twin stars, the two subjects have influenced each other greatly over the centuries, sometimes overlapping significantly, sometimes going their separate ways. In the fifties and sixties there was little contact — perhaps even some hostility. Physicists believed that too much mathematics hindered physical insight; some older ones still do. Mathematicians required more mathematical precision than physics deemed necessary and were developing abstract structures for their own sake.