Time correlation functions

In standard quantum mechanics, the calculation of any two-time correlation function has to include the evolution of the system between the two times considered; this evolution is contained in the unitary evolution operator U(t′, t), as for instance in relation (10.9). In Bohmian theory, it is important to take into account the effect of the first measurement, which correlates the system under study to a measurement apparatus and creates “empty waves” (§10.6.1.c). Otherwise one obtains contradictions with the standard predictions.

For instance the author of [443] considers a one-dimensional harmonic oscillator that is initially in a stationary state, and studies the correlation function of the position at times t and time t′, in the particular case where t′ − t is equal to half the period 2π/ω of the oscillator. In standard quantum mechanics, it is easy to show that the corresponding position operators 〉X(t) and X(t′) are then opposite, so that the correlation function 〉X(t)X(t′〈 is equal to − 〉[X(t)]2〈, therefore negative. In Bohmian mechanics, the particle is initially static since the wave function is real. If one ignores the effect of the first measurement, the position of the particle will remain at the same place, which corresponds to a correlation function equal to 〉[X(t)]2〈, therefore positive; one reaches an apparent complete contradiction. But, if one takes into account the effect of the first measurement on the particle, one finds that, just after this measurement, each position of the oscillator becomes correlated with a different Bohmian position of the pointer.

Introduction

The period between the years 1976 and 1984 shows very little activity in string theory. As we mentioned in the previous Part, a lot of work went into developing both perturbative and nonperturbative aspects of QCD, which established itself as the theory of strong interactions. Lattice gauge theory was formulated and the idea of confinement was developed. These were also the years when supersymmetry was used to construct the Minimal Supersymmetric Standard Model, and the various supergravities were obtained. The most fundamental of them, constructed by Cremmer, Julia and Scherk, was the eleven-dimensional one. The fact that the various supergravities showed better ultraviolet behaviour than the original gravity theory gave the hope that one could unify gauge interactions with gravity in the framework of supergravity without nonrenormalizable divergences. On the other hand, the machinery of string theory seemed too complicated and unnecessary for unification. We review these developments in Section 42.2.

Although research in string theory was very limited in these years, it led to three fundamental developments. The first one, discussed in the Chapter by Green, was the reformulation of the fermionic string in terms of a light-cone fermionic coordinate Sa, that is an SO(8) spinor, instead of the light-cone SO(8) vector ψi of the RNS model. This allowed the complete construction of type IIA, IIB and I superstring theories. It is described in Section 42.3.

Abstract Joël Scherk (1946–1980)was an important early contributor to the development of string theory. Together with various collaborators, he made numerous profound and influential contributions to the subject throughout the decade of the Seventies. On the occasion of a conference at the École Normale Supérieure in 2000 that was dedicated to the memory of Joël Scherk, I gave a talk entitled ‘Reminiscences of collaborations with Joël Scherk’ [Sch00]. The present Chapter, an expanded version of that presentation, also discusses work in which I was not involved.

Introduction

Joël Scherk was one of the most brilliant French theoretical physicists who emerged in the latter part of the Sixties. Together with André Neveu, he was educated at the École Normale Supérieure in Paris, and in Orsay. Together, they studied electromagnetic and final-state interaction corrections to nonleptonic kaon decays [NS70b] under the guidance of Claude Bouchiat and Philippe Meyer. They both defended their ‘thése de troisiéme cycle’ (the French equivalent of a PhD) in 1969, and they were hired together by the CNRS that year (tenure at age 23!). In September 1969 the two of them headed off to Princeton University.

In 1969 my duties as an Assistant Professor in Princeton included advising some assigned graduate students. The first advisees, who came together to see me, were André Neveu and Joël Scherk. I had no advance warning about them, and so I presumed they were just another pair of entering students.

I thought it might be interesting to mention here some personal recollections of the prehistory and early history of string theory. These reminiscences are presented in an informal manner, as if they were a contribution to oral history, without the usual footnotes and references of a scientific article.

I have always been a strong supporter of string theory, although (especially early on) I did not know exactly, any more than others did, how it would be useful.

During the Seventies and Eighties, in accordance with my role as an ardent conservationist, I set up at Caltech a nature reserve for endangered superstring theorists. I brought John Schwarz and Pierre Ramond to Caltech and encouraged André Neveu to visit.

Over the next few years we hosted Joël Scherk and Michael Green and a number of other brilliant long-term visitors. Some of our graduate students became distinguished superstring theorists. Between 1972 and 1984, a significant fraction of the work on superstrings was done at Caltech, but I myself did not carry out original research on superstrings. Earlier, however, I did have a connection with the prehistory of string theory.

During the Sixties I regarded somewhat favourably the bootstrap approach to the theory of hadrons and the strong interaction, as put forward by Chew and Frautschi. It was connected with the mass-shell formulation of quantum field theory. I had proposed that formulation in the mid-Fifties and described it at the Rochester Conference on High Energy Physics in 1956.

Introduction

The construction, from the axioms of S-matrix theory, of the Veneziano model and of its extension to N external particles, the Dual Resonance Model, made many people believe that this theory was completely different from quantum field theory. However, it soon became clear that the DRM was actually an extension of, rather than an alternative to, the various field theories, such as φ3 scalar theory [Sch71], gauge theories [NS72] and general relativity [Yon73, SS74, Yon74].

The DRM contains a parameter, the slope of the Regge trajectory α′ with dimension of (length)2, or its inverse, the string tension T = 1/(2πα′). The study of DRM amplitudes in the zero-slope (or infinite string tension) limit shows that these reduce to Feynman diagrams of specific field theories. In the string picture of the DRM, an intuitive way of understanding the zero-slope limit is the following. The string tension has the tendency to make the string collapse to a point, but, if the string is moving, for example rotating, there is also the centrifugal force, which has the opposite tendency. This means that a possible string motion results from the balance between these two forces. However, in the zero-slope limit the string tension is increasingly large, the centrifugal force cannot balance it anymore, and the string collapses to a point. In this limit, string theory becomes a theory of pointlike objects that is described by ordinary quantum field theory.

Introduction

Part V deals with the extensions of the Dual Resonance Model (DRM), i.e. the bosonic string, to include additional symmetries and degrees of freedom. These generalizations were originally motivated by the need to overcome the drawbacks of the DRM and obtain a more realistic model of hadrons. Such attempts were only partially successful, though, with hindsight, we can say that they added some essential elements for the construction of modern string theory.

One of the first modifications of the Koba–Nielsen amplitude aimed at incorporating the internal flavour symmetry of hadrons, and was proposed by Chan and Paton in 1969. As discussed in Section 27.2, these authors showed that an internal flavour symmetry can be introduced simply by multiplying the amplitudes by appropriate group theoretical factors. Such factors can be viewed as resulting from the presence of a quark–antiquark pair attached to the open string end-points, and carrying flavour quantum numbers.

However, the incorporation of flavour symmetry was not the only open issue. As discussed in the previous Parts, the main problems of the DRM were: (i) the presence of a tachyon; (ii) the absence of fermions, preventing the description of baryons; (iii) the presence of a critical dimension with an unrealistic value, d = 26. Attempts to solve these problems started very early, in fact immediately after the appearance of the Veneziano formula, and went on more or less in parallel with the understanding of the DRM and its reinterpretation as a quantum string (see Parts III and IV).