Linguistic change is a process which pervades all human languages. The extent of this change can be so radical that the intelligibility of former states of the language can be jeopardised. The language of Shakespeare causes some problems for the early twenty-first-century reader, but these are not insurmountable. However, if we go further back to the writings of Chaucer, we are faced with a much more alien, less easily recognised form of English. If we observe language change on a much smaller timescale, say that of the average life span of a human being, comprehension difficulties such as those confronting the reader of Chaucer do not arise. Languages actually change quite slowly, and hence the ability to communicate successfully with all generations of speakers of our own language variety is maintained. In this section, we will look at how the sounds of languages can change over time, both from a diachronic and synchronic perspective. Diachronic research on sound change has enabled us to chart changes that have taken place in earlier historical periods, while synchronic approaches allow us to observe language changes in progress today. In addition, we will examine sound change from the perspective of one of the principal problems of language change, namely the transition problem – what is the route by which sounds change?

Consonant change

In section 2, we saw that consonants can be largely classified according to a simple three-term description:

1. (a) voicing: do the vocal cords vibrate?

2. (b) place of articulation: where is the flow of air obstructed?

3. […]

So far, we have not attempted to develop any analytic account of the semantic representations which appear in lexical entries. Indeed, in the examples in (115) (section 10), what we see under the heading ‘semantics’ is taken directly from an ordinary dictionary. Whether such dictionary definitions can be regarded as supplying the meanings of words for the purposes of linguistic analysis is something we shall briefly consider later in this section after we have introduced some basic ideas.

As well as being concerned with the contents of lexical entries, a further matter which will arise in this section is that of the overall structure of the lexicon. In the Introduction (p. 4), we talked about the lexicon as a list of lexical entries, but it is at least conceivable that it has a more interesting structure than this. To say that the lexicon is no more than a list is to accept that there is no reason why items which are similar to each other in some linguistically relevant way are ‘close’ to each other in the mental lexicon. As we shall see, similarity of meaning is a rather rich notion, and as subsequent sections of this part of the book will show, it seems to play an important role in human cognitive processing. In such circumstances, it is important for our model of the lexicon to represent this notion properly.

There are two aspects to the real-time processing of language in which we all indulge on a day-to-day basis. One is hearing what others say to us, or in the case of written language and sign languages, seeing what others are saying to us. This is the problem of speech perception, and a fundamental part of it for spoken languages is the recognition of speech sounds. The other is producing language ourselves, speech production. For spoken varieties of language, this includes the problem of control of the muscles of the vocal tract (lungs, throat, tongue, lips) responsible for making the sounds. For sign languages, it is the problem of control of movements of the hands and face. In psychology, the organisation of movement is referred to as motor control.

Speech perception

Suppose you are singing a note on a certain pitch. If you wish to sing a different note, one option you have is to shift to the new note gradually and continuously (you can also jump straight to it, but this option doesn't concern us here). This indicates that the pitch of the human voice, determined by the rate at which the vocal cords vibrate, admits of any number of gradations. Now contrast this with someone playing two notes on a piano. A piano has a finite number of discrete notes, and as a consequence it isn't possible to play a note between C and C#; it is, however, perfectly feasible to sing such a note.

Up to this point, our discussion of syntax has focused largely on a variety of English which we will call Contemporary Standard English (CSE). But since we find numerous dimensions of variation in language (e.g. variation from one style to another, from one regional or social variety to another, from one period in the history of a language to another, and from one language to another), an important question to ask is what range of syntactic variation we find in the grammars of different languages or language varieties. Of course, having answered this question, further issues arise. For instance, if we are considering what are regarded as varieties of the same language, we might be concerned with understanding the social and contextual factors which determine when speakers use one variety or another. This is the sort of concern which our discussion of variation in parts I and II focused on, but here we shall adopt the less ambitious goal of seeing how our syntactic framework can come to terms with a small sample of within- and across-language variation.

Inversion in varieties of English

The most obvious manifestation of structural variation in syntax lies in word-order differences. If we suppose that the theory of Universal Grammar incorporated into the language faculty provides human beings with a genetically transmitted template for syntactic structure (so that clauses are universally CP+TP+VP structures, and nominal expressions are universally DPs), we should expect to find that word-order differences are attributable to differences in the movement operations which apply within a given type of structure.

Dp-branes and boundary conditions

A Dp-brane is an extended object with p spatial dimensions. In bosonic string theory, where the number of spatial dimensions is 25, a D25-brane is a space-filling brane. The letter D in Dp-brane stands for Dirichlet. In the presence of a D-brane, the endpoints of open strings must lie on the brane. As we will see in more detail below, this requirement imposes a number of Dirichlet boundary conditions on the motion of the open string endpoints.

Not all extended objects in string theory are D-branes. Strings, for example, are 1-branes because they are extended objects with one spatial dimension, but they are not D1-branes. Branes with p spatial dimensions are generically called p-branes. A 0-brane is some kind of particle. Just as the world-line of a particle is one-dimensional, the world-volume of a p-brane is (p + 1)-dimensional. Of these p + 1 dimensions, one is the time dimension and the other p are spatial dimensions.

Introduction

In this book, the quantization of strings was carried out using light-cone coordinates and the light-cone gauge. String theory is a Lorentz invariant theory, but Lorentz symmetry is not manifest in the light-cone quantum theory. Indeed, the choice of a particular coordinate X+ for special treatment hides from plain view the Lorentz symmetry of the theory. While hidden, the Lorentz symmetry is still a symmetry of the quantum theory, as we demonstrated by the construction of the Lorentz generator M−I. This generator has the expected properties when the spacetime has the critical dimension.

Since Lorentz symmetry is of central importance, it is natural to ask if we can quantize strings preserving manifest Lorentz invariance. It is indeed possible to do so. The Lorentz covariant quantization has some advantages over the light-cone quantization. Our lightcone quantization of open strings did not apply to D0-branes because the light-cone gauge requires that at least one spatial open string coordinate has Neumann boundary conditions. Covariant quantization applies to D0-branes.

String theory is one of the most exciting fields in theoretical physics. This ambitious and speculative theory offers the potential of unifying gravity and all the other forces of nature and all forms of matter into one unified conceptual structure.

String theory has the unfortunate reputation of being impossibly difficult to understand. To some extent this is because, even to its practitioners, the theory is so new and so ill understood. However, the basic concepts of string theory are quite simple and should be accessible to students of physics with only advanced undergraduate training.

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I strongly recommend this book to anyone who wants to learn the basics of string theory.

Light-cone Hamiltonian and commutators

We are at long last in a position to quantize the relativistic string. We have acquired considerable intuition for the dynamics of classical relativistic strings, and we have examined in detail how to quantize the simpler, but still nontrivial, relativistic point particle. Moreover, having taken a brief look into the basics of scalar, electromagnetic, and gravitational quantum fields in the light-cone gauge, we will be able to appreciate the implications of quantum open string theory. In this chapter we will deal with open strings. We will assume throughout the presence of a space-filling D-brane. In the next chapter we will quantize the closed string.

Maxwell fields coupling to open strings

Among the quantum states of open strings attached to a D-brane we found photon states with polarizations and momentum along the D-brane directions. We thus deduced that a Maxwell field lives on the world-volume of a D-brane. The existence of this Maxwell field was in fact necessary to preserve the gauge invariance of the term that couples the Kalb–Ramond field to the string in the presence of a D-brane. We also learned that the endpoints of open strings carry Maxwell charge.

Since any D-brane has a Maxwell field, it is physically reasonable to expect that background electromagnetic fields can exist: there may be electric or magnetic fields that permeate the D-brane.

Area functional for spatial surfaces

The action for a relativistic string must be a functional of the string trajectory. Just as a particle traces out a line in spacetime, a string traces out a surface. The line traced out by the particle in spacetime is called the world-line. The two-dimensional surface traced out by a string in spacetime will be called the world-sheet. A closed string, for example, will trace out a tube, while an open string will trace out a strip. These two-dimensional world-sheets are shown in the spacetime diagram of Figure 6.1. The lines of constant x0 in these surfaces are the strings. These are the objects an observer sees at the fixed time x0. They are open curves for the surface describing the open string evolution (left), and they are closed curves for the surface describing the closed string evolution (right).