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Introduction to String Field Theory (Advanced Series in Mathematical Physics)

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The author introduces the subject of his book as "the newest approach" to string theory, which he defines in analogy to the point particle theory, as an approach to the calculation of relevant quantities using field theory Lagrangians, instead of "off-shell" S-matrix computations, and which is done in 10 dimensions. The first five chapters of the book is not concerned directly with strings at all, but with the quantization of gauge theories, both pure and with the presence of matter (fermions). The author considers first point particle fields in the light cone gauge. In this gauge the field theory appears nonrelativistic, satisfying a non-relativistic "Schrodinger equation" with an imaginary Hamiltonian. The author then discusses the Yang-Mills theory in the light cone gauge, and derives the free Lagrangian for this theory. This motivates a detailed discussion of the conformal algebra since (nonlinear) representations of the Poincare group model the kinetic term of a free light-cone field theory, and one can obtain these, as the author shows, using the conformal group. He later generalizes to the case where interactions are present, and derives the Feynman rules. The reader can readily see the tension between the demands for covariance and unitarity, that is characteristic of gauge theories. The light-cone gauge is manifestly covariant, but in two less dimensions than the dimension of spacetime the fields are formulated in. This is apparent in the use of the Poincare group ISO(D-1,1), the representations of which are constructed for arbitrary massless and massive theories. The representations are nonlinear in the coordinates, and are constructed from irreducible representations of the SO(D-2) rotation group of the SO(D-1,1) Lorentz group. The conformal group is then SO(D, 2).

In these initial five chapters the reader also gets a detailed overview of the BRST formalism, which is very important in the quantization of gauge theories. This formalism is first introduced in the context of the Hamiltonian formalism, which is manifestly covariant in D - 1 dimensions. This involves as expected a separation of coordinates into space and time with the time components of the gauge fields set to zero. The famous Faddeev-Popov ghosts make their appearance here, since the quantization problem is a problem with constraints. The author gives several reasons for using the BRST formalism, and the reader sees the origin of the Slavnov-Identities, which are generalizations of the amazing Ward identities and are a consequence of the side constraint of unitarity.

The actual consideration of strings first takes place in chapter 6. The large amount of work done by the author in the first five chapters to find a general Poincare- and gauge-invariant action for any collection of fields is finally applied in this chapter and the rest of the book. The idea of viewing strings as 2-dimensional field theories is the main point behind the author's approach. The author quantizes the bosonic string in the light-cone gauge and derives the Poincare algebra, which can be viewed as a specialization of what was done in the first two chapters. This is generalized immediately to the case to the fermionic case by introducing a 2D supersymmetry on the world sheet, in complete analogy with the point particle case in chapter five. In this discussion the reader can see clearly the origin of the requirement that D be equal to 10. A manifestly covariant formalism is then discussed, which is a generalization of the bosonic string and the superparticle of chapter 5. This discussion is interesting in that it shows the origin of the Kac-Moody algebra in the covariant derivatives, and the Virasoro algebra. The BRST formalism is discussed later in the context of the first-quantization of the bosonic string as a constrained problem in the conformal gauge. The Feynman rules for interacting strings are then derived using first the external field formalism, and then using functional integration.

The author gets down to studying string field theory in the context of what was done early in chapter 2 in chapters 10 and 11, namely the light-cone gauge and the BRST formalism, with the goal to include the contributions of the string interactions. As expected, in the free field case the bosonic open strings satisfy a Schroedinger-like equation, and interactions are described by splitting and joining of strings, and as expected from a field-theoretic point of view, the graphs are composed of vertices and propagators. The BRST formalism is done only for the closed string case.The author introduces the reader to how to construct gauge-invariant actions for interacting strings in the last chapter of the book. He is careful to note that a string field theory of interacting strings does not exist, and gives explanations to the difficulties involved in constructing such theories. I have not followed the research on this topic since this book was published, so cannot comment on the present state of atttempts to construct these theories, except for those attempts to give an interpretation of open and closed strings in terms of algebraic topology, C*-algebras, and K-theory. These however do not permit any kind of Feynman rules to be derived. No doubt a perusal of the preprint servers will reveal that this problem has been absorbed in the current emphasis on D-brane and M-theories.