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1. Explicitly calculate the commutators [ + X ( , ), X ( , )] and [ X ( , ), X ( , )]. + 2. Consider the closed string and explicitly calculate [ x , p ] .
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3. Consider the rst excited state of the closed string ( k ) 1 1 0, k . Using the condition satis ed by physical states , in particular L1 = L1 = 0, nd k .
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i 4. Let 1 0, k be the state of an open string and suppose that the normal ordering constant a is undetermined. What is the mass of this state
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5. In the light-cone gauge, use [ x i , p j ] = i ij , [ X i ( ), P j ( )] = i ij ( ) i to nd a commutation relation for the transverse modes, [ m , nj ]. 6. Consider light-cone quantization for the closed string case. What additional commutation relation do you think should be imposed for the modes
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Conformal Field Theory Part I
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In this chapter we study conformal eld theory, an area of quantum eld theory that relies heavily on complex variables. In this chapter, we will introduce some of the basic concepts of conformal eld theory. The topic will be expanded and utilized in many areas in the rest of the book. In the next chapter, we discuss other aspects of conformal eld theory along with BRST quantization. Conformal eld theory is an important tool used in the analysis of perturbative string theory, so it plays a central role in our task at hand (understanding the physics of quantized strings). In particular, since the worldsheet can be described using two coordinates ( , ) two-dimensional conformal eld theory is used. The theory of complex variables plays an important role in the study of theoretical physics, and string theory is no exception. If you are not familiar with complex variables you should take time out to study the topic before proceeding any further. You can do so using my book Complex Variables Demysti ed, also published by McGraw-Hill.
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Copyright 2009 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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In the theory of complex variables, a conformal transformation is one that maps a region of the complex plane to a new more convenient region while preserving angles, but not lengths. For example, you can map the unit disk to the upper-half plane using a conformal transformation. The notion that angles are preserved is a geometric interpretation and leads us to the notion of a conformal transformation of space-time coordinates. Let us consider a transformation of the space-time coordinates such that x x . In general, the metric g ( x ) is found to transform in the following way: g ( x ) = x x g ( x ) x x (5.1)
Now consider a function of the space-time coordinates given by ( x ). If the metric transforms in the following way: g ( x ) = ( x ) g ( x ) (5.2)
Then we have a conformal transformation of the metric. Notice that ( x ) acts as a scaling factor, hence it will preserve angles but not lengths. If a metric is related to the at Minkowski metric as g = ( x ) ( x ) we say that the metric is conformally at. To see how a conformal transformation preserves angles, consider two tangent vectors u and v . Using the metric, the angle between them is given by cos = g(u , v) g(u , u ) g(v, v)
Now we apply the transformation given by Eq. (5.2) and nd cos g (u , v) = g (u , u ) g (v, v) ( x ) g(u , v) = ( x ) g(u , u ) ( x ) g(v, v) g(u , v) g(u , u ) g(v, v)
Hence a conformal transformation preserves angles. A conformal eld theory is a quantum eld theory that is invariant under conformal transformations. These theories are Euclidean quantum eld theories, meaning that a Euclidean metric is used. The symmetry group of such a theory will contain Euclidean symmetries (we will review those in a moment) along with local conformal transformations. It turns out that two-dimensional conformal eld theories are of particular use. Two-dimensional conformal eld theories have an in nite number of conserved charges.
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