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Printer QR Code 2d barcode in Java The Fourier expansion can be used to derive Poisson brackets for the modes. These are

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(2.69)
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The Poisson brackets will be the starting point to quantize the theory.
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1. Consider the lagrangian given in the Eq. (2.10) which includes the auxiliary eld. Use the Euler-Lagrange equations to derive the equation of motion. 2. Start with the Nambu-Goto lagrangian L = X 2 X 2 ( X X 2 ) and consider the gauge choice which gives the at metric h . Using the constraints X 2 + X 2 = 0, X X = 0, nd the equations of motion. 3. Consider the Polyakov action. A Weyl transformation is one of the form h e ( , ) h and X = 0. Determine the form of the Polyakov action ( , ) h ]. under a Weyl transformation [hint: h e 4. Using the Polyakov action, de ne the energy-momentum tensor on the worldsheet by varying the action with respect to the intrinsic metric T = 2 T
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String Theory Demysti ed
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Using the fact that h = 1 h h h nd an expression for the 2 induced metric so that it can be eliminated from the action. This should allow you to recover the Nambu-Goto action.
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5. Consider an open string with free endpoints. Find the variation of the center of mass position of the string with by calculating 1
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d X ( , )
6. Find the conserved momentum of the open string with free endpoints by calculating P = T d X ( , )
where the dot represents differentiation with respect to .
The Classical String II: Symmetries and Worldsheet Currents
In the last chapter we introduced some basic notions of classical string theory, including the equations of motion and boundary conditions. In this chapter we will expand our discussion of the classical string, discussing symmetries and introducing the energy-momentum tensor and conserved currents. This will nish the groundwork we need for the classical string, and in the next chapter we will quantize the string.
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String Theory Demysti ed
The Energy-Momentum Tensor
Let s quickly review a few things before getting started. Recall that the intrinsic distance on the worldsheet can be determined using the induced metric h . This is given by ds 2 = h d d (3.1)
where 0 = , 1 = are the coordinates which parameterize points on the worldsheet. A set of functions X ( , ) describe the shape of the worldsheet and the motion of the string with respect to the background space-time, where = 0, 1, ..., D 1 for a D-dimensional space-time. To nd the dynamics of the string, we can minimize the Polyakov action [Eq. (2.27)]: SP = T d 2 det(h ) h X X 2 (3.2)
Minimizing SP (by minimizing the area of the worldsheet) gives us the equations of motion for the X ( , ), and hence the dynamics of the string. In the quiz at the end of Chap. 2 in Prob. 4, you were invited to show that the Polyakov and NambuGoto actions were equivalent by considering the energy-momentum or stress-energy tensor T which is given by T = 2 T 1 SP h h (3.3)
In this book we ll go mostly by the name energy-momentum tensor. In a nutshell, the energy-momentum tensor describes the density and ux of energy and momentum in space-time. You should be familiar with the basics of what T is from some exposure to or study of quantum eld theory, so we re just going to go with that and describe how it works in string theory. When working out the solution to Prob. 4 in Chap. 2, you should have found that 1 T = X X h h X X 2
(3.4)
The rst property that we will establish for the energy-momentum tensor is that it has zero trace. We can calculate the trace using the induced metric: Tr (T ) = T = h T
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