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as R Reading QR Code In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. QR Generator In Java Using Barcode generation for Java Control to generate, create Denso QR Bar Code image in Java applications. The center of mass momentum Eq. (8.8) returns to the nonquantized, continuous momentum of the noncompacti ed case. Now let s think about the opposite limit where R 0. You might also expect that this is like returning to the noncompacti ed case. After all, taking the limit R 0 is like making the extra dimension go away. In quantum eld theory we might expect the elds to completely decouple from that unseen extra dimension. However, things don t quite work this way in string theory. As R 0, we nd that the KaluzaKlein modes become in nitely massive and decouple from the theory. Since these can be regarded as particle states, maybe this isn t so surprising. What s left behind for the center of mass momentum are the winding states. First note that as R 0 we obtain p R = pL Hence p 25 0 but the winding term behaves in the following way: w= 1 25 25 25 pL pR pL 2 Denso QR Bar Code Decoder In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Bar Code Creator In Java Using Barcode generation for Java Control to generate, create bar code image in Java applications. as R 0
Bar Code Scanner In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. QR Code Drawer In Visual C# Using Barcode encoder for Visual Studio .NET Control to generate, create QR Code image in .NET framework applications. 25 (or to pR if you like). Now the winding states, rather than the momentum states, form a continuum of states. This should not be so surprising, as R 0 the circle gets smaller and smaller. So it gets easier and easier to wrap a string around it that is, it costs less energy. When the circle is very small it doesn t require a lot of energy to wrap the string around it. So you see as the radius gets very large or very small there is a tradeoff between winding states and momentum. This tradeoff leads us to a discussion of Tduality, the topic of the next section. Print QR Code 2d Barcode In VS .NET Using Barcode printer for ASP.NET Control to generate, create QR Code image in ASP.NET applications. QR Code Encoder In VS .NET Using Barcode drawer for .NET framework Control to generate, create QR image in VS .NET applications. CHAPTER 8 Compacti cation and TDuality
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UPCA Encoder In Java Using Barcode drawer for Java Control to generate, create UPCA image in Java applications. Generate ANSI/AIM Code 128 In Java Using Barcode drawer for Java Control to generate, create USS Code 128 image in Java applications. Tduality is a symmetry which exists between different string theories. This symmetry relates small distances in one theory to large distances in another, seemingly different theory and shows that the two theories are in fact the same theory expressed from different viewpoints. This is an important recognition; before Tduality was discovered it was believed that there were ve different string theories, when in fact they were all different versions of the same theory that could be related to one another by transformations or dualities. One can transform between small and large distances when considering the compacti ed dimension in one theory, and arrive at another dual theory. This is the essence of Tduality. We will see later that other dualities exist in string theory as well. Tduality relates type IIA and type IIB string theories, as well as the heterotic string theories. It applies to the type of compacti cation that we have been studying in this chapter, namely the compacti cation of a spatial dimension to a circle of radius R. The transformation that is used in Tduality is to transform the radius to a new large radius R which is de ned by the exchange R Standard 2 Of 5 Encoder In Java Using Barcode creator for Java Control to generate, create 2 of 5 Industrial image in Java applications. Paint Code 39 In None Using Barcode generation for Online Control to generate, create Code39 image in Online applications. R Generate UPCA In .NET Framework Using Barcode generation for VS .NET Control to generate, create UPCA Supplement 5 image in VS .NET applications. Recognizing Code 128A In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. (8.29) Data Matrix ECC200 Creation In None Using Barcode maker for Online Control to generate, create Data Matrix image in Online applications. UPC Symbol Creation In None Using Barcode drawer for Word Control to generate, create UPCA image in Microsoft Word applications. The Tduality transformation also exchanges winding states characterized by a winding number n with highmomentum states in the other theory (KaluzaKlein excitations). That is, n K (8.30) Printing Bar Code In .NET Framework Using Barcode drawer for ASP.NET Control to generate, create bar code image in ASP.NET applications. Data Matrix ECC200 Maker In VB.NET Using Barcode creator for .NET Control to generate, create Data Matrix ECC200 image in .NET framework applications. The symmetry of Tduality, described by these exchanges, makes its appearance in the mass formula [Eq. (8.27)], which we reproduce here: 2 2 nR K m 2 = + + 2( N R + N L ) 4 R
Now exchange R / R and n K , then: K n nR = R R
(8.31) We also have: K nR R = K R
(8.32) String Theory Demysti ed
So we see that the mass formula Eq. (8.27) is invariant under the exchange R / R and n K . It assumes the form nR K + + 2( N R + N L ) 4 m = R 2 2 2 That is, it keeps the same form but now with the new radius R . That s the math of the transformation. The physics is that if we started with a theory with a small compacti ed dimension R, we have transformed to a dual theory with a large extra dimension R . What this means for the string is that a string in a type IIA theory (with small compacti ed dimension), which winds around the small compact dimension (with winding states) is dual to a string in type IIB theory (with the dimension transformed to a large dimension of radius R ), which has momentum along that dimension. Each time the string in type IIA theory winds around the compact dimension, this corresponds to increasing the momentum in type IIB theory by one unit. 25 25 Now let s examine how pL and pR , and by extension 0 and 0, transform under this symmetry. Recall Eq. (8.22) that states

