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code 128 font vb.net Adding together to get the total mode expansion gives in Java
Adding together to get the total mode expansion gives Reading QR Code JIS X 0510 In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Creating QR In Java Using Barcode generation for Java Control to generate, create QRCode image in Java applications. 25 25 X 25 = x 0 + p0 + i 25 n n e in( + ) e in( ) 2 n 0
QR Code 2d Barcode Scanner In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. Barcode Creation In Java Using Barcode drawer for Java Control to generate, create barcode image in Java applications. Now using Euler s famous formula: e in( + ) e in( ) e in( + ) e in( ) = 2i 2i ein e in in = 2ie in = 2ie sin n 2i This means that the mode expansion can be written as Barcode Reader In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Creating QR Code In Visual C# Using Barcode creator for .NET Control to generate, create Quick Response Code image in .NET framework applications. 25 25 X 25 = x 0 + p0 + 2 25 n in e sin n n 0 n
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ECC200 Generation In Java Using Barcode creator for Java Control to generate, create Data Matrix ECC200 image in Java applications. EAN13 Generator In Java Using Barcode maker for Java Control to generate, create European Article Number 13 image in Java applications. Now we can analyze this expression to discover the properties of open strings in the dual theory. The rst item to notice is: The expression for X 25 has no linear terms that contain the worldsheet time coordinate . Physically, This means that the dual string has no momentum in the 25th dimension. If the string carries no momentum for = 25, it must be xed. What does a xed vibrating string do The motion is oscillatory. Notice that the expansion contains a sin n term, which of course satis es sin n = 0 at = 0, . EAN8 Printer In Java Using Barcode printer for Java Control to generate, create European Article Number 8 image in Java applications. Drawing Bar Code In Java Using Barcode generation for Eclipse BIRT Control to generate, create barcode image in BIRT reports applications. String Theory Demysti ed
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25 X 25 ( = 0, ) = x 0
At = we have
25 X 25 ( = , ) = x 0 +
K 25 = x0 + 2K R R
Hence, X 25 ( = , ) X 25 ( = 0, ) = 2 K R This tells us that the dual string winds around the dual dimension of radius R with winding number K. Summarizing Tduality transforms Neumann boundary conditions into Dirichlet boundary conditions. Tduality transforms Dirichlet boundary conditions into Neumann boundary conditions. Tduality transforms a bosonic string with momentum but no winding into a string with winding but no momentum. For the dual string, the string endpoints are restricted to lie on a 25dimensional hyperplane in spacetime. The endpoints of the dual string can wind the circular dimension an integer number of times given by K. DBranes
The hyperplane that the open string is attached to carries special signi cance. A Dbrane is a hypersurface in spacetime. In the examples worked out in this chapter, it is a hyperplane with 24 spatial dimensions. The dimension which has been excluded in this example is the dimension which has been compacti ed. The D is short for Dirichlet which refers to the fact that the open strings in the theory have CHAPTER 8 Compacti cation and TDuality
endpoints that satisfy Dirichlet boundary conditions. In English this means that the endpoints of an open string are attached to a Dbrane. A Dbrane can be classi ed by the number of spatial dimensions it contains. A point is a zerodimensional object and therefore is a D0brane. A line, which is a onedimensional object is a D1brane (so strings can be thought of as D1branes). Later we will see that the physical world of three spatial dimensions and one time dimension that we can perceive directly is a D3brane contained in the larger world of 11dimensional hyperspace. In the example studied in this chapter, we considered a D24brane, with one spatial dimension compacti ed that leaves 24 dimensions for the hyperplane surface. Using the procedure outlined here, other dimensions can be compacti ed. If we choose to compactify n dimensions then that leaves behind a D(25n)brane. The procedure outlined here is essentially the same in superstring theory, but in that case compactifying n dimensions gives us a D(9n)brane. Note that: The ends of an open string are free to move in the noncompacti ed directions including time. So in bosonic theory, if we have compacti ed n directions, the endpoints of the string are free to move in the other 1 + (25n) directions. In superstring theory, the endpoints will be free to move in the other 1 + (9n) directions. In the example considered in this chapter where we compacti ed 1 dimension in bosonic string theory, the end points of the string are free to move in the other 1 + 24 dimensions. We can consider the existence of Dbranes to be a consequence of the symmetry of Tduality. The number, types, and arrangements of Dbranes restrict the open string states that can exist. We will have more to say about Dbranes and discuss Tduality in the context of superstrings in future chapters. Summary In this chapter we described compacti cation which involves taking a spatial dimension and compactifying it to a small circle of radius R. Going through this procedure, it was discovered that a symmetry emerges called Tduality, which relates theories with small R to equivalent theories with large R. An important consequence of Tduality was discovered when it was learned that open strings with Neumann boundary conditions are transformed into open strings with Dirichlet boundary conditions in the dual theory. The result is the endpoints of the string are xed to a hyperplane called a Dbrane.

