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java barcode generator source code ( k ) 1 1 0, k in Java
( k ) 1 1 0, k QR Reader In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Painting QRCode In Java Using Barcode encoder for Java Control to generate, create QR Code ISO/IEC18004 image in Java applications. The object ( k ) is a tensor which can be decomposed into symmetric ( ) ( k ) and antisymmetric ( ) ( k ) parts (see Relativity Demysti ed if you aren t sure about Decode QR Code In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. Bar Code Maker In Java Using Barcode encoder for Java Control to generate, create barcode image in Java applications. CHAPTER 4 String Quantization
Reading Bar Code In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. QRCode Printer In Visual C#.NET Using Barcode drawer for VS .NET Control to generate, create QR Code image in VS .NET applications. this). The symmetric part corresponds to a massless spin2 particle which is the graviton. The linearized metric g = + ( x ) satis es the linearized Einstein equations ( x ) = 0, and ( x ) = 0. By taking the Fourier transform of ( ) ( k ), it can be shown that these equations are satis ed. The trace ( k ), which de nes a scalar, is also important. This corresponds to a massless scalar particle called the dilaton. QR Code 2d Barcode Generator In VS .NET Using Barcode creation for ASP.NET Control to generate, create Quick Response Code image in ASP.NET applications. Printing Denso QR Bar Code In .NET Using Barcode creator for VS .NET Control to generate, create QR Code image in VS .NET applications. LightCone Quantization
Painting QR In VB.NET Using Barcode drawer for .NET Control to generate, create QR Code ISO/IEC18004 image in VS .NET applications. Bar Code Drawer In Java Using Barcode maker for Java Control to generate, create barcode image in Java applications. We now turn our attention to a different method of quantization. We began the chapter with a discussion of covariant quantization. This method is a straightforward application of the imposition of commutation relations. We took this approach rst because it may seem familiar from your studies of ordinary quantum mechanics. In addition, it preserves the Lorentz invariance of the theory. Physicists say that this approach is manifestly Lorentz invariant, colloquially meaning that the Lorentz invariance is obvious. The technique has the disadvantage in that negative norm states appear. Although this is a problem, it is instructive to go through the process of eliminating the negative norm states. Another approach is possible which avoids the negative norm states at the cost of losing manifest Lorentz invariance. This is called lightcone quantization. We brie y discuss it here, considering the open string case. We begin by using lightcone coordinates, which were introduced in Chap. 2 in Eq. (2.34): X = X 0 X D 1 2 (4.30) Encoding Linear Barcode In Java Using Barcode creation for Java Control to generate, create 1D image in Java applications. 2D Barcode Creator In Java Using Barcode creation for Java Control to generate, create Matrix Barcode image in Java applications. The remaining coordinates X i are transverse coordinates. The center of mass coordinate x and momentum p are also written as lightcone coordinates. In the lightcone gauge, we choose X + = x+ + Painting Leitcode In Java Using Barcode maker for Java Control to generate, create Leitcode image in Java applications. Making Code 39 Full ASCII In None Using Barcode creator for Online Control to generate, create Code39 image in Online applications. p +
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p p+
The Virasoro constraint becomes
String Theory Demysti ed
1 X X = ( X i X i )2 2 (4.32) So we have a relation between lightcone coordinates and the transverse coordinates. The mode expansion of the worldsheet coordinates for an open string is given by X = x + In particular X = x + 2 s 2 s
p + i
n in e cos n s n 0 n
p + i
n in e cos n s n 0 n
We can then solve for the nonzero modes of X in terms of the transverse oscillators. These are
n = 1 1 D 2 i i + 2 : n m m : a n , 0 p s i=1 m=
(4.33) In the case of the zeroth mode, we can derive an expression for the hamiltonian H = p = 1 i i 1 i i p p + n n + 2p n 0 (4.34) De ne a conjugate momentum P . The system is quantized by imposing commutation relations on the transverse components of position and momentum: [ x i , p j ] = i ij , [ X i ( ), P j ( )] = i ij ( ) The massshell condition becomes M 2 = 2 p+ p pi2 = i =1 D 2
(4.35) ( N a) (4.36) The number operator is given by
i i N = n n i =1 n =1 D 2
(4.37) CHAPTER 4 String Quantization
Normal ordering leads to 1 D 2 i i 1 D 2 i i D 2 n n n = 2 n : n n : + 2 n 2 i=1 = n =1 i =1 = The regularization trick can be applied to make the second term nite. We nd D 2 D 2 n = 24 2 n=1 Again taking a = 1, one nds D = 26. Summary In the previous two chapters, we constructed a relativistic theory of the string, the classical theory. In this chapter we have introduced the simplest possible quantum extension of the classical theory. This is a theory that consists only of bosons. While the theory is not realistic since it does not include fermionic states, it is easier to deal with and introduces important concepts and methods that will play a role in the full quantum theory. The classical theory was quantized using two different methods. The rst method, called covariant quantization, is a straightforward approach that imposes commutation relations on the X and their conjugate momenta. This leads to negative norm states. The Virasoro constraints are imposed to rid the theory of these states. When this is done, we nd that the theory must have 26 spacetime dimensions. We concluded the chapter with a different approach, known as lightcone quantization.

