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java barcode generator source code String Quantization in Java
CHAPTER 4 String Quantization QR Code Decoder In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. QR Code ISO/IEC18004 Generator In Java Using Barcode drawer for Java Control to generate, create QR image in Java applications. To obtain a formula for the mass operator in string theory, which is denoted by 2 M , we use the condition on physical states. Taking L0 = 1/ 2 0 + n n , using n =1 L0 a with a = 1 , we arrive at the condition Decoding QRCode In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Print Bar Code In Java Using Barcode generator for Java Control to generate, create barcode image in Java applications. 1 2 ( L0 a) = 0 0 + n n 1 = 0 2 n =1 1 2 0 + n n 1 = 0 2 n =1 The rst term in this expression is nothing other than the mass squared: 2 (1 / 2 ) 0 = M 2 where = 1 / (2 T ). So, in bosonic string theory the mass shell condition becomes M2 = 1 ( N 1) (4.22) Bar Code Scanner In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. QR Code JIS X 0510 Generation In C#.NET Using Barcode encoder for Visual Studio .NET Control to generate, create QR Code image in .NET applications. where N is the total number operator. The term 2 T sets the energy scale of the theory, it is taken to be on the order of the Planck mass. This is the origin of the high energy scale of string theory. It can be shown that M2 = 1 D 2 N 24 QR Code Creator In .NET Framework Using Barcode creation for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Drawing Quick Response Code In .NET Framework Using Barcode generation for .NET Control to generate, create QR Code JIS X 0510 image in VS .NET applications. Notice that setting a = 1forces us to take D = 26. The number operator acts on the ground state as N 0 =0 Hence the mass of the ground state is M2 0 = 1 D 2 1 D 2 1 0 = 0 N 0 = 24 24 QR Code Generator In Visual Basic .NET Using Barcode generation for Visual Studio .NET Control to generate, create Quick Response Code image in .NET applications. Creating Data Matrix ECC200 In Java Using Barcode encoder for Java Control to generate, create Data Matrix ECC200 image in Java applications. So the ground state of bosonic string theory in the open string case has negative mass. This means that the ground state is a Tachyon. This is an unphysical state which travels faster than the speed of light. Consistency of bosonic string theory requires that we choose a = 1, so the Tachyon cannot be removed from the theory. Data Matrix ECC200 Encoder In Java Using Barcode generation for Java Control to generate, create ECC200 image in Java applications. Make 1D Barcode In Java Using Barcode printer for Java Control to generate, create Linear 1D Barcode image in Java applications. String Theory Demysti ed
Identcode Drawer In Java Using Barcode encoder for Java Control to generate, create Identcode image in Java applications. Painting ECC200 In VS .NET Using Barcode creation for Reporting Service Control to generate, create DataMatrix image in Reporting Service applications. It will turn out that the introduction of supersymmetry (that is the introduction of fermionic states) into the theory will get rid of the Tachyon, giving us a realistic string theory. We will see that this also changes the number of spacetime dimensions. Now let s consider the mass of the rst excited state. The rst excited state is i i = 1 0 where i is a spatial index. Here, i = 1, ..., D 2, and the state transforms as a vector in spacetime. You will recall from your studies of quantum eld theory that a vector is a spin1 particle that in general has D 1 components, the fact that this state has D 2 components implies that it is a massless state. An example of a massless vector is the photon, it only has transverse components of spin. This explains why there are D 2 components rather than D 1. The mass of the state is Data Matrix Printer In None Using Barcode generator for Software Control to generate, create Data Matrix 2d barcode image in Software applications. Recognizing Code 128A In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. i M 2 1 0 =
EAN128 Printer In ObjectiveC Using Barcode maker for iPad Control to generate, create EAN / UCC  13 image in iPad applications. Matrix Barcode Creation In Visual C# Using Barcode generator for Visual Studio .NET Control to generate, create 2D Barcode image in VS .NET applications. 1 D 2 i 1 26 D i 1 1 0 = 1 0 24 24
ANSI/AIM Code 128 Printer In C# Using Barcode maker for VS .NET Control to generate, create USS Code 128 image in .NET applications. Data Matrix ECC200 Creation In ObjectiveC Using Barcode drawer for iPhone Control to generate, create DataMatrix image in iPhone applications. In order for the state to be massless, 26 D / 24 must vanish, once again setting the number of spacetime dimensions D to 26. Physicists refer to the bosonic string theory with a = 1, D = 26 as critical and call D = 26 the critical dimension. CLOSED STRING SPECTRUM
In the case of the closed string, things are a little more complicated than what you re used to from the harmonic oscillator in ordinary quantum theory due to the fact that we have a second commutation relation in addition to m , n = m m+n ,0 that must be satis ed, namely, m , n = m m+n ,0 . What this is going to mean is that we need to de ne two number operators. These are de ned by in nite sums over the modes: N R = m m m =1 N L = m m
m =1 (4.23) Together with the momentum operator p , the number operators N R and N L serve to characterize the state of a closed string. Let us denote a state by n, k . As in the open string case, the momentum operator will act according to p 0, k = k 0, k (4.24) Therefore, the state 0, k of the string carries momentum k . Turning our attention to the number operators, rst let s specify the action of the raising and

