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java barcode generator library X 25 ( , ) = x 25 + in Java
X 25 ( , ) = x 25 + Scanning QR Code In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Generating QR Code 2d Barcode In Java Using Barcode creation for Java Control to generate, create QR Code image in Java applications. 25 25 25 ( pL + pR ) + ( pL25 pR ) + modes 2 2
Quick Response Code Decoder In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Draw Barcode In Java Using Barcode creator for Java Control to generate, create bar code image in Java applications. (8.7) Bar Code Decoder In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. QR Code ISO/IEC18004 Encoder In C#.NET Using Barcode creator for .NET framework Control to generate, create Quick Response Code image in .NET applications. CHAPTER 8 Compacti cation and TDuality
QR Code Printer In .NET Using Barcode generator for ASP.NET Control to generate, create QR image in ASP.NET applications. Denso QR Bar Code Drawer In Visual Studio .NET Using Barcode drawer for VS .NET Control to generate, create QR image in .NET framework applications. The total center of mass momentum of the string is
QR Encoder In VB.NET Using Barcode printer for .NET Control to generate, create QR Code ISO/IEC18004 image in Visual Studio .NET applications. Print UPCA In Java Using Barcode creator for Java Control to generate, create UPCA image in Java applications. 25 25 p 25 = pL + pR
1D Printer In Java Using Barcode creator for Java Control to generate, create Linear image in Java applications. Printing DataMatrix In Java Using Barcode creator for Java Control to generate, create ECC200 image in Java applications. (8.8) Creating USPS Confirm Service Barcode In Java Using Barcode printer for Java Control to generate, create USPS PLANET Barcode image in Java applications. Encoding ECC200 In ObjectiveC Using Barcode printer for iPhone Control to generate, create ECC200 image in iPhone applications. Along the compacti ed dimension, the string acts like a particle moving on a circle. The momentum is quantized according to p 25 = K R (8.9) Print EAN / UCC  14 In Java Using Barcode drawer for Eclipse BIRT Control to generate, create UCC.EAN  128 image in BIRT applications. Data Matrix ECC200 Creator In .NET Using Barcode generator for Reporting Service Control to generate, create Data Matrix ECC200 image in Reporting Service applications. where K is an integer called the KaluzaKlein excitation number. This is an important result without the compacti ed dimension, the center of mass momentum of the string is continuous. Compactifying a dimension quantizes the center of mass momentum along that dimension. Looking at Eq. (8.7) then, the rst term involving the momenta is the total center of mass momentum of the string. We call this the momentum mode. The second term, however, also involves momentum. In fact this term is the winding mode of the string, which satis es Code 39 Extended Encoder In Java Using Barcode drawer for BIRT Control to generate, create Code 39 Full ASCII image in Eclipse BIRT applications. Draw EAN13 In None Using Barcode maker for Software Control to generate, create EAN13 image in Software applications. 25 25 p pR = nR 2 L
Drawing EAN13 Supplement 5 In ObjectiveC Using Barcode drawer for iPad Control to generate, create EAN13 Supplement 5 image in iPad applications. USS128 Printer In None Using Barcode generation for Software Control to generate, create GS1 128 image in Software applications. (8.10) Looking at Eq. (8.3), we see that the winding w can be de ned in terms of the momentum of the left and rightmoving modes as w= 1 nR 25 = p 25 pR 2 L (8.11) Modi ed Mass Spectrum
Compactifying a dimension will lead to a modi ed mass spectrum. To obtain the mass spectrum for the state with a compacti ed dimension, let us begin with the Virasoro operators. Recall that L0 = p R p R + n n 4 n =1
(8.12) Note the repeated index which is an upper and lower index on the rst term in the right so we have an implied sum. Here the index ranges over the entire String Theory Demysti ed
spacetime, that is, = 0, ..., 2. Now let s write L0 in such a way that we peel of the = 25 term. Then L0 = 25 25 24 pR pR + pR pR + n n 4 4 =0 n =1
(8.13) Similarly we can write L0 =
25 25 24 pL pL + pL pL + n n 4 4 =0 n =1
(8.14) With a single compacti ed dimension, the KaluzaKlein excitations on X 25 are considered to be distinct particles. Hence we can write down the mass operator as a mass term in the 25 noncompacti ed dimensions. That is, m 2 = p p =0 24 (8.15) Now, you should recognize the sums n =1 n n and n =1 n n as the number operators N R and N L . Using this fact together with Eq. (8.15) allows us to write the Virasoro operators as 25 25 2 p p m + NR 4 R R 4 25 25 2 L0 = p p m + NL 4 L L 4
L0 = (8.16) (8.17) Now we can utilize the massshell constraint. This is the condition that L0 1 and L0 1 annihilate physical states : ( L0 1) = 0 ( L0 1) = 0 (8.18) (8.19) The conditions in Eqs. (8.18) and (8.19) imply that L0 = 1 and L0 = 1. Applying the rst condition to Eq. (8.16) we get L0 = 1 25 25 2 pR pR m + N R 4 4 25 25 m2 = pR pR + 2 N R 2 2 2
(8.20) CHAPTER 8 Compacti cation and TDuality
Similarly, using L0 = 1 together with Eq. (8.17) we obtain
2 25 25 m = p p + 2NL 2 2 2 L L
Now using Eq. (8.8) together with Eqs. (8.9) and (8.10) we can write
25 pL =
(8.21) nR K + R
(8.22) Which of course allows us to compute
(p ) 25 L nK nR K nR K = + = + +2 R R
(8.23) 25 and similarly pR = ( K / R) (nR / ) so that 2 2 2
(p ) 25 R nK K nR nR K = = + 2 R R
(8.24) This allows us to obtain the sum and difference formulas: (p ) +(p ) 25 L 2 25 R 25 L 2 25 R
nR 2 K 2 = 2 + R =4 nK
(8.25) (8.26) (p ) (p ) Using Eqs. (8.25) and (8.26) we can add Eqs. (8.20) and (8.21) to obtain
2 2 nR K m 2 = + + 2( N R + N L ) 4 R
(8.27) and subtracting Eqs. (8.21) from (8.20) gives N R N L = nK (8.28) So notice we have extra terms in the formulas for mass [Eq. (8.27)] and the level matching condition [Eq. (8.28)] as compared to the formulas introduced for the String Theory Demysti ed
bosonic string in Chap. 2. The extra terms are due to two components, the KaluzaKlein excitations and the winding states of the string. The KaluzaKlein excitations can be regarded as particles and so cannot be thought of as due to strings in a general sense. However the winding excitations can only come from strings, because only strings can wrap around a compacti ed extra dimension. Now let s look at the mass formula in Eq. (8.27) together with the relations for the momenta in Eqs. (8.22) and (8.25). Our task here is to consider limiting behavior. First we consider the case where R . In this limit, the momentum goes to the continuum limit and the KaluzaKlein excitations disappear. It is simple to show that pL = pR and hence the winding state w= 1 25 25 p pR 0 2 L

