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java barcode generator source code J + = + + X J = X in Java
J + = + + X J = X QR Code ISO/IEC18004 Decoder In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Drawing QRCode In Java Using Barcode creation for Java Control to generate, create QR Code JIS X 0510 image in Java applications. i T = X X + 2
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Barcode Encoder In None Using Barcode creator for Font Control to generate, create bar code image in Font applications. EAN13 Creator In Java Using Barcode drawer for BIRT Control to generate, create EAN13 Supplement 5 image in Eclipse BIRT applications. We obtain conservation laws for the energymomentum tensor: T++ = +T = 0
Code39 Generator In None Using Barcode generator for Font Control to generate, create Code 39 Extended image in Font applications. Recognizing DataMatrix In Visual C#.NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET applications. (7.22) EXAMPLE 7.4 Show that the equations of motion for the fermion and boson elds lead to conservation of the supercurrent. SOLUTION We start with J + and consider the derivative J +. We have: J + = ( + + X ) = ( + ) + X + + ( + X ) =0 The result was readily obtained using Eqs. (7.20) and (7.21). Now taking J , we obtain a second conservation equation by calculating + J which gives + J = + ( X ) = + X + + X
= ( + X ) = 0 EXAMPLE 7.5 Show that T++ = 0. SOLUTION Using T++ = + X + X + i /2 + + + , we nd i T++ = + X + X + + + + 2 = ( + X ) + X + + X ( + X ) + i i = + + + = + + + = 0 2 2 To obtain this result, we applied Eqs. (7.20) and (7.21) together with the commutativity of partial derivatives. i i + 2 + + + 2 + + + String Theory Demysti ed
Mode Expansions and Boundary Conditions
The nal step in putting together the classical physics of the RNS superstring follows the program used in the bosonic case we need to apply boundary conditions and write down the mode expansions. Speci cally, we need to apply boundary conditions for the fermionic elds. It is simplest to continue working in lightcone coordinates and vary the fermionic part of the action. Before doing this, it can be helpful to review some elementary calculus. Recall integration by parts: f ( x) b df dg dx = fg b g( x ) dx a a dx dx
The product fg is called the boundary term. When we vary the fermionic action, we are going to obtain boundary terms for the elds , so we need to specify boundary conditions so that the variation in the action vanishes. The fermionic part of the action in lightcone coordinates, modulo a few constants and ignoring the spacetime index is SF = d 2 ( + + + + ) (7.23) For simplicity, let s consider one piece of this expression and vary it. We obtain
d 2 + + = d 2 [ + + + + ( + )] Following the usual procedure applied in eld theory, we want to move the derivative off the + term. This can be done using integration by parts. When this is done, we pick up a boundary term: d ( + ) = d + +
= =0 d 2 + +
A similar expression arises from the variation of the other term. All together, the boundary terms obtained by varying the action are SF = d ( + + ) = ( + + ) =0 (7.24) CHAPTER 7 RNS Superstrings
OPEN STRING BOUNDARY CONDITIONS
When varying the action, the boundary terms must vanish in order to maintain Lorentz invariance. In the case of open string, the boundary terms = 0 and = must both vanish independently. We can obtain + + = 0
at = 0 if we take
+ ( 0, ) = ( 0, ) (7.25) Now in general, + = will make the boundary terms vanish, but typical convention is to x the boundary condition at = 0 using Eq. (7.25). This leaves the choice of sign at = ambiguous. Depending on the sign we choose, we obtain two different boundary conditions. Ramond or R boundary conditions are given by the choice + ( , ) = ( , ) (Ramond) (7.26) The other choice we can make is known as NeveauSchwarz or NS boundary conditions: + ( , ) = ( , ) (NeveauSchwarz) (7.27) We often refer to the boundary conditions chosen as the sector. The choice of boundary conditions has dramatic consequences. In particular The R sector gives rise to string states that are spacetime fermions. The NS sector gives rise to string states that are spacetime bosons.

