String Theory Demysti ed

Recognizing QR Code In JavaUsing Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications.

QR-Code Creator In JavaUsing Barcode generator for Java Control to generate, create Denso QR Bar Code image in Java applications.

SOLUTION For simplicity, we consider motion in one spatial dimension. Now S = ds = dt 2 dx 2 = dt 1 dx 2 dt 2

QR Code Scanner In JavaUsing Barcode scanner for Java Control to read, scan read, scan image in Java applications.

Encode Barcode In JavaUsing Barcode creation for Java Control to generate, create barcode image in Java applications.

= dt 1 v 2 Now recall the binomial theorem. This tells us that 1 x 1 1 x 2

Barcode Scanner In JavaUsing Barcode scanner for Java Control to read, scan read, scan image in Java applications.

Denso QR Bar Code Printer In C#Using Barcode maker for Visual Studio .NET Control to generate, create Denso QR Bar Code image in VS .NET applications.

Hence, 1 1 v2 1 v2 2 Therefore S = dt 1 v 2 1 1 dt 1 v 2 = dt + dt v 2 2 2 Comparison of the second term in this expression with S0 = dt (1/ 2)mv 2 tells us that must be the mass of the particle. We can also determine the units of and deduce that it is the mass of the particle from dimensional analysis. First, what are the units of action Recall from your studies of quantum theory that the units of action from Planck s constant are mass times length squared per time: ML2 [ ]= T (2.6)

Generating QR Code In .NET FrameworkUsing Barcode printer for ASP.NET Control to generate, create QR Code ISO/IEC18004 image in ASP.NET applications.

Denso QR Bar Code Creation In VS .NETUsing Barcode maker for .NET Control to generate, create Quick Response Code image in VS .NET applications.

CHAPTER 2 Equations of Motion

Paint QR Code JIS X 0510 In VB.NETUsing Barcode drawer for VS .NET Control to generate, create QR Code image in .NET applications.

Create EAN-13 In JavaUsing Barcode maker for Java Control to generate, create EAN 13 image in Java applications.

Now let s look at S = ds. From the integral, we have length L, so we have ML2 = [ ]L T So it must be the case that [ ] = ML T

Data Matrix ECC200 Printer In JavaUsing Barcode generator for Java Control to generate, create Data Matrix 2d barcode image in Java applications.

1D Printer In JavaUsing Barcode creator for Java Control to generate, create Linear 1D Barcode image in Java applications.

We can obtain this result using the mass of the particle together with the speed of light c, which is of course a length over time. That is, m c ML [ ] = T

Identcode Maker In JavaUsing Barcode maker for Java Control to generate, create Identcode image in Java applications.

Generating Bar Code In Objective-CUsing Barcode maker for iPad Control to generate, create barcode image in iPad applications.

In units where c = = 1, which are commonly used in particle physics and string theory, the action is dimensionless. Hence mass is inverse length and

Recognize GS1 - 12 In Visual C#.NETUsing Barcode reader for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications.

Code 128C Encoder In JavaUsing Barcode generator for Eclipse BIRT Control to generate, create Code-128 image in Eclipse BIRT applications.

[ ] = M = 1 L (2.7)

Painting GS1 - 13 In VS .NETUsing Barcode generator for ASP.NET Control to generate, create EAN-13 image in ASP.NET applications.

Generating Barcode In VB.NETUsing Barcode maker for Visual Studio .NET Control to generate, create bar code image in .NET framework applications.

Now let s see how to write down the action and obtain the equations of motion from it. We start with the de nition of in nitesimal length given in Eq. (2.4). This gives the action as S = m dX dX Let s rewrite the integrand: d dX dX = dX dX d

UCC-128 Printer In NoneUsing Barcode encoder for Online Control to generate, create GTIN - 128 image in Online applications.

Scanning ANSI/AIM Code 39 In Visual C#Using Barcode recognizer for VS .NET Control to read, scan read, scan image in VS .NET applications.

(2.8)

= d

dX dX = d X X d d

String Theory Demysti ed

This allows us to write the action in the form S = m d X X (2.9)

This action is a nice compact form that allows us to derive the equations of motion. As you recall from your studies of classical mechanics or quantum eld theory, the quantity in the integrand is called the lagrangian: L = m X X = m X 2 There are two problems with the action so far developed in Eq. (2.9). First, think about what happens in the case of a massless particle. Setting m = 0 leaves us with S 0 and so there is nothing to vary to obtain the equations of motion. So this action isn t very helpful in the case of a massless particle. Also, it turns out that quantization is not easy when we have a square root in the action. For these reasons, we introduce an auxiliary eld that we will denote a( ) and consider the lagrangian L= 1 2 m2 X a 2a 2

We can use this to de ne an alternative expression for the action S = 1 1 2 2 d a X m a 2 (2.10)

We can vary this action to nd an equation of motion for the auxiliary eld a( ). We nd 1 1 S = d X 2 m 2 a a 2 = = 1 2 1 2 d a X (m a) 2 1 1 d X 2 m 2 2 a2

CHAPTER 2 Equations of Motion

Now we set S = 0. Since this means that the integrand must be 0, we obtain the equation 1 2 X m2 = 0 2 a

X 2 + m2a2 = 0 This is the equation of motion for the auxiliary eld. From this we nd an expression for the auxiliary eld given by X2 m2

a=

(2.11)

Using Eq. (2.11), we can show that the action in Eq. (2.10) is equivalent to Eq. (2.8), which we do in the next example. EXAMPLE 2.2 Show that if a = ( X 2 / m 2 )1/ 2 , the action S = 1 / 2 d [(1 / a ) X 2 m 2 a] can be recast