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If the momentum p and
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(1.9)
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String Theory Demysti ed
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then I in Eq. (1.8) is nite and calculations give answers that make sense. On the other hand, if the momentum p but
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the integral in Eq. (1.8) diverges. This leads to in nities in calculations. Now if I but does so slowly, then a mathematical technique called renormalization can be used to get nite results from calculations. Such is the case when working with established theories like quantum electrodynamics.
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The Standard Model
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In its nished form, the theoretical framework that describes known particle interactions with quantum eld theory is called the standard model. In the standard model, there are three basic types of particle interactions. These are Electromagnetic Weak Strong (nuclear) There are two basic types of particles in the standard model. These are Spin-1 gauge bosons that transmit particle interactions (they carry the force). These include the photon (electromagnetic interactions), W and Z (weak interactions), and gluons (strong interactions). Matter is made out of spin-1/2 fermions, such as electrons. In addition, the standard model requires the introduction of a spin-0 particle called the Higgs boson. Particles interact with the associated Higgs eld, and this interaction gives particles their mass.
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The general theory of relativity includes gravitational waves. They carry angular momentum J = 2, so we deduce that the quantum of the gravitational eld, known as the graviton, is a spin-2 particle. It turns out that string theory naturally includes
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CHAPTER 1 Introduction
a spin-2 boson, and so naturally includes the quantum of gravity. Returning to Eq. (1.8), if we let J = 2 and consider space-time as we know it D = 4, then 4 J 8 + D = 4 (2 ) 8 + 4 = 4 So in the case of the graviton, p 4 J 8 p 0 = 1 and I d4 p when integrated over all momenta. This means that gravity cannot be renormalized in the way that a theory like quantum electrodynamics can, because it diverges like p 4. In contrast, consider quantum electrodynamics. The spin of the photon is 1, so 4 J 8 + D = 4 (1) 8 + 4 = 0 and the loop integral goes as I p 4 J 8 d D p = p 4 d 4 p Goes like p0 = 1 This tells us that incorporating gravity into the standard quantum eld theory framework is an extremely problematic enterprise. The bottom line is nobody really had any idea how to do it for a very long time. String theory gets rid of this problem by getting rid of particle interactions that occur at a single point. Take a look at the uncertainty principle x p If momentum blow up, that is, p , this implies that x 0 . That is, large (in nite) momentum means small (zero) distance. Or put another way, pointlike
String Theory Demysti ed
Old quantum field theory: A particle is a mathematical point, with no extension
String theory: Particles are strings, with extension in one dimension. This gets rid of infinities
In string theory, particles are replaced by strings, spreading out interactions over space-time so that in nities don t result.
interactions (zero distance) imply in nite momentum. This leads to divergent loop integrals, and in nities in calculations. So in string theory, we replace a point particle by a one-dimensional string. This is illustrated in Fig. 1.4.
Some Basic Analysis in String Theory
In string theory, we don t go all the way to x 0 but instead cut it off at some small, but nonzero value. This means that there will be an upper limit to momentum and hence p . Instead momentum goes to a large, but nite value and the loop / integral divergences can be gotten rid of. If we have a cutoff de ned by the length of a string, then the uncertainty relations must be modi ed. It is found that in a string theory uncertainty in position x is approximately given by x = p + p (1.10)
A new term has been introduced into the uncertainty relation, ( p/ ) which can serve to x a minimum distance that exists in the theory. The parameter is related to the string tension TS as
=
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