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String Theory Demysti ed
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This is very easy to prove in our one-dimensional example. The calculation is dQ d dq = p dt dt ds = = = = d dq dp dq +p dt ds dt ds dp dq d dq +p dt ds ds dt d L dq L dq + dt q ds q ds d L dq L dq + dt dq ds q ds dt L dq L dq dL = =0 + q ds q ds ds (symmetry of the lagrangian) s (commutativity of partial derivatives) r (notation, let p L dq , q) q dt
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For a eld we de ne a Noether current which is a conserved quantity as j = L ( ) (3.6)
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We are done with our quick and dirty review of symmetries and conserved quantities. Now let s see what symmetries and conserved quantities we can describe for bosonic string theory in D at space-time dimensions.
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POINCAR TRANSFORMATIONS
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The Poincar group consists of the following transformations: Translations in space-time Lorentz transformations In at D-dimensional space-time, the Polyakov action is invariant under Poincar transformations. A space-time translation is a transformation of the form X X + b (3.7)
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CHAPTER 3 Symmetries and Worldsheet Currents
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where X = b . An in nitesimal Lorentz transformation is one of the form X X + X
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(3.8)
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In this case X = X . We can combine translations and in nitesimal Lorentz transformations as
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X = X + b
Under a Poincar transformation the worldsheet metric transforms as
(3.9)
h = 0
(3.10)
The Polyakov action of Eq. (3.2) is invariant under the transformations given in Eqs. (3.9) and (3.10). Invariance under Eq. (3.7) leads to conservation of energy and momentum (energy from time translational invariance and momentum from spatial translation invariance). Invariance of the Polyakov action under Eq. (3.8) leads to conservation of angular momentum. Recall the de nition of a global symmetry and notice that while the transformations in Eqs. (3.7) and (3.8) depend on the coordinates of the embedding space-time (the elds X ), they do not depend on the worldsheet coordinates ( , ). This means that on the worldsheet, these symmetries are global. Since the symmetry is global on the worldsheet and not over all of space-time, we say that this is a global internal symmetry. Put another way, in string theory a global internal symmetry is one that acts on the elds X but not on the two-dimensional space-time of the worldsheet, that is, the parameters of a global internal symmetry group are independent of the worldsheet coordinates ( , ).
REPARAMETERIZATIONS
Consider a coordinate transformation that takes ( , ) ( , ), which is a reparameterization of the worldsheet (also called a diffeomorphism). The metric h transforms as h = h ( , ) (3.11)
(note that in this context we are using primes not to denote differentiation, but rather to indicate quantities like the metric in the new coordinate system). Since / = ( / )( / ) and X ( , ) X ( , ) it follows that h ( , ) X X X X = h ( , )
String Theory Demysti ed
The jacobian for a change in coordinates is de ned by J = det The jacobian shows up in two places that turn out to cancel themselves to leave the form of the Polyakov action invariant. It shows up when calculating the determinant of the metric as det(h ) = J 2 det(h ) You may recall from calculus that it also shows up in the integration measure: d 2 = J d 2 These cancel out in the terms that appear in the Polyakov action [Eq. (3.2)]. That is, d 2 det h = d 2 det h Putting all of these results together, we see that a change of worldsheet coordinates (a reparameterization) leaves the Polyakov action invariant. Therefore a reparameterization is a symmetry of the action. Since a reparameterization depends on the worldsheet coordinates ( , ) , these are local symmetries.
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