java barcode generator library A SUPERSYMMETRIC POINT PARTICLE in Java

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A SUPERSYMMETRIC POINT PARTICLE
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We introduce the formalism by going back to the simplest case we can describe a point particle. This will allow us to go over the main ideas without getting bogged down by the formalism. It turns out this approach actually has some direct relevance to string theory anyway. In modern parlance, a point particle is called a D0-brane. So the physics we will lay out here is known as the D0-brane action (this is a Dp-brane with p = 0 ). This type of object can be found in the type IIA superstring theory. The action for a relativistic point particle of mass m can be written as S= 1 1 2 2 d e x em 2 (9.4)
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As noted in Chap. 2, e is called the auxiliary eld. The action written in this form is well suited to the study of massless particles. Letting m 0 gives S= 1 1 2 d e x 2 (9.5)
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To make the jump to superspace, we consider the space de ned by the pair of coordinates: x , Aa (9.6)
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CHAPTER 9 Superstring Theory Continued
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where Aa is anticommuting spinor coordinate. In the case we are studying here, for a point particle, these are functions of , that is, Aa = Aa ( ). The index A ranges over the number of supersymmetries in the theory. If there are N of them, then A = 1, ..., N Hence, if we have an N = 2 supersymmetry, then we have the two fermionic coordinates 1a and 2 a. You may be a little confused by the notation. We actually have a second index here. The second index is the spinor index. Consider a general Dirac spinor. In D dimensions it has 2 D/ 2 components. So, a = 1, ..., 2 D / 2 For Majorana spinors, this number is cut in half. Now, we are actually going to proceed in a manner which is not too different from what you learned for worldsheet supersymmetry. Once again, we consider a constant Majorana spinor that we denote by A (suppressing the spinor index) to emphasize that it is in nitesimal. Now we introduce the following SUSY transformations:
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x = i A A A = A A = A
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In addition, we have to worry about the auxiliary eld. We suppose that the SUSY transformation in this case is (9.7)
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e = 0
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(9.8)
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The simplest supersymmetric action that can be conceived of is an extension of the action in Eq. (9.5) written as follows: S= 1 1 A A 2 d e ( x i ) 2 (9.9)
Now, since A is a constant, it does not depend on and hence A = 0 . Given that plus Eq. (9.7), it s very easy to see that Eq. (9.9) is invariant under a SUSY transformation. First note that d A d d ( A ) = A = A = =0 d d d
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Now of course we can ignore the 1 e term when varying the action since the SUSY transformation is Eq. (9.8). Proceeding
S =
1 1 A A 2 d e ( x i ) 2 1 1 = d ( x i A A )2 2 e 1 = d ( x i A A ) ( x i A A ) e 1 = d ( x i A A ) x i ( A A ) e
Ok, now we have
( A A ) = ( A ) A + A ( A )
= ( A ) A = A A Using Eq. (9.7) then, we have
x i ( A A ) = i A A i A A = 0
Therefore, S = 0 and the action is invariant under a SUSY transformation. Since we are dealing with an enlargement of space-time coordinates, take a step back and recall that the actions described in Chap. 2 Are invariant under space-time translations a . Are invariant under Lorentz transformations x . We combine these two results in the Poincar group and note that the action in Eq. (9.4) is invariant under
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