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12. Using the result of Prob. 11, nd p0 , J . 13. Find [ J , J ] . 14. For the Virasoro operator Lm , nd [ Lm , J ]. 15. Consider the rst excited state of the open bosonic string. Let be a polarization vector and consider the action of L1 on the state 1 0, k . What condition on the polarization and momentum follows from the Virasoro constraint L1 = 0 for physical states 16. What condition cancels the conformal anomaly for the bosonic string 17. Consider the Polyakov action in the conformal gauge. State the constraints on the components of the energy momentum tensor. 18. State the Neumann boundary conditions for classical, open bosonic strings. 19. Consider a relativistic point particle with space-time coordinates x ( ) and dx . Use the usual variational action S = m d x x , where x = d procedure to nd the equations of motion, and write down these equations in terms of the conjugate momentum. 20. Consider your solution to Prob. 19. What is the condition on p 21. Consider your solution to Prob. 19. Take the static gauge, where x 0 = t . What is the action in this case if we use the usual de nition of the particle dx velocity v = dt 22. What is the momentum in this case (using the action from Prob. 21) 23. How do the spinors in the RNS formalism transform under Lorentz transformations

FIGURE 3-4

IBOutlet UIImageView *uiImageView; The result will look like this:#import <UIKit/UIKit.h> @interface testViewController : UIViewController { IBOutlet UILabel *label; IBOutlet UIImageView *uiImageView; } @property (nonatomic, retain) IBOutlet UILabel *label; @property (nonatomic, retain) IBOutlet UIImageView *uiImageView; @end

FIGURE 3-5

In problems 24 26, let be a gamma matrix in D = 10 space-time dimensions and following the text let 11 = 0 1 9. 24. Calculate { 11 , }. 25. Calculate 11 0 .

A sequence diagram describes how groups of objects collaborate in some behavior over time. It records the behavior of a single use case. It displays objects and the messages passed among these objects in the use case. A design can have lots of methods in different classes. This makes it difficult to determine the overall

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happening, and some have even proposed that microscopic black holes could be produced Again, you could imagine having extra dimensions without string theory, so discoveries like these would not prove string theory However, they would be major indirect evidence in its favor You will learn in this book that string theory predicts the existence of extra dimensions, so any evidence of this has to be taken as a serious indication that string theory is on the right path String theory has lots of problems it s a work in progress This time is akin to living in the era when the existence of atoms was postulated but unproven and skeptics abounded There are lots of skeptics out there.

Are we done with the header file yet Not quite. We need to add our IBActions. We ve handled our IBOutlets, both of them, but now we re going to utilize an IBAction for something we really need here. Can you guess

FIGURE 3-6

And string theory does seem a bit crazy there are several versions of the theory, and each has a myriad of particle states that have not been discovered (however, note that transformations called dualities have been discovered that relate the different string theories, and work is underway on an underlying theory believed to exist called M-theory) The only serious competitor right now for string theory is loop quantum gravity I want to emphasize I am not an expert, but I once took a seminar on it and to be honest I found it incredibly distasteful It seemed so abstract it almost didn t seem like physics at all It struck me more as mathematical philosophy String theory seems a lot more physical to me It makes outlandish predictions like the existence of extra dimensions, but general relativity and quantum theory make predictions that defy common sense as well.

sequence of behavior. This diagram is simple and logical, so as to make the sequence and flow of control obvious (see Figure 3-6).

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