qns 3 and 4

Question 3. (5 marks) Symmetry algebra. Use the Poisson bracket (X"(t,0), II'r, o')} = 78(0 - 0) (and all other brackets zero) and the definition of JH (see lecture notes) to compute {}",) (3) Question 4. (20 marks) Example of a string solution. Given the mode expansion of XM, solve the Virasoro constraints under the following simplifying assumptions. Assume that only a p" and the Enth (n fixed) modes d' and ce, as well as their left-moving purtners are turned on while the other modes for 0 vanish. Assume that the spatial oscillations of the string are restricted to the X! - XP-plane. Assume that x = 0 and p = 0, Go through the following steps: (a) Given the assumptions above, write down the relevant Virasoro constraints. (3 marks) (b) Solve the Virasoro constraints, (5 marks) (c) Up to signs and exchange of components there are two distinct solutions of the string equations of motion. Give them explicitly. You may choose suitable initial conditions to fix some undetermined phase factors in a (3 marks) convenient manner so that the final result matches with the lecture notes. (5 marks) (d) Plot and interpret these solutions (e) One of the solutions has in interpretation in terms of open string. For this case show that the ends of the string (man) move at the speed of light. Question 3. (5 marks) Symmetry algebra. Use the Poisson bracket (X"(t,0), II'r, o')} = 78(0 - 0) (and all other brackets zero) and the definition of JH (see lecture notes) to compute {}",) (3) Question 4. (20 marks) Example of a string solution. Given the mode expansion of XM, solve the Virasoro constraints under the following simplifying assumptions. Assume that only a p" and the Enth (n fixed) modes d' and ce, as well as their left-moving purtners are turned on while the other modes for 0 vanish. Assume that the spatial oscillations of the string are restricted to the X! - XP-plane. Assume that x = 0 and p = 0, Go through the following steps: (a) Given the assumptions above, write down the relevant Virasoro constraints. (3 marks) (b) Solve the Virasoro constraints, (5 marks) (c) Up to signs and exchange of components there are two distinct solutions of the string equations of motion. Give them explicitly. You may choose suitable initial conditions to fix some undetermined phase factors in a (3 marks) convenient manner so that the final result matches with the lecture notes. (5 marks) (d) Plot and interpret these solutions (e) One of the solutions has in interpretation in terms of open string. For this case show that the ends of the string (man) move at the speed of light