This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Homework Set No. 1, Physics 880.02 Deadline Thursday, January 22, 2009 1. Consider a free (real) scalar theory with the Lagrangian density L = 1 2 - m 2 2 2 . Define the Hamiltonian by H ( t ) = Z d 3 x [ ( ~x, t ) ( ~x, t )- L ] . a. (3 pts) Show that for classical field configurations d dt H ( t ) = 0 . b. (2 pts) Write H ( t ) in terms of and with no s appearing. c. (5 pts) Now imagine that the field is quantized. Use canonical quantization commu- tators [ ( ~x, t ) , ( ~x , t )] = i ( ~x- ~x ) (with all other commutators being zero) to show that H ( t ) (now an operator) generates time translations, i.e., show that i = [ , H ( t )] i = [ , H ( t )] . 2. The same as in problem 1, but now for Dirac field: start with a theory with Lagrangian density L = ( i - m ) . a. (3 pts) Construct a Hamiltonian H ( t ) and show that for classical field configurations d dt H ( t ) = 0 ....
View Full Document